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Neural style transfer(NST) refers to a class of software algorithms that manipulate digital images, or videos, in order to adopt the appearance or visual style of another image. NST algorithms are characterized by their use ofdeep neural networksfor the sake of image transformation. Common uses for NST are the creation... | https://en.wikipedia.org/wiki/Neural_style_transfer |
Computer facial animationis primarily an area ofcomputer graphicsthat encapsulates methods and techniques for generating and animating images or models of a character face. The character can be ahuman, a humanoid, ananimal, alegendary creatureor character, etc. Due to its subject and output type, it is also related to ... | https://en.wikipedia.org/wiki/Computer_facial_animation |
Digital cloningis an emerging technology, that involves deep-learning algorithms, which allows one to manipulate currently existingaudio,photos, andvideosthat are hyper-realistic.[1]One of the impacts of such technology is that hyper-realistic videos and photos makes it difficult for the human eye to distinguish what i... | https://en.wikipedia.org/wiki/Digital_cloning |
Digital face replacementis acomputer generated imageryeffect used inmotion picture post-production.[1]It is commonly used to make an actor'sbody doubleorstunt doublelook as if they are the original actor. Possibly the earliest use of face replacement was in the 1993 movieJurassic Park.[1]
Digital face replacementhas a... | https://en.wikipedia.org/wiki/Digital_face_replacement |
Facial motion captureis the process of electronically converting the movements of a person's face into a digital database using cameras orlaser scanners. This database may then be used to producecomputer graphics(CG),computer animationfor movies, games, or real-time avatars. Because the motion of CG characters is deriv... | https://en.wikipedia.org/wiki/Facial_motion_capture |
Fake nude photographyis the creation ofnude photographsdesigned to appear as genuine nudes of an individual.[1][2]The motivations for the creation of these modified photographs include curiosity, sexual gratification, the stigmatization or embarrassment of the subject, and commercial gain, such as through the sale of t... | https://en.wikipedia.org/wiki/Fake_nude_photography |
Fifth-generation warfare(5GW) is warfare that is conducted primarily throughnon-kinetic military action, such associal engineering,misinformation,cyberattacks, along with emerging technologies such asartificial intelligenceand fullyautonomous systems. Fifth generation warfare has been described by Daniel Abbot as a war... | https://en.wikipedia.org/wiki/Fifth-generation_warfare |
Hyperrealityis a concept inpost-structuralismthat refers to the process of the evolution of notions of reality, leading to a cultural state of confusion betweensignsand symbols invented to stand in for reality, and direct perceptions ofconsensus reality.[1]Hyperreality is seen as a condition in which, because of the co... | https://en.wikipedia.org/wiki/Hyperreality |
Identity replacement technologyis any technology that is used to cover up all or parts of a person's identity, either in real life orvirtually. This can include facemasks, face authentication technology, anddeepfakeson the Internet that spread fakeeditingof videos and images.[1]Face replacement and identity masking are... | https://en.wikipedia.org/wiki/Identity_replacement_technology |
Avirtual assistant(VA) is asoftware agentthat can perform a range of tasks or services for a user based on user input such as commands or questions, including verbal ones. Such technologies often incorporatechatbotcapabilities to streamline task execution. The interaction may be via text, graphical interface, or voice ... | https://en.wikipedia.org/wiki/Interactive_online_characters |
TheStyleGenerative Adversarial Network, orStyleGANfor short, is an extension to the GAN architecture introduced byNvidiaresearchers in December 2018,[1]and madesource availablein February 2019.[2][3]
StyleGAN depends on Nvidia'sCUDAsoftware, GPUs, andGoogle'sTensorFlow,[4]orMeta AI'sPyTorch, which supersedes TensorFlo... | https://en.wikipedia.org/wiki/StyleGAN |
Theuncanny valley(Japanese:不気味の谷,Hepburn:bukimi no tani)effect is a hypothesized psychological and aesthetic relation between an object's degree of resemblance to a human being and the emotional response to the object. The uncanny valley hypothesis predicts that an entity appearing almost human will risk eliciting eeri... | https://en.wikipedia.org/wiki/Uncanny_valley |
Avirtual actoror also known asvirtual human,virtual persona, ordigital cloneis the creation or re-creation of a human being in image and voice usingcomputer-generated imageryand sound, that is often indistinguishable from the real actor.
