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

2. coleman > head(coleman) salaryP fatherWc sstatus teacherSc motherLev Y 1 3.83 28.87 7.20 26.6 6.19 37.01 2 2.89 20.10 -11.71 24.4 5.17 26.51 3 2.86 69.05 12.32 25.7 7.04 36.51 4 2.92 65.40 14.28 25.7 7.10 40.70 5 3.06 29.59 6.31 25.4 6.15 37.10 6 2.07 44.82 6.16 21.6 6.41 33.90 2

3. 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,] 0.9880672 [2,] 0.9525881 [3,] 0.8989264 [4,] 1.0177694 [5,] 0.9860661 [6,] 1.8369717 [7,] 0.9550428 [8,] 1.0698466 [9,] 1.3568537 [10,] 0.8313474 3 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

4. Did you get this plot – how? > cvFits 5-fold CV results: Fit CV 1 LS 1.674485 2 MM 1.147130 3 LTS 1.291797 Best model: CV "MM" 4

5. 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 http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/ viewer.htm#statug_rreg_sect018.htm#statug.rreg.robustregfltsest http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/ viewer.htm#statug_rreg_sect018.htm#statug.rreg.robustregfltsest MM - http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/ viewer.htm#statug_rreg_sect019.htm http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/ viewer.htm#statug_rreg_sect019.htm 5

6. 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

7. cvFitsLts (50/75) > cvFitsLts 5-fold CV results: Fit reweighted raw 1 0.5 1.291797 1.640922 2 0.75 1.065495 1.232691 Best model: reweighted raw "0.75" "0.75" 7

8. 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

9. cvFitsLmrob 5-fold CV results: tuning.psi CV 1 3.14 1.179620 2 3.44 1.156674 3 3.88 1.169436 4 4.68 1.133975 Optimal tuning parameter: tuning.psi CV 4.68 9

10. Lab on Friday mammals.glm <- glm(log(brain) ~ log(body), data = mammals) (cv.err <- cv.glm(mammals, mammals.glm)$delta) [1] 0.4918650 0.4916571 > (cv.err.6 <- cv.glm(mammals, mammals.glm, K = 6)$delta) [1] 0.4967271 0.4938003 # 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] 0.491865 10

11. 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))$delta) [1] 0.1886792 0.1886792 > (cv.11.err <- cv.glm(nodal, nodal.glm, cost, K = 11)$delta) [1] 0.2264151 0.2228551 11

12. cvTools http://cran.r- 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

13. 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

14. Ozone 14 10 of 100 bootstrap samples average

15. 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

16. 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$confusion Observed Class Predicted Class benign malignant benign 142 2 malignant 8 81 16 BC.bagging.pred$error [1] 0.04291845

17. 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$confusion Observed Class Predicted Class benign malignant benign 147 1 malignant 7 78 > BC.bagging.pred$error [1] 0.03433476 17

18. 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$confusion Observed Class Predicted Class bus opel saab van bus 63 10 8 0 opel 1 42 27 0 saab 0 18 30 0 van 5 7 9 62 > Vehicle.bagging.pred$error [1] 0.3014184 18

19. 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

20. 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

21. 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. 22 ggplot(diamonds, aes(clarity)) + geom_freqpoly(aes(group = cut, colour = cut))

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24. 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

25. Using diamonds… boost (glm) > summary(mglmboost) #continued Number of boosting iterations: mstop = 100 Step size: 0.1 Offset: -1.339537 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 -1.5156330 1.5388715 0.1823241 attr(,"offset") [1] -1.339537 Selection frequencies: carat (Intercept) clarity.L 0.50 0.42 0.08 25

26. 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

27. 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

28. 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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30. 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

31. Compare linear models > coef(lm1) (Intercept) hipcirc kneebreadth anthro3a -75.2347840 0.5115264 1.9019904 8.9096375 > coef(glm1, off2int=TRUE) ## off2int adds the offset to the intercept (Intercept) hipcirc kneebreadth anthro3a -75.2073365 0.5114861 1.9005386 8.9071301 Conclusion? 31

32. > 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 -98.8166077 0.0136017 0.1897156 0.3516258 - 0.3841399 1.7365888 3.3268603 3.6565240 0.5953626 anthro4 0.0000000 attr(,"offset") [1] 30.78282 32

33. plot(glm2, off2int = TRUE) 33

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

35. > summary(bodyfat) age DEXfat waistcirc hipcirc elbowbreadth kneebreadth anthro3a Min. :19.00 Min. :11.21 Min. : 65.00 Min. : 88.00 Min. :5.200 Min. : 7.200 Min. :2.400 1st Qu.:42.00 1st Qu.:22.32 1st Qu.: 78.50 1st Qu.: 96.75 1st Qu.:6.200 1st Qu.: 8.600 1st Qu.:3.540 Median :56.00 Median :29.63 Median : 85.00 Median :103.00 Median :6.500 Median : 9.200 Median :3.970 Mean :50.86 Mean :30.78 Mean : 87.38 Mean :105.28 Mean :6.508 Mean : 9.301 Mean :3.869 3rd Qu.:62.00 3rd Qu.:39.33 3rd Qu.: 99.75 3rd Qu.:111.15 3rd Qu.:6.900 3rd Qu.: 9.800 3rd Qu.:4.155 Max. :67.00 Max. :62.02 Max. :117.00 Max. :132.00 Max. :7.400 Max. :11.800 Max. :4.680 anthro3b anthro3c anthro4 Min. :2.580 Min. :2.050 Min. :3.180 1st Qu.:4.060 1st Qu.:3.480 1st Qu.:5.040 Median :4.390 Median :3.990 Median :5.530 Mean :4.291 Mean :3.886 Mean :5.398 3rd Qu.:4.660 3rd Qu.:4.345 3rd Qu.:5.840 Max. :5.010 Max. :4.620 Max. :6.370 35

