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15,000 | in particularfor the squared-error loss we have ar er min (ka qe aka ( this is convex optimization problemand its solution is found by differentiating ( with respect to and and equating to zeroleading to the following system of ( mlinear equations>kkn gk kq ( qkqq as long as is of full column rankthe minimizing functio... |
15,001 | and on the square [- ] quasi-random point sets have better space-filling properties than either regular grid of points or set of pseudo-random points we refer to [ for details note that there is no observation noise in this particular problem - - - figure peaks function sampled at hammersley points the purpose of this ... |
15,002 | from matplotlib import cm from numpy linalg import norm import numpy as np def peaks ( , ) ( *( - )** np exp (-( ** ( + ** *( / ** ** np exp(- ** ** / np exp (-( + ** ** )return (zn - hammersley ([ , ,nz peaks ( [, , [, ]xx yy np mgrid - : : - : : jzz peaks (xx ,yyplt contour (xx ,yy ,zz levels = fig=plt figure (ax fig... |
15,003 | - - - - - figure contour plots for the prediction function (leftand the peaks function given in ( (rightorder derivatives consider functions [ that are twice differentiable and define kg : ( ( )) dx as measure of the size of the second derivative example (behavior of kg intuitivelythe larger kg isthe more "wigglythe fu... |
15,004 | imposing the condition that ( ( for functions in will ensure that where the null space contains only linear functionsas we will see to see that this is in fact an rkhswe derive its reproducing kernel using integration by parts (or directly from the taylor expansion above)write (xz (sds ( ( sds ( ( )ds if is kernelthen ... |
15,005 | figure various cubic smoothing splines for smoothing parameter /( { for the natural spline through the data points is obtainedfor the simple linear regression line is found the following code first computes the matrices and qand then solves the linear system ( finallythe smoothing curve is determined via ( )for selecte... |
15,006 | plot the curve xx np arange ( , + , reshape - , np zeros_like (xxqx np hstack (np ones_like (xx)xx) np zeros_like (xxn np shape (xx)[ kx np zeros (( , )for in range ( )for in range ( )kx[ ,jk( [ ]xx[ ] np hstack (kx tqx)ad plt ylim (( , )plt plot(xx glabel ' {}format ( )linewidth plt plot( , ' 'markersize = plt xlabel ... |
15,007 | iid where {ei ( to simplify the analysislet us assume that is knownso no prior needs to be specified for let [ ( ) (xn )]be the (unknownvector of regression values placing gp prior on the function is equivalent to placing multivariate gaussian prior on the vector gg ( )( where the covariance matrix of is gram matrix (i... |
15,008 | example (gp regressionsuppose the regression function is ( sin( px) [ we use gp regression to estimate gusing gaussian kernel of the form ( with bandwidth parameter the explanatory variables were drawn uniformly on the interval [ ]and the responses were obtained from ( )with noise level figure shows samples from the pr... |
15,009 | (xpredictive mean (xpredictive mean - - - - - - figure gp regression of synthetic data set with bandwidth (leftand (rightthe black dots represent the data and the blue curve is the latent function ( sin( pxthe red curve is the mean of the gp predictive distribution given by ( )and the shaded region is the confidence ba... |
15,010 | - - - figure contours of the marginal log-likelihood for the gp regression example the maximum is denoted by cross kernel pca in its basic formkernel pca (principal component analysiscan be thought of as pca in feature space the main motivation for pca introduced in section was as dimensionality reduction technique the... |
15,011 | point given by - / > xx - / > where we have (suggestivelydefined :[hx xihxn xi]the important point is that is completely determined by the vector of inner products and the principal eigenvalues and (righteigenvectors of the gram matrix note that each component zm of is of the form zm am, (xi ) ( = the preceding discuss... |
15,012 | figure first nine eigenfunctions using gaussian kernel for the two-dimensional data set formed by the red and cyan points - - - - - - figure projection of the data onto the first two principal components observe that already the projections of the inner and outer points are well separated |
15,013 | further reading for good overview of the ridge regression and the lassowe refer the reader to [ for overviews of the theory of rkhs we refer to [ ]and for in-depth background on splines and their connection to rkhss we refer to [ for further details on gp regression we refer to [ and for kernel pca in particular we ref... |