The idea of a virtual actor was first portrayed in the 1981 filmLooker, wherein m... | https://en.wikipedia.org/wiki/Virtual_actor |
Inprobability theoryandstatistics,diffusion processesare a class of continuous-timeMarkov processwithalmost surelycontinuoussample paths. Diffusion process isstochasticin nature and hence is used to model many real-life stochastic systems.Brownian motion,reflected Brownian motionandOrnstein–Uhlenbeck processesare examp... | https://en.wikipedia.org/wiki/Diffusion_process |
Variational Bayesian methodsare a family of techniques for approximating intractableintegralsarising inBayesian inferenceandmachine learning. They are typically used in complexstatistical modelsconsisting of observed variables (usually termed "data") as well as unknownparametersandlatent variables, with various sorts ... | https://en.wikipedia.org/wiki/Variational_inference |
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Fan conventions
15.aiis a free non-commercialweb applicationthat usesartificial intelligenceto generatetext-to-speechvoices of fictional characters f... | https://en.wikipedia.org/wiki/15.ai |
Artificial imaginationis anarrowsubcomponent ofartificial general intelligencewhichgenerates,simulates, andfacilitates[1]realorpossiblefictionmodelsto createpredictions,inventions,[2]orconscious experiences.
The term artificial imagination is also used to describe a property ofmachinesorprograms. Some of the traits th... | https://en.wikipedia.org/wiki/Artificial_imagination |
Automated journalism, also known asalgorithmic journalismorrobot journalism,[1][2][3]is a term that attempts to describe modern technological processes that have infiltrated the journalistic profession, such as news articles and videos generated by computer programs.[3][4][5]There are four main fields of application fo... | https://en.wikipedia.org/wiki/Automated_journalism |
Computer musicis the application ofcomputing technologyinmusic composition, to help human composers create new music or to have computers independently create music, such as withalgorithmic compositionprograms. It includes the theory and application of new and existing computer software technologies and basic aspects o... | https://en.wikipedia.org/wiki/Computer_music |
DALL-E,DALL-E 2, andDALL-E 3(stylisedDALL·E) aretext-to-image modelsdeveloped byOpenAIusingdeep learningmethodologies to generatedigital imagesfromnatural languagedescriptions known asprompts.
The first version of DALL-E was announced in January 2021. In the following year, its successor DALL-E 2 was released. DALL-E ... | https://en.wikipedia.org/wiki/DALL-E |
Generative Pre-trained Transformer 3(GPT-3) is alarge language modelreleased byOpenAIin 2020.
Like its predecessor,GPT-2, it is a decoder-only[2]transformer modelof deep neural network, which supersedes recurrence and convolution-based architectures with a technique known as "attention".[3]This attention mechanism all... | https://en.wikipedia.org/wiki/GPT-3 |
Human image synthesisis technology that can be applied to make believable and evenphotorealisticrenditions[1][2]of human-likenesses, moving or still. It has effectively existed since the early 2000s. Many films usingcomputer generated imageryhave featured synthetic images of human-like charactersdigitally compositedont... | https://en.wikipedia.org/wiki/Human_image_synthesis |
"AI slop", often simply "slop", is a derogatory term for low-quality media, including writing and images, made usinggenerative artificial intelligencetechnology, characterized by an inherent lack of effort, logic, or purpose.[1][4][5]Coined in the 2020s, the term has a pejorative connotation akin to "spam".[4]
It has ... | https://en.wikipedia.org/wiki/Slop_(artificial_intelligence) |
Atext-to-image modelis amachine learning modelwhich takes an inputnatural languagedescription and produces an image matching that description.
