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Peter Fox

Data Analytics – ITWS-4963/ITWS-6965

Week 13a, April 22, 2014

Boosting, dimension reduction and a preview of the return to

Big Data

Reading• http://data-informed.com/focus-predictive-ana

lytics/

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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?

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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.

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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))

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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)

}

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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

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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. 10

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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

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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?

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> 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 14

plot(glm2, off2int = TRUE)

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

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> 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

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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.

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> gam1 <- gamboost(DEXfat ~ bbs(hipcirc) + bbs(kneebreadth) + bbs(anthro3a),data = bodyfat)

> #Using plot() on a gamboost object delivers automatically the partial e ects of the di erent base-learners:ff ff> par(mfrow = c(1,3)) ## 3 plots in one device

> plot(gam1) ## get the partial effects

# bbs, bols, btree..

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> 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

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> 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 22

> 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 23

plot(cvm)

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> 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 25

> 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

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Compare to rpart> fattree<-rpart(DEXfat ~ ., data=bodyfat)

> plot(fattree)

> text(fattree)

> labels(fattree)

[1] "root" "waistcirc< 88.4" "anthro3c< 3.42" "anthro3c>=3.42" "hipcirc< 101.3" "hipcirc>=101.3"

[7] "waistcirc>=88.4" "hipcirc< 109.9" "hipcirc>=109.9"

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cars

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iris

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cars

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OptimizingCoefficients:

(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] 35.9999934

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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

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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

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Dimension reduction..• Principle component analysis (PCA) and

metaPCA (in R)• Singular Value Decomposition• Feature selection, reduction• Clustering• Why?

– Curse of dimensionality – or – some subset of the data should not be used as it adds noise

• What is it?– Various methods to reach an optimal subset

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Feature selection• The goodness of a feature/feature subset is

dependent on measures

• Various measures– Information measures – Distance measures – Dependence measures – Consistency measures – Accuracy measures

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Libraries in R that you used…• MetaPCA (prcomp, metaPCA)• EDR (effective dimension reduction)• dr

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Lab 6

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prostate data (lab 7).

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Demo (lab 8)library(EDR) # effective dimension reduction

###install.packages("edrGraphicalTools")

###library(edrGraphicalTools)

demo(edr_ex1)

demo(edr_ex2)

demo(edr_ex3)

demo(edr_ex4)

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Lab 9library(dr)

data(ais)

# default fitting method is "sir"

s0 <- dr(LBM~log(SSF)+log(Wt)+log(Hg)+log(Ht)+log(WCC)+log(RCC)+

log(Hc)+log(Ferr),data=ais)

# Refit, using a different function for slicing to agree with arc.

summary(s1 <- update(s0,slice.function=dr.slices.arc))

# Refit again, using save, with 10 slices; the default is max(8,ncol+3)

summary(s2<-update(s1,nslices=10,method="save"))

# Refit, using phdres. Tests are different for phd, and not

# Fit using phdres; output is similar for phdy, but tests are not justifiable.

summary(s3<- update(s1,method="phdres"))

# fit using ire:

summary(s4 <- update(s1,method="ire"))

# fit using Sex as a grouping variable.

s5 <- update(s4,group=~Sex)45

> s0

dr(formula = LBM ~ log(SSF) + log(Wt) + log(Hg) + log(Ht) +

log(WCC) + log(RCC) + log(Hc) + log(Ferr), data = ais)

Estimated Basis Vectors for Central Subspace:

Dir1 Dir2 Dir3 Dir4

log(SSF) 0.150963358 -0.0501785457 0.10898336 -0.002210206

log(Wt) -0.916480522 -0.1942298625 -0.20123696 -0.089722026

log(Hg) -0.131538894 0.6854750758 0.71997546 -0.663097774

log(Ht) -0.093358860 -0.0433408964 0.46445398 0.290838658

log(WCC) 0.004467838 0.0001833808 0.04497590 0.071904557

log(RCC) -0.188973540 0.3475652934 0.29496908 0.037056363

log(Hc) 0.274758965 -0.6058301419 -0.34196615 0.678877114

log(Ferr) -0.005631238 0.0130588502 -0.08702709 0.015547214

Eigenvalues:

[1] 0.95766163 0.24504161 0.10707594 0.0904130546

> summary(s1 <- update(s0,slice.function=dr.slices.arc))

Call:

dr(formula = LBM ~ log(SSF) + log(Wt) + log(Hg) + log(Ht) + log(WCC) +

log(RCC) + log(Hc) + log(Ferr), data = ais, slice.function = dr.slices.arc)

Method:

sir with 11 slices, n = 202.

