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Company (renamed Alcoa in 1907) and Aluminium Industrie AG. The British Aluminium Company, Produits Chimiques d’Alais et de la Camargue, and Société Electro-Métallurgique de Froges also joined the cartel. By the mid-20th century, aluminium had become a part of everyday life and an essential component of housewares. In 1954, production of aluminium surpassed that of copper, historically second in production only to iron, making it the most produced non-ferrous metal. During the mid-20th century, aluminium emerged as a civil engineering material, with building applications in both basic construction and interior finish work, and increasingly being used in military engineering, for both airplanes and land armor vehicle engines. Earth's first artificial satellite, launched in 1957, consisted of two separate aluminium semi-spheres joined and all subsequent space vehicles have used aluminium to some extent. The aluminium can was invented in 1956 and employed as a storage for drinks in 1958. Throughout the 20th century, the production of aluminium rose rapidly: while the world production of aluminium in 1900 was 6,800 metric tons, the annual production first exceeded 100,000 metric tons in 1916; 1,000,000 tons in 1941; 10,000,000 tons in 1971. In the 1970s, the increased demand for aluminium made it an exchange commodity; it entered the London Metal Exchange, the oldest industrial metal exchange in the world, in 1978. The output continued to grow: the annual production of aluminium exceeded 50,000,000 metric tons in 2013. The real price for aluminium declined from $14,000 per metric ton in 1900 to $2,340 in 1948 (in 1998 United States dollars). Extraction and processing costs were lowered over technological progress and the scale of the economies. However, the need to exploit lower-grade poorer quality deposits and the use of fast increasing input costs (above all, energy) increased the net cost of aluminium; the real price began
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to grow in the 1970s with the rise of energy cost. Production moved from the industrialized countries to countries where production was cheaper. Production costs in the late 20th century changed because of advances in technology, lower energy prices, exchange rates of the United States dollar, and alumina prices. The BRIC countries' combined share in primary production and primary consumption grew substantially in the first decade of the 21st century. China is accumulating an especially large share of the world's production thanks to an abundance of resources, cheap energy, and governmental stimuli; it also increased its consumption share from 2% in 1972 to 40% in 2010. In the United States, Western Europe, and Japan, most aluminium was consumed in transportation, engineering, construction, and packaging. In 2021, prices for industrial metals such as aluminium have soared to near-record levels as energy shortages in China drive up costs for electricity. == Etymology == The names aluminium and aluminum are derived from the word alumine, an obsolete term for alumina, the primary naturally occurring oxide of aluminium. Alumine was borrowed from French, which in turn derived it from alumen, the classical Latin name for alum, the mineral from which it was collected. The Latin word alumen stems from the Proto-Indo-European root *alu- meaning "bitter" or "beer". === Origins === British chemist Humphry Davy, who performed a number of experiments aimed to isolate the metal, is credited as the person who named the element. The first name proposed for the metal to be isolated from alum was alumium, which Davy suggested in an 1808 article on his electrochemical research, published in Philosophical Transactions of the Royal Society. It appeared that the name was created from the English word alum and the Latin suffix -ium; but it was customary then to give elements names originating
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in Latin, so this name was not adopted universally. This name was criticized by contemporary chemists from France, Germany, and Sweden, who insisted the metal should be named for the oxide, alumina, from which it would be isolated. The English name alum does not come directly from Latin, whereas alumine/alumina comes from the Latin word alumen (upon declension, alumen changes to alumin-). One example was Essai sur la Nomenclature chimique (July 1811), written in French by a Swedish chemist, Jöns Jacob Berzelius, in which the name aluminium is given to the element that would be synthesized from alum. (Another article in the same journal issue also refers to the metal whose oxide is the basis of sapphire, i.e. the same metal, as to aluminium.) A January 1811 summary of one of Davy's lectures at the Royal Society mentioned the name aluminium as a possibility. The next year, Davy published a chemistry textbook in which he used the spelling aluminum. Both spellings have coexisted since. Their usage is currently regional: aluminum dominates in the United States and Canada; aluminium is prevalent in the rest of the English-speaking world. === Spelling === In 1812, British scientist Thomas Young wrote an anonymous review of Davy's book, in which he proposed the name aluminium instead of aluminum, which he thought had a "less classical sound". This name persisted: although the -um spelling was occasionally used in Britain, the American scientific language used -ium from the start. Ludwig Wilhelm Gilbert had proposed Thonerde-metall, after the German "Thonerde" for alumina, in his Annalen der Physik but that name never caught on at all even in Germany. Joseph W. Richards in 1891 found just one occurrence of argillium in Swedish, from the French "argille" for clay. The French themselves had used aluminium from the start. However, in
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England and Germany Davy's spelling aluminum was initially used; until German chemist Friedrich Wöhler published his account of the Wöhler process in 1827 in which he used the spelling aluminium, which caused that spelling's largely wholesale adoption in England and Germany, with the exception of a small number of what Richards characterized as "patriotic" English chemists that were "averse to foreign innovations" who occasionally still used aluminum. Most scientists throughout the world used -ium in the 19th century; and it was entrenched in several other European languages, such as French, German, and Dutch. In 1828, an American lexicographer, Noah Webster, entered only the aluminum spelling in his American Dictionary of the English Language. In the 1830s, the -um spelling gained usage in the United States; by the 1860s, it had become the more common spelling there outside science. In 1892, Hall used the -um spelling in his advertising handbill for his new electrolytic method of producing the metal, despite his constant use of the -ium spelling in all the patents he filed between 1886 and 1903. It is unknown whether this spelling was introduced by mistake or intentionally, but Hall preferred aluminum since its introduction because it resembled platinum, the name of a prestigious metal. By 1890, both spellings had been common in the United States, the -ium spelling being slightly more common; by 1895, the situation had reversed; by 1900, aluminum had become twice as common as aluminium; in the next decade, the -um spelling dominated American usage. In 1925, the American Chemical Society adopted this spelling. The International Union of Pure and Applied Chemistry (IUPAC) adopted aluminium as the standard international name for the element in 1990. In 1993, they recognized aluminum as an acceptable variant; the most recent 2005 edition of the IUPAC nomenclature of inorganic chemistry
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also acknowledges this spelling. IUPAC official publications use the -ium spelling as primary, and they list both where it is appropriate. == Production and refinement == The production of aluminium starts with the extraction of bauxite rock from the ground. The bauxite is processed and transformed using the Bayer process into alumina, which is then processed using the Hall–Héroult process, resulting in the final aluminium. Aluminium production is highly energy-consuming, and so the producers tend to locate smelters in places where electric power is both plentiful and inexpensive. Production of one kilogram of aluminium requires 7 kilograms of oil energy equivalent, as compared to 1.5 kilograms for steel and 2 kilograms for plastic. As of 2024, the world's largest producers of aluminium were China, Russia, India, Canada, and the United Arab Emirates, while China is by far the top producer of aluminium with a world share of over 55%. According to the International Resource Panel's Metal Stocks in Society report, the global per capita stock of aluminium in use in society (i.e. in cars, buildings, electronics, etc.) is 80 kg (180 lb). Much of this is in more-developed countries (350–500 kg (770–1,100 lb) per capita) rather than less-developed countries (35 kg (77 lb) per capita). === Bayer process === Bauxite is converted to alumina by the Bayer process. Bauxite is blended for uniform composition and then is ground fine. The resulting slurry is mixed with a hot solution of sodium hydroxide; the mixture is then treated in a digester vessel at a pressure well above atmospheric, dissolving the aluminium hydroxide in bauxite while converting impurities into relatively insoluble compounds: After this reaction, the slurry is at a temperature above its atmospheric boiling point. It is cooled by removing steam as pressure is reduced. The bauxite residue is separated from the
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solution and discarded. The solution, free of solids, is seeded with small crystals of aluminium hydroxide; this causes decomposition of the [Al(OH)4]− ions to aluminium hydroxide. After about half of aluminium has precipitated, the mixture is sent to classifiers. Small crystals of aluminium hydroxide are collected to serve as seeding agents; coarse particles are converted to alumina by heating; the excess solution is removed by evaporation, (if needed) purified, and recycled. === Hall–Héroult process === The conversion of alumina to aluminium is achieved by the Hall–Héroult process. In this energy-intensive process, a solution of alumina in a molten (940 and 970 °C (1,720 and 1,780 °F)) mixture of cryolite (Na3AlF6) with calcium fluoride is electrolyzed to produce metallic aluminium. The liquid aluminium sinks to the bottom of the solution and is tapped off, and usually cast into large blocks called aluminium billets for further processing. Anodes of the electrolysis cell are made of carbon—the most resistant material against fluoride corrosion—and either bake at the process or are prebaked. The former, also called Söderberg anodes, are less power-efficient and fumes released during baking are costly to collect, which is why they are being replaced by prebaked anodes even though they save the power, energy, and labor to prebake the cathodes. Carbon for anodes should be preferably pure so that neither aluminium nor the electrolyte is contaminated with ash. Despite carbon's resistivity against corrosion, it is still consumed at a rate of 0.4–0.5 kg per each kilogram of produced aluminium. Cathodes are made of anthracite; high purity for them is not required because impurities leach only very slowly. The cathode is consumed at a rate of 0.02–0.04 kg per each kilogram of produced aluminium. A cell is usually terminated after 2–6 years following a failure of the cathode. The Hall–Heroult process produces
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aluminium with a purity of above 99%. Further purification can be done by the Hoopes process. This process involves the electrolysis of molten aluminium with a sodium, barium, and aluminium fluoride electrolyte. The resulting aluminium has a purity of 99.99%. Electric power represents about 20 to 40% of the cost of producing aluminium, depending on the location of the smelter. Aluminium production consumes roughly 5% of electricity generated in the United States. Because of this, alternatives to the Hall–Héroult process have been researched, but none has turned out to be economically feasible. === Recycling === Recovery of the metal through recycling has become an important task of the aluminium industry. Recycling was a low-profile activity until the late 1960s, when the growing use of aluminium beverage cans brought it to public awareness. Recycling involves melting the scrap, a process that requires only 5% of the energy used to produce aluminium from ore, though a significant part (up to 15% of the input material) is lost as dross (ash-like oxide). An aluminium stack melter produces significantly less dross, with values reported below 1%. White dross from primary aluminium production and from secondary recycling operations still contains useful quantities of aluminium that can be extracted industrially. The process produces aluminium billets, together with a highly complex waste material. This waste is difficult to manage. It reacts with water, releasing a mixture of gases including, among others, acetylene, hydrogen sulfide and significant amounts of ammonia. Despite these difficulties, the waste is used as a filler in asphalt and concrete. Its potential for hydrogen production has also been considered and researched. == Applications == === Metal === The global production of aluminium in 2016 was 58.8 million metric tons. It exceeded that of any other metal except iron (1,231 million metric tons). Aluminium is
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almost always alloyed, which markedly improves its mechanical properties, especially when tempered. For example, the common aluminium foils and beverage cans are alloys of 92% to 99% aluminium. The main alloying agents for both wrought and cast aluminium are copper, zinc, magnesium, manganese, and silicon (e.g., duralumin) with the levels of other metals in a few percent by weight. The major uses for aluminium are in: Transportation (automobiles, aircraft, trucks, railway cars, marine vessels, bicycles, spacecraft, etc.). Aluminium is used because of its low density; Packaging (cans, foil, frame, etc.). Aluminium is used because it is non-toxic (see below), non-adsorptive, and splinter-proof; Building and construction (windows, doors, siding, building wire, sheathing, roofing, etc.). Since steel is cheaper, aluminium is used when lightness, corrosion resistance, or engineering features are important; Electricity-related uses (conductor alloys, motors, and generators, transformers, capacitors, etc.). Aluminium is used because it is relatively cheap, highly conductive, has adequate mechanical strength and low density, and resists corrosion; A wide range of household items, from cooking utensils to furniture. Low density, good appearance, ease of fabrication, and durability are the key factors of aluminium usage; Machinery and equipment (processing equipment, pipes, tools). Aluminium is used because of its corrosion resistance, non-pyrophoricity, and mechanical strength. === Compounds === The great majority (about 90%) of aluminium oxide is converted to metallic aluminium. Being a very hard material (Mohs hardness 9), alumina is widely used as an abrasive; being extraordinarily chemically inert, it is useful in highly reactive environments such as high pressure sodium lamps. Aluminium oxide is commonly used as a catalyst for industrial processes; e.g. the Claus process to convert hydrogen sulfide to sulfur in refineries and to alkylate amines. Many industrial catalysts are supported by alumina, meaning that the expensive catalyst material is dispersed over a surface of the
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inert alumina. Another principal use is as a drying agent or absorbent. Several sulfates of aluminium have industrial and commercial application. Aluminium sulfate (in its hydrate form) is produced on the annual scale of several millions of metric tons. About two-thirds is consumed in water treatment. The next major application is in the manufacture of paper. It is also used as a mordant in dyeing, in pickling seeds, deodorizing of mineral oils, in leather tanning, and in production of other aluminium compounds. Two kinds of alum, ammonium alum and potassium alum, were formerly used as mordants and in leather tanning, but their use has significantly declined following availability of high-purity aluminium sulfate. Anhydrous aluminium chloride is used as a catalyst in chemical and petrochemical industries, the dyeing industry, and in synthesis of various inorganic and organic compounds. Aluminium hydroxychlorides are used in purifying water, in the paper industry, and as antiperspirants. Sodium aluminate is used in treating water and as an accelerator of solidification of cement. Many aluminium compounds have niche applications, for example: Aluminium acetate in solution is used as an astringent. Aluminium phosphate is used in the manufacture of glass, ceramic, pulp and paper products, cosmetics, paints, varnishes, and in dental cement. Aluminium hydroxide is used as an antacid, and mordant; it is used also in water purification, the manufacture of glass and ceramics, and in the waterproofing of fabrics. Lithium aluminium hydride is a powerful reducing agent used in organic chemistry. Organoaluminiums are used as Lewis acids and co-catalysts. Methylaluminoxane is a co-catalyst for Ziegler–Natta olefin polymerization to produce vinyl polymers such as polyethene. Aqueous aluminium ions (such as aqueous aluminium sulfate) are used to treat against fish parasites such as Gyrodactylus salaris. In many vaccines, certain aluminium salts serve as an immune adjuvant (immune response booster)
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to allow the protein in the vaccine to achieve sufficient potency as an immune stimulant. Until 2004, most of the adjuvants used in vaccines were aluminium-adjuvanted. == Biology == Despite its widespread occurrence in the Earth's crust, aluminium has no known function in biology. At pH 6–9 (relevant for most natural waters), aluminium precipitates out of water as the hydroxide and is hence not available; most elements behaving this way have no biological role or are toxic. Aluminium sulfate has an LD50 of 6207 mg/kg (oral, mouse), which corresponds to 435 grams (about one pound) for a 70 kg (150 lb) mouse. === Toxicity === Aluminium is classified as a non-carcinogen by the United States Department of Health and Human Services. A review published in 1988 said that there was little evidence that normal exposure to aluminium presents a risk to healthy adult, and a 2014 multi-element toxicology review was unable to find deleterious effects of aluminium consumed in amounts not greater than 40 mg/day per kg of body mass. Most aluminium consumed will leave the body in feces; most of the small part of it that enters the bloodstream, will be excreted via urine; nevertheless some aluminium does pass the blood-brain barrier and is lodged preferentially in the brains of Alzheimer's patients. Evidence published in 1989 indicates that, for Alzheimer's patients, aluminium may act by electrostatically crosslinking proteins, thus down-regulating genes in the superior temporal gyrus. === Effects === Aluminium, although rarely, can cause vitamin D-resistant osteomalacia, erythropoietin-resistant microcytic anemia, and central nervous system alterations. People with kidney insufficiency are especially at a risk. Chronic ingestion of hydrated aluminium silicates (for excess gastric acidity control) may result in aluminium binding to intestinal contents and increased elimination of other metals, such as iron or zinc; sufficiently high doses (>50 g/day)
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can cause anemia. During the 1988 Camelford water pollution incident, people in Camelford had their drinking water contaminated with aluminium sulfate for several weeks. A final report into the incident in 2013 concluded it was unlikely that this had caused long-term health problems. Aluminium has been suspected of being a possible cause of Alzheimer's disease, but research into this for over 40 years has found, as of 2018, no good evidence of causal effect. Aluminium increases estrogen-related gene expression in human breast cancer cells cultured in the laboratory. In very high doses, aluminium is associated with altered function of the blood–brain barrier. A small percentage of people have contact allergies to aluminium and experience itchy red rashes, headache, muscle pain, joint pain, poor memory, insomnia, depression, asthma, irritable bowel syndrome, or other symptoms upon contact with products containing aluminium. Exposure to powdered aluminium or aluminium welding fumes can cause pulmonary fibrosis. Fine aluminium powder can ignite or explode, posing another workplace hazard. === Exposure routes === Food is the main source of aluminium. Drinking water contains more aluminium than solid food; however, aluminium in food may be absorbed more than aluminium from water. Major sources of human oral exposure to aluminium include food (due to its use in food additives, food and beverage packaging, and cooking utensils), drinking water (due to its use in municipal water treatment), and aluminium-containing medications (particularly antacid/antiulcer and buffered aspirin formulations). Dietary exposure in Europeans averages to 0.2–1.5 mg/kg/week but can be as high as 2.3 mg/kg/week. Higher exposure levels of aluminium are mostly limited to miners, aluminium production workers, and dialysis patients. Consumption of antacids, antiperspirants, vaccines, and cosmetics provide possible routes of exposure. Consumption of acidic foods or liquids with aluminium enhances aluminium absorption, and maltol has been shown to increase the accumulation
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of aluminium in nerve and bone tissues. === Treatment === In case of suspected sudden intake of a large amount of aluminium, the only treatment is deferoxamine mesylate which may be given to help eliminate aluminium from the body by chelation therapy. However, this should be applied with caution as this reduces not only aluminium body levels, but also those of other metals such as copper or iron. == Environmental effects == High levels of aluminium occur near mining sites; small amounts of aluminium are released to the environment at coal-fired power plants or incinerators. Aluminium in the air is washed out by the rain or normally settles down but small particles of aluminium remain in the air for a long time. Acidic precipitation is the main natural factor to mobilize aluminium from natural sources and the main reason for the environmental effects of aluminium; however, the main factor of presence of aluminium in salt and freshwater are the industrial processes that also release aluminium into air. In water, aluminium acts as a toxiс agent on gill-breathing animals such as fish when the water is acidic, in which aluminium may precipitate on gills, which causes loss of plasma- and hemolymph ions leading to osmoregulatory failure. Organic complexes of aluminium may be easily absorbed and interfere with metabolism in mammals and birds, even though this rarely happens in practice. Aluminium is primary among the factors that reduce plant growth on acidic soils. Although it is generally harmless to plant growth in pH-neutral soils, in acid soils the concentration of toxic Al3+ cations increases and disturbs root growth and function. Wheat has developed a tolerance to aluminium, releasing organic compounds that bind to harmful aluminium cations. Sorghum is believed to have the same tolerance mechanism. Aluminium production possesses its own challenges to
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the environment on each step of the production process. The major challenge is the emission of greenhouse gases. These gases result from electrical consumption of the smelters and the byproducts of processing. The most potent of these gases are perfluorocarbons, namely CF4 and C2F6, from the smelting process. Biodegradation of metallic aluminium is extremely rare; most aluminium-corroding organisms do not directly attack or consume the aluminium, but instead produce corrosive wastes. The fungus Geotrichum candidum can consume the aluminium in compact discs. The bacterium Pseudomonas aeruginosa and the fungus Cladosporium resinae are commonly detected in aircraft fuel tanks that use kerosene-based fuels (not avgas), and laboratory cultures can degrade aluminium. == See also == Aluminium granules Aluminium joining Aluminium–air battery Aluminized steel, for corrosion resistance and other properties Aluminized screen, for display devices Aluminized cloth, to reflect heat Aluminized mylar, to reflect heat Panel edge staining Quantum clock == Notes == == References == == Bibliography == Davis, J. R. (1999). Corrosion of Aluminum and Aluminum Alloys. ASM International. ISBN 978-1-61503-238-9. Dean, J. A. (1999). Lange's handbook of chemistry (15 ed.). McGraw-Hill. ISBN 978-0-07-016384-3. OCLC 40213725. Drozdov, A. (2007). Aluminium: The Thirteenth Element. RUSAL Library. ISBN 978-5-91523-002-5. King, R. B. (1995). Inorganic Chemistry of Main Group Elements. Wiley-VCH. ISBN 978-0-471-18602-1. Lide, D. R., ed. (2004). Handbook of Chemistry and Physics (84 ed.). CRC Press. ISBN 978-0-8493-0566-5. Nappi, C. (2013). The global aluminium industry 40 years from 1972 (PDF) (Report). International Aluminium Institute. Archived (PDF) from the original on 9 October 2022. Richards, J. W. (1896). Aluminium: Its history, occurrence, properties, metallurgy and applications, including its alloys (3 ed.). Henry Carey Baird & Co. Schmitz, C. (2006). Handbook of Aluminium Recycling. Vulkan-Verlag GmbH. ISBN 978-3-8027-2936-2. == Further reading == Mimi Sheller, Aluminum Dream: The Making of Light Modernity. Cambridge, Mass.: Massachusetts
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Institute of Technology Press, 2014. == External links == Aluminium at The Periodic Table of Videos (University of Nottingham) Toxicological Profile for Aluminum (PDF) (September 2008) – 357-page report from the United States Department of Health and Human Services, Public Health Service, Agency for Toxic Substances and Disease Registry Aluminum entry (last reviewed 30 October 2019) in the NIOSH Pocket Guide to Chemical Hazards published by the CDC's National Institute for Occupational Safety and Health Current and historical prices (1998–present) for aluminum futures on the global commodities market The short film Aluminum is available for free viewing and download at the Internet Archive.