36. 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

37. > 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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39. > gam2 <- gamboost(DEXfat ~., baselearner = "bbs", data = bodyfat,control = boost_control(trace = TRUE)) [ 1].................................................. -- risk: 515.5713 [ 53].............................................. Final risk: 460.343 > set.seed(123) ## set seed to make results reproducible > cvm <- cvrisk(gam2) ## default method is 25-fold bootstrap cross-validation 39

40. > cvm Cross-validated Squared Error (Regression) gamboost(formula = DEXfat ~., data = bodyfat, baselearner = "bbs", control = boost_control(trace = TRUE)) 1 2 3 4 5 6 7 8 9 10 109.44043 93.90510 80.59096 69.60200 60.13397 52.59479 46.11235 40.80175 36.32637 32.66942 11 12 13 14 15 16 17 18 19 20 29.66258 27.07809 24.99304 23.11263 21.55970 20.40313 19.16541 18.31613 17.59806 16.96801 21 22 23 24 25 26 27 28 29 30 16.48827 16.07595 15.75689 15.47100 15.21898 15.06787 14.96986 14.86724 14.80542 14.74726 31 32 33 34 35 36 37 38 39 40 14.68165 14.68648 14.64315 14.67862 14.68193 14.68394 14.75454 14.80268 14.81760 14.87570 41 42 43 44 45 46 47 48 49 50 14.90511 14.92398 15.00389 15.03604 15.07639 15.10671 15.15364 15.20770 15.23825 15.30189 51 52 53 54 55 56 57 58 59 60 15.31950 15.35630 15.41134 15.46079 15.49545 15.53137 15.57602 15.61894 15.66218 15.71172 61 62 63 64 65 66 67 68 69 70 15.72119 15.75424 15.80828 15.84097 15.89077 15.90547 15.93003 15.95715 15.99073 16.03679 71 72 73 74 75 76 77 78 79 80 16.06174 16.10615 16.12734 16.15830 16.18715 16.22298 16.27167 16.27686 16.30944 16.33804 81 82 83 84 85 86 87 88 89 90 16.36836 16.39441 16.41587 16.43615 16.44862 16.48259 16.51989 16.52985 16.54723 16.58531 91 92 93 94 95 96 97 98 99 100 16.61028 16.61020 16.62380 16.64316 16.64343 16.68386 16.69995 16.73360 16.74944 16.75756 Optimal number of boosting iterations: 33 40

41. > 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: 30.78282 Number of baselearners: 9 41

42. plot(cvm) 42

43. > 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: 423.9261 [ 153].................................................. -- risk: 397.4189 [ 205].................................................. -- risk: 377.0872 [ 257].................................................. -- risk: 360.7946 [ 309].................................................. -- risk: 347.4504 [ 361].................................................. -- risk: 336.1172 [ 413].................................................. -- risk: 326.277 [ 465].................................................. -- risk: 317.6053 [ 517].................................................. -- risk: 309.9062 [ 569].................................................. -- risk: 302.9771 [ 621].................................................. -- risk: 296.717 [ 673].................................................. -- risk: 290.9664 [ 725].................................................. -- risk: 285.683 [ 777].................................................. -- risk: 280.8266 [ 829].................................................. -- risk: 276.3009 [ 881].................................................. -- risk: 272.0859 [ 933].................................................. -- risk: 268.1369 [ 985].............. Final risk: 266.9768 43

44. > 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 -63.5164304 0.5331394 0.7699975 7.8350858 44

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46. 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

47. 47

48. cars 48

49. iris 49

50. cars 50

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52. Optimizing Coefficients: (Intercept) speed -60.331204 3.918359 attr(,"offset") [1] 42.98 Call: glmboost.formula(formula = dist ~ speed, data = cars, control = boost_control(mstop = 1000), family = Laplace()) Coefficients: (Intercept) speed -47.631025 3.402015 attr(,"offset") [1] 35.99999 52

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54. Sparse matrix example > coef(mod, which = which(beta > 0)) V306 V1052 V1090 V3501 V4808 V5473 V7929 V8333 V8799 V9191 2.1657532 0.0000000 4.8756163 4.7068006 0.4429911 5.4029763 3.6435648 0.0000000 3.7843504 0.4038770 attr(,"offset") [1] 2.90198 54

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56. 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

57. 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

58. 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: 42.98 Number of baselearners: 1 58