Text-to-image models began to be developed in the mid-2010s during the beginnings of theAI boom, as a result of advances indeep neural networks. In 2022, the output of state-of... | https://en.wikipedia.org/wiki/Text-to-image_model |
Atext-to-video modelis amachine learning modelthat uses anatural languagedescription as input to produce avideorelevant to the input text.[1]Advancements during the 2020s in the generation of high-quality, text-conditioned videos have largely been driven by the development of videodiffusion models.[2]
There are differ... | https://en.wikipedia.org/wiki/Text-to-video_model |
WaveNetis a deepneural networkfor generating raw audio. It was created by researchers at London-basedAIfirmDeepMind. The technique, outlined in a paper in September 2016,[1]is able to generate relatively realistic-sounding human-like voices by directly modelling waveforms using aneural networkmethod trained with record... | https://en.wikipedia.org/wiki/WaveNet |
Instatistical mechanicsandmathematics, aBoltzmann distribution(also calledGibbs distribution[1]) is aprobability distributionorprobability measurethat gives the probability that a system will be in a certainstateas a function of that state's energy and the temperature of the system. The distribution is expressed in the... | https://en.wikipedia.org/wiki/Boltzmann_distribution |
Instatistical mechanics, theGibbs algorithm, introduced byJ. Willard Gibbsin 1902, is a criterion for choosing aprobability distributionfor thestatistical ensembleofmicrostatesof athermodynamic systemby minimizing the average log probability
subject to the probability distributionpisatisfying a set of constraints (usu... | https://en.wikipedia.org/wiki/Gibbs_algorithm |
Instatistics,Gibbs samplingor aGibbs sampleris aMarkov chain Monte Carlo(MCMC)algorithmfor sampling from a specifiedmultivariateprobability distributionwhen direct sampling from the joint distribution is difficult, but sampling from theconditional distributionis more practical. This sequence can be used to approximate... | https://en.wikipedia.org/wiki/Gibbs_sampling |
Inprobability theory, aninteracting particle system(IPS) is astochastic process(X(t))t∈R+{\displaystyle (X(t))_{t\in \mathbb {R} ^{+}}}on some configuration spaceΩ=SG{\displaystyle \Omega =S^{G}}given by a site space, acountably-infinite-ordergraphG{\displaystyle G}and a local state space, acompactmetric spaceS{\displa... | https://en.wikipedia.org/wiki/Interacting_particle_system |
Ingame theory, a game is said to be apotential gameif the incentive of all players to change theirstrategycan be expressed using a single global function called thepotential function. The concept originated in a 1996 paper by Dov Monderer andLloyd Shapley.[1]
The properties of several types of potential games have sin... | https://en.wikipedia.org/wiki/Potential_game#Bounded_Rational_Models |
Stochastic cellular automataorprobabilistic cellular automata(PCA) orrandom cellular automataorlocally interactingMarkov chains[1][2]are an important extension ofcellular automaton. Cellular automata are a discrete-timedynamical systemof interacting entities, whose state is discrete.
The state of the collection of ent... | https://en.wikipedia.org/wiki/Stochastic_cellular_automata |
ThePearson distributionis a family ofcontinuousprobability distributions. It was first published byKarl Pearsonin 1895 and subsequently extended by him in 1901 and 1916 in a series of articles onbiostatistics.
The Pearson system was originally devised in an effort to model visiblyskewedobservations. It was well known ... | https://en.wikipedia.org/wiki/Pearson_distribution |
Inmathematics, aSheffer sequenceorpoweroidis apolynomial sequence, i.e., asequence(pn(x) :n= 0, 1, 2, 3, ...)ofpolynomialsin which the index of each polynomial equals itsdegree, satisfying conditions related to theumbral calculusincombinatorics. They are named forIsador M. Sheffer.
Fix a polynomial sequence (pn). Defi... | https://en.wikipedia.org/wiki/Sheffer_sequence |
Inmathematics, anorthogonal polynomial sequenceis a family ofpolynomialssuch that any two different polynomials in the sequence areorthogonalto each other under someinner product.
The most widely used orthogonal polynomials are theclassical orthogonal polynomials, consisting of theHermite polynomials, theLaguerre poly... | https://en.wikipedia.org/wiki/Orthogonal_polynomials |
Acompound Poisson processis a continuous-timestochastic processwith jumps. The jumps arrive randomly according to aPoisson processand the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rateλ>0{\displaystyle \lambda >0}and jump s... | https://en.wikipedia.org/wiki/Compound_Poisson_process |
Inprobability theoryandstatistics, thegeometric Poisson distribution(also called thePólya–Aeppli distribution) is used for describing objects that come in clusters, where the number of clusters follows aPoisson distributionand the number of objects within a cluster follows ageometric distribution.[1]It is a particular ... | https://en.wikipedia.org/wiki/Geometric_Poisson_distribution |
Aphase-type distributionis aprobability distributionconstructed by a convolution or mixture ofexponential distributions.[1]It results from a system of one or more inter-relatedPoisson processesoccurring insequence, or phases. The sequence in which each of the phases occurs may itself be astochastic process. The distrib... | https://en.wikipedia.org/wiki/Phase-type_distribution#Coxian_distribution |
Inqueueing theory, theEngset formulais used to determine the blocking probability of an M/M/c/c/N queue (inKendall's notation).