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A generalized compound is a mixture of chemical compounds of constant composition, despite possible changes in the total amount. The concept is used in the Dynamic Energy Budget theory, where biomass is partitioned into a limited set of generalised compounds, which contain a high percentage of organic compounds. The amount of generalized compound can be quantified in terms of weight, but more conveniently in terms of C-moles. The concept of strong homeostasis has an intimate relationship with that of generalised compound. == References ==
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This page provides supplementary chemical data on carbon monoxide. == Material safety data sheet == The handling of this chemical may incur notable safety precautions. It is highly recommended that you seek the material safety data sheet (MSDS) for this chemical from a reliable source such as SIRI, and follow its directions. MSDS from Advanced Gas Technologies in the SDSdata.org database == Structure and properties == == Thermodynamic properties == == Spectral data == == References ==
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This page provides supplementary chemical data on n-hexane. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. It is highly recommend that you seek the Material Safety Datasheet (MSDS) for this chemical from a reliable source and follow its directions. eChemPortal Science Stuff Fisher Scientific. == Structure and properties == == Thermodynamic properties == == Vapor pressure of liquid == Table data obtained from CRC Handbook of Chemistry and Physics 44th ed. == Distillation data == == Spectral data == == References == Linstrom, Peter (1997). "NIST Standard Reference Database". National Institute of Standards and Technology. doi:10.18434/T4D303. {{cite journal}}: Cite journal requires |journal= (help)
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Semi-deciduous or semi-evergreen is a botanical term which refers to plants that lose their foliage for a very short period, when old leaves fall off and new foliage growth is starting. This phenomenon occurs in tropical and sub-tropical woody species, for example in Dipteryx odorata. Semi-deciduous or semi-evergreen may also describe some trees, bushes or plants that normally only lose part of their foliage in autumn/winter or during the dry season, but might lose all their leaves in a manner similar to deciduous trees in an especially cold autumn/winter or severe dry season (drought). == See also == Brevideciduous Evergreen Marcescence Hedera == References ==
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Nano spray dryers refer to using spray drying to create particles in the nanometer range. Spray drying is a gentle method for producing powders with a defined particle size out of solutions, dispersions, and emulsions which is widely used for pharmaceuticals, food, biotechnology, and other industrial materials synthesis. In the past, the limitations of spray drying were the particle size (minimum 2 micrometres), the yield (maximum around 70%), and the sample volume (minimum 50 ml for devices in lab scale). Recently, minimum particle sizes have been reduced to 300 nm, yields up to 90% are possible, and the sample amount can be as small as 1 ml. These expanded limits are possible due to new technological developments to the spray head, the heating system, and the electrostatic particle collector. To emphasize the small particle sizes possible with this new technology, it has been described as "nano" spray drying. However, the smallest particles produced are in the sub-micrometre range common to fine particles rather than the nanometer scale of ultrafine particles. == Functional principle == The functional principle is basically the same as with normal spray dryers. There are just different technologies that are used to do similar things. The drying gas enters the system via the heater. A new kind of heater system allows for laminar air flow. The spray head sprays the fine droplets with a narrow size distribution into the drying chamber. The droplets dry and become solid particles. The solid particles are separated in the electrostatic particle collector. The [exhaust gas] is filtered and sent to a fume hood or the environment. The inlet temperature is controlled by a temperature sensor. and can be very dangerous also due to particulate matter == Applications == Pharmaceuticals: This technique is widely used in the pharma market. Because of the
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small sample amounts and the high yields, it is ideal for spray drying expensive substances in basic research. The following list shows examples of what is possible: Inhalable drugs for dry powder inhalers (DPI‘s) Nano- and microencapsulation of liposomes Stabilization of heat-sensitive vaccines, insulin, growth hormones Encapsulation of nanoparticle drugs for high bioavailability Nanocapsules of biodegradable polymers (lactides, glycolides) Porous drug carriers for nanoparticle suspensions Excipients for controlled drug release studies: trehalose, mannitol, lactose, HPMC, PVA, chitosan, dextrin, PLGA, starch, gelatin Materials science: This new technique offers new prospects in materials science, specially in the nanomaterial field. Now it is possible to spray dry fine particles. The following list shows examples of what is possible: Fine metal particles for novel catalysts Fine magnetic powders Carbon nanotubes as additives High performance ceramics with novel structures and high specific surface area Titanium oxide particles Nanoparticle suspensions for agglomeration Silicon oxide nanoparticle agglomerates Finest pigments for paints and coatings Food: Also in the field of food science this technology offers new possibilities. Especially in the currently vibrant field of functional food, the following list shows examples of what is possible: Nano food – Functional additives Encapsulation of fruit aromas, flavours, or perfumes Spray drying of fine powder aromas for pet food Encapsulation of fish oil for smell protection Vitamins, other food additives, etc. == Spray head == The spray head is one of the three new technologies that make "nano" spray drying possible. A piezoelectric system precisely vibrates a fine mesh. Vibration produces fine droplets with a narrow size distribution. == Heating system == In the field of "nano" spray drying a new heating system is used to provide the drying gas to produce the particles. The gas flow in the system is laminar and not turbulent as in common spray drying. The
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advantage of a laminar flow is that the particles fall straight down from the spray head and do not stick to the glass wall. The laminar flow is produced by pressing the air through a porous metal foam. == Electrostatic particle collector == To collect the very fine particles a new technology is used in the field of "nano" spray drying. The reason is that common cyclone technology depends on the particle mass; particles smaller than 2 μm can't be separated and instead exit the system along with the exhaust gas. The electrostatic particle collector charges the dry particles' surface and deflects them with an electrical field. To produce the electrical field, a high voltage (16 kV) is applied to a round collector tube. The electrical field builds up between the inner wall of the collector tube and the tips of a grounded star electrode. To have a low level of energy in the system the current is very low. After getting deflected the particles stay at the inner wall of the particle collector tube and are completely uncharged. This separation method works fine for all kinds of materials. The efficiency of the electrostatic particle collector is very high: 99% of all particles that enter the system are collected. == References ==
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In machine learning, the kernel perceptron is a variant of the popular perceptron learning algorithm that can learn kernel machines, i.e. non-linear classifiers that employ a kernel function to compute the similarity of unseen samples to training samples. The algorithm was invented in 1964, making it the first kernel classification learner. == Preliminaries == === The perceptron algorithm === The perceptron algorithm is an online learning algorithm that operates by a principle called "error-driven learning". It iteratively improves a model by running it on training samples, then updating the model whenever it finds it has made an incorrect classification with respect to a supervised signal. The model learned by the standard perceptron algorithm is a linear binary classifier: a vector of weights w (and optionally an intercept term b, omitted here for simplicity) that is used to classify a sample vector x as class "one" or class "minus one" according to y ^ = sgn ( w ⊤ x ) {\displaystyle {\hat {y}}=\operatorname {sgn}(\mathbf {w} ^{\top }\mathbf {x} )} where a zero is arbitrarily mapped to one or minus one. (The "hat" on ŷ denotes an estimated value.) In pseudocode, the perceptron algorithm is given by: Initialize w to an all-zero vector of length p, the number of predictors (features). For some fixed number of iterations, or until some stopping criterion is met: For each training example xi with ground truth label yi ∈ {-1, 1}: Let ŷ = sgn(wT xi). If ŷ ≠ yi, update w ← w + yi xi. === Kernel Methods === By contrast with the linear models learned by the perceptron, a kernel method is a classifier that stores a subset of its training examples xi, associates with each a weight αi, and makes decisions for new samples x' by evaluating sgn ∑
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"page_id": 41485209,
"source": null,
"title": "Kernel perceptron"
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i α i y i K ( x i , x ′ ) {\displaystyle \operatorname {sgn} \sum _{i}\alpha _{i}y_{i}K(\mathbf {x} _{i},\mathbf {x'} )} . Here, K is some kernel function. Formally, a kernel function is a non-negative semidefinite kernel (see Mercer's condition), representing an inner product between samples in a high-dimensional space, as if the samples had been expanded to include additional features by a function Φ: K(x, x') = Φ(x) · Φ(x'). Intuitively, it can be thought of as a similarity function between samples, so the kernel machine establishes the class of a new sample by weighted comparison to the training set. Each function x' ↦ K(xi, x') serves as a basis function in the classification. == Algorithm == To derive a kernelized version of the perceptron algorithm, we must first formulate it in dual form, starting from the observation that the weight vector w can be expressed as a linear combination of the n training samples. The equation for the weight vector is w = ∑ i n α i y i x i {\displaystyle \mathbf {w} =\sum _{i}^{n}\alpha _{i}y_{i}\mathbf {x} _{i}} where αi is the number of times xi was misclassified, forcing an update w ← w + yi xi. Using this result, we can formulate the dual perceptron algorithm, which loops through the samples as before, making predictions, but instead of storing and updating a weight vector w, it updates a "mistake counter" vector α. We must also rewrite the prediction formula to get rid of w: y ^ = sgn ( w T x ) = sgn ( ∑ i n α i y i x i ) T x = sgn ∑ i n α i y i ( x i ⋅ x ) {\displaystyle {\begin{aligned}{\hat {y}}&=\operatorname {sgn}(\mathbf {w} ^{\mathsf {T}}\mathbf
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{
"page_id": 41485209,
"source": null,
"title": "Kernel perceptron"
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{x} )\\&=\operatorname {sgn} \left(\sum _{i}^{n}\alpha _{i}y_{i}\mathbf {x} _{i}\right)^{\mathsf {T}}\mathbf {x} \\&=\operatorname {sgn} \sum _{i}^{n}\alpha _{i}y_{i}(\mathbf {x} _{i}\cdot \mathbf {x} )\end{aligned}}} Plugging these two equations into the training loop turn it into the dual perceptron algorithm. Finally, we can replace the dot product in the dual perceptron by an arbitrary kernel function, to get the effect of a feature map Φ without computing Φ(x) explicitly for any samples. Doing this yields the kernel perceptron algorithm: Initialize α to an all-zeros vector of length n, the number of training samples. For some fixed number of iterations, or until some stopping criterion is met: For each training example xj, yj: Let y ^ = sgn ∑ i n α i y i K ( x i , x j ) {\displaystyle {\hat {y}}=\operatorname {sgn} \sum _{i}^{n}\alpha _{i}y_{i}K(\mathbf {x} _{i},\mathbf {x} _{j})} If ŷ ≠ yj, perform an update by incrementing the mistake counter: αj ← αj + 1 == Variants and extensions == One problem with the kernel perceptron, as presented above, is that it does not learn sparse kernel machines. Initially, all the αi are zero so that evaluating the decision function to get ŷ requires no kernel evaluations at all, but each update increments a single αi, making the evaluation increasingly more costly. Moreover, when the kernel perceptron is used in an online setting, the number of non-zero αi and thus the evaluation cost grow linearly in the number of examples presented to the algorithm. The forgetron variant of the kernel perceptron was suggested to deal with this problem. It maintains an active set of examples with non-zero αi, removing ("forgetting") examples from the active set when it exceeds a pre-determined budget and "shrinking" (lowering the weight of) old examples as new ones are promoted to non-zero αi. Another problem
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{
"page_id": 41485209,
"source": null,
"title": "Kernel perceptron"
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with the kernel perceptron is that it does not regularize, making it vulnerable to overfitting. The NORMA online kernel learning algorithm can be regarded as a generalization of the kernel perceptron algorithm with regularization. The sequential minimal optimization (SMO) algorithm used to learn support vector machines can also be regarded as a generalization of the kernel perceptron. The voted perceptron algorithm of Freund and Schapire also extends to the kernelized case, giving generalization bounds comparable to the kernel SVM. == References ==
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{
"page_id": 41485209,
"source": null,
"title": "Kernel perceptron"
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Charusita Chakravarty (5 May 1964 – 29 March 2016) was an Indian academic and scientist. She was a professor of chemistry at the Indian Institute of Technology, Delhi since 1999. In 2009 she was conferred Shanti Swarup Bhatnagar Prize for Science and Technology in the field of chemical science. In 1999, she received B.M. Birla Science Award. She was an Associate Member of the Centre for Computational Material Science, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore. On 29 March 2016, Chakravarty passed after a long and arduous battle with breast cancer. == Early life and education == Chakravarty was born in Cambridge, Massachusetts, U.S. on 5 May 1964 as the only daughter of Sukhamoy and Lalita Chakravarty. She was raised in Delhi, India and chose to give up her American citizenship in her twenties. Chakravarty was selected as the National Science Talent Scholar and went on to clear the Joint Entrance Exam (JEE) of the Indian Institutes of Technology (IIT). She did her BSc Chemistry program from St. Stephen's College, University of Delhi. Having graduated from Delhi University with a gold medal, she went on to do the Natural Science Tripos from Cambridge University, UK. Following this, she joined the Doctorate of Philosophy program at Cambridge under the guidance of David Clary. Her thesis was on the spectra and dynamics of Ar–OH, an open shell system that involved a lot of nuances. Charusita then became a Post Doctoral Scholar at the University of California at Santa Barbara, under Professor Horia Metiu. After a brief visit to India, she returned to Cambridge as a Gulbenkian junior research fellow in an independent post-Doctoral position. == Career == In 1994 Chakravarty returned to India for good. The IITs hesitated to give her a teaching position as she did not have a master's
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{
"page_id": 42206106,
"source": null,
"title": "Charusita Chakravarty"
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degree, even though she had a PhD from Cambridge. She did get an offer from IIT Kanpur, and then went on to accept a position in IIT Delhi's Department of Chemistry, where she continued to teach till her death. Soon after joining IIT Delhi, she submitted a research proposal to the Department of Science and Technology and having received funding easily, carried on with her research. Her initial work was related to atomic and molecular clusters and over the course of her career, she became famous for her specialised application of path integral Monte Carlo simulation to unravel quantum mechanical effects in the properties of atomic and molecular clusters. Her fields of interest also included theoretical chemistry and chemical physics, the structure and dynamics of Liquids, water and hydration, nucleation and self-assembly. International and national journals have published her articles and she was widely known for her single-author papers, published extensively over the course of her career. A few of her co-written works include, Multiple Time-scale Behaviour of the Hydrogen Bond Network in Water (2004), Estimating the entropy of liquids from atom-atom radial distribution functions: silica, beryllium fluoride and water (2008), and Excess entropy scaling of transport properties in network-forming ionic melts (2011). == Research fields == Chakravarty worked in the following fields— Theoretical chemistry and chemical physics Classical and quantum Monte Carlo Molecular dynamics Structure and Dynamics of Liquids Water and hydration Nucleation Self-assembly == Selected publications == Here is a list of selected publications and collaborative research works where Chakravarty has worked— Agarwal, M., Singh, M., Jabes, S. B., and Charusita Chakravarty, Excess entropy scaling of transport properties in network-forming ionic melts (SiO2 and BeF2). J. Chem. Phys. 2011, 134, 014502 Sharma, R., Agarwal, M. and Charusita, C. Estimating the entropy of liquids from atom-atom radial distribution functions:
|
{
"page_id": 42206106,
"source": null,
"title": "Charusita Chakravarty"
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silica, beryllium fluoride and water. Mol. Phys. 2008, 106, 1925. Agarwal, M. and Chakravarty, C. Waterlike structural and excess entropy anomalies in liquid beryllium fluoride. J. Phys. Chem. B, 2007, 111, 13294. Sharma, R., Nath, Chakraborty, S. N. and Charusita C. Entropy, Diffusivity and Structural Order in Liquids with Water-like Anomalies. J. Chem. Phys. 2006, 125, 204501. Mudi, A.; Chakravarty, C. Multiple Time-scale Behaviour of the Hydrogen Bond Network in Water. J. Phys. Chem. B, 2004, 108, 19607. == Awards and achievements == Medal for Young Scientists from the Indian National Science Academy (INSA) (1996) Shanti Swarup Bhatnagar Prize for Science and Technology (2009) B.M. Birla Science Award (1999) Indian National Science Academy Medal for Young Scientists (1996) Anil Kumar Bose Memorial Award of Indian National Science Academy (1999) Fellowship of Indian Academy of Sciences (2006) Swarnajayanti Fellowship of the Department of Science and Technology (India) (2004) Served as a member of the Abdus Salam International Center for Theoretical Physics, Trieste, Italy (1996–2003) Associate Member of the Centre for Computational Material Science, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore. == References == == Other sources == Autobiographical article by Charusita Chakravarty in Lilavati's Daughters "Profile of Top 25 scientists in India". India Today. 11 September 2011. Retrieved 17 January 2018.
