1

Peter Fox

Data Analytics – ITWS-4963/ITWS-6965

Week 7b, March 13, 2015

Interpreting weighted kNN, decision trees, cross-validation,

dimension reduction and scaling

Contents

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Numeric v. non-numeric

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In R – data frame and types• Almost always at input R sees categorical

data as “strings” or “character”

• You can test for membership (as a type) using is.<x> (x=number, factor, etc.)

• You can “coerce” it (i.e. change the type) using as.<x> (same x)

• To tell R you have categorical types (also called enumerated types), R calls them “factor”…. Thus – as.factor()

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In Rfactor(x = character(), levels, labels = levels,

exclude = NA, ordered = is.ordered(x), nmax = NA),

ordered(x, ...), is.factor(x), is.ordered(x), as.factor(x), as.ordered(x), addNA(x, ifany = FALSE)

levels - values that x might have taken.

labels - either an optional vector of labels for the levels (in the same order as levels after removing those in exclude), or a character string of length 1.

Exclude -

Ordered -

Nmax - upper bound on the number of levels;

Ifany -

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Relate to the datasets…• Abalone - Sex = {F, I, M} • Eye color?• EPI - GeoRegions?

• Sample v. population – levels and names IN the dataset versus all possible levels/names

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Weighted KNNrequire(kknn)

data(iris)

m <- dim(iris)[1]

val <- sample(1:m, size = round(m/3), replace = FALSE, prob = rep(1/m, m))

iris.learn <- iris[-val,] # train

iris.valid <- iris[val,] # test

iris.kknn <- kknn(Species~., iris.learn, iris.valid, distance = 1, kernel = "triangular") # Possible choices are "rectangular" (which is standard unweighted knn), "triangular", "epanechnikov" (or beta(2,2)), "biweight" (or beta(3,3)), "triweight" (or beta(4,4)), "cos", "inv", "gaussian", "rank" and "optimal".

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names(iris.kknn)• fitted.values Vector of predictions.• CL Matrix of classes of the k nearest neighbors.• W Matrix of weights of the k nearest neighbors.• D Matrix of distances of the k nearest neighbors.• C Matrix of indices of the k nearest neighbors.• prob Matrix of predicted class probabilities.• response Type of response variable, one of

continuous, nominal or ordinal.• distance Parameter of Minkowski distance.• call The matched call.• termsThe 'terms' object used. 8

Look at the output> head(iris.kknn$W)

[,1] [,2] [,3] [,4] [,5] [,6] [,7]

[1,] 0.4493696 0.2306555 0.1261857 0.1230131 0.07914805 0.07610159 0.014184110

[2,] 0.7567298 0.7385966 0.5663245 0.3593925 0.35652546 0.24159191 0.004312408

[3,] 0.5958406 0.2700476 0.2594478 0.2558161 0.09317996 0.09317996 0.042096849

[4,] 0.6022069 0.5193145 0.4229427 0.1607861 0.10804205 0.09637177 0.055297983

[5,] 0.7011985 0.6224216 0.5183945 0.2937705 0.16230921 0.13964231 0.053888244

[6,] 0.5898731 0.5270226 0.3273701 0.1791715 0.15297478 0.08446215 0.010180454

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Look at the output> head(iris.kknn$D)

[,1] [,2] [,3] [,4] [,5] [,6] [,7]

[1,] 0.7259100 1.0142464 1.1519716 1.1561541 1.2139825 1.2179988 1.2996261

[2,] 0.2508639 0.2695631 0.4472127 0.6606040 0.6635606 0.7820818 1.0267680

[3,] 0.6498131 1.1736274 1.1906700 1.1965092 1.4579977 1.4579977 1.5401298

[4,] 0.2695631 0.3257349 0.3910409 0.5686904 0.6044323 0.6123406 0.6401741

[5,] 0.7338183 0.9272845 1.1827617 1.7344095 2.0572618 2.1129288 2.3235298

[6,] 0.5674645 0.6544263 0.9306719 1.1357241 1.1719707 1.2667669 1.3695454

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Look at the output> head(iris.kknn$C)

