1 Peter Fox Data Analytics – ITWS-4963/ITWS-6965 Week 11a, April 14, 2015 Interpreting cross-validation, bootstrapping, bagging, boosting, etc.

coleman > head(coleman) salaryP fatherWc sstatus teacherSc motherLev Y

What were you doing? > call <- call("lmrob", formula = Y ~.) > # set up folds for cross-validation > folds <- cvFolds(nrow(coleman), K = 5, R = 10) > # perform cross-validation > cvTool(call, data = coleman, y = coleman$Y, cost = rtmspe, + folds = folds, costArgs = list(trim = 0.1)) CV [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] Warning messages: 1: In lmrob.S(x, y, control = control) : S refinements did not converge (to refine.tol=1e-07) in 200 (= k.max) steps 2: In lmrob.S(x, y, control = control) : S refinements did not converge (to refine.tol=1e-07) in 200 (= k.max) steps 3: In lmrob.S(x, y, control = control) : find_scale() did not converge in 'maxit.scale' (= 200) iterations 4: In lmrob.S(x, y, control = control) : find_scale() did not converge in 'maxit.scale' (= 200) iterations

Did you get this plot – how? > cvFits 5-fold CV results: Fit CV 1 LS MM LTS Best model: CV "MM" 4

LS, LTS, MM? The breakdown value of an estimator is defined as the smallest fraction of contamination that can cause the estimator to take on values arbitrarily far from its value on the uncontaminated data. The breakdown value of an estimator can be used as a measure of the robustness of the estimator. Rousseeuw and Leroy (1987) and others introduced high breakdown value estimators for linear regression. LTS – see viewer.htm#statug_rreg_sect018.htm#statug.rreg.robustregfltsest viewer.htm#statug_rreg_sect018.htm#statug.rreg.robustregfltsest MM - viewer.htm#statug_rreg_sect019.htm viewer.htm#statug_rreg_sect019.htm 5

50 and 75% subsets fitLts50 <- ltsReg(Y ~., data = coleman, alpha = 0.5) cvFitLts50 <- cvLts(fitLts50, cost = rtmspe, folds = folds, fit = "both", trim = 0.1) # 75% subsets fitLts75 <- ltsReg(Y ~., data = coleman, alpha = 0.75) cvFitLts75 <- cvLts(fitLts75, cost = rtmspe, folds = folds, fit = "both", trim = 0.1) # combine and plot results cvFitsLts <- cvSelect("0.5" = cvFitLts50, "0.75" = cvFitLts75) 6

cvFitsLts (50/75) > cvFitsLts 5-fold CV results: Fit reweighted raw Best model: reweighted raw "0.75" "0.75" 7

Tuning tuning <- list(tuning.psi=c(3.14, 3.44, 3.88, 4.68)) # perform cross-validation cvFitsLmrob <- cvTuning(fitLmrob$call, data = coleman, y = coleman$Y, tuning = tuning, cost = rtmspe, folds = folds, costArgs = list(trim = 0.1)) 8

cvFitsLmrob 5-fold CV results: tuning.psi CV Optimal tuning parameter: tuning.psi CV

Lab on Friday mammals.glm <- glm(log(brain) ~ log(body), data = mammals) (cv.err <- cv.glm(mammals, mammals.glm)$delta) [1] > (cv.err.6 <- cv.glm(mammals, mammals.glm, K = 6)$delta) [1] # As this is a linear model we could calculate the leave-one-out # cross-validation estimate without any extra model-fitting. muhat <- fitted(mammals.glm) mammals.diag <- glm.diag(mammals.glm) (cv.err <- mean((mammals.glm$y - muhat)^2/(1 - mammals.diag$h)^2)) [1]

Cost functions, etc. # leave-one-out and 11-fold cross-validation prediction error for # the nodal data set. Since the response is a binary variable # an appropriate cost function is > cost 0.5) > nodal.glm <- glm(r ~ stage+xray+acid, binomial, data = nodal) > (cv.err <- cv.glm(nodal, nodal.glm, cost, K = nrow(nodal))$delta) [1] > (cv.11.err <- cv.glm(nodal, nodal.glm, cost, K = 11)$delta) [1]

cvTools project.org/web/packages/cvTools/cvTools.pd fhttp://cran.r- project.org/web/packages/cvTools/cvTools.pd f Very powerful and flexible package for CV (regression) but very much a black box! If you use it, become very, very familiar with the outputs and be prepared to experiment… 12