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{
"page_id": 42206106,
"source": null,
"title": "Charusita Chakravarty"
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The Bacterial protein tyrosine-kinase database (BYKdb) is a specialized database of computer-annotated bacterial tyrosine-kinases that share no resemblance with their eukaryotic counterparts. == See also == Tyrosine-kinases == References == == External links == http://bykdb.ibcp.fr
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{
"page_id": 34472859,
"source": null,
"title": "BYKdb"
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Detached eddy simulation (DES) is a modification of a Reynolds-averaged Navier–Stokes equations (RANS) model in which the model switches to a subgrid scale formulation in regions fine enough for large eddy simulation (LES) calculations. == Details == Regions near solid boundaries and where the turbulent length scale is less than the maximum grid dimension are assigned the RANS mode of solution. As the turbulent length scale exceeds the grid dimension, the regions are solved using the LES mode. Therefore, the grid resolution is not as demanding as pure LES, thereby considerably cutting down the cost of the computation. Though DES was initially formulated for the Spalart-Allmaras model, it can be implemented with other RANS models (Strelets, 2001), by appropriately modifying the length scale which is explicitly or implicitly involved in the RANS model. So while Spalart-Allmaras model based DES acts as LES with a wall model, DES based on other models (like two equation models) behave as a hybrid RANS-LES model. Grid generation is more complicated than for a simple RANS or LES case due to the RANS-LES switch. DES is a non-zonal approach and provides a single smooth velocity field across the RANS and the LES regions of the solution. == References == == External links == CFD wiki article on DES technique Article comparing RANS and DES for Automotive Applications.
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{
"page_id": 2884506,
"source": null,
"title": "Detached eddy simulation"
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In molecular biology mir-216 microRNA is a short RNA molecule. MicroRNAs function to regulate the expression levels of other genes by several mechanisms. == See also == MicroRNA == References == == Further reading == == External links == Page for mir-216 microRNA precursor family at Rfam
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{
"page_id": 36373405,
"source": null,
"title": "Mir-216 microRNA precursor family"
}
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A list of particle accelerators used for particle physics experiments. Some early particle accelerators that more properly did nuclear physics, but existed prior to the separation of particle physics from that field, are also included. Although a modern accelerator complex usually has several stages of accelerators, only accelerators whose output has been used directly for experiments are listed. == Early accelerators == These all used single beams with fixed targets. They tended to have very briefly run, inexpensive, and unnamed experiments. === Cyclotrons === [1] The magnetic pole pieces and return yoke from the 60-inch cyclotron were later moved to UC Davis and incorporated into a 76-inch isochronous cyclotron which is still in use today === Other early accelerator types === === Synchrotrons === == Fixed-target accelerators == More modern accelerators that were also run in fixed target mode; often, they will also have been run as colliders, or accelerated particles for use in subsequently built colliders. === High intensity hadron accelerators (Meson and neutron sources) === === Electron and low intensity hadron accelerators === == Colliders == === Electron–positron colliders === === Hadron colliders === === Electron-proton colliders === == Light sources == == Hypothetical accelerators == Besides the real accelerators listed above, there are hypothetical accelerators often used as hypothetical examples or optimistic projects by particle physicists. Eloisatron (Eurasiatic Long Intersecting Storage Accelerator) was a project of INFN headed by Antonio Zichichi at the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Sicily. The center-of-mass energy was planned to be 200 TeV, and the size was planned to span parts of Europe and Asia. Fermitron was an accelerator sketched by Enrico Fermi on a notepad in the 1940s proposing an accelerator in stable orbit around the Earth. The undulator radiation collider is a design for an
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{
"page_id": 3343262,
"source": null,
"title": "List of accelerators in particle physics"
}
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accelerator with a center-of-mass energy around the GUT scale. It would be light-weeks across and require the construction of a Dyson swarm around the Sun. Planckatron is an accelerator with a center-of-mass energy of the order of the Planck scale. It is estimated that the radius of the Planckatron would have to be roughly the radius of the Milky Way. It would require so much energy to run that it could only be built by at least a Kardashev Type II civilization. Arguably also in this category falls the Zevatron, a hypothetical source for observed ultra-high-energy cosmic rays. == See also == List of accelerator mass spectrometry facilities List of synchrotron radiation facilities == References == == External links == Judy Goldhaber. October 9, 1992. Bevalac Had 40-Year Record of Historic Discoveries Archived 2011-05-14 at the Wayback Machine High-energy collider parameters from the Particle Data Group Particle accelerators around the world Lawrence and his laboratory Archived 2018-01-18 at the Wayback Machine – a history of the early years of accelerator physics at Lawrence Berkeley Laboratory A brief history and review of accelerators (11 pgs, PDF file) SLAC beamlines over time Accelerators and detectors named Mark at SLAC Lawson, J. D. (1997), "Early British Synchrotrons, An Informal History", [accessed 17 May 2009] A FEW QUICK FACTS ABOUT THE TRIUMF CYCLOTRON
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{
"page_id": 3343262,
"source": null,
"title": "List of accelerators in particle physics"
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The rangeland health refers to the degree to which the integrity of the soil and ecological processes of rangeland ecosystems are sustained. The attributes evaluated during rangeland health assessments are 1) Soil and Site Stability 2) Hydrologic Function 3) Biotic Integrity. == References ==
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{
"page_id": 72745888,
"source": null,
"title": "Rangeland health"
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Epis (, Haitian Creole: epis) is a blend of peppers, garlic, and herbs that is used as a flavor base for many foods in Haitian cuisine. Some refer to it as a pesto sauce. It is also known as epise and zepis. It is essential for Haitian cuisine. == Background == Epis has Taino and African origins. It also has similarities to sofrito which is used in Hispanic cuisine. This use of a flavor base is common in Caribbean cuisine. == Ingredients == Epis often contains parsley, scallions, garlic, citrus juice, and Scotch bonnet peppers. Numerous recipes for epis exist, as traditionally, Haitian women would cook and have their personal epis recipe. Also, various regions have different recipes. == Preparation == Traditionally, epis is made with a large wooden mortar and pestle (called munsh pilon). Today, it is often made with a blender. The ingredients are blended until the consistency is as smooth as desired. == Use == It can be used as a marinade for meat. It can also marinate fish. It also is added to flavor a number of Haitian dishes. This includes rice and beans, soups, and stews. It is a convenient way to utilize flavors from fresh herbs and spices in everyday cooking. Many Haitians have epis available on hand to be used for various dishes. === Dishes === Griot Haitian spaghetti == Storage == Epis can last up to three months in the refrigerator, but this time will vary depending on the ingredients that are used. The acidity helps keep the ingredients from spoiling. Epis will last indefinitely in the freezer and will not transfer its odor to other freezer items. The epis can be distributed in an ice cube tray and frozen, so that the frozen cubes can be used in future cooking. ==
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{
"page_id": 67371936,
"source": null,
"title": "Epis"
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|
See also == Sofrito Holy trinity Tempering (spices) Mirepoix Sauce == References ==
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{
"page_id": 67371936,
"source": null,
"title": "Epis"
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Tellurium oxide may refer to: Tellurium monoxide, TeO Tellurium dioxide, TeO2 Tellurium trioxide, TeO3
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{
"page_id": 27591586,
"source": null,
"title": "Tellurium oxide"
}
|
V. Parmeswaran Nair is a physicist, currently a Distinguished Professor at City College of New York, holding it since September 2011, and is also a published author. == Early life and education == Nair received a bachelor degree in 1976 and a masters degree in 1978 at the University of Kerala in India. He also received a doctorate in 1983 at Syracuse University in New York. == References == ORCID profile
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{
"page_id": 52429737,
"source": null,
"title": "V. Parmeswaran Nair"
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|
A pheromone trap is a type of insect trap that uses pheromones to lure insects. Sex pheromones and aggregating pheromones are the most common types used. A pheromone-impregnated lure is encased in a conventional trap such as a bottle trap, delta trap, water-pan trap, or funnel trap. Pheromone traps are used both to count insect populations by sampling, and to trap pests such as clothes moths to destroy them. == Sensitivity == Pheromone traps are very sensitive, meaning they attract insects present at very low densities. They are often used to detect presence of exotic pests, or for sampling, monitoring, or to determine the first appearance of a pest in an area. They can be used for legal control, and are used to monitor the success of the Boll Weevil Eradication Program and the spread of the spongy moth. The high species-specificity of pheromone traps can also be an advantage, and they tend to be inexpensive and easy to implement. This sensitivity is especially suited to some investigations of invasive species: Flying males are easily blown off course by winds. Rather than introducing noise, Frank et al. 2013 find this can actually help detect isolated nests or populations and determine the length of time necessary between introduction and establishment. (Although any trap can answer the same questions, high sensitivity such as provided by pheromone traps does so more accurately.) However, it is impractical in most cases to completely remove or "trap out" pests using a pheromone trap. Some pheromone-based pest control methods have been successful, usually those designed to protect enclosed areas such as households or storage facilities. There has also been some success in mating disruption. In one form of mating disruption, males are attracted to a powder containing female attractant pheromones. The pheromones stick to the males' bodies,
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{
"page_id": 24118186,
"source": null,
"title": "Pheromone trap"
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|
and when they fly off, the pheromones make them attractive to other males. It is hoped that if enough males chase other males instead of females, egg-laying will be severely impeded. Some difficulties surrounding pheromone traps include sensitivity to bad weather, their ability to attract pests from neighboring areas, and that they generally only attract adults, although it is the juveniles in many species that are pests. They are also generally limited to one sex. == Targets == Though certainly not all insect pheromones have been discovered, many are known and many more are discovered every year. Some sites curate large lists of insect pheromones. Pheromones are frequently used to monitor and control lepidopteran and coleopteran species, with many available commercially. Pheromones are available for insects including: == Gallery == == References ==
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{
"page_id": 24118186,
"source": null,
"title": "Pheromone trap"
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CompHEP is a software package for automatic computations in high energy physics from Lagrangians to collision events or particle decays. CompHEP is based on quantum theory of gauge fields, namely it uses the technique of squared Feynman diagrams at the tree-level approximation. By default, CompHEP includes the Standard Model Lagrangian in the unitarity and 't Hooft-Feynman gauges and several MSSM models. However users can create new physical models, based on different Lagrangians. There is a special tool for that - LanHEP. CompHEP is able to compute basically the LO cross sections and distributions with several particles in the final state (up to 6-7). It can take into account, if necessary, all QCD and EW diagrams, masses of fermions and bosons and widths of unstable particles. Processes computed by means of CompHEP can be interfaced to the Monte-Carlo generators PYTHIA and HERWIG as new external processes. The CompHEP project started in 1989 in Skobeltsyn Institute of Nuclear Physics (SINP) of Moscow State University. During the 1990s this package was developed, and now it is a powerful tool for automatic computations of collision processes. The CompHEP program has been used in the past for many studies in many experimental groups as shown schematically in the scheme Due to an intuitive graphical interface CompHEP is a very useful tool for education in particle and nuclear physics. == External links == official CompHEP page manual for version 3.3 Boos, E.; Bunichev, V.; Dubinin, M.; Dudko, L.; Edneral, V.; Ilyin, V.; Kryukov, A.; Savrin, V.; Semenov, A.; Sherstnev, A. (2004). "CompHEP 4.4—automatic computations from Lagrangians to events". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 534 (1–2). Elsevier BV: 250–259. arXiv:hep-ph/0403113. doi:10.1016/j.nima.2004.07.096. ISSN 0168-9002. Skobeltsyn Institute of Nuclear Physics (SINP)
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{
"page_id": 12125099,
"source": null,
"title": "CompHEP"
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In mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence), denoted D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} , is a type of statistical distance: a measure of how much a model probability distribution Q is different from a true probability distribution P. Mathematically, it is defined as D KL ( P ∥ Q ) = ∑ x ∈ X P ( x ) log P ( x ) Q ( x ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {P(x)}{Q(x)}}.} A simple interpretation of the KL divergence of P from Q is the expected excess surprise from using Q as a model instead of P when the actual distribution is P. While it is a measure of how different two distributions are and is thus a distance in some sense, it is not actually a metric, which is the most familiar and formal type of distance. In particular, it is not symmetric in the two distributions (in contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence, a generalization of squared distance, and for certain classes of distributions (notably an exponential family), it satisfies a generalized Pythagorean theorem (which applies to squared distances). Relative entropy is always a non-negative real number, with value 0 if and only if the two distributions in question are identical. It has diverse applications, both theoretical, such as characterizing the relative (Shannon) entropy in information systems, randomness in continuous time-series, and information gain when comparing statistical models of inference; and practical, such as applied statistics, fluid mechanics, neuroscience, bioinformatics, and machine learning. == Introduction and context == Consider two probability distributions P and Q. Usually, P represents the data, the observations, or
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"page_id": 1115052,
"source": null,
"title": "Kullback–Leibler divergence"
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a measured probability distribution. Distribution Q represents instead a theory, a model, a description or an approximation of P. The Kullback–Leibler divergence D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is then interpreted as the average difference of the number of bits required for encoding samples of P using a code optimized for Q rather than one optimized for P. Note that the roles of P and Q can be reversed in some situations where that is easier to compute, such as with the expectation–maximization algorithm (EM) and evidence lower bound (ELBO) computations. == Etymology == The relative entropy was introduced by Solomon Kullback and Richard Leibler in Kullback & Leibler (1951) as "the mean information for discrimination between H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} per observation from μ 1 {\displaystyle \mu _{1}} ", where one is comparing two probability measures μ 1 , μ 2 {\displaystyle \mu _{1},\mu _{2}} , and H 1 , H 2 {\displaystyle H_{1},H_{2}} are the hypotheses that one is selecting from measure μ 1 , μ 2 {\displaystyle \mu _{1},\mu _{2}} (respectively). They denoted this by I ( 1 : 2 ) {\displaystyle I(1:2)} , and defined the "'divergence' between μ 1 {\displaystyle \mu _{1}} and μ 2 {\displaystyle \mu _{2}} " as the symmetrized quantity J ( 1 , 2 ) = I ( 1 : 2 ) + I ( 2 : 1 ) {\displaystyle J(1,2)=I(1:2)+I(2:1)} , which had already been defined and used by Harold Jeffreys in 1948. In Kullback (1959), the symmetrized form is again referred to as the "divergence", and the relative entropies in each direction are referred to as a "directed divergences" between two distributions; Kullback preferred the term discrimination information. The term "divergence" is in contrast to a distance (metric), since the
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"page_id": 1115052,
"source": null,
"title": "Kullback–Leibler divergence"