[,1] [,2] [,3] [,4] [,5] [,6] [,7]

[1,] 86 38 43 73 92 85 60

[2,] 31 20 16 21 24 15 7

[3,] 48 80 44 36 50 63 98

[4,] 4 21 25 6 20 26 1

[5,] 68 79 70 65 87 84 75

[6,] 91 97 100 96 83 93 81

> head(iris.kknn$prob)

setosa versicolor virginica

[1,] 0 0.3377079 0.6622921

[2,] 1 0.0000000 0.0000000

[3,] 0 0.8060743 0.1939257

[4,] 1 0.0000000 0.0000000

[5,] 0 0.0000000 1.0000000

[6,] 0 0.0000000 1.000000011

Look at the output> head(iris.kknn$fitted.values)

[1] virginica setosa versicolor setosa virginica virginica

Levels: setosa versicolor virginica

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Contingency tables

fitiris <- fitted(iris.kknn)

table(iris.valid$Species, fitiris)

fitiris

setosa versicolor virginica

setosa 17 0 0

versicolor 0 18 2

virginica 0 1 12

# rectangular – no weight

fitiris2

setosa versicolor virginica

setosa 17 0 0

versicolor 0 18 2

virginica 0 2 1113

(Weighted) kNN• Advantages

– Robust to noisy training data (especially if we use inverse square of weighted distance as the “distance”)

– Effective if the training data is large

• Disadvantages– Need to determine value of parameter K (number of

nearest neighbors)– Distance based learning is not clear which type of

distance to use and which attribute to use to produce the best results. Shall we use all attributes or certain attributes only?

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Additional factors• Dimensionality – with too many dimensions

the closest neighbors are too far away to be considered close

• Overfitting – does closeness mean right classification (e.g. noise or incorrect data, like wrong street address -> wrong lat/lon) – beware of k=1!

• Correlated features – double weighting• Relative importance – including/ excluding

features15

More factors• Sparseness – the standard distance measure

(Jaccard) loses meaning due to no overlap• Errors – unintentional and intentional• Computational complexity• Sensitivity to distance metrics – especially

due to different scales (recall ages, versus impressions, versus clicks and especially binary values: gender, logged in/not)

• Does not account for changes over time• Model updating as new data comes in 16

Glasslibrary(e1071)

library(rpart)

data(Glass, package="mlbench")

index <- 1:nrow(Glass)

testindex <- sample(index, trunc(length(index)/3))

testset <- Glass[testindex,]

trainset <- Glass[-testindex,]

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Now what?# now what happens?

> rpart.model <- rpart(Type ~ ., data = trainset)

> rpart.pred <- predict(rpart.model, testset[,-10], type = "class”)

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General idea behind trees• Although the basic philosophy of all the classifiers

based on decision trees is identical, there are many possibilities for its construction.

• Among all the key points in the selection of an algorithm to build decision trees some of them should be highlighted for their importance:– Criteria for the choice of feature to be used in each

node– How to calculate the partition of the set of examples– When you decide that a node is a leaf– What is the criterion to select the class to assign to

each leaf19

• Some important advantages can be pointed to the decision trees, including:– Can be applied to any type of data– The final structure of the classifier is quite simple and can

be stored and handled in a graceful manner– Handles very efficiently conditional information,

subdividing the space into sub-spaces that are handled individually

– Reveal normally robust and insensitive to misclassification in the training set

– The resulting trees are usually quite understandable and can be easily used to obtain a better understanding of the phenomenon in question. This is perhaps the most important of all the advantages listed

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Stopping – leaves on the tree• A number of stopping conditions can be used

to stop the recursive process. The algorithm stops when any one of the conditions is true:– All the samples belong to the same class, i.e.

have the same label since the sample is already "pure"

– Stop if most of the points are already of the same class. This is a generalization of the first approach, with some error threshold

– There are no remaining attributes on which the samples may be further partitioned

– There are no samples for the branch test attribute21

Recursive partitioning• Recursive partitioning is a fundamental tool in data mining. It

helps us explore the structure of a set of data, while developing easy to visualize decision rules for predicting a categorical (classification tree) or continuous (regression tree) outcome.

