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variant herewe apply different aggregation functions on two different attributes the agg(function takes dictionary as input containing attributes as keys and aggregation functions as values (see figure - figure - groupby with different aggregation functions for different attributes variant herewe do combination of vari...
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data visualization is thus the process of visually representing information in the form of chartsgraphspicturesand so on for better and universally consistent understanding we mention universally consistent understanding to point out very common issue with human languages human languages are inherently complex and depe...
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the plt alias is for matplotlib pyplot we will discuss this more in the coming sectionfor now assume we require this to add-on enhancements to plots generated by pandas in this case we use it to add title to our plot the plot generated is depicted in figure - figure - line chart showing price trend for user though we c...
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figure - price trend over time for given user this time our visualization clearly helps us see the purchase pattern for this user though we can discuss about insights from this visualization at lengtha quick inference is clearly visible as the plot showsthe user seems to have purchased high valued items in the starting...
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figure - bar plot representing quantities purchased at weekly level histograms one of the most important aspects of exploratory data analysis (edais to understand distribution of various numerical attributes of given dataset the simplest and the most common way of visualizing distribution is through histograms we plot ...
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the output shown in figure - clearly shows skewed and tailed distribution this information will be useful while using such attributes in our algorithms more will be clear when we work on actual use cases we can take this step further and try to visualize the price distribution on per week basis we do so by using the pa...
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the previous snippet uses groupby(to extract series representing number of transactions on per user_class level we then use the pie(function to plot the percentage distribution we use the autopct parameter to annotate the plot with actual percentage contribution by each use_class figure - depicts the output pie chart f...
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we'll look at the attributes quantity_purchased and purchase_week using box plots the following snippet generates the required plot for us df[['quantity_purchased','purchase_week']plot box(plt title('quantity and week value distribution'nowwe'll look at the plots generated (see figure - the bottom edge of the box in bo...
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bubble_df df[['enc_uclass''purchase_week''price','product_id']groupby(['purchase_week'enc_uclass']agg{'price':'mean''product_id':'count'reset_index(bubble_df rename(columns={'product_id':'total_transactions'},inplace=truefigure - dataframe aggregated on per week per user_class level figure - showcases the resultant dat...
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this generates the plot in figure - showcasing an almost random spread of data across weeks and average price with some slight concentration in the top left of the plot figure - scatter plot showing spread of data across purchase_week and price scatter plot also provides us the capability to visualize more than the bas...
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the parameters are self-explanatory-- represents color while stands for size of the bubble such plots are also called bubble charts the output generated is shown in figure - figure - scatter plot visualizing multi dimensional data in this sectionwe utilized pandas to plot all sorts of visualizations these were some of ...
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figures and subplots first things first the base of any matplotlib style visualization begins with figure and subplot objects the figure module helps matplotlib generate the plotting window object and its associated elements in shortit is the top-level container for all visualization components in matplotlib syntaxa fi...
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now that we have sample plot donelet' look at how different objects interact in the matplotlib universe as mentionedthe figure object is the top-most container of all elements before complicating thingswe begin by first plotting different figuresi each figure containing only single plot the following snippet plots sine...
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figure - multiple figures using matplotlib we plot multiple figures while telling the data story for use case yetthere are cases where we need multiple plots in the same figure this is where the concept of subplots comes into the picture subplot divides figure into grid of specified rows and columns and also provides i...
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ax figure_obj add_subplot( , , ax plot( + ,ythis snippet first defines figure object using plt figure(we then get an axes object pointing to the first subplot generated using the statement figure_obj add_subplot( , , this statement is actually dividing the figure into two rows and two columns the last parameter (value ...
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to have the same -axis this allows us to view data on the same scale along with aesthetic improvements the output is depicted in figure - showcasing sine and cosine curves on the same -axis figure - subplots using subplots(method another variant is the subplot(functionwhich is also exposed through the pyplot module dir...
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the subplot grid(function takes number of parametersexplained as followsshapea tuple representing rows and columns in the grid as (rowscolumnsloca tuple representing the location of subplot this parameter is indexed rowspanthis parameter represents the number of rows the subplot covers colspanthis parameter represents ...
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alpha[ set_alpha( ax set_title('line alpha'plt setp(ax get_yticklabels()visible=falsefigure - setting color and alpha properties of plot along the same lineswe also have options to use different shapes to mark data points as well as different styles to plot lines these options come in handy while representing different...