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symmetrized divergence does not satisfy the triangle inequality. Numerous references to earlier uses of the symmetrized divergence and to other statistical distances are given in Kullback (1959, pp. 6–7, §1.3 Divergence). The asymmetric "directed divergence" has come to be known as the Kullback–Leibler divergence, while the symmetrized "divergence" is now referred to as the Jeffreys divergence. == Definition == For discrete probability distributions P and Q defined on the same sample space, X {\displaystyle {\mathcal {X}}} , the relative entropy from Q to P is defined to be D KL ( P ∥ Q ) = ∑ x ∈ X P ( x ) log P ( x ) Q ( x ) , {\displaystyle D_{\text{KL}}(P\parallel Q)=\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {P(x)}{Q(x)}}\,,} which is equivalent to D KL ( P ∥ Q ) = − ∑ x ∈ X P ( x ) log Q ( x ) P ( x ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=-\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {Q(x)}{P(x)}}\,.} In other words, it is the expectation of the logarithmic difference between the probabilities P and Q, where the expectation is taken using the probabilities P. Relative entropy is only defined in this way if, for all x, Q ( x ) = 0 {\displaystyle Q(x)=0} implies P ( x ) = 0 {\displaystyle P(x)=0} (absolute continuity). Otherwise, it is often defined as + ∞ {\displaystyle +\infty } , but the value + ∞ {\displaystyle \ +\infty \ } is possible even if Q ( x ) ≠ 0 {\displaystyle Q(x)\neq 0} everywhere, provided that X {\displaystyle {\mathcal {X}}} is infinite in extent. Analogous comments apply to the continuous and general measure cases defined below. Whenever P ( x ) {\displaystyle P(x)} is zero the contribution of the corresponding term is interpreted as zero because lim x
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{
"page_id": 1115052,
"source": null,
"title": "Kullback–Leibler divergence"
}
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→ 0 + x log ( x ) = 0 . {\displaystyle \lim _{x\to 0^{+}}x\,\log(x)=0\,.} For distributions P and Q of a continuous random variable, relative entropy is defined to be the integral D KL ( P ∥ Q ) = ∫ − ∞ ∞ p ( x ) log p ( x ) q ( x ) d x , {\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{-\infty }^{\infty }p(x)\,\log {\frac {p(x)}{q(x)}}\,dx\,,} where p and q denote the probability densities of P and Q. More generally, if P and Q are probability measures on a measurable space X , {\displaystyle {\mathcal {X}}\,,} and P is absolutely continuous with respect to Q, then the relative entropy from Q to P is defined as D KL ( P ∥ Q ) = ∫ x ∈ X log P ( d x ) Q ( d x ) P ( d x ) , {\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{x\in {\mathcal {X}}}\log {\frac {P(dx)}{Q(dx)}}\,P(dx)\,,} where P ( d x ) Q ( d x ) {\displaystyle {\frac {P(dx)}{Q(dx)}}} is the Radon–Nikodym derivative of P with respect to Q, i.e. the unique Q almost everywhere defined function r on X {\displaystyle {\mathcal {X}}} such that P ( d x ) = r ( x ) Q ( d x ) {\displaystyle P(dx)=r(x)Q(dx)} which exists because P is absolutely continuous with respect to Q. Also we assume the expression on the right-hand side exists. Equivalently (by the chain rule), this can be written as D KL ( P ∥ Q ) = ∫ x ∈ X P ( d x ) Q ( d x ) log P ( d x ) Q ( d x ) Q ( d x ) , {\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{x\in {\mathcal {X}}}{\frac {P(dx)}{Q(dx)}}\ \log {\frac {P(dx)}{Q(dx)}}\ Q(dx)\,,} which is
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the entropy of P relative to Q. Continuing in this case, if μ {\displaystyle \mu } is any measure on X {\displaystyle {\mathcal {X}}} for which densities p and q with P ( d x ) = p ( x ) μ ( d x ) {\displaystyle P(dx)=p(x)\mu (dx)} and Q ( d x ) = q ( x ) μ ( d x ) {\displaystyle Q(dx)=q(x)\mu (dx)} exist (meaning that P and Q are both absolutely continuous with respect to μ {\displaystyle \mu } ), then the relative entropy from Q to P is given as D KL ( P ∥ Q ) = ∫ x ∈ X p ( x ) log p ( x ) q ( x ) μ ( d x ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{x\in {\mathcal {X}}}p(x)\,\log {\frac {p(x)}{q(x)}}\ \mu (dx)\,.} Note that such a measure μ {\displaystyle \mu } for which densities can be defined always exists, since one can take μ = 1 2 ( P + Q ) {\textstyle \mu ={\frac {1}{2}}\left(P+Q\right)} although in practice it will usually be one that applies in the context like counting measure for discrete distributions, or Lebesgue measure or a convenient variant thereof like Gaussian measure or the uniform measure on the sphere, Haar measure on a Lie group etc. for continuous distributions. The logarithms in these formulae are usually taken to base 2 if information is measured in units of bits, or to base e if information is measured in nats. Most formulas involving relative entropy hold regardless of the base of the logarithm. Various conventions exist for referring to D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} in words. Often it is referred to as the divergence between P and Q, but this fails to convey the fundamental asymmetry
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in the relation. Sometimes, as in this article, it may be described as the divergence of P from Q or as the divergence from Q to P. This reflects the asymmetry in Bayesian inference, which starts from a prior Q and updates to the posterior P. Another common way to refer to D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is as the relative entropy of P with respect to Q or the information gain from P over Q. == Basic example == Kullback gives the following example (Table 2.1, Example 2.1). Let P and Q be the distributions shown in the table and figure. P is the distribution on the left side of the figure, a binomial distribution with N = 2 {\displaystyle N=2} and p = 0.4 {\displaystyle p=0.4} . Q is the distribution on the right side of the figure, a discrete uniform distribution with the three possible outcomes x = 0, 1, 2 (i.e. X = { 0 , 1 , 2 } {\displaystyle {\mathcal {X}}=\{0,1,2\}} ), each with probability p = 1 / 3 {\displaystyle p=1/3} . Relative entropies D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} and D KL ( Q ∥ P ) {\displaystyle D_{\text{KL}}(Q\parallel P)} are calculated as follows. This example uses the natural log with base e, designated ln to get results in nats (see units of information): D KL ( P ∥ Q ) = ∑ x ∈ X P ( x ) ln P ( x ) Q ( x ) = 9 25 ln 9 / 25 1 / 3 + 12 25 ln 12 / 25 1 / 3 + 4 25 ln 4 / 25 1 / 3 = 1 25 ( 32 ln 2 + 55
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ln 3 − 50 ln 5 ) ≈ 0.0852996 , {\displaystyle {\begin{aligned}D_{\text{KL}}(P\parallel Q)&=\sum _{x\in {\mathcal {X}}}P(x)\,\ln {\frac {P(x)}{Q(x)}}\\&={\frac {9}{25}}\ln {\frac {9/25}{1/3}}+{\frac {12}{25}}\ln {\frac {12/25}{1/3}}+{\frac {4}{25}}\ln {\frac {4/25}{1/3}}\\&={\frac {1}{25}}\left(32\ln 2+55\ln 3-50\ln 5\right)\\&\approx 0.0852996,\end{aligned}}} D KL ( Q ∥ P ) = ∑ x ∈ X Q ( x ) ln Q ( x ) P ( x ) = 1 3 ln 1 / 3 9 / 25 + 1 3 ln 1 / 3 12 / 25 + 1 3 ln 1 / 3 4 / 25 = 1 3 ( − 4 ln 2 − 6 ln 3 + 6 ln 5 ) ≈ 0.097455. {\displaystyle {\begin{aligned}D_{\text{KL}}(Q\parallel P)&=\sum _{x\in {\mathcal {X}}}Q(x)\,\ln {\frac {Q(x)}{P(x)}}\\&={\frac {1}{3}}\,\ln {\frac {1/3}{9/25}}+{\frac {1}{3}}\,\ln {\frac {1/3}{12/25}}+{\frac {1}{3}}\,\ln {\frac {1/3}{4/25}}\\&={\frac {1}{3}}\left(-4\ln 2-6\ln 3+6\ln 5\right)\\&\approx 0.097455.\end{aligned}}} == Interpretations == === Statistics === In the field of statistics, the Neyman–Pearson lemma states that the most powerful way to distinguish between the two distributions P and Q based on an observation Y (drawn from one of them) is through the log of the ratio of their likelihoods: log P ( Y ) − log Q ( Y ) {\displaystyle \log P(Y)-\log Q(Y)} . The KL divergence is the expected value of this statistic if Y is actually drawn from P. Kullback motivated the statistic as an expected log likelihood ratio. === Coding === In the context of coding theory, D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} can be constructed by measuring the expected number of extra bits required to code samples from P using a code optimized for Q rather than the code optimized for P. === Inference === In the context of machine learning, D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is
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often called the information gain achieved if P would be used instead of Q which is currently used. By analogy with information theory, it is called the relative entropy of P with respect to Q. Expressed in the language of Bayesian inference, D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is a measure of the information gained by revising one's beliefs from the prior probability distribution Q to the posterior probability distribution P. In other words, it is the amount of information lost when Q is used to approximate P. === Information geometry === In applications, P typically represents the "true" distribution of data, observations, or a precisely calculated theoretical distribution, while Q typically represents a theory, model, description, or approximation of P. In order to find a distribution Q that is closest to P, we can minimize the KL divergence and compute an information projection. While it is a statistical distance, it is not a metric, the most familiar type of distance, but instead it is a divergence. While metrics are symmetric and generalize linear distance, satisfying the triangle inequality, divergences are asymmetric and generalize squared distance, in some cases satisfying a generalized Pythagorean theorem. In general D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} does not equal D KL ( Q ∥ P ) {\displaystyle D_{\text{KL}}(Q\parallel P)} , and the asymmetry is an important part of the geometry. The infinitesimal form of relative entropy, specifically its Hessian, gives a metric tensor that equals the Fisher information metric; see § Fisher information metric. Fisher information metric on the certain probability distribution let determine the natural gradient for information-geometric optimization algorithms. Its quantum version is Fubini-study metric. Relative entropy satisfies a generalized Pythagorean theorem for exponential families (geometrically interpreted as dually flat manifolds), and this allows
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one to minimize relative entropy by geometric means, for example by information projection and in maximum likelihood estimation. The relative entropy is the Bregman divergence generated by the negative entropy, but it is also of the form of an f-divergence. For probabilities over a finite alphabet, it is unique in being a member of both of these classes of statistical divergences. The application of Bregman divergence can be found in mirror descent. === Finance (game theory) === Consider a growth-optimizing investor in a fair game with mutually exclusive outcomes (e.g. a “horse race” in which the official odds add up to one). The rate of return expected by such an investor is equal to the relative entropy between the investor's believed probabilities and the official odds. This is a special case of a much more general connection between financial returns and divergence measures. Financial risks are connected to D KL {\displaystyle D_{\text{KL}}} via information geometry. Investors' views, the prevailing market view, and risky scenarios form triangles on the relevant manifold of probability distributions. The shape of the triangles determines key financial risks (both qualitatively and quantitatively). For instance, obtuse triangles in which investors' views and risk scenarios appear on “opposite sides” relative to the market describe negative risks, acute triangles describe positive exposure, and the right-angled situation in the middle corresponds to zero risk. Extending this concept, relative entropy can be hypothetically utilised to identify the behaviour of informed investors, if one takes this to be represented by the magnitude and deviations away from the prior expectations of fund flows, for example. == Motivation == In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value x i {\displaystyle x_{i}} out of a set of possibilities X can be seen
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as representing an implicit probability distribution q ( x i ) = 2 − ℓ i {\displaystyle q(x_{i})=2^{-\ell _{i}}} over X, where ℓ i {\displaystyle \ell _{i}} is the length of the code for x i {\displaystyle x_{i}} in bits. Therefore, relative entropy can be interpreted as the expected extra message-length per datum that must be communicated if a code that is optimal for a given (wrong) distribution Q is used, compared to using a code based on the true distribution P: it is the excess entropy. D KL ( P ∥ Q ) = ∑ x ∈ X p ( x ) log 1 q ( x ) − ∑ x ∈ X p ( x ) log 1 p ( x ) = H ( P , Q ) − H ( P ) {\displaystyle {\begin{aligned}D_{\text{KL}}(P\parallel Q)&=\sum _{x\in {\mathcal {X}}}p(x)\log {\frac {1}{q(x)}}-\sum _{x\in {\mathcal {X}}}p(x)\log {\frac {1}{p(x)}}\\[5pt]&=\mathrm {H} (P,Q)-\mathrm {H} (P)\end{aligned}}} where H ( P , Q ) {\displaystyle \mathrm {H} (P,Q)} is the cross entropy of Q relative to P and H ( P ) {\displaystyle \mathrm {H} (P)} is the entropy of P (which is the same as the cross-entropy of P with itself). The relative entropy D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} can be thought of geometrically as a statistical distance, a measure of how far the distribution Q is from the distribution P. Geometrically it is a divergence: an asymmetric, generalized form of squared distance. The cross-entropy H ( P , Q ) {\displaystyle H(P,Q)} is itself such a measurement (formally a loss function), but it cannot be thought of as a distance, since H ( P , P ) =: H ( P ) {\displaystyle H(P,P)=:H(P)} is not zero. This can be fixed by subtracting H (
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P ) {\displaystyle H(P)} to make D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} agree more closely with our notion of distance, as the excess loss. The resulting function is asymmetric, and while this can be symmetrized (see § Symmetrised divergence), the asymmetric form is more useful. See § Interpretations for more on the geometric interpretation. Relative entropy relates to "rate function" in the theory of large deviations. Arthur Hobson proved that relative entropy is the only measure of difference between probability distributions that satisfies some desired properties, which are the canonical extension to those appearing in a commonly used characterization of entropy. Consequently, mutual information is the only measure of mutual dependence that obeys certain related conditions, since it can be defined in terms of Kullback–Leibler divergence. == Properties == Relative entropy is always non-negative, D KL ( P ∥ Q ) ≥ 0 , {\displaystyle D_{\text{KL}}(P\parallel Q)\geq 0,} a result known as Gibbs' inequality, with D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} equals zero if and only if P = Q {\displaystyle P=Q} as measures. In particular, if P ( d x ) = p ( x ) μ ( d x ) {\displaystyle P(dx)=p(x)\mu (dx)} and Q ( d x ) = q ( x ) μ ( d x ) {\displaystyle Q(dx)=q(x)\mu (dx)} , then p ( x ) = q ( x ) {\displaystyle p(x)=q(x)} μ {\displaystyle \mu } -almost everywhere. The entropy H ( P ) {\displaystyle \mathrm {H} (P)} thus sets a minimum value for the cross-entropy H ( P , Q ) {\displaystyle \mathrm {H} (P,Q)} , the expected number of bits required when using a code based on Q rather than P; and the Kullback–Leibler divergence therefore represents the expected number of extra bits that must
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be transmitted to identify a value x drawn from X, if a code is used corresponding to the probability distribution Q, rather than the "true" distribution P. No upper-bound exists for the general case. However, it is shown that if P and Q are two discrete probability distributions built by distributing the same discrete quantity, then the maximum value of D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} can be calculated. Relative entropy remains well-defined for continuous distributions, and furthermore is invariant under parameter transformations. For example, if a transformation is made from variable x to variable y ( x ) {\displaystyle y(x)} , then, since P ( d x ) = p ( x ) d x = p ~ ( y ) d y = p ~ ( y ( x ) ) | d y d x ( x ) | d x {\displaystyle P(dx)=p(x)\,dx={\tilde {p}}(y)\,dy={\tilde {p}}(y(x))\left|{\tfrac {dy}{dx}}(x)\right|\,dx} and Q ( d x ) = q ( x ) d x = q ~ ( y ) d y = q ~ ( y ) | d y d x ( x ) | d x {\displaystyle Q(dx)=q(x)\,dx={\tilde {q}}(y)\,dy={\tilde {q}}(y)\left|{\tfrac {dy}{dx}}(x)\right|dx} where | d y d x ( x ) | {\displaystyle \left|{\tfrac {dy}{dx}}(x)\right|} is the absolute value of the derivative or more generally of the Jacobian, the relative entropy may be rewritten: D KL ( P ∥ Q ) = ∫ x a x b p ( x ) log p ( x ) q ( x ) d x = ∫ x a x b p ~ ( y ( x ) ) | d y d x | log p ~ ( y ( x ) ) | d y d x | q ~ ( y ( x )
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) | d y d x | d x = ∫ y a y b p ~ ( y ) log p ~ ( y ) q ~ ( y ) d y {\displaystyle {\begin{aligned}D_{\text{KL}}(P\parallel Q)&=\int _{x_{a}}^{x_{b}}p(x)\,\log {\frac {p(x)}{q(x)}}\,dx\\[6pt]&=\int _{x_{a}}^{x_{b}}{\tilde {p}}(y(x))\left|{\frac {dy}{dx}}\right|\log {\frac {{\tilde {p}}(y(x))\,\left|{\frac {dy}{dx}}\right|}{{\tilde {q}}(y(x))\,\left|{\frac {dy}{dx}}\right|}}\,dx\\&=\int _{y_{a}}^{y_{b}}{\tilde {p}}(y)\,\log {\frac {{\tilde {p}}(y)}{{\tilde {q}}(y)}}\,dy\end{aligned}}} where y a = y ( x a ) {\displaystyle y_{a}=y(x_{a})} and y b = y ( x b ) {\displaystyle y_{b}=y(x_{b})} . Although it was assumed that the transformation was continuous, this need not be the case. This also shows that the relative entropy produces a dimensionally consistent quantity, since if x is a dimensioned variable, p ( x ) {\displaystyle p(x)} and q ( x ) {\displaystyle q(x)} are also dimensioned, since e.g. P ( d x ) = p ( x ) d x {\displaystyle P(dx)=p(x)\,dx} is dimensionless. The argument of the logarithmic term is and remains dimensionless, as it must. It can therefore be seen as in some ways a more fundamental quantity than some other properties in information theory (such as self-information or Shannon entropy), which can become undefined or negative for non-discrete probabilities. Relative entropy is additive for independent distributions in much the same way as Shannon entropy. If P 1 , P 2 {\displaystyle P_{1},P_{2}} are independent distributions, and P ( d x , d y ) = P 1 ( d x ) P 2 ( d y ) {\displaystyle P(dx,dy)=P_{1}(dx)P_{2}(dy)} , and likewise Q ( d x , d y ) = Q 1 ( d x ) Q 2 ( d y ) {\displaystyle Q(dx,dy)=Q_{1}(dx)Q_{2}(dy)} for independent distributions Q 1 , Q 2 {\displaystyle Q_{1},Q_{2}} then D KL ( P ∥ Q ) = D KL ( P 1 ∥ Q 1 ) + D
