Lecture 13: Validation

Transcription

1 Lecture 3: Validatio Resampli methods Holdout Cross Validatio Radom Subsampli -Fold Cross-Validatio Leave-oe-out The Bootstrap Bias ad variace estimatio Three-way data partitioi Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity

2 Itroductio Almost ivariably, all the patter recoitio techiques that we have itroduced have oe or more free parameters The umber of eihbors i a k classificatio rule (or the k desity estimatio method) The badwidth of the kerel fuctio i kerel desity estimatio The etwork size, leari parameters ad weihts i Multilayer Perceptros Two questios arise at this poit How do we select the optimal parameter(s) for a ive classificatio problem? Oce we have chose a model, how do we estimate its true error rate? The true error rate is the classifier s error rate whe tested o the ETIRE POPULATIO If we had access to a ulimited umber of examples these questios have a straihtforward aswer Choose the model that provides the lowest error rate o the etire populatio Ad, of course, that error rate is the true error rate I real applicatios we oly have access to a fiite set of examples, usually smaller tha we wated Oe approach is to use the etire traii data to select our classifier ad estimate the error rate This aïve approach has two fudametal problems The fial model will ormally overfit the traii data: it will ot be able to eeralize to ew data The problem of overfitti is more proouced with models that have a lare umber of parameters The error rate estimate will be overly optimistic (lower tha the true error rate) I fact, it is ot ucommo to have 00% correct classificatio o traii data A much better approach is to split the traii data ito disjoit subsets: the holdout method Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 2

3 The holdout method () Split dataset ito two roups Traii set: used to trai the classifier Test set: used to estimate the error rate of the traied classifier Total umber of examples Traii Set Test Set A typical applicatio the holdout method is determii a stoppi poit for the back propaatio error MSE Stoppi poit Test set error Traii set error Epochs Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 3

4 The holdout method (2) The holdout method has two basic drawbacks I problems where we have a sparse dataset we may ot be able to afford the luxury of setti aside a portio of the dataset for testi Sice it is a sile trai-ad-test experimet, the holdout estimate of error rate will be misleadi if we happe to et a ufortuate split The limitatios of the holdout ca be overcome with a family of resampli methods at the expese of hiher computatioal cost Cross Validatio Radom Subsampli -Fold Cross-Validatio Leave-oe-out Cross-Validatio Bootstrap Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 4

5 Radom Subsampli Radom Subsampli performs data splits of the etire dataset Each data split radomly selects a (fixed) umber of examples without replacemet For each data split we retrai the classifier from scratch with the traii examples ad the estimate E i with the test examples Total umber of examples Experimet Test example Experimet 2 Experimet 3 The true error estimate is obtaied as the averae of the separate estimates E i This estimate is siificatly better tha the holdout estimate E = i= E i Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 5

6 -Fold Cross-validatio Create a -fold partitio of the the dataset For each of experimets, use - folds for traii ad a differet fold for testi This procedure is illustrated i the followi diaram for =4 Total umber of examples Experimet Experimet 2 Experimet 3 Experimet 4 Test examples -Fold Cross validatio is similar to Radom Subsampli The advatae of -Fold Cross validatio is that all the examples i the dataset are evetually used for both traii ad testi As before, the true error is estimated as the averae error rate o test examples E = E i i= Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 6

7 Leave-oe-out Cross Validatio Leave-oe-out is the deeerate case of -Fold Cross Validatio, where is chose as the total umber of examples For a dataset with examples, perform experimets For each experimet use - examples for traii ad the remaii example for testi Total umber of examples Experimet Experimet 2 Experimet 3 Sile test example Experimet As usual, the true error is estimated as the averae error rate o test examples E = E i i= Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 7

8 How may folds are eeded? With a lare umber of folds + The bias of the true error rate estimator will be small (the estimator will be very accurate) - The variace of the true error rate estimator will be lare - The computatioal time will be very lare as well (may experimets) With a small umber of folds + The umber of experimets ad, therefore, computatio time are reduced + The variace of the estimator will be small - The bias of the estimator will be lare (coservative or smaller tha the true error rate) I practice, the choice of the umber of folds depeds o the size of the dataset For lare datasets, eve 3-Fold Cross Validatio will be quite accurate For very sparse datasets, we may have to use leave-oe-out i order to trai o as may examples as possible A commo choice for -Fold Cross Validatio is =0 Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 8

9 The bootstrap () The bootstrap is a resampli techique with replacemet From a dataset with examples Radomly select (with replacemet) examples ad use this set for traii The remaii examples that were ot selected for traii are used for testi This value is likely to chae from fold to fold Repeat this process for a specified umber of folds () Complete dataset X X 2 X 3 X 5 Experimet X 3 X X 3 X 3 X 5 X 2 Experimet 2 X 5 X 5 X 3 X X 2 Experimet 3 X 5 X 5 X X 2 X X 3 Experimet X X 2 X 3 X 5 Traii sets Validatio sets As usual, the true error is estimated as the averae error rate o test examples E = i= E i Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 9