The formula is named after its developer,T. O. Engset.
Consider a fleet ofc{\displaystyle c}vehicles andN{\displaystyle N}operators. Operators enter the system randomly to request the use of... | https://en.wikipedia.org/wiki/Engset_calculation |
Theerlang(symbolE[1]) is adimensionless unitthat is used intelephonyas a measure ofoffered loador carried load on service-providing elements such as telephone circuits or telephone switching equipment. A singlecord circuithas the capacity to be used for 60 minutes in one hour. Full utilization of that capacity, 60 minu... | https://en.wikipedia.org/wiki/Erlang_B |
Aphase-type distributionis aprobability distributionconstructed by a convolution or mixture ofexponential distributions.[1]It results from a system of one or more inter-relatedPoisson processesoccurring insequence, or phases. The sequence in which each of the phases occurs may itself be astochastic process. The distrib... | https://en.wikipedia.org/wiki/Phase-type_distribution |
For detection systems that record discrete events, such asparticleandnucleardetectors, thedead timeis the time after each event during which the system is not able to record another event.[1]An everyday life example of this is what happens when someone takes a photo using a flash – another picture cannot be taken immed... | https://en.wikipedia.org/wiki/Dead_time |
In applied statistics, theMarshall–Olkin exponential distributionis any member of a certain family of continuous multivariate probability distributions with positive-valued components. It was introduced by Albert W. Marshall andIngram Olkin.[1]One of its main uses is in reliability theory, where the Marshall–Olkin copu... | https://en.wikipedia.org/wiki/Marshall%E2%80%93Olkin_exponential_distribution |
Inprobability theory, abeta negative binomial distributionis theprobability distributionof adiscreterandom variableX{\displaystyle X}equal to the number of failures needed to getr{\displaystyle r}successes in a sequence ofindependentBernoulli trials. The probabilityp{\displaystyle p}of success on each trial stays const... | https://en.wikipedia.org/wiki/Beta_negative_binomial_distribution |
Inprobabilityandstatisticstheextended negative binomial distributionis adiscrete probability distributionextending thenegative binomial distribution. It is atruncatedversion of the negative binomial distribution[1]for which estimation methods have been studied.[2]
In the context ofactuarial science, the distribution a... | https://en.wikipedia.org/wiki/Extended_negative_binomial_distribution |
Inprobability theoryandstatistics, thenegative multinomial distributionis a generalization of thenegative binomial distribution(NB(x0,p)) to more than two outcomes.[1]
As with the univariate negative binomial distribution, if the parameterx0{\displaystyle x_{0}}is a positive integer, the negative multinomial distribut... | https://en.wikipedia.org/wiki/Negative_multinomial_distribution |
Instatistics,Poisson regressionis ageneralized linear modelform ofregression analysisused to modelcount dataandcontingency tables.[1]Poisson regression assumes the response variableYhas aPoisson distribution, and assumes thelogarithmof itsexpected valuecan be modeled by a linear combination of unknownparameters. A Pois... | https://en.wikipedia.org/wiki/Negative_binomial_regression |
Instatistics, the class ofvector generalized linear models(VGLMs) was proposed to
enlarge the scope of models catered for bygeneralized linear models(GLMs).
In particular, VGLMs allow for response variables outside the classicalexponential familyand for more than one parameter. Each parameter (not necessarily a mean) ... | https://en.wikipedia.org/wiki/Vector_generalized_linear_model |
Instatistics,burstinessis the intermittent increases and decreases in activity orfrequencyof an event.[1][2]One measure of burstiness is theFano factor—a ratio between thevarianceandmeanof counts.