15,014 | (bshow that every evaluation functional on is continuous at the function that ise kg ( ) ( continuity of at all functions then follows automatically from linearity (cshow that is completethat isevery cauchy sequence fn converges in the norm ||| if and are kernels on and ythen ((xy)( ): (xx (yy and kx ((xy)( : (xx ) (yy... |
15,015 | for the smoothing cubic spline of section show that (xumax{ , } min{ ,umin{ , } let be an model matrix and let be the unit-length vector with -th entry equal to one (uk kuk suppose that the -th column of is and that it is replaced with new predictor wso that we obtain the new model matrixe ( )ux (adenoting : ( vkw vk s... |
15,016 | algorithm ridge regression coefficients via sherman-morrison formula inputtraining set {xyand regularization parameter outputsolution ( xx)- xy - set to be an matrix of zeros and ( for do set to be the -th column of update {abvia algorithm with inputs {abjw (xy return consider example with diag( for some nonnegative ve... |
15,017 | (exercise continued consider again example with diag( for some nonnegative model-selection parameter bayesian choice for is the maximizer of the marginal likelihood ( )that isl argmax (bs ldb ds > where ln (bs np ky xbk bd- ln |dln( ps ln to maximize ( )one can use the em algorithm with and acting as latent variables i... |
15,018 | in this exercise we explore how the early stopping of the gradient descent iterations (see example )xt+ xt (xt ) is (approximatelyequivalent to the global minimization of ( gkxk for certain values of the ridge regularization parameter (see example we illustrate the early stopping idea on the quadratic function ( ( ) ( ... |
15,019 | lassification the purpose of this is to explain the mathematical ideas behind well-known classification techniques such as the naive bayes methodlinear and quadratic discriminant analysislogistic/softmax classificationthe -nearest neighbors methodand support vector machines introduction classification methods are super... |
15,020 | introduction (xx xwith respect to ( )for every fixed in other wordstake (xto be equal to the class label for which [ xis maximal bayes error rate the formulation ( allows for "ties"when there is an equal probability between optimal classes for feature vector assigning one of these tied classes arbitrarily (or randomlyt... |
15,021 | estimate the conditional pdf ( xfrom the training datausing only feature vectors in the neighborhood of in section we explain the support vector methodology for classificationthis is based on the same reproducing kernel hilbert space ideas that proved successful for regression analysis in section finallya versatile way... |
15,022 | true positive true negative false positive false negative in the spirit of table for hypothesis testingit is sometimes useful to divide the elements of confusion matrix into four groups the diagonal elements are the true positive countsthat isthe numbers of correct classifications for each class the true positive count... |
15,023 | from ( )we conclude that the accuracy of classification is equal to accuracy tp tn tp tn fp fn howeverwe can see that in this particular caseaccuracy is problematic metricsince the algorithm allowed non-authorized personnel to enter the facility one way to deal with this issue is to modify the loss function to give muc... |
15,024 | table confusion matrix for authorized personnel classificationusing different classifier (classifier predicted actual authorized non-authorized authorized non-authorized , table comparing the metrics for the confusion matrices in tables and metric multilabel classification hierarchical classification classifier classif... |
15,025 | defined as pn exact match ratio = {yi yi the exact match ratio is rather stringentas it requires full match in order to consider partial correctnessthe following metrics could be used instead the accuracy is defined as the ratio of correctly predicted labels and the total number of predicted and actual labels the formu... |
15,026 | bayes optimal decision rule ( all classes are priori equally likelyand the likelihood functionone obtains the posterior pdf via bayesformula ( class is then assigned to feature vector according to the highest posterior probabilitythat iswe classify according to the bayes optimal decision ruleb argmax ( )( naive bayes w... |
15,027 | table feature parameters feature feature feature class naivebayes py import numpy as np np array ([ , , ]reshape ( , mu np array ([ ]reshape ( , sig np array ([ ]reshape ( , lambda np prod(sig[ ,:]np exp- np sum (( -mu[ ,:]** sig[ ,:]** )for in range ( , )print ('{: }format ( ( )) + - + - linear and quadratic discrimin... |
15,028 | quadratic discriminant function linear discriminant function linear and quadratic discriminant analysis andaccording to the bayes optimal decision rule ( )we classify to come from class if ( th ( th orequivalently (by taking logarithmsif ln | ( ) - ln ln | ( ) - ( ln ( the function rp ( dy (xln ay ln |sy ( uy ) - ( uy ... |