Bootstrap aggregation (bagging) Improve the stability and accuracy of machine learning algorithms used in statistical classification and regression. Also reduces variance and helps to avoid overfitting. Usually applied to decision tree methods, but can be used with any type of method. –Bagging is a special case of the model averaging approach. Harder to interpret – why? 13

Ozone of 100 bootstrap samples average

Shows improvements for unstable procedures (Breiman, 1996): e.g. neural nets, classification and regression trees, and subset selection in linear regression … can mildly degrade the performance of stable methods such as K-nearest neighbors 15

Bagging (bootstrapping aggregation)* library(mlbench) data(BreastCancer) l <- length(BreastCancer[,1]) sub <- sample(1:l,2*l/3) BC.bagging <- bagging(Class ~., data=BreastCancer[,-1], mfinal=20, control=rpart.control(maxdepth=3)) BC.bagging.pred <-predict.bagging( BC.bagging, newdata=BreastCancer[-sub,-1]) BC.bagging.pred$confusion Observed Class Predicted Class benign malignant benign malignant BC.bagging.pred$error [1]

A little later > data(BreastCancer) > l <- length(BreastCancer[,1]) > sub <- sample(1:l,2*l/3) > BC.bagging <- bagging(Class ~.,data=BreastCancer[,-1],mfinal=20, + control=rpart.control(maxdepth=3)) > BC.bagging.pred <- predict.bagging(BC.bagging,newdata=BreastCancer[- sub,-1]) > BC.bagging.pred$confusion Observed Class Predicted Class benign malignant benign malignant 7 78 > BC.bagging.pred$error [1]

Bagging (Vehicle) > data(Vehicle) > l <- length(Vehicle[,1]) > sub <- sample(1:l,2*l/3) > Vehicle.bagging <- bagging(Class ~.,data=Vehicle[sub, ],mfinal=40, + control=rpart.control(maxdepth=5)) > Vehicle.bagging.pred <- predict.bagging(Vehicle.bagging, newdata=Vehicle[-sub, ]) > Vehicle.bagging.pred$confusion Observed Class Predicted Class bus opel saab van bus opel saab van > Vehicle.bagging.pred$error [1]

Weak models … A weak learner: a classifier which is only slightly correlated with the true classification (it can label examples better than random guessing) A strong learner: a classifier that is arbitrarily well-correlated with the true classification. Can a set of weak learners create a single strong learner? 19

Boosting … reducing bias in supervised learning most boosting algorithms consist of iteratively learning weak classifiers with respect to a distribution and adding them to a final strong classifier. –typically weighted in some way that is usually related to the weak learners' accuracy. After a weak learner is added, the data is reweighted: examples that are misclassified gain weight and examples that are classified correctly lose weight Thus, future weak learners focus more on the examples that previous weak learners misclassified. 20

Diamonds require(ggplot2) # or load package first data(diamonds) head(diamonds) # look at the data! # ggplot(diamonds, aes(clarity, fill=cut)) + geom_bar() ggplot(diamonds, aes(clarity)) + geom_bar() + facet_wrap(~ cut) ggplot(diamonds) + geom_histogram(aes(x=price)) + geom_vline(xintercept=12000) ggplot(diamonds, aes(clarity)) + geom_freqpoly(aes(group = cut, colour = cut)) 21

22 ggplot(diamonds, aes(clarity)) + geom_freqpoly(aes(group = cut, colour = cut))

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Using diamonds… boost (glm) > mglmboost<-glmboost(as.factor(Expensive) ~., data=diamonds, family=Binomial(link="logit")) > summary(mglmboost) Generalized Linear Models Fitted via Gradient Boosting Call: glmboost.formula(formula = as.factor(Expensive) ~., data = diamonds, family = Binomial(link = "logit")) Negative Binomial Likelihood Loss function: { f <- pmin(abs(f), 36) * sign(f) p <- exp(f)/(exp(f) + exp(-f)) y <- (y + 1)/2 -y * log(p) - (1 - y) * log(1 - p) } 24

Using diamonds… boost (glm) > summary(mglmboost) #continued Number of boosting iterations: mstop = 100 Step size: 0.1 Offset: Coefficients: NOTE: Coefficients from a Binomial model are half the size of coefficients from a model fitted via glm(..., family = 'binomial'). See Warning section in ?coef.mboost (Intercept) carat clarity.L attr(,"offset") [1] Selection frequencies: carat (Intercept) clarity.L