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figure - setting marker and line style properties of plot though there are many more fine tuning options availableyou are encouraged to go through the documentation for detailed information we conclude the formatting section with final two tricks related to line width and shorthand notation to do it all quicklyas shown...
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figure - example to show line_width and shorthand notation legends graph legend is key that helps us map color/shape or other attributes to different attributes being visualized by the plot though in most cases matplotlib does wonderful job at preparing and showing the legendsthere are times when we require finer level...
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figure - sample plot with legend one of the primary goals of matplotlib is to provide publication quality visualizations matplotlib supports latex style formatting of the legends to cleanly visualize mathematical symbols and equations the symbol is used to mark the start and end of latex style formatting the same is sh...
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xis controls the next feature from matplotlib is the ability to control the xand -axes of plot apart from basic features like setting the axis labels and colors using the methods set_xlabel(and set_ylabel()there are finer controls available as well let' first see how to add secondary -axis there are many scenarios when...
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by defaultmatplotlib identifies the range of values being plotted and adjusts the ticks and the range of both xand -axes it also provides capability to manually set these through the axis(function through this functionwe can set the axis range using predefined keywords like tightscaledand equalalong with passing list s...
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the output is plot with -axis having labels marked only between - and while -axis has range of to with labels changed manually the output plot is shown in figure - figure - plot showcasing axes with manual ticks before we move on to next set of features/capabilitiesit is worth noting that with matplotlibwe also get the...
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nnotations the text(interface from the pyplot module exposes the annotation capabilities of matplotlib we can annotate any part of the figure/plot/subplot using this interface it takes the and coordinatesthe text to be displayedalignmentand fontsize parameters as inputs to place the annotations at the desired place on ...
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global parameters to maintain consistencywe usually try to keep the plot sizesfontsand colors the same across the visual story setting each of these attributes creates complexity and difficulties in maintaining the code base to overcome these issueswe can set formatting settings globallyas shown in the following snippe...
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feature engineering and selection building machine learning systems and pipelines take significant effortwhich is evident from the knowledge you gained in the previous in the first we presented some high-level architecture for building machine learning pipelines the path from data to insights and information is not an ...
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this covers essential concepts for all the three major areas mentioned above techniques for feature engineering will be covered in detail for diverse data types including numericcategoricaltemporaltext and image data we would like to thank our good friend and fellow data scientistgabriel moreira for helping us with som...
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features raw data is hardly used to build any machine learning modelmostly because algorithms can' work with data which is not properly processed and wrangled in desired format features are attributes or properties obtained from raw data each feature is specific representation on top of the raw data typicallyeach featu...
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figure - revisiting our standard machine learning pipeline the figure clearly depicts the main components in the pipelinewhich you should already be well-versed on by now these components are mentioned once more for ease of understanding data retrieval data preparation modeling model evaluation and tuning model deploym...
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it is quite evident that based on the sequence of steps depicted in the figurefeatures are first crafted and engineeringnecessary normalization and scaling is performed and finally the most relevant features are selected to give us the final set of features we will cover these three components in detail in subsequent s...
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we will now look at definition of feature engineering by dr jason brownleedata scientist and ml practitioner who provides lot of excellent resources over at regard to machine learning and data science dr brownlee defines feature engineering as follows "feature engineering is the process of transforming raw data into fe...
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feature engineering is indeed both an art and science to transform data into features for feeding into models sometimes you need combination of domain knowledgeexperienceintuitionand mathematical transformations to give you the features you need by solving more problems over timeyou will gain the experience you need to...
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more flexibility on data typeswhile is it definitely easier to use numeric data types directly with machine learning algorithms with little or no data transformationsthe real challenge is to build models on more complex data types like textimagesand even videos feature engineering helps us build models on diverse data ...
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feature engineering on numeric data numeric datafieldsvariablesor features typically represent data in the form of scalar information that denotes an observationrecordingor measurement of coursenumeric data can also be represented as vector of scalars where each specific entity in the vector is numeric data point in it...
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values usuallyscalar values in its raw form indicate specific measurementmetricor observation belonging to specific variable or field the semantics of this field is usually obtained from the field name itself or data dictionary if present let' load dataset now about pokemonthis dataset is also available on kaggle if yo...