15,029 | we used the following python code to make this figure ldamixture py import numpy as np matplotlib pyplot as plt from scipy stats import multivariate_normal from mpl_toolkits mplot import axes from matplotlib colors import lightsource mu mu np array ([ , ]np array ([ , ]sigma np array ([[ , ,[ ]]xy np mgrid - : : - : : ... |
15,030 | invsigma (mu -mu ) mu reshape ( , invsigma mu reshape ( , mu reshape ( , invsigma @mu reshape ( , xx np linspace - , , yy (-( [ ]xx + )/ [ ][ plt plot(xx ,yy ,' 'plt show ( figure the linear discriminant boundary lies between the two modes of the mixture density and is linear to illustrate the difference between the li... |
15,031 | qda py import numpy as np import matplotlib pyplot as plt from scipy stats import multivariate_normal mu np array ([ , ]mu np array ([ , ]sigma np array ([[ , ,[ ]]sigma np array ([[ , ,[ ]]xy np mgrid - : : - : : jmvn multivariate_normal mu sigma mvn multivariate_normal mu sigma xy np hstack (( reshape - , , reshape -... |
15,032 | sphere the data for the linear discriminant case (that iswhen sy for all )it is convenient to first "whitenor sphere the data as follows let be an invertible matrix such that bbobtainedfor examplevia the cholesky method we linearly transform each data point to : - and each mean uy to : - uy let the random vector be dis... |
15,033 | datared py import numpy as np from numpy random import randn import matplotlib pyplot as plt from mpl_toolkits mplot import axes = mu np array ([ , - ]mu np array ([ - , ]mu np array ([ , , ] randn ( , mu randn ( , mu randn ( , mu fig plt figure (ax fig gcaprojection =' ,ax plot( [, [, [, ' ',alpha = markersize = ax pl... |
15,034 | dataproj py from datared import from numpy linalg import svd pinv mu (mu mu reshape ( , mu (mu mu reshape ( , np hstack (mu mu ) , , svd(wwe only need pinv(wr rx ( tt rx ( tt rx ( tt plt plot(rx [, rx [, ,' ',alpha = markersize = plt plot(rx [, rx [, ,' ',alpha = markersize = plt plot(rx [, rx [, ,' ',alpha = markersiz... |
15,035 | to replace ( with ln ( wbxxb ( wbxj ( where the matrix ( - ) ( - and vector rc- reparameterize all such that (recall [ ])we [ bc- ] observe that the random response is assumed to have conditional probability distribution for which the log-odds ratio with respect to class and "referenceclass (in this case is linear the ... |
15,036 | -nearest neighbors -nearest neighbors classification let {(xi yi )}ni= be the training setwith yi { }and let be new feature vector define ( ( (nas the feature vectors ordered by closeness to in some distance dist(xxi ) the euclidean distance kxx let ( :{( ( ( ( (ky( )be the subset of that contains feature vectors xi th... |
15,037 | np random seed ( randn ( , np zeros (mpre allocate list for in range ( )if rand (< [ , [ix[ , np absrandn ()) elsex[ , [ix[ , np absrandn ()) vor voronoi (xplt_options {'show_vertices ':false 'show_points ':false 'line_alpha ': fig voronoi_plot_ (vor *plt_options plt plot( [ == , [ == , 'bo' [ == , [ == , 'rs'markersiz... |
15,038 | optimal decision boundary thereforea feature vector is classified according to or - depending on whether gt ( or respectively the optimal decision boundary is given by the set of for which gt ( similar to the cubic smoothing spline or rkhs setting in ( )we can consider finding the best classifiergiven the training data... |
15,039 | note thatfrom ( )the optimal pre-classifier (xand the classifier sign (xonly depend on vectors xi for which li these vectors are called the support vectors of the support vector machine it is also important to note that the quadratic function in ( depends on the regularization parameter by defining ni :li /gi nwe can r... |
15,040 | points for which li these pointswhich are also support vectorslie strictly inside the margins (points and in the figuresuch points may or may not be correctly classified points for which li these are the non-support vectorswhich all lie outside the margins every such point is correctly classified if the classes of poin... |
15,041 | figure in feature space the points can be separated by plane we wish to find separating plane in using the transformed features the following python code uses the svc function of the sklearn module to solve the quadratic optimization problem ( (with the results are summarized in table the data is available from the boo... |
15,042 | table optimal support vector machine parameters for the data zy ny - - - - - - - - - - it follows that the normal vector of the plane is bai zi [- - ]is where is the set of indices of the support vectors we see that the plane is almost perpendicular to the plane the bias term can also be found from the table above in p... |