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KL ( P 2 ∥ Q 2 ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=D_{\text{KL}}(P_{1}\parallel Q_{1})+D_{\text{KL}}(P_{2}\parallel Q_{2}).} Relative entropy D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is convex in the pair of probability measures ( P , Q ) {\displaystyle (P,Q)} , i.e. if ( P 1 , Q 1 ) {\displaystyle (P_{1},Q_{1})} and ( P 2 , Q 2 ) {\displaystyle (P_{2},Q_{2})} are two pairs of probability measures then D KL ( λ P 1 + ( 1 − λ ) P 2 ∥ λ Q 1 + ( 1 − λ ) Q 2 ) ≤ λ D KL ( P 1 ∥ Q 1 ) + ( 1 − λ ) D KL ( P 2 ∥ Q 2 ) for 0 ≤ λ ≤ 1. {\displaystyle D_{\text{KL}}(\lambda P_{1}+(1-\lambda )P_{2}\parallel \lambda Q_{1}+(1-\lambda )Q_{2})\leq \lambda D_{\text{KL}}(P_{1}\parallel Q_{1})+(1-\lambda )D_{\text{KL}}(P_{2}\parallel Q_{2}){\text{ for }}0\leq \lambda \leq 1.} D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} may be Taylor expanded about its minimum (i.e. P = Q {\displaystyle P=Q} ) as D KL ( P ∥ Q ) = ∑ n = 2 ∞ 1 n ( n − 1 ) ∑ x ∈ X ( Q ( x ) − P ( x ) ) n Q ( x ) n − 1 {\displaystyle D_{\text{KL}}(P\parallel Q)=\sum _{n=2}^{\infty }{\frac {1}{n(n-1)}}\sum _{x\in {\mathcal {X}}}{\frac {(Q(x)-P(x))^{n}}{Q(x)^{n-1}}}} which converges if and only if P ≤ 2 Q {\displaystyle P\leq 2Q} almost surely w.r.t Q {\displaystyle Q} . == Duality formula for variational inference == The following result, due to Donsker and Varadhan, is known as Donsker and Varadhan's variational formula. == Examples == === Multivariate normal distributions === Suppose that we have two multivariate normal distributions, with means μ 0 , μ 1 {\displaystyle \mu _{0},\mu _{1}} and with (non-singular) covariance matrices Σ
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0 , Σ 1 . {\displaystyle \Sigma _{0},\Sigma _{1}.} If the two distributions have the same dimension, k, then the relative entropy between the distributions is as follows: D KL ( N 0 ∥ N 1 ) = 1 2 [ tr ( Σ 1 − 1 Σ 0 ) − k + ( μ 1 − μ 0 ) T Σ 1 − 1 ( μ 1 − μ 0 ) + ln det Σ 1 det Σ 0 ] . {\displaystyle D_{\text{KL}}\left({\mathcal {N}}_{0}\parallel {\mathcal {N}}_{1}\right)={\frac {1}{2}}\left[\operatorname {tr} \left(\Sigma _{1}^{-1}\Sigma _{0}\right)-k+\left(\mu _{1}-\mu _{0}\right)^{\mathsf {T}}\Sigma _{1}^{-1}\left(\mu _{1}-\mu _{0}\right)+\ln {\frac {\det \Sigma _{1}}{\det \Sigma _{0}}}\right].} The logarithm in the last term must be taken to base e since all terms apart from the last are base-e logarithms of expressions that are either factors of the density function or otherwise arise naturally. The equation therefore gives a result measured in nats. Dividing the entire expression above by ln ( 2 ) {\displaystyle \ln(2)} yields the divergence in bits. In a numerical implementation, it is helpful to express the result in terms of the Cholesky decompositions L 0 , L 1 {\displaystyle L_{0},L_{1}} such that Σ 0 = L 0 L 0 T {\displaystyle \Sigma _{0}=L_{0}L_{0}^{T}} and Σ 1 = L 1 L 1 T {\displaystyle \Sigma _{1}=L_{1}L_{1}^{T}} . Then with M and y solutions to the triangular linear systems L 1 M = L 0 {\displaystyle L_{1}M=L_{0}} , and L 1 y = μ 1 − μ 0 {\displaystyle L_{1}y=\mu _{1}-\mu _{0}} , D KL ( N 0 ∥ N 1 ) = 1 2 ( ∑ i , j = 1 k ( M i j ) 2 − k + | y | 2 + 2 ∑ i = 1 k ln ( L 1 )
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i i ( L 0 ) i i ) . {\displaystyle D_{\text{KL}}\left({\mathcal {N}}_{0}\parallel {\mathcal {N}}_{1}\right)={\frac {1}{2}}\left(\sum _{i,j=1}^{k}{\left(M_{ij}\right)}^{2}-k+|y|^{2}+2\sum _{i=1}^{k}\ln {\frac {(L_{1})_{ii}}{(L_{0})_{ii}}}\right).} A special case, and a common quantity in variational inference, is the relative entropy between a diagonal multivariate normal, and a standard normal distribution (with zero mean and unit variance): D KL ( N ( ( μ 1 , … , μ k ) T , diag ( σ 1 2 , … , σ k 2 ) ) ∥ N ( 0 , I ) ) = 1 2 ∑ i = 1 k [ σ i 2 + μ i 2 − 1 − ln ( σ i 2 ) ] . {\displaystyle D_{\text{KL}}\left({\mathcal {N}}\left(\left(\mu _{1},\ldots ,\mu _{k}\right)^{\mathsf {T}},\operatorname {diag} \left(\sigma _{1}^{2},\ldots ,\sigma _{k}^{2}\right)\right)\parallel {\mathcal {N}}\left(\mathbf {0} ,\mathbf {I} \right)\right)={\frac {1}{2}}\sum _{i=1}^{k}\left[\sigma _{i}^{2}+\mu _{i}^{2}-1-\ln \left(\sigma _{i}^{2}\right)\right].} For two univariate normal distributions p and q the above simplifies to D KL ( p ∥ q ) = log σ 1 σ 0 + σ 0 2 + ( μ 0 − μ 1 ) 2 2 σ 1 2 − 1 2 {\displaystyle D_{\text{KL}}\left({\mathcal {p}}\parallel {\mathcal {q}}\right)=\log {\frac {\sigma _{1}}{\sigma _{0}}}+{\frac {\sigma _{0}^{2}+{\left(\mu _{0}-\mu _{1}\right)}^{2}}{2\sigma _{1}^{2}}}-{\frac {1}{2}}} In the case of co-centered normal distributions with k = σ 1 / σ 0 {\displaystyle k=\sigma _{1}/\sigma _{0}} , this simplifies to: D KL ( p ∥ q ) = log 2 k + ( k − 2 − 1 ) / 2 / ln ( 2 ) b i t s {\displaystyle D_{\text{KL}}\left({\mathcal {p}}\parallel {\mathcal {q}}\right)=\log _{2}k+(k^{-2}-1)/2/\ln(2)\mathrm {bits} } === Uniform distributions === Consider two uniform distributions, with the support of p = [ A , B ] {\displaystyle p=[A,B]} enclosed within q = [ C , D ] {\displaystyle q=[C,D]} ( C ≤ A
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< B ≤ D {\displaystyle C\leq A<B\leq D} ). Then the information gain is: D KL ( p ∥ q ) = log D − C B − A {\displaystyle D_{\text{KL}}\left({\mathcal {p}}\parallel {\mathcal {q}}\right)=\log {\frac {D-C}{B-A}}} Intuitively, the information gain to a k times narrower uniform distribution contains log 2 k {\displaystyle \log _{2}k} bits. This connects with the use of bits in computing, where log 2 k {\displaystyle \log _{2}k} bits would be needed to identify one element of a k long stream. === Exponential family === The exponential family of distribution is given by p X ( x | θ ) = h ( x ) exp ( θ T T ( x ) − A ( θ ) ) {\displaystyle p_{X}(x|\theta )=h(x)\exp \left(\theta ^{\mathsf {T}}T(x)-A(\theta )\right)} where h ( x ) {\displaystyle h(x)} is reference measure, T ( x ) {\displaystyle T(x)} is sufficient statistics, θ {\displaystyle \theta } is canonical natural parameters, and A ( θ ) {\displaystyle A(\theta )} is the log-partition function. The KL divergence between two distributions p ( x | θ 1 ) {\displaystyle p(x|\theta _{1})} and p ( x | θ 2 ) {\displaystyle p(x|\theta _{2})} is given by D KL ( θ 1 ∥ θ 2 ) = ( θ 1 − θ 2 ) T μ 1 − A ( θ 1 ) + A ( θ 2 ) {\displaystyle D_{\text{KL}}(\theta _{1}\parallel \theta _{2})={\left(\theta _{1}-\theta _{2}\right)}^{\mathsf {T}}\mu _{1}-A(\theta _{1})+A(\theta _{2})} where μ 1 = E θ 1 [ T ( X ) ] = ∇ A ( θ 1 ) {\displaystyle \mu _{1}=E_{\theta _{1}}[T(X)]=\nabla A(\theta _{1})} is the mean parameter of p ( x | θ 1 ) {\displaystyle p(x|\theta _{1})} . For example, for the Poisson distribution with mean λ {\displaystyle \lambda } ,
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the sufficient statistics T ( x ) = x {\displaystyle T(x)=x} , the natural parameter θ = log λ {\displaystyle \theta =\log \lambda } , and log partition function A ( θ ) = e θ {\displaystyle A(\theta )=e^{\theta }} . As such, the divergence between two Poisson distributions with means λ 1 {\displaystyle \lambda _{1}} and λ 2 {\displaystyle \lambda _{2}} is D KL ( λ 1 ∥ λ 2 ) = λ 1 log λ 1 λ 2 − λ 1 + λ 2 . {\displaystyle D_{\text{KL}}(\lambda _{1}\parallel \lambda _{2})=\lambda _{1}\log {\frac {\lambda _{1}}{\lambda _{2}}}-\lambda _{1}+\lambda _{2}.} As another example, for a normal distribution with unit variance N ( μ , 1 ) {\displaystyle N(\mu ,1)} , the sufficient statistics T ( x ) = x {\displaystyle T(x)=x} , the natural parameter θ = μ {\displaystyle \theta =\mu } , and log partition function A ( θ ) = μ 2 / 2 {\displaystyle A(\theta )=\mu ^{2}/2} . Thus, the divergence between two normal distributions N ( μ 1 , 1 ) {\displaystyle N(\mu _{1},1)} and N ( μ 2 , 1 ) {\displaystyle N(\mu _{2},1)} is D KL ( μ 1 ∥ μ 2 ) = ( μ 1 − μ 2 ) μ 1 − μ 1 2 2 + μ 2 2 2 = ( μ 2 − μ 1 ) 2 2 . {\displaystyle D_{\text{KL}}(\mu _{1}\parallel \mu _{2})=\left(\mu _{1}-\mu _{2}\right)\mu _{1}-{\frac {\mu _{1}^{2}}{2}}+{\frac {\mu _{2}^{2}}{2}}={\frac {{\left(\mu _{2}-\mu _{1}\right)}^{2}}{2}}.} As final example, the divergence between a normal distribution with unit variance N ( μ , 1 ) {\displaystyle N(\mu ,1)} and a Poisson distribution with mean λ {\displaystyle \lambda } is D KL ( μ ∥ λ ) = ( μ − log λ ) μ − μ 2 2 + λ
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. {\displaystyle D_{\text{KL}}(\mu \parallel \lambda )=(\mu -\log \lambda )\mu -{\frac {\mu ^{2}}{2}}+\lambda .} == Relation to metrics == While relative entropy is a statistical distance, it is not a metric on the space of probability distributions, but instead it is a divergence. While metrics are symmetric and generalize linear distance, satisfying the triangle inequality, divergences are asymmetric in general and generalize squared distance, in some cases satisfying a generalized Pythagorean theorem. In general D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} does not equal D KL ( Q ∥ P ) {\displaystyle D_{\text{KL}}(Q\parallel P)} , and while this can be symmetrized (see § Symmetrised divergence), the asymmetry is an important part of the geometry. It generates a topology on the space of probability distributions. More concretely, if { P 1 , P 2 , … } {\displaystyle \{P_{1},P_{2},\ldots \}} is a sequence of distributions such that lim n → ∞ D KL ( P n ∥ Q ) = 0 , {\displaystyle \lim _{n\to \infty }D_{\text{KL}}(P_{n}\parallel Q)=0,} then it is said that P n → D Q . {\displaystyle P_{n}\xrightarrow {D} \,Q.} Pinsker's inequality entails that P n → D P ⇒ P n → T V P , {\displaystyle P_{n}\xrightarrow {D} P\Rightarrow P_{n}\xrightarrow {TV} P,} where the latter stands for the usual convergence in total variation. === Fisher information metric === Relative entropy is directly related to the Fisher information metric. This can be made explicit as follows. Assume that the probability distributions P and Q are both parameterized by some (possibly multi-dimensional) parameter θ {\displaystyle \theta } . Consider then two close by values of P = P ( θ ) {\displaystyle P=P(\theta )} and Q = P ( θ 0 ) {\displaystyle Q=P(\theta _{0})} so that the parameter θ {\displaystyle \theta } differs by only
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"page_id": 1115052,
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a small amount from the parameter value θ 0 {\displaystyle \theta _{0}} . Specifically, up to first order one has (using the Einstein summation convention) P ( θ ) = P ( θ 0 ) + Δ θ j P j ( θ 0 ) + ⋯ {\displaystyle P(\theta )=P(\theta _{0})+\Delta \theta _{j}\,P_{j}(\theta _{0})+\cdots } with Δ θ j = ( θ − θ 0 ) j {\displaystyle \Delta \theta _{j}=(\theta -\theta _{0})_{j}} a small change of θ {\displaystyle \theta } in the j direction, and P j ( θ 0 ) = ∂ P ∂ θ j ( θ 0 ) {\displaystyle P_{j}\left(\theta _{0}\right)={\frac {\partial P}{\partial \theta _{j}}}(\theta _{0})} the corresponding rate of change in the probability distribution. Since relative entropy has an absolute minimum 0 for P = Q {\displaystyle P=Q} , i.e. θ = θ 0 {\displaystyle \theta =\theta _{0}} , it changes only to second order in the small parameters Δ θ j {\displaystyle \Delta \theta _{j}} . More formally, as for any minimum, the first derivatives of the divergence vanish ∂ ∂ θ j | θ = θ 0 D KL ( P ( θ ) ∥ P ( θ 0 ) ) = 0 , {\displaystyle \left.{\frac {\partial }{\partial \theta _{j}}}\right|_{\theta =\theta _{0}}D_{\text{KL}}(P(\theta )\parallel P(\theta _{0}))=0,} and by the Taylor expansion one has up to second order D KL ( P ( θ ) ∥ P ( θ 0 ) ) = 1 2 Δ θ j Δ θ k g j k ( θ 0 ) + ⋯ {\displaystyle D_{\text{KL}}(P(\theta )\parallel P(\theta _{0}))={\frac {1}{2}}\,\Delta \theta _{j}\,\Delta \theta _{k}\,g_{jk}(\theta _{0})+\cdots } where the Hessian matrix of the divergence g j k ( θ 0 ) = ∂ 2 ∂ θ j ∂ θ k | θ = θ 0 D KL ( P (
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θ ) ∥ P ( θ 0 ) ) {\displaystyle g_{jk}(\theta _{0})=\left.{\frac {\partial ^{2}}{\partial \theta _{j}\,\partial \theta _{k}}}\right|_{\theta =\theta _{0}}D_{\text{KL}}(P(\theta )\parallel P(\theta _{0}))} must be positive semidefinite. Letting θ 0 {\displaystyle \theta _{0}} vary (and dropping the subindex 0) the Hessian g j k ( θ ) {\displaystyle g_{jk}(\theta )} defines a (possibly degenerate) Riemannian metric on the θ parameter space, called the Fisher information metric. ==== Fisher information metric theorem ==== When p ( x , ρ ) {\displaystyle p_{(x,\rho )}} satisfies the following regularity conditions: ∂ log ( p ) ∂ ρ , ∂ 2 log ( p ) ∂ ρ 2 , ∂ 3 log ( p ) ∂ ρ 3 {\displaystyle {\frac {\partial \log(p)}{\partial \rho }},{\frac {\partial ^{2}\log(p)}{\partial \rho ^{2}}},{\frac {\partial ^{3}\log(p)}{\partial \rho ^{3}}}} exist, | ∂ p ∂ ρ | < F ( x ) : ∫ x = 0 ∞ F ( x ) d x < ∞ , | ∂ 2 p ∂ ρ 2 | < G ( x ) : ∫ x = 0 ∞ G ( x ) d x < ∞ | ∂ 3 log ( p ) ∂ ρ 3 | < H ( x ) : ∫ x = 0 ∞ p ( x , 0 ) H ( x ) d x < ξ < ∞ {\displaystyle {\begin{aligned}\left|{\frac {\partial p}{\partial \rho }}\right|&<F(x):\int _{x=0}^{\infty }F(x)\,dx<\infty ,\\\left|{\frac {\partial ^{2}p}{\partial \rho ^{2}}}\right|&<G(x):\int _{x=0}^{\infty }G(x)\,dx<\infty \\\left|{\frac {\partial ^{3}\log(p)}{\partial \rho ^{3}}}\right|&<H(x):\int _{x=0}^{\infty }p(x,0)H(x)\,dx<\xi <\infty \end{aligned}}} where ξ is independent of ρ ∫ x = 0 ∞ ∂ p ( x , ρ ) ∂ ρ | ρ = 0 d x = ∫ x = 0 ∞ ∂ 2 p ( x , ρ ) ∂ ρ 2 | ρ = 0 d x = 0 {\displaystyle
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{
"page_id": 1115052,
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\left.\int _{x=0}^{\infty }{\frac {\partial p(x,\rho )}{\partial \rho }}\right|_{\rho =0}\,dx=\left.\int _{x=0}^{\infty }{\frac {\partial ^{2}p(x,\rho )}{\partial \rho ^{2}}}\right|_{\rho =0}\,dx=0} then: D ( p ( x , 0 ) ∥ p ( x , ρ ) ) = c ρ 2 2 + O ( ρ 3 ) as ρ → 0. {\displaystyle {\mathcal {D}}(p(x,0)\parallel p(x,\rho ))={\frac {c\rho ^{2}}{2}}+{\mathcal {O}}\left(\rho ^{3}\right){\text{ as }}\rho \to 0.} === Variation of information === Another information-theoretic metric is variation of information, which is roughly a symmetrization of conditional entropy. It is a metric on the set of partitions of a discrete probability space. === MAUVE Metric === MAUVE is a measure of the statistical gap between two text distributions, such as the difference between text generated by a model and human-written text. This measure is computed using Kullback–Leibler divergences between the two distributions in a quantized embedding space of a foundation model. == Relation to other quantities of information theory == Many of the other quantities of information theory can be interpreted as applications of relative entropy to specific cases. === Self-information === The self-information, also known as the information content of a signal, random variable, or event is defined as the negative logarithm of the probability of the given outcome occurring. When applied to a discrete random variable, the self-information can be represented as I ( m ) = D KL ( δ im ∥ { p i } ) , {\displaystyle \operatorname {\operatorname {I} } (m)=D_{\text{KL}}\left(\delta _{\text{im}}\parallel \{p_{i}\}\right),} is the relative entropy of the probability distribution P ( i ) {\displaystyle P(i)} from a Kronecker delta representing certainty that i = m {\displaystyle i=m} — i.e. the number of extra bits that must be transmitted to identify i if only the probability distribution P ( i ) {\displaystyle P(i)} is available to the receiver,
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"page_id": 1115052,
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not the fact that i = m {\displaystyle i=m} . === Mutual information === The mutual information, I ( X ; Y ) = D KL ( P ( X , Y ) ∥ P ( X ) P ( Y ) ) = E X { D KL ( P ( Y ∣ X ) ∥ P ( Y ) ) } = E Y { D KL ( P ( X ∣ Y ) ∥ P ( X ) ) } {\displaystyle {\begin{aligned}\operatorname {I} (X;Y)&=D_{\text{KL}}(P(X,Y)\parallel P(X)P(Y))\\[5pt]&=\operatorname {E} _{X}\{D_{\text{KL}}(P(Y\mid X)\parallel P(Y))\}\\[5pt]&=\operatorname {E} _{Y}\{D_{\text{KL}}(P(X\mid Y)\parallel P(X))\}\end{aligned}}} is the relative entropy of the joint probability distribution P ( X , Y ) {\displaystyle P(X,Y)} from the product P ( X ) P ( Y ) {\displaystyle P(X)P(Y)} of the two marginal probability distributions — i.e. the expected number of extra bits that must be transmitted to identify X and Y if they are coded using only their marginal distributions instead of the joint distribution. Equivalently, if the joint probability P ( X , Y ) {\displaystyle P(X,Y)} is known, it is the expected number of extra bits that must on average be sent to identify Y if the value of X is not already known to the receiver. === Shannon entropy === The Shannon entropy, H ( X ) = E [ I X ( x ) ] = log N − D KL ( p X ( x ) ∥ P U ( X ) ) {\displaystyle {\begin{aligned}\mathrm {H} (X)&=\operatorname {E} \left[\operatorname {I} _{X}(x)\right]\\&=\log N-D_{\text{KL}}{\left(p_{X}(x)\parallel P_{U}(X)\right)}\end{aligned}}} is the number of bits which would have to be transmitted to identify X from N equally likely possibilities, less the relative entropy of the uniform distribution on the random variates of X, P U ( X )
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"page_id": 1115052,
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{\displaystyle P_{U}(X)} , from the true distribution P ( X ) {\displaystyle P(X)} — i.e. less the expected number of bits saved, which would have had to be sent if the value of X were coded according to the uniform distribution P U ( X ) {\displaystyle P_{U}(X)} rather than the true distribution P ( X ) {\displaystyle P(X)} . This definition of Shannon entropy forms the basis of E.T. Jaynes's alternative generalization to continuous distributions, the limiting density of discrete points (as opposed to the usual differential entropy), which defines the continuous entropy as lim N → ∞ H N ( X ) = log N − ∫ p ( x ) log p ( x ) m ( x ) d x , {\displaystyle \lim _{N\to \infty }H_{N}(X)=\log N-\int p(x)\log {\frac {p(x)}{m(x)}}\,dx,} which is equivalent to: log ( N ) − D KL ( p ( x ) | | m ( x ) ) {\displaystyle \log(N)-D_{\text{KL}}(p(x)||m(x))} === Conditional entropy === The conditional entropy, H ( X ∣ Y ) = log N − D KL ( P ( X , Y ) ∥ P U ( X ) P ( Y ) ) = log N − D KL ( P ( X , Y ) ∥ P ( X ) P ( Y ) ) − D KL ( P ( X ) ∥ P U ( X ) ) = H ( X ) − I ( X ; Y ) = log N − E Y [ D KL ( P ( X ∣ Y ) ∥ P U ( X ) ) ] {\displaystyle {\begin{aligned}\mathrm {H} (X\mid Y)&=\log N-D_{\text{KL}}(P(X,Y)\parallel P_{U}(X)P(Y))\\[5pt]&=\log N-D_{\text{KL}}(P(X,Y)\parallel P(X)P(Y))-D_{\text{KL}}(P(X)\parallel P_{U}(X))\\[5pt]&=\mathrm {H} (X)-\operatorname {I} (X;Y)\\[5pt]&=\log N-\operatorname {E} _{Y}\left[D_{\text{KL}}\left(P\left(X\mid Y\right)\parallel P_{U}(X)\right)\right]\end{aligned}}} is the number
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"page_id": 1115052,
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of bits which would have to be transmitted to identify X from N equally likely possibilities, less the relative entropy of the product distribution P U ( X ) P ( Y ) {\displaystyle P_{U}(X)P(Y)} from the true joint distribution P ( X , Y ) {\displaystyle P(X,Y)} — i.e. less the expected number of bits saved which would have had to be sent if the value of X were coded according to the uniform distribution P U ( X ) {\displaystyle P_{U}(X)} rather than the conditional distribution P ( X | Y ) {\displaystyle P(X|Y)} of X given Y. === Cross entropy === When we have a set of possible events, coming from the distribution p, we can encode them (with a lossless data compression) using entropy encoding. This compresses the data by replacing each fixed-length input symbol with a corresponding unique, variable-length, prefix-free code (e.g.: the events (A, B, C) with probabilities p = (1/2, 1/4, 1/4) can be encoded as the bits (0, 10, 11)). If we know the distribution p in advance, we can devise an encoding that would be optimal (e.g.: using Huffman coding). Meaning the messages we encode will have the shortest length on average (assuming the encoded events are sampled from p), which will be equal to Shannon's Entropy of p (denoted as H ( p ) {\displaystyle \mathrm {H} (p)} ). However, if we use a different probability distribution (q) when creating the entropy encoding scheme, then a larger number of bits will be used (on average) to identify an event from a set of possibilities. This new (larger) number is measured by the cross entropy between p and q. The cross entropy between two probability distributions (p and q) measures the average number of bits needed to identify an event from