Cluster boosting Assessment of the clusterwise stability of a clustering of data, which can be cases x variables or dissimilarity data. The data is resampled using several schemes (bootstrap, subsetting, jittering, replacement of points by noise) and the Jaccard similarities of the original clusters to the most similar clusters in the resampled data are computed. The mean over these similarities is used as an index of the stability of a cluster (other statistics can be computed as well). 26

Cluster boosting Quite general clustering methods are possible, i.e. methods estimating or fixing the number of clusters, methods producing overlapping clusters or not assigning all cases to clusters (but declaring them as "noise"). In R – clustermethod = X is used to select the method, e.g. Kmeans Lab on Friday… (iris, etc..) 27

Example - bodyfat The response variable is the body fat measured by DXA (DEXfat), which can be seen as the gold standard to measure body fat. However, DXA measurements are too expensive and complicated for a broad use. Anthropometric measurements as waist or hip circumferences are in comparison very easy to measure in a standard screening. A prediction formula only based on these measures could therefore be a valuable alternative with high clinical relevance for daily usage. 28

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bodyfat ## regular linear model using three variables lm1 <- lm(DEXfat ~ hipcirc + kneebreadth + anthro3a, data = bodyfat) ## Estimate same model by glmboost glm1 <- glmboost(DEXfat ~ hipcirc + kneebreadth + anthro3a, data = bodyfat) # We consider all available variables as potential predictors. glm2 <- glmboost(DEXfat ~., data = bodyfat) # or one could essentially call: preds <- names(bodyfat[, names(bodyfat) != "DEXfat"]) ## names of predictors fm <- as.formula(paste("DEXfat ~", paste(preds, collapse = "+"))) ## build formula 30

Compare linear models > coef(lm1) (Intercept) hipcirc kneebreadth anthro3a > coef(glm1, off2int=TRUE) ## off2int adds the offset to the intercept (Intercept) hipcirc kneebreadth anthro3a Conclusion? 31

> fm DEXfat ~ age + waistcirc + hipcirc + elbowbreadth + kneebreadth + anthro3a + anthro3b + anthro3c + anthro4 > coef(glm2, which = "") ## select all. (Intercept) age waistcirc hipcirc elbowbreadth kneebreadth anthro3a anthro3b anthro3c anthro attr(,"offset") [1]

plot(glm2, off2int = TRUE) 33

plot(glm2, ylim = range(coef(glm2, which = preds))) 34

> summary(bodyfat) age DEXfat waistcirc hipcirc elbowbreadth kneebreadth anthro3a Min. :19.00 Min. :11.21 Min. : Min. : Min. :5.200 Min. : Min. : st Qu.: st Qu.: st Qu.: st Qu.: st Qu.: st Qu.: st Qu.:3.540 Median :56.00 Median :29.63 Median : Median : Median :6.500 Median : Median :3.970 Mean :50.86 Mean :30.78 Mean : Mean : Mean :6.508 Mean : Mean : rd Qu.: rd Qu.: rd Qu.: rd Qu.: rd Qu.: rd Qu.: rd Qu.:4.155 Max. :67.00 Max. :62.02 Max. : Max. : Max. :7.400 Max. : Max. :4.680 anthro3b anthro3c anthro4 Min. :2.580 Min. :2.050 Min. : st Qu.: st Qu.: st Qu.:5.040 Median :4.390 Median :3.990 Median :5.530 Mean :4.291 Mean :3.886 Mean : rd Qu.: rd Qu.: rd Qu.:5.840 Max. :5.010 Max. :4.620 Max. :

Other forms of boosting Gamboost = Generalized Additive Model - Gradient boosting for optimizing arbitrary loss functions, where component-wise smoothing procedures are utilized as (univariate) base- learners. 36

> gam1 <- gamboost(DEXfat ~ bbs(hipcirc) + bbs(kneebreadth) + bbs(anthro3a),data = bodyfat) > #Using plot() on a gamboost object delivers automatically the partial e ff ects of the di ff erent base-learners: > par(mfrow = c(1,3)) ## 3 plots in one device > plot(gam1) ## get the partial effects # bbs, bols, btree.. 37