Burstiness is observable in natural phenomena, such asnatural disasters, or other phenomena, such asnetwork/data/emailnetw... | https://en.wikipedia.org/wiki/Burstiness |
Theclustering illusionis the tendency to erroneously consider the inevitable "streaks" or "clusters" arising in small samples from random distributions to be non-random. The illusion is caused by a human tendency to underpredict the amount ofvariabilitylikely to appear in a small sample of random orpseudorandomdata.[1]... | https://en.wikipedia.org/wiki/Clustering_illusion |
Forstatisticsinprobability theory, theBoolean-Poisson modelor simplyBoolean modelfor a random subset of the plane (or higher dimensions, analogously) is one of the simplest and most tractable models instochastic geometry. Take aPoisson point processof rateλ{\displaystyle \lambda }in the plane and make each point be th... | https://en.wikipedia.org/wiki/Boolean_model_(probability_theory) |
Inprobability theory, aCox process, also known as adoubly stochastic Poisson processis apoint processwhich is a generalization of aPoisson processwhere the intensity that varies across the underlying mathematical space (often space or time) is itself a stochastic process. The process is named after thestatisticianDavid... | https://en.wikipedia.org/wiki/Cox_process |
Instatisticsandprobability theory, apoint processorpoint fieldis a set of a random number ofmathematical pointsrandomly located on a mathematical space such as thereal lineorEuclidean space.[1][2]
Point processes on the real line form an important special case that is particularly amenable to study,[3]because the poin... | https://en.wikipedia.org/wiki/Point_process |
In mathematics,stochastic geometryis the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory ofspatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting ofrandom measures.
There are various models f... | https://en.wikipedia.org/wiki/Stochastic_geometry |
Inmathematicsandtelecommunications,stochastic geometry models of wireless networksrefer tomathematical modelsbased onstochastic geometrythat are designed to represent aspects ofwireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks... | https://en.wikipedia.org/wiki/Stochastic_geometry_models_of_wireless_networks |
Inqueueing theory, a discipline within the mathematicaltheory of probability, aMarkovian arrival process(MAPorMArP[1]) is a mathematical model for the time between job arrivals to a system. The simplest such process is aPoisson processwhere the time between each arrival isexponentially distributed.[2][3]
The processes... | https://en.wikipedia.org/wiki/Markovian_arrival_processes |
Instatistics, afixed-effect Poisson modelis aPoisson regressionmodel used for staticpanel datawhen the outcome variable iscount data. Hausman, Hall, and Griliches pioneered the method in the mid 1980s. Their outcome of interest was the number of patents filed by firms, where they wanted to develop methods to control fo... | https://en.wikipedia.org/wiki/Fixed-effect_Poisson_model |
Partial (pooled) likelihood estimation forpanel datais aquasi-maximum likelihoodmethod forpanel analysisthat assumes that density ofyit{\displaystyle y_{it}}givenxit{\displaystyle x_{it}}is correctly specified for each time period but it allows for misspecification in the conditional density ofyi=(yi1,…,yiT){\displayst... | https://en.wikipedia.org/wiki/Partial_likelihood_methods_for_panel_data#Pooled_QMLE_for_Poisson_models |
Control functions(also known astwo-stage residual inclusion) are statistical methods to correct forendogeneityproblems by modelling the endogeneity in theerror term. The approach thereby differs in important ways from other models that try to account for the sameeconometricproblem.Instrumental variables, for example, a... | https://en.wikipedia.org/wiki/Control_function_(econometrics)#Endogeneity_in_Poisson_regression |
In the theory of finite population sampling,Bernoulli samplingis a sampling process where each element of thepopulationis subjected to anindependentBernoulli trialwhich determines whether the element becomes part of the sample. An essential property of Bernoulli sampling is that all elements of the population have equa... | https://en.wikipedia.org/wiki/Bernoulli_sampling |
a0,t+(a0,t)2−(a0,t)2=a0,t{\displaystyle a_{0,t}+(a_{0,t})^{2}-(a_{0,t})^{2}=a_{0,t}}sinceRx(t1,t2)=a0,min(t1,t2)+a0,t1a0,t2{\displaystyle R_{x}(t_{1},t_{2})=a_{0,min(t_{1},t_{2})}+a_{0,t_{1}}a_{0,t_{2}}}
Inprobability theory,statisticsand related fields, aPoisson point process(also known as:Poisson random measure,Pois... | https://en.wikipedia.org/wiki/Poisson_process |
In the theory of finitepopulationsampling, asampling designspecifies for every possiblesampleitsprobabilityof being drawn.