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"page_id": 1115052,
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a set of possibilities, if a coding scheme is used based on a given probability distribution q, rather than the "true" distribution p. The cross entropy for two distributions p and q over the same probability space is thus defined as follows. H ( p , q ) = E p [ − log q ] = H ( p ) + D KL ( p ∥ q ) . {\displaystyle \mathrm {H} (p,q)=\operatorname {E} _{p}[-\log q]=\mathrm {H} (p)+D_{\text{KL}}(p\parallel q).} For explicit derivation of this, see the Motivation section above. Under this scenario, relative entropies (kl-divergence) can be interpreted as the extra number of bits, on average, that are needed (beyond H ( p ) {\displaystyle \mathrm {H} (p)} ) for encoding the events because of using q for constructing the encoding scheme instead of p. == Bayesian updating == In Bayesian statistics, relative entropy can be used as a measure of the information gain in moving from a prior distribution to a posterior distribution: p ( x ) → p ( x ∣ I ) {\displaystyle p(x)\to p(x\mid I)} . If some new fact Y = y {\displaystyle Y=y} is discovered, it can be used to update the posterior distribution for X from p ( x ∣ I ) {\displaystyle p(x\mid I)} to a new posterior distribution p ( x ∣ y , I ) {\displaystyle p(x\mid y,I)} using Bayes' theorem: p ( x ∣ y , I ) = p ( y ∣ x , I ) p ( x ∣ I ) p ( y ∣ I ) {\displaystyle p(x\mid y,I)={\frac {p(y\mid x,I)p(x\mid I)}{p(y\mid I)}}} This distribution has a new entropy: H ( p ( x ∣ y , I ) ) = − ∑ x p ( x ∣ y , I ) log
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"page_id": 1115052,
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p ( x ∣ y , I ) , {\displaystyle \mathrm {H} {\big (}p(x\mid y,I){\big )}=-\sum _{x}p(x\mid y,I)\log p(x\mid y,I),} which may be less than or greater than the original entropy H ( p ( x ∣ I ) ) {\displaystyle \mathrm {H} (p(x\mid I))} . However, from the standpoint of the new probability distribution one can estimate that to have used the original code based on p ( x ∣ I ) {\displaystyle p(x\mid I)} instead of a new code based on p ( x ∣ y , I ) {\displaystyle p(x\mid y,I)} would have added an expected number of bits: D KL ( p ( x ∣ y , I ) ∥ p ( x ∣ I ) ) = ∑ x p ( x ∣ y , I ) log p ( x ∣ y , I ) p ( x ∣ I ) {\displaystyle D_{\text{KL}}{\big (}p(x\mid y,I)\parallel p(x\mid I){\big )}=\sum _{x}p(x\mid y,I)\log {\frac {p(x\mid y,I)}{p(x\mid I)}}} to the message length. This therefore represents the amount of useful information, or information gain, about X, that has been learned by discovering Y = y {\displaystyle Y=y} . If a further piece of data, Y 2 = y 2 {\displaystyle Y_{2}=y_{2}} , subsequently comes in, the probability distribution for x can be updated further, to give a new best guess p ( x ∣ y 1 , y 2 , I ) {\displaystyle p(x\mid y_{1},y_{2},I)} . If one reinvestigates the information gain for using p ( x ∣ y 1 , I ) {\displaystyle p(x\mid y_{1},I)} rather than p ( x ∣ I ) {\displaystyle p(x\mid I)} , it turns out that it may be either greater or less than previously estimated: ∑ x p ( x ∣ y 1 , y 2 , I ) log
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p ( x ∣ y 1 , y 2 , I ) p ( x ∣ I ) {\displaystyle \sum _{x}p(x\mid y_{1},y_{2},I)\log {\frac {p(x\mid y_{1},y_{2},I)}{p(x\mid I)}}} may be ≤ or > than ∑ x p ( x ∣ y 1 , I ) log p ( x ∣ y 1 , I ) p ( x ∣ I ) {\textstyle \sum _{x}p(x\mid y_{1},I)\log {\frac {p(x\mid y_{1},I)}{p(x\mid I)}}} and so the combined information gain does not obey the triangle inequality: D KL ( p ( x ∣ y 1 , y 2 , I ) ∥ p ( x ∣ I ) ) {\displaystyle D_{\text{KL}}{\big (}p(x\mid y_{1},y_{2},I)\parallel p(x\mid I){\big )}} may be <, = or > than D KL ( p ( x ∣ y 1 , y 2 , I ) ∥ p ( x ∣ y 1 , I ) ) + D KL ( p ( x ∣ y 1 , I ) ∥ p ( x ∣ I ) ) {\displaystyle D_{\text{KL}}{\big (}p(x\mid y_{1},y_{2},I)\parallel p(x\mid y_{1},I){\big )}+D_{\text{KL}}{\big (}p(x\mid y_{1},I)\parallel p(x\mid I){\big )}} All one can say is that on average, averaging using p ( y 2 ∣ y 1 , x , I ) {\displaystyle p(y_{2}\mid y_{1},x,I)} , the two sides will average out. === Bayesian experimental design === A common goal in Bayesian experimental design is to maximise the expected relative entropy between the prior and the posterior. When posteriors are approximated to be Gaussian distributions, a design maximising the expected relative entropy is called Bayes d-optimal. == Discrimination information == Relative entropy D KL ( p ( x ∣ H 1 ) ∥ p ( x ∣ H 0 ) ) {\textstyle D_{\text{KL}}{\bigl (}p(x\mid H_{1})\parallel p(x\mid H_{0}){\bigr )}} can also be interpreted as the expected discrimination information for H 1 {\displaystyle H_{1}} over
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"page_id": 1115052,
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H 0 {\displaystyle H_{0}} : the mean information per sample for discriminating in favor of a hypothesis H 1 {\displaystyle H_{1}} against a hypothesis H 0 {\displaystyle H_{0}} , when hypothesis H 1 {\displaystyle H_{1}} is true. Another name for this quantity, given to it by I. J. Good, is the expected weight of evidence for H 1 {\displaystyle H_{1}} over H 0 {\displaystyle H_{0}} to be expected from each sample. The expected weight of evidence for H 1 {\displaystyle H_{1}} over H 0 {\displaystyle H_{0}} is not the same as the information gain expected per sample about the probability distribution p ( H ) {\displaystyle p(H)} of the hypotheses, D KL ( p ( x ∣ H 1 ) ∥ p ( x ∣ H 0 ) ) ≠ I G = D KL ( p ( H ∣ x ) ∥ p ( H ∣ I ) ) . {\displaystyle D_{\text{KL}}(p(x\mid H_{1})\parallel p(x\mid H_{0}))\neq IG=D_{\text{KL}}(p(H\mid x)\parallel p(H\mid I)).} Either of the two quantities can be used as a utility function in Bayesian experimental design, to choose an optimal next question to investigate: but they will in general lead to rather different experimental strategies. On the entropy scale of information gain there is very little difference between near certainty and absolute certainty—coding according to a near certainty requires hardly any more bits than coding according to an absolute certainty. On the other hand, on the logit scale implied by weight of evidence, the difference between the two is enormous – infinite perhaps; this might reflect the difference between being almost sure (on a probabilistic level) that, say, the Riemann hypothesis is correct, compared to being certain that it is correct because one has a mathematical proof. These two different scales of loss function for uncertainty are both useful, according
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"page_id": 1115052,
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to how well each reflects the particular circumstances of the problem in question. === Principle of minimum discrimination information === The idea of relative entropy as discrimination information led Kullback to propose the Principle of Minimum Discrimination Information (MDI): given new facts, a new distribution f should be chosen which is as hard to discriminate from the original distribution f 0 {\displaystyle f_{0}} as possible; so that the new data produces as small an information gain D KL ( f ∥ f 0 ) {\displaystyle D_{\text{KL}}(f\parallel f_{0})} as possible. For example, if one had a prior distribution p ( x , a ) {\displaystyle p(x,a)} over x and a, and subsequently learnt the true distribution of a was u ( a ) {\displaystyle u(a)} , then the relative entropy between the new joint distribution for x and a, q ( x ∣ a ) u ( a ) {\displaystyle q(x\mid a)u(a)} , and the earlier prior distribution would be: D KL ( q ( x ∣ a ) u ( a ) ∥ p ( x , a ) ) = E u ( a ) { D KL ( q ( x ∣ a ) ∥ p ( x ∣ a ) ) } + D KL ( u ( a ) ∥ p ( a ) ) , {\displaystyle D_{\text{KL}}(q(x\mid a)u(a)\parallel p(x,a))=\operatorname {E} _{u(a)}\left\{D_{\text{KL}}(q(x\mid a)\parallel p(x\mid a))\right\}+D_{\text{KL}}(u(a)\parallel p(a)),} i.e. the sum of the relative entropy of p ( a ) {\displaystyle p(a)} the prior distribution for a from the updated distribution u ( a ) {\displaystyle u(a)} , plus the expected value (using the probability distribution u ( a ) {\displaystyle u(a)} ) of the relative entropy of the prior conditional distribution p ( x ∣ a ) {\displaystyle p(x\mid a)} from the new conditional distribution q
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"page_id": 1115052,
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( x ∣ a ) {\displaystyle q(x\mid a)} . (Note that often the later expected value is called the conditional relative entropy (or conditional Kullback–Leibler divergence) and denoted by D KL ( q ( x ∣ a ) ∥ p ( x ∣ a ) ) {\displaystyle D_{\text{KL}}(q(x\mid a)\parallel p(x\mid a))} ) This is minimized if q ( x ∣ a ) = p ( x ∣ a ) {\displaystyle q(x\mid a)=p(x\mid a)} over the whole support of u ( a ) {\displaystyle u(a)} ; and we note that this result incorporates Bayes' theorem, if the new distribution u ( a ) {\displaystyle u(a)} is in fact a δ function representing certainty that a has one particular value. MDI can be seen as an extension of Laplace's Principle of Insufficient Reason, and the Principle of Maximum Entropy of E.T. Jaynes. In particular, it is the natural extension of the principle of maximum entropy from discrete to continuous distributions, for which Shannon entropy ceases to be so useful (see differential entropy), but the relative entropy continues to be just as relevant. In the engineering literature, MDI is sometimes called the Principle of Minimum Cross-Entropy (MCE) or Minxent for short. Minimising relative entropy from m to p with respect to m is equivalent to minimizing the cross-entropy of p and m, since H ( p , m ) = H ( p ) + D KL ( p ∥ m ) , {\displaystyle \mathrm {H} (p,m)=\mathrm {H} (p)+D_{\text{KL}}(p\parallel m),} which is appropriate if one is trying to choose an adequate approximation to p. However, this is just as often not the task one is trying to achieve. Instead, just as often it is m that is some fixed prior reference measure, and p that one is attempting to optimise by minimising D
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"page_id": 1115052,
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KL ( p ∥ m ) {\displaystyle D_{\text{KL}}(p\parallel m)} subject to some constraint. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by redefining cross-entropy to be D KL ( p ∥ m ) {\displaystyle D_{\text{KL}}(p\parallel m)} , rather than H ( p , m ) {\displaystyle \mathrm {H} (p,m)} . == Relationship to available work == Surprisals add where probabilities multiply. The surprisal for an event of probability p is defined as s = − k ln p {\displaystyle s=-k\ln p} . If k is { 1 , 1 / ln 2 , 1.38 × 10 − 23 } {\displaystyle \left\{1,1/\ln 2,1.38\times 10^{-23}\right\}} then surprisal is in { {\displaystyle \{} nats, bits, or J / K } {\displaystyle J/K\}} so that, for instance, there are N bits of surprisal for landing all "heads" on a toss of N coins. Best-guess states (e.g. for atoms in a gas) are inferred by maximizing the average surprisal S (entropy) for a given set of control parameters (like pressure P or volume V). This constrained entropy maximization, both classically and quantum mechanically, minimizes Gibbs availability in entropy units A ≡ − k ln Z {\displaystyle A\equiv -k\ln Z} where Z is a constrained multiplicity or partition function. When temperature T is fixed, free energy ( T × A {\displaystyle T\times A} ) is also minimized. Thus if T , V {\displaystyle T,V} and number of molecules N are constant, the Helmholtz free energy F ≡ U − T S {\displaystyle F\equiv U-TS} (where U is energy and S is entropy) is minimized as a system "equilibrates." If T and P are held constant (say during processes in your body), the Gibbs free energy G = U + P V − T S
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"page_id": 1115052,
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{\displaystyle G=U+PV-TS} is minimized instead. The change in free energy under these conditions is a measure of available work that might be done in the process. Thus available work for an ideal gas at constant temperature T o {\displaystyle T_{o}} and pressure P o {\displaystyle P_{o}} is W = Δ G = N k T o Θ ( V / V o ) {\displaystyle W=\Delta G=NkT_{o}\Theta (V/V_{o})} where V o = N k T o / P o {\displaystyle V_{o}=NkT_{o}/P_{o}} and Θ ( x ) = x − 1 − ln x ≥ 0 {\displaystyle \Theta (x)=x-1-\ln x\geq 0} (see also Gibbs inequality). More generally the work available relative to some ambient is obtained by multiplying ambient temperature T o {\displaystyle T_{o}} by relative entropy or net surprisal Δ I ≥ 0 , {\displaystyle \Delta I\geq 0,} defined as the average value of k ln ( p / p o ) {\displaystyle k\ln(p/p_{o})} where p o {\displaystyle p_{o}} is the probability of a given state under ambient conditions. For instance, the work available in equilibrating a monatomic ideal gas to ambient values of V o {\displaystyle V_{o}} and T o {\displaystyle T_{o}} is thus W = T o Δ I {\displaystyle W=T_{o}\Delta I} , where relative entropy Δ I = N k [ Θ ( V V o ) + 3 2 Θ ( T T o ) ] . {\displaystyle \Delta I=Nk\left[\Theta {\left({\frac {V}{V_{o}}}\right)}+{\frac {3}{2}}\Theta {\left({\frac {T}{T_{o}}}\right)}\right].} The resulting contours of constant relative entropy, shown at right for a mole of Argon at standard temperature and pressure, for example put limits on the conversion of hot to cold as in flame-powered air-conditioning or in the unpowered device to convert boiling-water to ice-water discussed here. Thus relative entropy measures thermodynamic availability in bits. == Quantum information theory ==
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"page_id": 1115052,
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For density matrices P and Q on a Hilbert space, the quantum relative entropy from Q to P is defined to be D KL ( P ∥ Q ) = Tr ( P ( log P − log Q ) ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=\operatorname {Tr} (P(\log P-\log Q)).} In quantum information science the minimum of D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} over all separable states Q can also be used as a measure of entanglement in the state P. == Relationship between models and reality == Just as relative entropy of "actual from ambient" measures thermodynamic availability, relative entropy of "reality from a model" is also useful even if the only clues we have about reality are some experimental measurements. In the former case relative entropy describes distance to equilibrium or (when multiplied by ambient temperature) the amount of available work, while in the latter case it tells you about surprises that reality has up its sleeve or, in other words, how much the model has yet to learn. Although this tool for evaluating models against systems that are accessible experimentally may be applied in any field, its application to selecting a statistical model via Akaike information criterion are particularly well described in papers and a book by Burnham and Anderson. In a nutshell the relative entropy of reality from a model may be estimated, to within a constant additive term, by a function of the deviations observed between data and the model's predictions (like the mean squared deviation) . Estimates of such divergence for models that share the same additive term can in turn be used to select among models. When trying to fit parametrized models to data there are various estimators which attempt to minimize relative entropy, such as maximum
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"page_id": 1115052,
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"title": "Kullback–Leibler divergence"
}
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likelihood and maximum spacing estimators. == Symmetrised divergence == Kullback & Leibler (1951) also considered the symmetrized function: D KL ( P ∥ Q ) + D KL ( Q ∥ P ) {\displaystyle D_{\text{KL}}(P\parallel Q)+D_{\text{KL}}(Q\parallel P)} which they referred to as the "divergence", though today the "KL divergence" refers to the asymmetric function (see § Etymology for the evolution of the term). This function is symmetric and nonnegative, and had already been defined and used by Harold Jeffreys in 1948; it is accordingly called the Jeffreys divergence. This quantity has sometimes been used for feature selection in classification problems, where P and Q are the conditional pdfs of a feature under two different classes. In the Banking and Finance industries, this quantity is referred to as Population Stability Index (PSI), and is used to assess distributional shifts in model features through time. An alternative is given via the λ {\displaystyle \lambda } -divergence, D λ ( P ∥ Q ) = λ D KL ( P ∥ λ P + ( 1 − λ ) Q ) + ( 1 − λ ) D KL ( Q ∥ λ P + ( 1 − λ ) Q ) , {\displaystyle D_{\lambda }(P\parallel Q)=\lambda D_{\text{KL}}(P\parallel \lambda P+(1-\lambda )Q)+(1-\lambda )D_{\text{KL}}(Q\parallel \lambda P+(1-\lambda )Q),} which can be interpreted as the expected information gain about X from discovering which probability distribution X is drawn from, P or Q, if they currently have probabilities λ {\displaystyle \lambda } and 1 − λ {\displaystyle 1-\lambda } respectively. The value λ = 0.5 {\displaystyle \lambda =0.5} gives the Jensen–Shannon divergence, defined by D JS = 1 2 D KL ( P ∥ M ) + 1 2 D KL ( Q ∥ M ) {\displaystyle D_{\text{JS}}={\tfrac {1}{2}}D_{\text{KL}}(P\parallel M)+{\tfrac {1}{2}}D_{\text{KL}}(Q\parallel M)} where M is the average of
|
{
"page_id": 1115052,
"source": null,
"title": "Kullback–Leibler divergence"
}
|