15,043 | on which in turn gives rise to (uniquerkhs the optimal prediction function ( is now of the form gt (xa where and yi li ph(xi )ph(xb bph( ) = ( byi li ph(xi = the decision boundary{ gt ( }is again circle in the following code determines the fitted model parameters and the decision boundary figure shows the optimal decis... |
15,044 | clf predict (np c_[xx ravel (yy ravel (] reshape (xx shape plt contour (xx yy zcolors =" "plt show (finallywe illustrate the use of the gaussian kernel (xx - kx- ( where is some tuning constant this is an example of radial basis function kernelwhich are reproducing kernels of the form (xx (kx )for some positive realval... |
15,045 | remark (scaling and penalty parameterswhen using radial basis function in svc in sklearnthe scaling ( can be set via the parameter gamma note that large values of gamma lead to highly peaked predicted functionsand small values lead to highly smoothed predicted functions the parameter in svc refers / in ( classification... |
15,046 | fig plt figure (ax fig gcaprojection ' 'ax scatter ( [bidx , [bidx , [bidx , =' 'marker ='^'label ='benign 'ax scatter ( [midx , [midx , [midx , =' 'marker =' 'label ='malignant 'ax legend (ax set_xlabel ('mean radius 'ax set_ylabel ('mean texture 'ax set_zlabel ('mean concavity 'plt show (benign mean co ncavity malign... |
15,047 | classifiers logisticregression ( = )gaussiannb (da lineardiscriminantanalysis (da quadraticdiscriminantanalysis (kneighborsclassifier n_neighbors = svckernel ='rbf 'gamma - )print ('name accuracy \ '+ '-'for name clf in zip(names classifiers )clf fit(x_train y_train y_pred clf predict x_test print ('{: {: }format (name... |
15,048 | where (xy this such thatln (xy thln ay ln |sy ( uy ) - ln( py ( uy adapt the code in example to plot the estimated decision boundary instead of the true one in figure compare the true and estimated decision boundaries recall from equation ( that the decision boundaries of the multi-logit classifier are linearand that t... |
15,049 | in factmany such lines are possible svm gives the best separationin the sense that the gap (marginbetween the points is maximal - - - - - figure separate the points by straight line so that the separation between the two groups is maximal table data for figure - - - - - - - - - - - - - - - - - (aidentify from the figur... |
15,050 | penalty function (bb kbk = min ( bxi yi for some positive penalty constant (efind the solution the dual optimization problem ( by using sklearn' scv method note thatas the two point sets are separablethe constraint may be removedand the value of can be set to in example we used the feature map ph( [ ]to classify the po... |
15,051 | the purpose of this exercise is to derive the dual program ( from the primal program ( the starting point is to introduce vector of auxiliary variables :[ xn ]and write the primal program as xi aka , , = min subject tox yi ( {ka} xi ( (aapply the lagrangian optimization theory from section to obtain the lagrangian func... |
15,052 | (afor li ( show that (xi yi lies exactly on the decision border (bfor li show that (xi yi lies strictly inside the margins (cshow that for li the point (xi yi lies outside the margins and is correctly classified well-known data set is the mnist handwritten digit databasecontaining many thousands of digitalized numbers ... |
15,053 | sklearn model_selection to obtain five-fold cross-validation score as an estimate of the probability that the predicted class matches the expert' class consider the credit approval data set crx data from uci' credit approval website the data set is concerned with credit card applications the last column in the data set... |
15,054 | (btrain the logisticregression classifier from the sklearn linear_model package ( "traina naive classifier that always returns that isthe naive classifier identifies each instance as being not (dcompare the zero-one test losses of the logistic regression and the naive classifiers (efind the confusion matrixthe precisio... |
15,055 | ecision rees and nsemble ethods statistical learning methods based on decision trees have gained tremendous popularity due to their simplicityintuitive representationand predictive accuracy this gives an introduction to the construction and use of such trees we also discuss two key ensemble methodsnamely bootstrap aggr... |
15,056 | decision tree it is not possible to linearly separate the training setbut we can partition the feature space into rectangular regions and assign class (colorto each regionas shown in the right panel of figure points in these regions are classified accordingly as blue or red the partition thus defines classifier (predic... |