Mathematically, a sampling design is denoted by the functionP(S){\displaystyle P(S)}which gives the probability of drawing a sampleS.{\displaystyle S.}
DuringBernoulli sampling,P(S){\displaystyle... | https://en.wikipedia.org/wiki/Sampling_design |
Inprobability theoryandstatistics,Campbell's theoremor theCampbell–Hardy theoremis either a particularequationor set of results relating to theexpectationof afunctionsummed over apoint processto anintegralinvolving themean measureof the point process, which allows for the calculation ofexpected valueandvarianceof thera... | https://en.wikipedia.org/wiki/Campbell%27s_theorem_(probability) |
Acontinuous-time Markov chain(CTMC) is a continuousstochastic processin which, for each state, the process will change state according to anexponential random variableand then move to a different state as specified by the probabilities of astochastic matrix. An equivalent formulation describes the process as changing s... | https://en.wikipedia.org/wiki/Continuous-time_Markov_process |
In mathematicalqueueing theory,Little's law(alsoresult,theorem,lemma, orformula[1][2]) is a theorem byJohn Littlewhich states that the long-term average numberLof customers in astationarysystem is equal to the long-term average effective arrival rateλmultiplied by the average timeWthat a customer spends in the system. ... | https://en.wikipedia.org/wiki/Little%27s_lemma |
In the study of age-structured population growth, probably one of the most important equations is theEuler–Lotka equation. Based on the age demographic of females in the population and female births (since in many cases it is the females that are more limited in the ability to reproduce), this equation allows for an es... | https://en.wikipedia.org/wiki/Lotka%27s_integral_equation |
Inprobability theory, thePalm–Khintchine theorem, the work ofConny PalmandAleksandr Khinchin, expresses that a large number ofrenewal processes, not necessarilyPoissonian, when combined ("superimposed") will have Poissonian properties.[1]
It is used to generalise the behaviour of users or clients inqueuing theory. It ... | https://en.wikipedia.org/wiki/Palm%E2%80%93Khintchine_theorem |
In the theory ofrenewal processes, a part of the mathematical theory of probability, theresidual timeor theforward recurrence timeis the time between any given timet{\displaystyle t}and the nextepochof the renewal process under consideration. In the context of random walks, it is also known asovershoot. Another way to ... | https://en.wikipedia.org/wiki/Residual_time |
TheMcKendrick–von Foerster equationis a linear first-orderpartial differential equationencountered in several areas ofmathematical biology– for example,demography[1]andcell proliferationmodeling; it is applied when age structure is an important feature in themathematical model.[2]It was first presented byAnderson Gray ... | https://en.wikipedia.org/wiki/Von_Foerster_equation |
Aratio distribution(also known as aquotient distribution) is aprobability distributionconstructed as the distribution of theratioofrandom variableshaving two other known distributions.
Given two (usuallyindependent) random variablesXandY, the distribution of the random variableZthat is formed as the ratioZ=X/Yis aratio... | https://en.wikipedia.org/wiki/Ratio_distribution#Poisson_and_truncated_Poisson_distributions |
Ahurdle modelis a class ofstatistical modelswhere a random variable is modelled using two parts, the first of which is the probability of attaining the value 0, and the second part models the probability of the non-zero values. The use of hurdle models is often motivated by an excess of zeroes in the data that is not s... | https://en.wikipedia.org/wiki/Hurdle_model |
Also known as the(Moran-)Gamma Process,[1]thegamma processis a random process studied inmathematics,statistics,probability theory, andstochastics. The gamma process is astochastic or random processconsisting of independently distributedgamma distributionswhereN(t){\displaystyle N(t)}represents the number of event occur... | https://en.wikipedia.org/wiki/Gamma_process |
Thegeneralized normal distribution(GND) orgeneralized Gaussian distribution(GGD) is either of two families ofparametriccontinuous probability distributionson therealline. Both families add ashape parameterto thenormal distribution. To distinguish the two families, they are referred to below as "symmetric" and "asymme... | https://en.wikipedia.org/wiki/Generalized_normal_distribution#Symmetric_version |
In the mathematical theory of probability,multivariate Laplace distributionsare extensions of theLaplace distributionand theasymmetric Laplace distributionto multiple variables. Themarginal distributionsof symmetric multivariate Laplace distribution variables are Laplace distributions. The marginal distributions of a... | https://en.wikipedia.org/wiki/Multivariate_Laplace_distribution |
Inmathematics— specifically, in the fields ofprobability theoryandinverse problems—Besov measuresand associatedBesov-distributed random variablesare generalisations of the notions ofGaussian measuresandrandom variables,Laplace distributions, and other classical distributions. They are particularly useful in the study ... | https://en.wikipedia.org/wiki/Besov_measure |
Inmathematics, afunction spaceis asetoffunctionsbetween two fixed sets. Often, thedomainand/orcodomainwill have additionalstructurewhich is inherited by the function space. For example, the set of functions from any setXinto avector spacehas anaturalvector space structure given bypointwiseaddition and scalar multiplica... | https://en.wikipedia.org/wiki/Function_space |
Inprobabilityandstatistics, acompound probability distribution(also known as amixture distributionorcontagious distribution) is theprobability distributionthat results from assuming that arandom variableis distributed according to some parametrized distribution, with (some of) the parameters of that distribution themse... | https://en.wikipedia.org/wiki/Scale_mixture |
Infinance, thecapital asset pricing model(CAPM) is a model used to determine a theoretically appropriate requiredrate of returnof anasset, to make decisions about adding assets to awell-diversifiedportfolio.