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> gam2 <- gamboost(DEXfat ~., baselearner = "bbs", data = bodyfat,control = boost_control(trace = TRUE)) [ 1] risk: [ 53] Final risk: > set.seed(123) ## set seed to make results reproducible > cvm <- cvrisk(gam2) ## default method is 25-fold bootstrap cross-validation 39

> cvm Cross-validated Squared Error (Regression) gamboost(formula = DEXfat ~., data = bodyfat, baselearner = "bbs", control = boost_control(trace = TRUE)) Optimal number of boosting iterations: 33 40

> mstop(cvm) ## extract the optimal mstop [1] 33 > gam2[ mstop(cvm) ] ## set the model automatically to the optimal mstop Model-based Boosting Call: gamboost(formula = DEXfat ~., data = bodyfat, baselearner = "bbs", control = boost_control(trace = TRUE)) Squared Error (Regression) Loss function: (y - f)^2 Number of boosting iterations: mstop = 33 Step size: 0.1 Offset: Number of baselearners: 9 41

plot(cvm) 42

> names(coef(gam2)) ## displays the selected base-learners at iteration 30 [1] "bbs(waistcirc, df = dfbase)" "bbs(hipcirc, df = dfbase)" "bbs(kneebreadth, df = dfbase)" [4] "bbs(anthro3a, df = dfbase)" "bbs(anthro3b, df = dfbase)" "bbs(anthro3c, df = dfbase)" [7] "bbs(anthro4, df = dfbase)" > gam2[1000, return = FALSE] # return = FALSE just supresses "print(gam2)" [ 101] risk: [ 153] risk: [ 205] risk: [ 257] risk: [ 309] risk: [ 361] risk: [ 413] risk: [ 465] risk: [ 517] risk: [ 569] risk: [ 621] risk: [ 673] risk: [ 725] risk: [ 777] risk: [ 829] risk: [ 881] risk: [ 933] risk: [ 985] Final risk:

> names(coef(gam2)) ## displays the selected base-learners, now at iteration 1000 [1] "bbs(age, df = dfbase)" "bbs(waistcirc, df = dfbase)" "bbs(hipcirc, df = dfbase)" [4] "bbs(elbowbreadth, df = dfbase)" "bbs(kneebreadth, df = dfbase)" "bbs(anthro3a, df = dfbase)" [7] "bbs(anthro3b, df = dfbase)" "bbs(anthro3c, df = dfbase)" "bbs(anthro4, df = dfbase)” > glm3 <- glmboost(DEXfat ~ hipcirc + kneebreadth + anthro3a, data = bodyfat,family = QuantReg(tau = 0.5), control = boost_control(mstop = 500)) > coef(glm3, off2int = TRUE) (Intercept) hipcirc kneebreadth anthro3a

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Compare to rpart > fattree<-rpart(DEXfat ~., data=bodyfat) > plot(fattree) > text(fattree) > labels(fattree) [1] "root" "waistcirc =3.42" "hipcirc =101.3" [7] "waistcirc>=88.4" "hipcirc =109.9" 46

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cars 48

iris 49

cars 50

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Optimizing Coefficients: (Intercept) speed attr(,"offset") [1] Call: glmboost.formula(formula = dist ~ speed, data = cars, control = boost_control(mstop = 1000), family = Laplace()) Coefficients: (Intercept) speed attr(,"offset") [1]

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Sparse matrix example > coef(mod, which = which(beta > 0)) V306 V1052 V1090 V3501 V4808 V5473 V7929 V8333 V8799 V attr(,"offset") [1]

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Aside: Boosting and SVM… Remember “margins” from the SVM? Partitioning the “linear” or transformed space? In boosting we are effectively (not explicitly) attempting to maximize the minimum margin of any training example 56

Variants on boosting – loss fn cars.gb <- blackboost(dist ~ speed, data = cars, control = boost_control(mstop = 50)) ### plot fit plot(dist ~ speed, data = cars) lines(cars$speed, predict(cars.gb), col = "red") 57

Blackboosting (cf. brown) Gradient boosting for optimizing arbitrary loss functions where regression trees are utilized as base-learners. > cars.gb Model-based Boosting Call: blackboost(formula = dist ~ speed, data = cars, control = boost_control(mstop = 50)) Squared Error (Regression) Loss function: (y - f)^2 Number of boosting iterations: mstop = 50 Step size: 0.1 Offset: Number of baselearners: 1 58