15,057 | the same predicted value of coursedifferent regions will usually have different predicted values constructing tree with training set {(xi yi )}}ni= amounts to minimizing the training loss ` ( loss(yi (xi ) = ( for some loss functionsee with of the form ( )we can write ` (gn xx loss(yi (xi ) {xi rw loss(yi (xi ) = = ww ... |
15,058 | algorithm construct_subtree inputa node and subset of the training datas outputa (subdecision tree tv if termination criterion is met then / is leaf node train regional prediction function using the training data else /split the node find the best splitting rule sv for node create successors vt and vf of st {(xys sv (x... |
15,059 | where nw is again the number of feature vectors in rw it is not difficult to show that gw (xyrw minimizes the squared-error loss with respect to all constant functionsin the region rw see exercise splitting rules in line in algorithm we divide region rv into two setsusing splitting rule (functionsv consequentlythe data... |
15,060 | for classification problemusing the indicator loss and constant regional prediction function as in ( )the aim is to choose splitting rule that minimizes { * { * } ( , ) ( , ) ( where * gt (xis the most prevalent class (majority votein the data set st and * is the most prevalent class in sf if the feature space is and t... |
15,061 | cross-entropy gini index misclassification impurity figure entropyginiand misclassification impurities for binary classificationwith class frequencies and the entropy impurity was normalized (divided by )to ensure that all impurity measures attain the same maximum value of / at / significant advantagein terms of traini... |
15,062 | to create figure we used the python method make_blobs from the sklearn module to produce training set of size with ten-dimensional feature vectors (thusp and )each of which is classified into one of classes the full code is given below treedepthcv py import numpy as np from sklearn datasets import make_blobs from sklea... |
15,063 | basictree py import numpy as np from sklearn datasets import make_friedman from sklearn model_selection import train_test_split def makedata ()n_points number of samples xy make_friedman n_samples =n_points n_features = noise = random_state = return train_test_split (xytest_size = random_state = the "mainmethod calls t... |
15,064 | self none def calculateloss (self)if(len(self )== )return return np sum(np power (self self mean (, )the function below implements the training (tree-buildingalgorithm def construct_subtree (node max_depth )if(node depth =max_depth or len(node = )node node mean (elsejxi calculateoptimalsplit (nodenode node xi xi xt yt ... |
15,065 | for in range ( , )xi [ ,jxt yt xf yf datasplit ( , , ,xitmpt tnode ( xt yttmpf tnode ( xf yfloss_t tmpt calculateloss (loss_f tmpf calculateloss (curr_val loss_t loss_f if curr_val best_split_val )best_split_val curr_val best_var best_xi xi return best_var best_xi finallywe implement the recursive method for prediction... |
15,066 | additional considerations binary versus non-binary trees while it is possible to split tree node into more than two groups (multiway splits)it generally produces inferior results compared to the simple binary split the major reason is that multiway splits can lead to too many nodes near the tree root that have only few... |
15,067 | in some casessuch as the one just discussedit may be useful to use splitting rule that involves several variablesas opposed to single one the decision regarding the split type clearly depends on the problem domain for examplefor logical (binaryvariables our domain knowledge may indicate that different behavior is expec... |
15,068 | {salary }and {height table example data with three variables (ageheightand salaryid age height salary the {salary surrogate rule completely mimics the primary rulein the sense that the data splits induced by these rules are identical namelyboth rules partition the data into two sets (by id{ and { on the other handthe {... |
15,069 | train cv loss tree depth figure the cross-validation and the training loss as function of the tree depth for binary classification problem smaller than according to the proposed procedurethis node will not be split further this mayhoweverbe sub-optimalbecause it could happen that one of the node' descendantsif splitcou... |
15,070 | figure the node is descendant of and is an ancestor of { }but is not descendant of (at (btv (ct tv figure the pruned tree tv in (cis the result of pruning the tv branch in (bfrom the original tree in (aalgorithm decision tree pruning inputtraining set outputsequence of decision trees build large decision tree via algor... |