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we can see multiple statistical measures like countaveragestandard deviationand quartiles for each of the numeric features in this output try plotting their distributions if possiblec ounts raw numeric measures can also indicate countsfrequencies and occurrences of specific attributes let' look at sample of data from t...
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in [ ]from sklearn preprocessing import binarizer bn binarizer(threshold= pd_watched bn transform([popsong_df['listen_count']])[ popsong_df['pd_watched'pd_watched popsong_df head( figure - binarizing song counts you can clearly see from figure - that both the methods have produced the same results depicted in features ...
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it_ it_ it_ thus after our rounding operationsyou can see the new features in the data depicted in the previous dataframe basically we tried two forms of rounding the features depict the item popularities now both on scale of - and on scale of - you can use these values both as numerical or categorical features based o...
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we can clearly see from this output that we have total of five features including the new interaction features we can see the degree of each feature in the matrixusing the following snippet in [ ]pd dataframe(pf powers_columns=['attack_degree''defense_degree']out[ ]attack_degree defense_degree now that we know what eac...
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attack defense attack^ attack defense defense^ thus you can see that we have successfully obtained the necessary interaction features for the new dataset try building interaction features on three or more features nowbinning often when working with numeric datayou might come across features or attributes which depict ...
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fixed-width binning in fixed-width binningas the name indicateswe have specific fixed widths for each of the binswhich are usually pre-defined by the user analyzing the data each bin has pre-fixed range of values which should be assigned to that bin on the basis of some business or custom logicrulesor necessary transfo...
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and so on we can easily do this using what we learned in the "roundingsection earlier where we round off these raw age values by taking the floor value after dividing it by the following code depicts the same in [ ]fcc_survey_df['age_bin_round'np array(np floor(np array(fcc_survey_df['age'] )fcc_survey_df[['id ''age''a...
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figure - custom age binning for developer ages we can see from the dataframe output in figure - that the custom bins based on our scheme have been assigned for each developer' age try out some of your own binning schemesa daptive binning so farwe have decided the bin width and ranges in fixed-width binning howeverthis ...
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figure - histogram depicting developer income distribution we can see from the distribution depicted in figure - that as expected there is right skew with lesser developers earning more money and vice versa let' take -quantile or quartile based adaptive binning scheme the following snippet helps us obtain the income va...
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figure - histogram depicting developer income distribution with quartile values the -quantile values for the income attribute are depicted by red vertical lines in figure - let' now use quantile binning to bin each of the developer income values into specific bins using the following code in [ ]quantile_labels [' - '' ...
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the result dataframe depicted in figure - clearly shows the quantile based bin range and corresponding label assigned for each developer income value in the income_quantile_range and income_quantile_labels featuresrespectively statistical transformations let' look at different strategy of feature engineering on numeric...
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figure - histogram depicting developer income distribution after log transform thus we can clearly see that the original developer income distribution that was right skewed in figure - is more gaussian or normal-like in figure - after applying the log transform ox-cox transform let' now look at the box-cox transformano...
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in [ ]get optimal lambda value from non null income values income np array(fcc_survey_df['income']income_clean income[~np isnan(income)lopt_lambda spstats boxcox(income_cleanprint('optimal lambda value:'opt_lambdaoptimal lambda value now that we have obtained the optimal valuelet' use the box-cox transform for two valu...
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figure - histogram depicting developer income distribution after box-cox transform ( loptimalthe distribution of the transformed numeric values for developer income after the box-cox distribution also look similar to the one we had obtained after the log transform such that it is more normal-like and the extreme right ...
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once you have these dependencies loadedlet' get started and engineer some features from categorical data transforming nominal features nominal features or attributes are categorical variables that usually have finite set of distinct discrete values often these values are in string or text format and machine learning al...
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from the outputwe can see that mapping scheme has been generated where each genre value is mapped to number with the help of the labelencoder object gle the transformed labels are stored in the genre_labels value let' write it back to the original dataframe and view the results in [ ]vg_df['genrelabel'genre_labels vg_d...
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deoxysdefense forme gen rapidash gen swanna gen thusyou can see that it is really easy to build your own transformation mapping scheme with the help of python dictionaries and use the mapfunction from pandas to transform the ordinal feature encoding categorical features we have mentioned several times in the past that ...
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gen_labels gen_le fit_transform(poke_df['generation']poke_df['gen_label'gen_labels transform and map pokemon legendary status leg_le labelencoder(leg_labels leg_le fit_transform(poke_df['legendary']poke_df['lgnd_label'leg_labels poke_df_sub poke_df[['name''generation''gen_label''legendary''lgnd_label']poke_df_sub iloc[...