Slice Sizes:

19 19 19 19 19 19 19 18 18 18 15

Estimated Basis Vectors for Central Subspace:

Dir1 Dir2 Dir3 Dir4

log(SSF) 0.143177 -0.0476079 -0.02815 0.003785

log(Wt) -0.879504 -0.1425841 0.23303 -0.094970

log(Hg) -0.195963 0.6318503 0.24483 -0.509424

log(Ht) -0.058923 -0.1100757 -0.87893 0.217803

log(WCC) -0.007276 -0.0029772 -0.05309 0.043056

log(RCC) -0.167736 0.3924936 -0.19711 -0.213689

log(Hc) 0.368652 -0.6418658 -0.26373 0.796849

log(Ferr) -0.002697 0.0002593 0.03492 0.03911647

Dir1 Dir2 Dir3 Dir4Eigenvalues 0.9572 0.2275 0.09368 0.07319R^2(OLS|dr) 0.9980 0.9981 0.99839 0.99864

Large-sample Marginal Dimension Tests: Stat df p.value0D vs >= 1D 284.78 80 0.000001D vs >= 2D 91.43 63 0.011132D vs >= 3D 45.48 48 0.576903D vs >= 4D 26.55 35 0.84694

> summary(s2<-update(s1,nslices=10,method="save"))

Call:

dr(formula = LBM ~ log(SSF) + log(Wt) + log(Hg) + log(Ht) + log(WCC) +

log(RCC) + log(Hc) + log(Ferr), data = ais, slice.function = dr.slices.arc,

nslices = 10, method = "save")

Method:

save with 10 slices, n = 202.

Slice Sizes:

21 21 20 20 20 25 24 22 20 9

Estimated Basis Vectors for Central Subspace:

Dir1 Dir2 Dir3 Dir4

log(SSF) 0.127709 -0.00907 0.01018 -0.06144

log(Wt) -0.905004 -0.07107 -0.15734 0.25774

log(Hg) -0.056187 0.50674 -0.34064 -0.38087

log(Ht) 0.399868 0.36613 0.68439 -0.54216

log(WCC) 0.032608 0.02733 0.02277 0.03474

log(RCC) -0.008463 0.15137 -0.24136 -0.47219

log(Hc) -0.021630 -0.76164 0.57591 0.51526

log(Ferr) 0.002116 -0.01670 0.01631 -0.03360

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Dir1 Dir2 Dir3 Dir4Eigenvalues 0.9389 0.6611 0.5129 0.4653R^2(OLS|dr) 0.9936 0.9950 0.9985 0.9989

Large-sample Marginal Dimension Tests: Stat df(Nor) p.value(Nor) p.value(Gen)0D vs >= 1D 378.3 324 0.02012 0.10711D vs >= 2D 279.6 252 0.11214 0.31162D vs >= 3D 179.9 189 0.67101 0.51603D vs >= 4D 134.3 135 0.50176 0.2786

S0 v. S2

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S3 and S4

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Remember - Assignment 2?

• General assignment – read EPI_data, specify a new data subset, create data frames in R and save them into a database

• In R Studio– Install package – “rmongodb” (activate it)– http://www.r-tutor.com/r-introduction/

• MongoDB - http://www.mongodb.org/ – http://kkovacs.eu/cassandra-vs-mongodb-vs-cou

chdb-vs-redis 52

We’ll revisit these• http://projects.apache.org/indexes/category.ht

ml#database – Hadoop (MapReduce)– Pig (http://wiki.apache.org/pig/RunPig )– HIVE (http://hive.apache.org/releases.html )

• https://cwiki.apache.org/confluence/display/Hive/GettingStarted

• https://cwiki.apache.org/confluence/display/Hive/Tutorial

• https://cwiki.apache.org/confluence/display/Hive/LanguageManual

– Spark53