The model takes into account the asset's sensitivity to non-diversifiable risk (also known assystematic riskorm... | https://en.wikipedia.org/wiki/Capital_asset_pricing_model |
Instatistics,ordinary least squares(OLS) is a type oflinear least squaresmethod for choosing the unknownparametersin alinear regressionmodel (with fixed level-one[clarification needed]effects of alinear functionof a set ofexplanatory variables) by the principle ofleast squares: minimizing the sum of the squares of the ... | https://en.wikipedia.org/wiki/Ordinary_least_squares#Estimation |
Numerical linear algebra, sometimes calledapplied linear algebra, is the study of howmatrix operationscan be used to createcomputer algorithmswhichefficientlyand accurately provide approximate answers to questions incontinuousmathematics. It is a subfield ofnumerical analysis, and a type oflinear algebra.Computersusefl... | https://en.wikipedia.org/wiki/Numerical_linear_algebra |
Non-linear least squaresis the form ofleast squaresanalysis used to fit a set ofmobservations with a model that is non-linear innunknown parameters (m≥n). It is used in some forms ofnonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive ite... | https://en.wikipedia.org/wiki/Numerical_methods_for_non-linear_least_squares |
Inmathematics, theHessian matrix,Hessianor (less commonly)Hesse matrixis asquare matrixof second-orderpartial derivativesof a scalar-valuedfunction, orscalar field. It describes the localcurvatureof a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematicianLudwig Otto... | https://en.wikipedia.org/wiki/Hessian_matrix#Bordered_Hessian |
Inmathematics, adifferentiable functionof onerealvariable is afunctionwhosederivativeexists at each point in itsdomain. In other words, thegraphof a differentiable function has a non-verticaltangent lineat each interior point in its domain. A differentiable function issmooth(the function is locally well approximated as... | https://en.wikipedia.org/wiki/Differentiability |
Inmathematics, theinterior extremum theorem, also known asFermat's theorem, is a theorem which states that at thelocal extremaof adifferentiable function, itsderivativeis always zero. It belongs to the mathematical field ofreal analysisand is named after French mathematicianPierre de Fermat.
By using the interior extr... | https://en.wikipedia.org/wiki/Fermat%27s_theorem_(stationary_points) |
Indifferential calculusanddifferential geometry, aninflection point,point of inflection,flex, orinflection(rarelyinflexion) is a point on asmooth plane curveat which thecurvaturechanges sign. In particular, in the case of thegraph of a function, it is a point where the function changes from beingconcave(concave downwar... | https://en.wikipedia.org/wiki/Inflection_point |
Inmathematical analysis, themaximumandminimum[a]of afunctionare, respectively, the greatest and least value taken by the function. Known generically asextremum,[b]they may be defined either within a givenrange(thelocalorrelativeextrema) or on the entiredomain(theglobalorabsoluteextrema) of a function.[1][2][3]Pierre de... | https://en.wikipedia.org/wiki/Maxima_and_minima |
Inmathematics, aphase lineis a diagram that shows the qualitative behaviour of anautonomousordinary differential equationin a single variable,dydx=f(y){\displaystyle {\tfrac {dy}{dx}}=f(y)}. The phase line is the 1-dimensional form of the generaln{\displaystyle n}-dimensionalphase space, and can be readily analyzed.
A... | https://en.wikipedia.org/wiki/Phase_line_(mathematics) |
Inmathematics, thesecond partial derivative testis a method inmultivariable calculusused to determine if acritical pointof a function is alocal minimum, maximum orsaddle point.