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figure - feature set depicting one hot encoded features for pokemon generation and legendary status from the result feature set depicted in figure - we can clearly see the new one hot encoded features for gen_label and lgnd_label each of these one hot encoded features is binary in nature and if they contain the value i...
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in [ ]new_gen_feature_arr gen_ohe transform(new_poke_df[['gen_label']]toarray(new_gen_features pd dataframe(new_gen_feature_arrcolumns=gen_feature_labelsnew_leg_feature_arr leg_ohe transform(new_poke_df[['lgnd_label']]toarray(new_leg_features pd dataframe(new_leg_feature_arrcolumns=leg_feature_labelsnew_poke_ohe pd con...
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the following code depicts the dummy coding scheme on pokemon generation by dropping the first level binary encoded feature (gen in [ ]gen_dummy_features pd get_dummies(poke_df['generation']drop_first=truepd concat([poke_df[['name''generation']]gen_dummy_features]axis= iloc[ : out[ ]name generation gen gen gen gen gen ...
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deoxysdefense forme gen rapidash gen swanna gen we can clearly see from the output feature set that all have been replaced by - in case of values which were previously all in the dummy coding scheme bin-counting scheme the encoding schemes discovered so far work quite well on categorical data in generalbut they start c...
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total game genres ['action'adventure'fighting'misc'platform'puzzle'racing'role-playing'shooter'simulation'sports'strategy'we can clearly see from the output that there are distinct genres and if we used one hot encoding scheme on the genre featurewe would end up having binary features insteadwe will now use feature has...
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in [ ]import pandas as pd import numpy as np import re import nltk let' now load some sample text documentsdo some basic pre-processingand learn about various feature engineering strategies to deal with text data the following code creates our sample text corpus ( collection of text documents)which we will use in this ...
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for more details on these topicsyou can jump ahead to of this book or refer to the section "text normalization,page of text analytics with python (apressdipanjan sarkar which covers each of these techniques in detail we will be normalizing our text here by lowercasingremoving special characterstokenizingand removing st...
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cv_matrix cv fit_transform(norm_corpuscv_matrix cv_matrix toarray(cv_matrix out[ ]array([[ ][ ][ ][ ][ ][ ]]dtype=int the output represents numeric term frequency based feature vector for each document like we mentioned before to understand it betterwe can represent it using the feature names and view it as dataframe i...
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figure - bi-gram feature vectors for our corpus based on bag of -grams model figure - clearly shows our bi-gram feature vectors where each feature is bi-gram of two contiguous words and the values depict the frequency of that bi-gram in each document you can use the ngram_range parameter to extend the -gram range to ge...
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out[ ]beautiful blue brown dog fox jumps lazy love quick sky today thusthe preceding output depicts the tf-idf based feature vectors for each of our text documents notice how this is scaled and normalized version as compared to the raw bag of words model interested readers who might want to dive into further details of...
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from figure - we can clearly see that feature vectors having similar orientation will be very close to one another and the angle between them will be closer to deg and thus cosine similarity would be cos deg when cosine similarity is close to cos deg the angle between the documents is closer to deg indicating they are ...
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topic models besides document termsphrases and similaritieswe can also use some summarization techniques to extract topic or concept based features from text documents the idea of topic models revolves around the process of extracting key themes or concepts from corpus of documents which are represented as topics each...
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topic [item for item in topic if item[ print(topicprint([('fox' )('quick' )('dog' )('brown' )('lazy' )('jumps' )('blue' )[('sky' )('beautiful' )('blue' )('love' )('today' )the preceding output represents each of the two topics as collection of terms and their importance is depicted by the corresponding weight it is def...
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sizerepresents the feature vector size for each word in the corpus when transformed windowsets the context window size specifying the length of the window of words to be taken into account as belonging to singlesimilar context when training min_countspecifies the minimum word frequency value needed across the corpus to...
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feature_vector np add(feature_vectormodel[word]if nwordsfeature_vector np divide(feature_vectornwordsreturn feature_vector def averaged_word_vectorizer(corpusmodelnum_features)vocabulary set(model wv index wordfeatures [average_word_vectors(tokenized_sentencemodelvocabularynum_featuresfor tokenized_sentence in corpusre...