the two distributions, M = 1 2 ( P + Q ) . {\displaystyle M={\tfrac {1}{2}}\left(P+Q\right).} We can also interpret D JS {\displaystyle D_{\text{JS}}} as the capacity of a noisy information channel with two inputs giving the output distributions P and Q. The Jensen–Shannon divergence, like all f-divergences, is locally proportional to the Fisher information metric. It is similar to the Hellinger metric (in the sense that it induces the same affine connection on a statistical manifold). Furthermore, the Jensen–Shannon divergence can be generalized using abstract statistical M-mixtures relying on an abstract mean M. == Relationship to other probability-distance measures == There are many other important measures of probability distance. Some of these are particularly connected with relative entropy. For example: The total-variation distance, δ ( p , q ) {\displaystyle \delta (p,q)} . This is connected to the divergence through Pinsker's inequality: δ ( P , Q ) ≤ 1 2 D KL ( P ∥ Q ) . {\displaystyle \delta (P,Q)\leq {\sqrt {{\tfrac {1}{2}}D_{\text{KL}}(P\parallel Q)}}.} Pinsker's inequality is vacuous for any distributions where D K L ( P ∥ Q ) > 2 {\displaystyle D_{\mathrm {KL} }(P\parallel Q)>2} , since the total variation distance is at most 1. For such distributions, an alternative bound can be used, due to Bretagnolle and Huber (see, also, Tsybakov): δ ( P , Q ) ≤ 1 − e − D K L ( P ∥ Q ) . {\displaystyle \delta (P,Q)\leq {\sqrt {1-e^{-D_{\mathrm {KL} }(P\parallel Q)}}}.} The family of Rényi divergences generalize relative entropy. Depending on the value of a certain parameter, α {\displaystyle \alpha } , various inequalities may be deduced. Other notable measures of distance include the Hellinger distance, histogram intersection, Chi-squared statistic, quadratic form distance, match distance, Kolmogorov–Smirnov distance, and earth mover's distance. == Data differencing == Just
|
{
"page_id": 1115052,
"source": null,
"title": "Kullback–Leibler divergence"
}
|
as absolute entropy serves as theoretical background for data compression, relative entropy serves as theoretical background for data differencing – the absolute entropy of a set of data in this sense being the data required to reconstruct it (minimum compressed size), while the relative entropy of a target set of data, given a source set of data, is the data required to reconstruct the target given the source (minimum size of a patch). == See also == == References == == External links == Information Theoretical Estimators Toolbox Ruby gem for calculating Kullback–Leibler divergence Jon Shlens' tutorial on Kullback–Leibler divergence and likelihood theory Matlab code for calculating Kullback–Leibler divergence for discrete distributions Sergio Verdú, Relative Entropy, NIPS 2009. One-hour video lecture. A modern summary of info-theoretic divergence measures
|
{
"page_id": 1115052,
"source": null,
"title": "Kullback–Leibler divergence"
}
|
A polar organic chemical integrative sampler (POCIS) is a passive sampling device which allows for the in situ collection of a time-integrated average of hydrophilic organic contaminants developed by researchers with the United States Geological Survey in Columbia, Missouri. POCIS provides a means for estimating the toxicological significance of waterborne contaminants. The POCIS sampler mimics the respiratory exposure of organisms living in the aquatic environment and can provide an understanding of bioavailable contaminants present in the system. POCIS can be deployed in a wide range of aquatic environments and is commonly used to assist in environmental monitoring studies. == Background == The first passive sampling devices were developed in the 1970s to determine concentrations of contaminants in the air. In 1980 this technology was first adapted for the monitoring of organic contaminants in water. The initial type of passive sampler developed for aquatic monitoring purposes was the semipermeable membrane device (SPMD). SPMD samplers are most effective at absorbing hydrophobic pollutants with an octanol-water partition coefficient (Kow) ranging from 4-8. As the global emission of bioconcentratable persistent organic pollutants (POPs) was shown to result in adverse ecological effects, industry developed a wide range of increasing water-soluble, polar hydrophilic organic compounds (HpOCs) to replace them. These compounds generally have lower bioconcentration factors. However, there is evidence that large fluxes of these HpOCs into aquatic environments may be responsible for a number of adverse effects to aquatic organisms, such as altered behavior, neurotoxicity, endocrine disruption, and impaired reproduction. In the late 1990s research was underway to develop a new passive sampler in order to monitor HpOCs with a log Kow value of less than 3. In 1999 the POCIS sampler was under development at the University of Missouri-Columbia. It gathered more support in the early 2000s as concern increased regarding the effects of
|
{
"page_id": 42927021,
"source": null,
"title": "Polar organic chemical integrative sampler"
}
|
pharmaceutical and personal care products in surface waters. The United States Geological Survey (USGS) has been heavily involved in the development of passive samplers and has articles in their database regarding the development of POCIS as early as 2000. The USGS Columbia Environmental Research Center (CERC) is a self-proclaimed international leader in the field of passive sampling. There have been recent efforts by the USGS to connect people who have an interest in passive sampling. An international workshop and symposium on passive sampling was held by the USGS in 2013 to connect developers, policy makers and end users in order to discuss ways of monitoring environmental pollution. == Fundamentals == The POCIS device was developed and patented by Jimmie D. Petty, James N. Huckins, and David A. Alvarez, of the Columbia Environmental Research Center. Integrative passive samplers are an effective way to monitor the concentration of organic contaminants in aquatic systems over time. Most aquatic monitoring programs rely on collecting individual samples, often called grab samples, at a specific time. The grab sampling method is associated with many disadvantages that can be resolved by passive sampling techniques. When contaminants are present in trace amounts, grab sampling may require the collection of large volumes of water. Also, lab analysis of the sample can only provide a snapshot of contaminant levels at the time of collection. This approach therefore has drawbacks when monitoring in environments where water contamination varies over time and episodic contamination events occur. Passive sampling techniques have been able to provide a time-integrated sample of water contamination with low detection limits and in situ extraction of analytes. === POCIS set-up === The POCIS sampler consists of an array of sampling disks mounted on a support rod. Each disk consists of a solid sorbent sandwiched between two polyethersoulfone (PES) microporous
|
{
"page_id": 42927021,
"source": null,
"title": "Polar organic chemical integrative sampler"
}
|
membranes which are then compressed between two stainless steel rings which expose a sampling area. A standard POCIS disk consists of a sampling surface area to sorbent mass ratio of approximately 180 cm2g. Because the amount of chemical sampled is directly related to the sample surface area, it is sometimes necessary to combine extracts from multiple POCIS disks into one sample. Stainless steel rings, or other rigid inert material, are essential to prevent sorbent loss as the PES membranes are not able to be heat sealed. The POCIS array is then inserted and deployed within a protective canister. This canister is usually made of stainless steel or PVC and works to deflect debris that may displace the POCIS array during its deployment. The PES membrane acts as a semipermeable barrier between the sorbent and surrounding aquatic environment. It allows dissolved contaminants to pass through the sorbent while selectively excluding any particles larger than 100 nm. The membrane resists biofouling because the polyethersulphone used in the design is less prone than other materials. The POCIS is versatile in that the sorbents can be changed to target different classes of contaminants. However, only two sorbent classes are considered as standards of all POCIS deployments to date. === Theory and modeling === Each POCIS disk will sample a certain volume of water per day. The volume of water sampled varies from chemical to chemical and is dependent on the physical and chemical properties of the compound as well as the duration of sampling. The sampling rate of POCIS can vary with changes in the water flow, turbulence, temperature, and the buildup of solids on the sampler’s surface. The accumulation of contaminants into a POCIS device is the result of three successive process occurring at the same time. First, the contaminants have to diffuse
|
{
"page_id": 42927021,
"source": null,
"title": "Polar organic chemical integrative sampler"
}
|
across the water boundary layer. The thickness of this layer is dependent on water flow and turbulence around the sampler and can significantly alter sampling rates. Second, the contaminant must transport across the membrane either through the water-filled pores or through the membrane itself. Finally, contaminants transfer from the membrane into the sorbent material mainly through adsorption. These last two steps make the modeling, understanding, and prediction of accumulation by a POCIS device challenging. To date, a limited number of chemical sampling rates have been determined. Accumulation of chemicals by a POCIS device generally follows first order kinetics. The kinetics are characterized by an initial integrative phase, followed by an equilibrium partitioning phase. During the integrative phase of uptake, a passive sampling device accumulates residues linearly relative to time, assuming constant exposure concentrations. Based on current results, the POCIS sampler remains in a linear phase for at least 30 days, and has been observed up to 56 days. Therefore, both laboratory and field data justify the use of a linear uptake model for the calculation of sample rates. In order to estimate the ambient water concentration of contaminants sampled by a POCIS device, there must be available calibration data applicable for in situ conditions regarding the target compound. Currently, this information is limited. == Applicability == POCIS can be deployed in a wide range of aquatic environments including stagnant pools, rivers, springs, estuarine systems, and wastewater streams. However, there has been little research into the use of POCIS in strictly marine environments. Prior to deployment of a POCIS device, it is essential to select a study site that will maximize the effectiveness of the sampler. Selecting an area that is shaded will help prevent light sensitive chemicals from being degrading. The site should also allow the sampler to be submerged
|
{
"page_id": 42927021,
"source": null,
"title": "Polar organic chemical integrative sampler"
}
|
in the water without being buried in the sediment. It is ideal to place the sampler in moving water in order to increase sampling rates, however, areas with an extremely turbulent water flow should be avoided as to prevent damage to the POCIS device. Passive samplers are very vulnerable to vandalism and it is therefore important to secure the sampler in areas that are not easily visible and that are away from areas frequently used by people. POCIS samplers can be deployed for a period of time ranging from weeks to months. The shortest deployment lengths are typically 7 days but average 2–3 months. It is important to have a long enough deployment period to allow for adequate detection of contaminants at ambient environmental concentrations. Often, the two different types of POCIS devices will be deployed together in order to provide the greatest understanding of contamination. It is also important to deploy enough POCIS devices to ensure a large enough sample of contaminant is recovered for chemical analysis. An estimate or the number of samplers needed at a given site can be determined by the following equation. Rs x t x n x Cc x Pr x Et > MQL x Vi where Cc is the predicted environmental concentration of the contaminant t is the deployment time in days Rs is sampling rate in liters of water extracted by the passive sampler per day(L/day) Pr is the overall method recovery for the analyte (expressed as a factor of one; ::therefore 0.9 is used for 90 percent recovery), n is the number of passive samplers combined into a single sample, Et is the fraction of the total sample extract which is injected into the ::instrument for quantification MQL is the method quantification limit Vi is the volume of standard injection (commonly
|
{
"page_id": 42927021,
"source": null,
"title": "Polar organic chemical integrative sampler"
}
|
1 μL). === Relevant contaminants === Any compound with a log Kow of less than or equal to 3 can concentrate in a POCIS sampler. Applicable classes of contaminants measured by POCIS are pharmaceuticals, household and industrial products, hormones, herbicides, and polar pesticides (Table 1). Currently, there are two POCIS configurations that are targeted for different classes of contaminants. A general POCIS design contains a sorbent that is used to collect pesticides, natural as well as synthetic hormones, and wastewater related chemicals. The pharmaceutical POCIS configuration contains a sorbent that is designed to specifically target classes of pharmaceuticals. Applicable contaminants that concentrate in a POCIS device. Not to be considered a complete list. === POCIS processing === Before the POCIS is constructed, all the hardware as well as the sorbents and membrane must be thoroughly cleaned so that any potential interference is removed. During and after sampling the only cleaning necessary is the removal of any sediment that has adhered to the surface of the sampler. After assembly, and prior to deployment, the samplers are stored in frozen airtight containers to avoid any contamination. The samplers should be kept in airtight containers during transportation both to and from the sampling site so that airborne contaminants do not contaminate the sampler. It is ideal to keep the samplers cold while transporting them in order to preserve the integrity of the samples. After the POCIS is retrieved from the field, the membrane is gently cleaned to reduce the possibility of any contamination to the sorbent. The sorbent is placed into a chromatography column so that the chemicals that samples can be recovered using an organic solvent. The solvent used is specifically chosen based on the type of sorbent and chemicals sampled. The sample can go through further processing such as cleanup or
|
{
"page_id": 42927021,
"source": null,
"title": "Polar organic chemical integrative sampler"
}
|
fractionation depending on the desired use of the sample. === Data analysis === After the sample has been processed, the extract can be analysed using a variety of data analysis techniques. The chemical analysis and analytical instrumentation used depends on the goal of the study. Many analyses require multiple samples, although in some cases a single POCIS sample can be used for multiple analyses. It is vital to use quality control (QC) procedures when using passive samplers. It is common practice for 10% to 50% of the total number of samples to be used for QC purposes. The number of QC samples depends on the study objectives. The QC samples are used to address issues such as sample contamination and analyte recovery. The types of QC samples commonly used include; reagent blanks, field blanks, matrix spikes, and procedural spikes. A large number of studies have been performed in which POCIS data was combined with bioassays to measure biological endpoints. Testing POCIS extracts in biological assays is useful as a POCIS device samples over its entire deployment period, and biologically active compounds can be effectively monitored. It can also be argued that the use of POCIS is a more relevant from an ecotoxicological perspective as the use of a passive sampler mimics the uptake of compounds by organisms. Another strength in using bioassays to test environmental samples is that they can provide an integrative measure of the toxic potential of a group of chemical compounds, rather than a single contaminant. == Other passive samplers == There are many types of passive samplers used that specialize in absorbing different classes of aquatic contaminants found in the environment. Chemcatcher and SMPD are two types of passive samplers that are also commonly used. Monitoring programs use SMPDs to measure to hydrophobic organic contaminants. SPMDs
|
{
"page_id": 42927021,
"source": null,
"title": "Polar organic chemical integrative sampler"
}
|
are designed to mimic the bioconcentration of contaminants in fatty tissues (ITRC, 2006). Contaminants applicable to the use of an SPMD include, but are not limited to, polychlorinated biphenyls (PCBs), polycyclic aromatic hydrocarbons (PAHs), organochlorine pesticides, dioxins, and furans. The SPMD consist of a thin-walled, nonporous, polyethylene membrane tube that is filled with high molecular weight lipid. These tubes are approximately 90 cm long and wrap around the inside of a stainless steel deployment canister. SMPDs are efficient at absorbing pollutants with a log Kow of 4-8. This slightly overlaps with the range of contaminants absorbed by POCIS. Because of this, SMPDs and POCIS devices are often used together in monitoring studies to achieve a more representative understanding of contamination. == Future development == The POCIS system is continually evaluated for the potential to sample a wide range of contaminants. Calibration data and analyte recovery methods are currently being generated by researchers around the world. Techniques to merge the POCIS device with bioassays are also under development. The POCIS sampler already serves as a versatile, economical, and robust tool for monitoring studies and observing trends in both space and time. However, sampling rates are not yet robust enough to supply reliable contaminant concentrations, particularly when regarding environmental quality standards. A limited number of sampling rates have been determined for chemicals and the determination of additional sampling rate data is necessary for the advancement of passive sampling technology. == References ==
|
{
"page_id": 42927021,
"source": null,
"title": "Polar organic chemical integrative sampler"
}
|
The marsh organ is a collection of plastic pipes attached to a wooden framework that is placed in marshes to measure the effects of inundation time and flood frequency on the productivity of marsh vegetation. The information is used for scientific research purposes. The marsh organ was developed by James Morris from the University of South Carolina with support from the National Science Foundation and NOAA's National Centers for Coastal Ocean Science. Their objective was to quantify short-term and long-term effects of sea level rise on coastal processes such as plant productivity, decomposition of organic matter in soil, sedimentation that contribute to the structuring of wetland stability. == Climate change == The marsh organ is used to determine how well various coastal processes will respond to sea level rise. Climate change impacts such as accelerated sea level rise causes coastal marshes to experience higher water levels than normal, which leads to higher salinity inland, sediment and elevation loss, and change to the plant community structure. These consequences will affect stress-gradients that are imposed on coastal vegetation, but the tolerances of these plant species and the trade-offs they may experience are unclear. This device is a way to directly manipulate what marsh vegetation may experience in the future and provide better insight into the restoration efforts needed to prevent detrimental consequences to coastal marshes. == Design == The marsh organ is a structure with rows of pipes at different vertical elevations. These pipes are filled with mud and marsh plant species are planted into each pipe. The various vertical levels represent varying water-level "elevation" that the marsh plants would experience. As the tides ebb and flow, the pipes are exposed to rising and falling water levels. Scientists can adjust various factors, such as the total elevation of the setup, flooding duration,