15,071 | let be the initial (deeptree and let tk be the tree obtained after the -th pruning operationfor as soon as the sequence of trees tk is availk ableone can choose the best tree of {tk } = according to the smallest generalization risk specificallywe can split the data into training and validation sets in this casealgorith... |
15,072 | advantages and limitations of decision trees we list number of advantages and disadvantages of decision treesas compared with other supervised learning methods such as were discussed in and advantages the tree structure can handle both categorical and numerical features in natural and straightforward way specificallyth... |
15,073 | limitations despite the fact that the decision trees are extremely interpretablethe predictive accuracy is generally inferior to other established statistical learning methods in additiondecision treesand in particular very deep trees that were not subject to pruningare heavily reliant on their training set small chang... |
15,074 | proofwe have " gt (xxy [ xye[gt (xxyy ( where the inequality follows from eu (eu) for any (conditionalexpectation consequentlyby the tower property ii gt (xe gt (xxy ( bagged estimator unfortunatelymultiple independent data sets are rarely available but we can substitute them by bootstrapped ones specifically... |
15,075 | similar to ( compare this with the same decomposition for the average prediction function gbag in ( as egbag (xegt ( )we see that any possible improvement in the generalization risk must be due to the expected variance term averaging and bagging are thus only useful for predictors with large expected variancerelative t... |
15,076 | example (bagging for regression treewe next proceed with basic bagging example for regression treein which we compare the decision tree estimator with the corresponding bagged estimator we use the metric (coefficient of determinationfor comparison baggingexample py import numpy as np from sklearn datasets import make_f... |
15,077 | oob_pred_arr np zeros (lenx_train )for in range (lenx_train )) x_train [ireshape ( - [for in range n_estimators )if(np isin(ibootstrap_ds_arr [ ]=false ) append (bfor pred in bag[ ]oob_pred_arr [ioob_pred_arr [ (pred predict ( )/len( )l_oob _score (y_train oob_pred_arr print (decisiontreeregressor ^ score ", _score (y_... |
15,078 | the major idea of random forests is to perform bagging in combination with "decorrelationof the trees by including only subset of features during the tree construction for each bootstrapped training set tbwe build decision tree using randomly selected subset of features for the splitting rules this simple but powerful ... |
15,079 | max_features = random_state = rf fit(x_train y_train yhatrf rf predict x_test print ("rf ^ score " _score (y_test yhatrf )"\nrf oob ^ score "rf oob_score_ rf ^ score rf oob ^ score remark (the optimal number of subset features mthe default values for are bp/ and for regression and classification settingrespectively how... |
15,080 | varimportance py import numpy as np from sklearn datasets import make_classification from sklearn ensemble import randomforestclassifier import matplotlib pyplot as plt pylab n_points create regression data with data points xy make_classification n_samples =n_points n_features = n_informative = n_redundant = n_repeated... |
15,081 | clearlyit is hard to visualize and understand the prediction process based on trees howeverfigure shows that the features and were correctly identified as being important boosting boosting is powerful idea that aims to improve the accuracy of any learning algorithmespecially when involving weak learners -simple predict... |
15,082 | algorithm regression boosting with squared-error loss inputtraining set {(xi yi )}ni= the number of boosting rounds band shrinkage step-size parameter outputboosted prediction function - pn set (xn = yi for to do (bn set (by ( for nand let - = fit prediction function hb on the training data tb set gb (xgb- (xg hb ( s... |
15,083 | residuals yalpha *g_ list of basic regressor g_boost [for in range boostingrounds )h_i decisiontreeregressor max_depth = h_i fit(xresiduals residuals residuals alpha *h_i predict (xg_boost append (h_ireturn g_ g_boost def predict (g_ g_boost ,alpha )yhat alpha *g_ *np ones(len( )for in range (leng_boost ))yhat yhatalph... |
15,084 | gradient boosting resulting algorithm is called gradient boosting the general gradient boosting algorithm is summarized in algorithm the main idea is to mimic gradient descent algorithm in the following sense at each stage of the boosting procedurewe calculate negative gradient on training points xn (lines - thenwe fit... |
15,085 | create regression problem n_points points xy make_friedman n_samples =n_points n_features = noise = random_state = split to train /test set x_train x_test y_train y_test train_test_split (xytest_size = random_state = boosting sklearn from sklearn ensemble import gradientboostingregressor breg gradientboostingregressor ... |