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the brown fox is quick and the blue dog is lazyanimals the sky is very blue and the sky is very beaut weather the dog is lazy but the brown fox is quickanimals the preceding output uses the averaged word vectors based on word embeddings to cluster the documents in our corpus and we can clearly see that it has obtained ...
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timestamp( : :'tz='pytz fixedoffset(- )')timestamp( : : + 'tz='pytz fixedoffset( )')timestamp( : : + 'tz='pytz fixedoffset( )')]dtype=objectyou can clearly see from the temporal values that we have multiple components for each timestamp object which include datetimeand even time based offsetwhich can be used to identif...
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time-based features each temporal value also has time component that can be used to extract useful information and features pertaining to the time these include attributes like hourminutesecondmicrosecondutc offsetand more the following code snippet extracts some of the previously mentioned time-based features from ou...
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in [ ]df['tz_info'df['ts_obj'apply(lambda dd tzinfodf['timezones'df['ts_obj'apply(lambda dlist({ astimezone(tztzname(for tz in map(pytz timezonepytz all_timezones_setif astimezone(tzutcoffset(= utcoffset()})df[['time''utc_offset''tz_info''timezones']figure - time zone relevant features in temporal data thus as we menti...
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figure - deriving elapsed time difference from current time based on our computationseach new derived feature should give us the elapsed time difference between the current time and the time value in the time column (actually timeutc since conversion to utc is necessaryboth the values are almost equal to one anotherwhi...
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image metadata features there are tons of useful features obtainable from the image metadata itself without even processing the image most of this information can be found from the exif datawhich is usually recorded for each image by the device when the picture is being taken following are some of the popular features...
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figure - our two sample color images we can clearly see from figure - that we have two images of cat and dog having dimensions pixels where each row and column denotes specific pixel of the image the third dimension indicates these are color images having three color channels let' now try to use numpy indexing to slice...
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[ ][ ][ ]]dtype=uint this image pixel matrix is two-dimensional matrix so you can extract features from this further or even flatten it to one-dimensional vector to use as inputs for any machine learning algorithm grayscale image pixels if you are dealing with color imagesit might get difficult working with multiple ch...
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in [ ]fig plt figure(figsize ( , )ax fig add_subplot( , ax imshow(cgscmap="gray"ax fig add_subplot( , ax imshow(dgscmap='gray'ax fig add_subplot( , c_freqc_binsc_patches ax hist(cgs flatten()bins= ax fig add_subplot( , d_freqd_binsd_patches ax hist(dgs flatten()bins= figure - binning image intensity distributions with ...
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cs describe(cat_rgbaxis= ds describe(dog_rgbaxis= cat_rgb_range cs minmax[ cs minmax[ dog_rgb_range ds minmax[ ds minmax[ rgb_range_df pd dataframe([cat_rgb_rangedog_rgb_range]columns=['r_range''g_range''b_range']pd concat([dfrgb_range_df]axis= out[ ]image r_range g_range b_range cat dog we can then use these range fea...
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potential edges are thinned down to curves with width of pixel and hysteresis based thresholding is used to label all points above specific high threshold as edges and then recursively use the low threshold value to label points above the low threshold as edges connected to any of the previously labeled points the foll...
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fd_catcat_hog hog(cgsorientations= pixels_per_cell=( )cells_per_block=( )visualise=truefd_dogdog_hog hog(dgsorientations= pixels_per_cell=( )cells_per_block=( )visualise=truerescaling intensity to get better plots cat_hogs exposure rescale_intensity(cat_hogin_range=( )dog_hogs exposure rescale_intensity(dog_hogin_range...
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in [ ]from mahotas features import surf import mahotas as mh cat_mh mh colors rgb gray(catdog_mh mh colors rgb gray(dogcat_surf surf surf(cat_mhnr_octaves= nr_scales= initial_step_size= threshold= max_points= dog_surf surf surf(dog_mhnr_octaves= nr_scales= initial_step_size= threshold= max_points= fig plt figure(figsiz...
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visual bag of words model we have seen the effectiveness of the popular bag of words model in extracting meaningful features from unstructured text documents bag of words refers to the document being broken down into its constituentswords and computing frequency of occurrences or other measures like tf-idf similarlyin ...