Suppose thatf(x,y)is a differentiablereal functionof two variables whose secondpartial derivativesexist and arecontinuous. TheHessian matrixH... | https://en.wikipedia.org/wiki/Second_partial_derivative_test |
Inmathematics, particularly incalculus, astationary pointof adifferentiable functionof one variable is a point on thegraphof the function where the function'sderivativeis zero.[1][2][3]Informally, it is a point where the function "stops" increasing or decreasing (hence the name).
For a differentiablefunction of sever... | https://en.wikipedia.org/wiki/Stationary_point |
Inmathematics, the concepts ofessential infimumandessential supremumare related to the notions ofinfimum and supremum, but adapted tomeasure theoryandfunctional analysis, where one often deals with statements that are not valid forallelements in aset, but ratheralmost everywhere, that is, except on aset of measure zero... | https://en.wikipedia.org/wiki/Essential_supremum_and_essential_infimum |
Inmathematics, especially inorder theory, thegreatest elementof a subsetS{\displaystyle S}of apartially ordered set(poset) is an element ofS{\displaystyle S}that is greater than every other element ofS{\displaystyle S}. The termleast elementis defineddually, that is, it is an element ofS{\displaystyle S}that is smaller... | https://en.wikipedia.org/wiki/Greatest_element_and_least_element |
Inmathematics, especially inorder theory, amaximal elementof asubsetS{\displaystyle S}of somepreordered setis an element ofS{\displaystyle S}that is not smaller than any other element inS{\displaystyle S}. Aminimal elementof a subsetS{\displaystyle S}of some preordered set is definedduallyas an element ofS{\displaystyl... | https://en.wikipedia.org/wiki/Maximal_and_minimal_elements |
Inmathematics, particularly inorder theory, anupper boundormajorant[1]of asubsetSof somepreordered set(K, ≤)is anelementofKthat isgreater than or equal toevery element ofS.[2][3]Dually, alower boundorminorantofSis defined to be an element ofKthat is less than or equal to every element ofS.
A set with an upper (respect... | https://en.wikipedia.org/wiki/Upper_and_lower_bounds |
Incombinatorics, theinclusion–exclusion principleis a counting technique which generalizes the familiar method of obtaining the number of elements in theunionof twofinite sets; symbolically expressed as
whereAandBare two finite sets and |S| indicates thecardinalityof a setS(which may be considered as the number of ele... | https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle |
Inmathematical analysis, themaximumandminimum[a]of afunctionare, respectively, the greatest and least value taken by the function. Known generically asextremum,[b]they may be defined either within a givenrange(thelocalorrelativeextrema) or on the entiredomain(theglobalorabsoluteextrema) of a function.[1][2][3]Pierre de... | https://en.wikipedia.org/wiki/Maxima_and_minima#In_relation_to_sets |
In thephysicalscience ofdynamics,rigid-body dynamicsstudies the movement ofsystemsof interconnectedbodiesunder the action of externalforces. The assumption that the bodies arerigid(i.e. they do notdeformunder the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration o... | https://en.wikipedia.org/wiki/Dynamic_equilibrium_(mechanics) |
Applied mechanicsis the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments.[1]In short, when mechanics concepts surpass being theoretical and are applied and executed, general mechanics becomes applied mechanics. It is this stark d... | https://en.wikipedia.org/wiki/Engineering_mechanics |
Inchemistryandphysics,metastabilityis an intermediateenergetic statewithin adynamical systemother than the system'sstate of least energy.
A ball resting in a hollow on a slope is a simple example of metastability. If the ball is only slightly pushed, it will settle back into its hollow, but a stronger push may start th... | https://en.wikipedia.org/wiki/Metastability |
Instaticsandstructural mechanics, a structure isstatically indeterminatewhen theequilibriumequations – force and moment equilibrium conditions – are insufficient for determining theinternal forcesandreactionson that structure.[1][2]
Based onNewton's laws of motion, the equilibrium equations available for a two-dimen... | https://en.wikipedia.org/wiki/Statically_indeterminate |
Staticsis the branch ofclassical mechanicsthat is concerned with the analysis offorceandtorqueacting on aphysical systemthat does not experience anacceleration, but rather is inequilibriumwith its environment.
IfF{\displaystyle {\textbf {F}}}is the total of the forces acting on the system,m{\displaystyle m}is the mass... | https://en.wikipedia.org/wiki/Statics |
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