15,086 | where ` ( ( :(bi= wi { (xi yi pn (bi= wi pn can be interpreted as the weighted zero-one training loss at iteration for any the program ( is minimized by classifier that minimizes this weighted training lossthat iscb (xargmin ` ( ( cc substituting ( into ( and solving for the optimal gives ` ( (cb ab ln ` ( (cb ( this g... |
15,087 | the step-size parameter ab found by the adaboost algorithm in line can be viewed as an optimal step-size in the sense of training loss minimization howeversimilar to the regression settingone can slow down the adaboost algorithm by setting ab to be fixed (smallvalue ab as usualwhen the latter is done in practiceit is t... |
15,088 | for in range ( )iftrain_pred [ ]!= [ ])err_b err_b + [ierr_b err_b /np sum(walpha_b np log (( err_b )err_b alpha_b_arr append alpha_b for in range ( ) [iw[ ]np exp(- [ ]alpha_b train_pred [ ]yhat_boost np zeros (len( )for in range boostingrounds )yhat_boost yhat_boost alpha_b_arr [ ]learner [jpredict (xyhat np zeros (n... |
15,089 | further reading breiman' book on decision trees[ ]serves as great starting point some additional advances can be found in [ from the computational point of viewthere exists an efficient recursive procedure for tree pruningsee and in [ several advantages and disadvantages of using decision trees are debated in [ detaile... |
15,090 | suppose is training set with elements and talso of size nis obtained from by bootstrappingthat isresampling with replacement show that for large ntdoes not contain fraction of about - of the points from prove equation ( consider the following training/test split of the data construct random forest regressor and identif... |
15,091 | eep earning in this we show how one can construct rich class of approximating functions called neural networks the learners belonging to the neural-network class of functions have attractive properties that have made them ubiquitous in modern machine learning applications -their training is computationally feasible and... |
15,092 | alternativelyin ( we will define the output of neural network as the repeated composition of linear and (componentwisenonlinear functions as we shall seethis representation of the output will provide flexible class of nonlinear functions that can be easily differentiated as resultthe training of learners via gradient o... |
15,093 | in particulareach of the components of the input is represented as node in the input layer ( in the hidden layer ( there are : nodeseach of which is associated with pair of variables (zawith values :hi (xi and : ( hidden layer = link between nodes ( and xi with weight hi signifies that the value of depends on the value... |
15,094 | feed-forward weight matrix bias vector feed-forward neural networks in neural network with + layersthe zero or input layer ( encodes the input feature vector xand the last or output layer ( lencodes the (multivaluedoutput function (xthe remaining layers are called hidden layers each layer has number of nodessay pl node... |
15,095 | herethe (ij)-th element of the weight matrix wl [wl, is the weight that connects the -th node in the ( )-st layer with the -th node in the -th layer the name given to (the number of layers without the input layeris the network depth and maxl pl is called the network width while we mostly study networks that have an equ... |
15,096 | example (nonlinear multi-output regressiongiven the input and an activation function rthe output ( :[ ( ) ( )]of nonlinear multioutput regression model can be computed via neural network withz where xp , ( , ) (xw where xp which is neural network with one hidden layer and output function (zz in the special case where a... |
15,097 | example (density estimationestimating the density of some random feature is the prototypical unsupervised learning taskwhich we tackled in section using gaussian mixture models we can view gaussian mixture model with components and common scale parameter as neural network with two hidden layerssimilar to the one on fig... |
15,098 | of all possible input matrix regions and the kernel matrix(see example in particularby noting that there are ( ( possible regions in the original imagewe conclude that the convolution layer output size is ( ( in practicewe frequently define several kernel matricesgiving an output layer of size ( ( (the number of kernel... |
15,099 | the following theorem provides us with the formulas needed to compute the gradient of typical ci (ththeorem gradient of training loss for given (inputoutputpair (xy)let ( thbe the output of algorithm and let (thloss(yg( th)be an almost-everywhere differentiable loss funcl tion suppose {zl al } = are the vectors obtaine... |
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