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figure - transforming an image into vbow vector (courtesy of ian londonimage classification in pythonwith visual bag of wordsthus you can see from figure - how two-dimensional image and its corresponding feature descriptors can be easily transformed into one-dimensional vbow vector [ going into extensive details of the...
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you can see how easy it is to transform complex two-dimensional surf feature descriptor matrices into easy to interpret vbow vectors let' now take new image and think about how we could apply the vbow pipeline first we would need to extract the surf feature descriptors from the image using the following snippet (this i...
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in [ ]from sklearn metrics pairwise import euclidean_distancescosine_similarity eucdis euclidean_distances(new_vbow reshape( ,- vbow_featurescossim cosine_similarity(new_vbow reshape( ,- vbow_featuresresult_df pd dataframe({'euclideandistance'eucdis[ ]'cosinesimilarity'cossim[ ]}pd concat([dfresult_df]axis= out[ ]image...
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you can use theano or tensorflow as your backend deep learning framework for keras to work on am using tensorflow in this scenario let' build basic two-layer cnn now with max pooling layer between them in [ ]model sequential(model add(conv ( ( )input_shape=( )activation='relu'kernel_initializer='glorot_uniform')model a...
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in [ ]first_conv_layer function([model layers[ inputk learning_phase()][model layers[ output]second_conv_layer function([model layers[ inputk learning_phase()][model layers[ output]let' now use these functions to extract the feature representations learned in the convolutional layers and visualize these features to see...
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the feature map visualizations depicted in figure - are definitely interesting you can clearly see that each feature matrix produced by the convolutional neural network is trying to learn something about the image like its texturecornersedgesilluminationhuebrightnessand so on this should give you an idea of how these a...
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standardized scaling the standard scaler tries to standardize each value in feature column by removing the mean and scaling the variance to be from the values this is also known as centering and scaling and can be denoted mathematically as ss mx sx where each value in feature is subtracted by the mean mx and the result...
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views out[ ]views zscore minmax - - - - - the preceding output shows the min-max scaled values in the minmax column and as expectedthe maximum viewed video in row index has value of and the minimum viewed video in row index has value of you can also compute this mathematically using the following code (sample computati...
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iqr quartiles[ quartiles[ (vw[ np median(vw)iqr out[ ] there are several other techniques for feature scaling and normalizationbut these should be sufficient to get you started and are used extensively in building machine learning systems always remember to check if you need to scale and standardize features whenever y...
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we will now look at various ways of selecting features including statistical and model based techniques by using some sample datasets threshold-based methods this is filter based feature selection strategywhere you can use some form of cut-off or thresholding for limiting the total number of features during feature sel...
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in [ ]from sklearn feature_selection import variancethreshold vt variancethreshold(threshold vt fit(poke_genout[ ]variancethreshold(threshold= to view the variances as well as which features were finally selected by this algorithmwe can use the variances_ property and the get_supportfunction respectively the following ...
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build featureset and response class labels bc_x np array(bc_featuresbc_y np array(bc_classest[ print('feature set shape:'bc_x shapeprint('response class shape:'bc_y shapefeature set shape( response class shape( ,we can clearly see thatas we mentioned beforethere are total of features in this dataset and total of rows o...
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in [ ]from sklearn feature_selection import chi selectkbest skb selectkbest(score_func=chi = skb fit(bc_xbc_yout[ ]selectkbest( = score_func=you can see that we have passed our input features (bc_xand corresponding response class outputs (bc_yto the fitfunction when computing the necessary metrics the chi-square test w...
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figure - selected feature subset of the wisconsin diagnostic breast cancer dataset using chi-square tests the dataframe with the top scoring features is depicted in figure - let' now build simple classification model using logistic regression on the original feature set of features and compare the model accuracy perfor...
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the remaining features to obtain the new weights or scores this process is recursively carried out multiple times and each time features with the lowest scores/weights are eliminateduntil the pruned feature subset contains the desired number of features that the user wanted to select (this is taken as an input paramete...
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at the cost of slightly increasing the bias overall this produces better and more generalized model we will cover the bias-variance tradeoff in more detail in let' now use the random forest model to score and rank features based on their importance in [ ]from sklearn ensemble import randomforestclassifier rfc randomfor...
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feature extraction with principal component analysis principal component analysispopularly known as pcais statistical method that uses the process of linearorthogonal transformation to transform higher-dimensional set of features that could be possibly correlated into lower-dimensional set of linearly uncorrelated fea...