10 The bootstrap (2) Compared to basic cross-validatio, the bootstrap icreases the variace that ca occur i each fold [Efro ad Tibshirai, 993] This is a desirable property sice it is a more realistic simulatio of the real-life experimet from which we obtaied out dataset Cosider a classificatio problem with C classes, a total of examples ad i examples for each class ω i The a priori probability of choosi a example from class ω i is i / Oce we choose a example from class ω i, if we do ot replace it for the ext selectio, the the a priori probabilities will have chaed sice the probability of choosi a example from class ω i will ow be ( i -)/ Sampli with replacemet preserves the a priori probabilities of the classes throuhout the radom selectio process A additioal beefit of the bootstrap is its ability to obtai accurate measures of BOTH the bias ad variace of the true error estimate Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 0

11 Bias ad variace of a statistical estimate Cosider the problem of estimati a parameter α of ukow distributio G To emphasize the fact that α cocers G we will refer to it as α(g) We collect examples X={x, x 2,, x } from the distributio G These examples defie a discrete distributio G with mass / at each of the examples We compute the statistic α =α(g ) as a estimator of α(g) I the cotext of this lecture, α(g ) is the estimate of the true error rate for our classifier How ood is this estimator? The oodess of a statistical estimator is measured by BIAS: How much it deviates from the true value VARIACE: How much variability it shows for differet samples X={x, x 2,, x } of the populatio G Var = E E 2 If we are tryi to estimate the mea of the populatio with the sample mea = E [ ( G) ] ( G) where E [ X] x ( x) Bias G [( [ ]) ] G G The bias of the sample mea is kow to be ZERO The stadard deviatio of the sample mea is, from elemetary statistics, equal to 2 std( x) = ( x i x) ( ) i= This term is also kow i statistics as the STADARD ERROR Ufortuately, there is o such a eat alebraic formula for almost ay estimate other tha the sample mea G + = dx Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity

12 Bias ad variace estimates with the bootstrap The bootstrap, with its eleat simplicity, allows us to estimate bias ad variace for practically ay statistical estimate, be it a scalar or vector (matrix) Here we will oly describe the estimatio procedure Timothy Masters has a excellet itroductio to the bootstrap i his textbook Advaced alorithms for eural etworks if you are iterested i the details The bootstrap estimate of bias ad variace Cosider a dataset of examples X={x, x 2,, x } from the distributio G This dataset defies a discrete distributio G Compute α =α(g ) as our iitial estimate of α(g) Let {x *, x 2 *,, x *} be a bootstrap dataset draw from X={x, x 2,, x } Estimate the parameter α usi this bootstrap dataset α*(g*) Geerate bootstrap datasets ad obtai estimates {α* (G*), α* 2 (G*),, α* (G*)} The ratioale i the bootstrap method is that the effect of eerati a bootstrap dataset from the distributio G is similar to the effect of obtaii the dataset X={x, x 2,, x } from the oriial distributio I other words, the distributio {α* (G*), α* 2 (G*),, α* (G*)} is related to the iitial estimate α i the same fashio as multiple estimates α are related to the true value α, so the bias ad variace estimates of α are Bias Var ( ) = i= i i ( ) = i= i= i Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 2

13 Three-way data splits () If model selectio ad true error estimates are to be computed simultaeously, the data eeds to be divided ito three disjoit sets [Ripley, 996] Traii set: a set of examples used for leari: to fit the parameters of the classifier I the MLP case, we would use the traii set to fid the optimal weihts with the back-prop rule Validatio set: a set of examples used to tue the parameters of of a classifier I the MLP case, we would use the validatio set to fid the optimal umber of hidde uits or determie a stoppi poit for the back propaatio alorithm Test set: a set of examples used oly to assess the performace of a fully-traied classifier I the MLP case, we would use the test to estimate the error rate after we have chose the fial model (MLP size ad actual weihts) After assessi the fial model with the test set, YOU MUST OT further tue the model Why separate test ad validatio sets? The error rate estimate of the fial model o validatio data will be biased (smaller tha the true error rate) sice the validatio set is used to select the fial model After assessi the fial model with the test set, YOU MUST OT tue the model ay further Procedure outlie. Divide the available data ito traii, validatio ad test set 2. Select architecture ad traii parameters 3. Trai the model usi the traii set 4. Evaluate the model usi the validatio set 5. Repeat steps 2 throuh 4 usi differet architectures ad traii parameters 6. Select the best model ad trai it it usi data from the traii ad validatio set 7. Assess this fial model usi the test set This outlie assumes a holdout method If CV or Bootstrap are used, steps 3 ad 4 have to be repeated for each of the folds Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 3

14 Three-way data splits (2) Test set Traii set Validatio set Model Error Model 2 Model 3 Error 2 Error 3 Mi + Fial Model Expected Error Model M Error M Model selectio True error rate estimatio Itroductio to Patter Recoitio Ricardo Gutierrez-Osua Wriht State Uiversity 4