|
{
"page_id": 67568559,
"source": null,
"title": "Marsh organ"
}
|
added nutrients and much more. Over time, scientists can gather information such as total plant biomass accumulation, total organic matter, peat formation, decomposition rates, and sedimentation. The data can be used to forecast the future health of the marsh being studied, and to infer how the marsh will respond to sea level rise in the future. == Species-specific effects == With many different marsh organ study designs and approaches, researchers have found that marsh plants may respond to future sea level rise differently, therefore it is entirely species-specific. Researchers studying marsh plant responses in Northeast Pacific tidal marshes utilized a marsh organ and found that species typically found in the high marsh (flooded only during high tide or extreme weather events) like Salicornia pacifica and Juncus balticus were sensitive to increased flooding. Other species such as Bolboschoenus maritimus and Carex lyngbyei were abundant in marshes at or above the elevation corresponding with their maximum productivity. Another group using the marsh organ also found that increased inundation reduced biomass for species commonly found at higher marsh elevations. The presence of neighbors reduced total biomass even more. A group of researchers used a marsh organ to evaluate the effects of an invasive grass to the native plant communities of an estuary in China. They found that the invasive grass survived well in optimal elevations, and not very well in extreme high and low elevations. When mixed with native species, the invasive grass suppressed the native biomass by 90% at intermediate elevation where biomass was typically the greatest. Another group who used the marsh organ in the Pacific Northwest of North America to study its role in field testing seed recruitment niches found that species common to the area like S. tabernaemontani exhibited nearly significant higher germination rates around the average tidal height,
|
{
"page_id": 67568559,
"source": null,
"title": "Marsh organ"
}
|
while the species Carex lyngbyei survived significantly better around the highest tidal height. Both species also showed sensitivity to competitors, with S. tabernaemontani being the only species to germinate in the presence of competition. == External stressors == Along with the stress of rising sea levels, marsh vegetation is also influenced by many outside sources such as storms, drought, nutrient enrichment, and elevation change with subsidence. The responses of marsh plants to these stressors have been tracked in various studies using the marsh organ. A group using the marsh organ to study Spartina alterniflora, an abundant low marsh (typically flooded throughout the day) grass found that storm and drought stressors led to significantly less above-ground biomass and below-ground biomass than those planted in ambient rain conditions. Plants flooded at high inundations additionally had finer roots and shoots resulting in a plant that is structurally weaker. Nutrient addition has the potential to aid in the growth of many plant species, but excess nutrients can have the reverse affect and be detrimental to the success of many marsh plants. In one marsh organ study, researchers found a positive relationship where added nitrogen enhanced plant growth at sea levels where plants are most stressed by flooding, and the effects were larger in combination with elevated carbon dioxide. However, they noted that chronic nitrogen addition from pollution reduces the availability of propagules (a bud of a new plant) of flood-tolerant species which would shift species dominance making marshes more susceptible to collapse. This trade-off has also been found in marsh organ studies where nutrient addition has to potential to increase primary productivity, but can adversely impact organic matter accumulation and peat formation. Marsh plants can be sensitive to elevation change accompanying sea level rise due to the altering of their desired habitats. Using the
|
{
"page_id": 67568559,
"source": null,
"title": "Marsh organ"
}
|
marsh organ setup, researchers discovered that for marsh elevations higher than optimum expected at low sea level rise rates, acceleration in the rate of sea level rise will enhance root growth, organic accretion and wetland stability altogether. But, for sub-optimum marsh elevations expected at rapid sea level rise rates with low sediment supply, increases in water level will lead to reduced root growth and a decrease in the rate of elevation gain. This could lead to a rapidly deteriorating marsh. Coinciding with elevation, a group of researchers utilizing the marsh organ found that soil subsided less in planted treatments than unplanted control treatments suggested that plants potentially help alleviate the loss of marsh elevation due to sea level rise Researchers have also found that water level variability in a specified time period affects the growth of coastal marshes, with emphasis on anomalies in sea level. These consist of slow changes that do not affect sediment transport, but do affect marsh flooding and vegetation growth. == References ==
|
{
"page_id": 67568559,
"source": null,
"title": "Marsh organ"
}
|
T790M, also known as Thr790Met, is a gatekeeper mutation of the epidermal growth factor receptor (EGFR). The mutation substitutes a threonine (T) with a methionine (M) at position 790 of exon 20, affecting the ATP binding pocket of the EGFR kinase domain. Threonine is a small polar amino acid; methionine is a larger nonpolar amino acid. Rather than directly blocking inhibitor binding to the active site, T790M increases the affinity for ATP so that the inhibitors are outcompeted; irreversible covalent inhibitors such as osimertinib can overcome this resistance. == Clinical == Over 50% of acquired resistance to EGFR tyrosine kinase inhibitors (TKI) is caused by a mutation in the ATP binding pocket of the EGFR kinase domain involving substitution of a small polar threonine residue with a large nonpolar methionine residue, T790M. In November 2015, the US FDA granted accelerated approval to osimertinib (Tagrisso) for the treatment of patients with metastatic epidermal growth factor receptor (EGFR) T790M mutation-positive non-small cell lung cancer (NSCLC), as detected by an FDA-approved test, which progressed on or after EGFR TKI therapy. == References ==
|
{
"page_id": 48563120,
"source": null,
"title": "T790M"
}
|
Chlorodifluoroethane has the following isomers: 2-Chloro-1,1-difluoroethane (R-142) 1-Chloro-1,2-difluoroethane (R-142a) 1-Chloro-1,1-difluoroethane (R-142b)
|
{
"page_id": 40698799,
"source": null,
"title": "Chlorodifluoroethane"
}
|
In molecular biology mir-241 microRNA is a short RNA molecule. MicroRNAs function to regulate the expression levels of other genes by several mechanisms. == See also == MicroRNA == References == == Further reading == == External links == Page for mir-241 microRNA precursor family at Rfam
|
{
"page_id": 36373427,
"source": null,
"title": "Mir-241 microRNA precursor family"
}
|
Ubiquitin-conjugating enzymes, also known as E2 enzymes and more rarely as ubiquitin-carrier enzymes, perform the second step in the ubiquitination reaction that targets a protein for degradation via the proteasome. The ubiquitination process covalently attaches ubiquitin, a short protein of 76 amino acids, to a lysine residue on the target protein. Once a protein has been tagged with one ubiquitin molecule, additional rounds of ubiquitination form a polyubiquitin chain that is recognized by the proteasome's 19S regulatory particle, triggering the ATP-dependent unfolding of the target protein that allows passage into the proteasome's 20S core particle, where proteases degrade the target into short peptide fragments for recycling by the cell. == Relationships == A ubiquitin-activating enzyme, or E1, first activates the ubiquitin by covalently attaching the molecule to its active site cysteine residue. The activated ubiquitin is then transferred to an E2 cysteine. Once conjugated to ubiquitin, the E2 molecule binds one of several ubiquitin ligases or E3s via a structurally conserved binding region. The E3 molecule is responsible for binding the target protein substrate and transferring the ubiquitin from the E2 cysteine to a lysine residue on the target protein. A particular cell usually contains only a few types of E1 molecule, a greater diversity of E2s, and a very large variety of E3s. In humans, there are about 30 E2s which can bind with one of the 600+ E3s. The E3 molecules responsible for substrate identification and binding are thus the mechanisms of substrate specificity in proteasomal degradation. Each type of E2 can associate with many E3s. E2s can also be used to study protein folding mechanisms. Since the ubiquitylation system is shared across all organisms, studies can use modified E2 proteins in order to understand the overall system for how all organisms process proteins. There are also some
|
{
"page_id": 8258485,
"source": null,
"title": "Ubiquitin-conjugating enzyme"
}
|
proteins which can act as both and E2 and an E3 containing domains which cover both E2 and E3 functionality. == Isozymes == The following human genes encode ubiquitin-conjugating enzymes: == See also == Ubiquitin Ubiquitin-activating enzyme Ubiquitin ligase == References == == External links == Eukaryotic Linear Motif resource motif class MOD_SUMO Ubiquitin-Conjugating+Enzymes at the U.S. National Library of Medicine Medical Subject Headings (MeSH)
|
{
"page_id": 8258485,
"source": null,
"title": "Ubiquitin-conjugating enzyme"
}
|
The regulation of science refers to use of law, or other ruling, by academic or governmental bodies to allow or restrict science from performing certain practices, or researching certain scientific areas. Science could be regulated by legislation if areas are seen as harmful, immoral, or dangerous. For these reasons science regulation may be closely related to religion, culture and society. Science regulation is often a bioethical issue related to practices such as abortion and euthanasia, and areas of research such as stem-cell research and cloning synthetic biology. == United States == === Biomedical research === Unjust events such as the St. Louis tragedy or the Tuskegee syphilis experiment have prompted regulations in biomedical research. Over the years, regulations have been extended to encompass animal welfare and research misconduct. The federal government also monitors the production and sale of the results of biomedical research such as drugs and biopharmaceuticals. The FDA and the Department of Health and Human Services oversee the implementation of these regulations. The Dickey–Wicker Amendment prohibits the Department of Health and Human Services (HHS) from using appropriated funds for the creation of human embryos for research purposes or for research in which human embryos are destroyed. ==== Human subject research ==== The issue of experimentation on human subjects gained prominence after World War II and the revelation of atrocities committed in the name of science. In the United States, the 1962 Kefauver-Harris amendments to the FDA included for the first time a requirement for informed consent of participants. In 1966, a policy statement by the U.S Surgeon General required that all human subject research go through independent prior review. The National Research Act of 1974 institutionalized this review process by requiring that research centers establish Institutional Review Boards (IRBs). Universities, hospitals, and other research institutions set up these
|
{
"page_id": 1311669,
"source": null,
"title": "Regulation of science"
}
|
IRBs to review all the research done at the institution. These boards, generally composed of both scientific peers from the institution and lay persons, are tasked with assessing the risks and benefits associated with the use of human subjects, in addition to the adequacy of the protection and consent of the participants. The IRBs can approve research proposals, make modifications, or disapprove them entirely. Research projects cannot receive federal funding without approval from an IRB. Noncompliance can also induce sanctions from the institution, such as revoked access to facilities and subjects, suspension, and dismissal. The National Research Act of 1974 also set up the National Commission for the Protection of Human Subjects of Biomedical and Behavioral Research, which produced the Belmont Report (Report on Ethical Principles and Guidelines for the Protection of Human Subjects of Research) in 1979. This report established a moral framework for the regulation of research involving human subjects. ==== Animal welfare ==== The Animal Welfare Act of 1966 sets standards of treatment of animals in research experiments. It requires all research facilities to register with the USDA and allows officials to conduct unannounced facility inspections. The Health Research Extension Act of 1985 requires that all research facilities using animals establish Institutional Animal Care and Use Committees (IACUCs) to evaluate twice a year the institutions' activities involving animals. The IACUCs report to the NIH Office of Laboratory Animal Welfare annually. ==== Research misconduct ==== The Health Research Extension Act of 1985 led to the establishment of the Office of Research Integrity (ORI) within the Department of Health and Human Services. ORI is responsible for reviewing research misconduct allegations and developing policies to improve the responsible conduct of research. ==== Commercialization ==== Two divisions of the Food and Drug Administration (FDA) are in charge of monitoring the production
|
{
"page_id": 1311669,
"source": null,
"title": "Regulation of science"
}
|
and sale of drugs. The Center for Drug Evaluation and Research (CDER) is responsible for reviewing new drug applications and requires clinical trials as proof of effectiveness. The Center for Biologics Evaluation and Research (CBER) is responsible for implementing federal regulations of biopharmaceuticals such as vaccines, blood components, gene therapies, etc. They approve new drugs on the basis of safety and effectiveness, and issue licenses, which allow companies to market their products. === Nuclear energy research === Nuclear energy is historically linked to issues of national security. From 1942 to 1946, nuclear research was controlled by the military, which conducted research in secrecy. In 1946, the Atomic Energy Act handed over control to civilians, although the government retained a tight monopoly over nuclear energy. The 1954 amendment to this act enabled private industry to pursue non-military applications of nuclear research. The Energy Reorganization Act of 1974 established the Nuclear Regulatory Commission (NRC), in charge of licensing and safety. The Chernobyl and Fukushima accidents raised concerns and public apprehension over the safety of nuclear power. As a result, the NRC strengthened safety regulations for nuclear power plants. === Teaching === Science education is a controversial subject in the United States. Several states banned the teaching of evolution in the 20th century, most notably the state of Tennessee with the Butler Act of 1925. It was followed by the Scopes Trial, in which the state of Tennessee accused Scopes, a high school teacher, of teaching evolution. Although he was found guilty and fined, the trial showed declining public support for Fundamentalists. The Scopes Trial had an important impact in the larger creation versus evolution debate. In the following decades, the term "evolution" was omitted in many biology textbooks, even when the text discusses it. These bans on teaching evolution were overturned by
|
{
"page_id": 1311669,
"source": null,
"title": "Regulation of science"
}
|
a Supreme Court ruling in Epperson v. Arkansas in 1968. Since 2001, there has been a resurgence of anti-evolution bills, one of which, the Louisiana Science Education Act, was passed. This Act allows public schools to use supplementary material that is critical of the scientific theories such as evolution and global warming in science classrooms. The U.S. government and state legislatures have also enacted regulations promoting science education. The National Defense Education Act of 1958 was passed soon after the Soviet Union's launch of Sputnik 1 and linked education with issues of national security. This law provided funding for scholarships and science programs. In 2013, 26 state governments worked together to produce the Next Generation Science Standards, which sets expectations for K–12 science education. == International regulations == The Nuremberg Code was written as part of the trials of Nazi doctors after World War II. It introduced ten ethical principles regarding human experimentation, the first of which requires informed consent from human subjects. It also states that experimentation on humans must be necessary to society, be preceded by studies on animals, and protect subjects from injury, disability and death. The Nuremberg Code was very influential in shaping regulations of scientific research across the world. For example, the Helsinki Declaration of 1964 was developed by the World Medical Association and establishes ethical principles for the medical community. == See also == Ethics committee Intelligent design in politics Lysenkoism Politicization of science Right to science and culture Scientific freedom == References ==
|
{
"page_id": 1311669,
"source": null,
"title": "Regulation of science"
}
|
Kovács reagent is a biochemical reagent consisting of isoamyl alcohol, para-dimethylaminobenzaldehyde (DMAB), and concentrated hydrochloric acid. It is used for the diagnostical indole test, to determine the ability of the organism to split indole from the amino acid tryptophan. The indole produced yields a red complex with para-dimethylaminobenzaldehyde under the given conditions. This was invented by the Hungarian physician Nicholas Kovács and was published in 1928. This reagent is used in the confirmation of E. coli and many other pathogenic microorganisms. == See also == Ehrlich's reagent is similar but uses ethyl alcohol or 1-propyl alcohol. == References == 2. Kovacs, N. (1928): Eine vereinfachte Methode zum Nachweis der Indolbildung durch Bakterien. Zeitschrift für Immunitätsforschung und Experimentelle Therapie, 55, 311.
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{
"page_id": 34407360,
"source": null,
"title": "Kovac's reagent"
}
|
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