• The rpart programs build classification or regression models of a very general structure using a two stage procedure; the resulting models can be represented as binary trees.

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Recursive partitioning• The tree is built by the following process:

– first the single variable is found which best splits the data into two groups ('best' will be defined later). The data is separated, and then this process is applied separately to each sub-group, and so on recursively until the subgroups either reach a minimum size or until no improvement can be made.

– second stage of the procedure consists of using cross-validation to trim back the full tree.

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Why are we careful doing this?• Because we will USE these trees, i.e. apply

them to make decisions about what things are and what to do with them!

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> printcp(rpart.model)

Classification tree:

rpart(formula = Type ~ ., data = trainset)

Variables actually used in tree construction:

[1] Al Ba Mg RI

Root node error: 92/143 = 0.64336

n= 143

CP nsplit rel error xerror xstd

1 0.206522 0 1.00000 1.00000 0.062262

2 0.195652 1 0.79348 0.92391 0.063822

3 0.050725 2 0.59783 0.63043 0.063822

4 0.043478 5 0.44565 0.64130 0.063990

5 0.032609 6 0.40217 0.57609 0.062777

6 0.010000 7 0.36957 0.51087 0.06105625

plotcp(rpart.model)

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> rsq.rpart(rpart.model)

Classification tree:

rpart(formula = Type ~ ., data = trainset)

Variables actually used in tree construction:

[1] Al Ba Mg RI

Root node error: 92/143 = 0.64336

n= 143

CP nsplit rel error xerror xstd

1 0.206522 0 1.00000 1.00000 0.062262

2 0.195652 1 0.79348 0.92391 0.063822

3 0.050725 2 0.59783 0.63043 0.063822

4 0.043478 5 0.44565 0.64130 0.063990

5 0.032609 6 0.40217 0.57609 0.062777

6 0.010000 7 0.36957 0.51087 0.061056

Warning message:

In rsq.rpart(rpart.model) : may not be applicable for this method 27

rsq.rpart

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> print(rpart.model)

n= 143

node), split, n, loss, yval, (yprob)

* denotes terminal node

1) root 143 92 2 (0.3 0.36 0.091 0.056 0.049 0.15)

2) Ba< 0.335 120 70 2 (0.35 0.42 0.11 0.058 0.058 0.0083)

4) Al< 1.42 71 33 1 (0.54 0.28 0.15 0.014 0.014 0)

8) RI>=1.517075 58 22 1 (0.62 0.28 0.086 0.017 0 0)

16) RI< 1.518015 21 1 1 (0.95 0 0.048 0 0 0) *

17) RI>=1.518015 37 21 1 (0.43 0.43 0.11 0.027 0 0)

34) RI>=1.51895 25 10 1 (0.6 0.2 0.16 0.04 0 0)

68) Mg>=3.415 18 4 1 (0.78 0 0.22 0 0 0) *

69) Mg< 3.415 7 2 2 (0.14 0.71 0 0.14 0 0) *

35) RI< 1.51895 12 1 2 (0.083 0.92 0 0 0 0) *

9) RI< 1.517075 13 7 3 (0.15 0.31 0.46 0 0.077 0) *

5) Al>=1.42 49 19 2 (0.082 0.61 0.041 0.12 0.12 0.02)

10) Mg>=2.62 33 6 2 (0.12 0.82 0.061 0 0 0) *

11) Mg< 2.62 16 10 5 (0 0.19 0 0.37 0.37 0.062) *

3) Ba>=0.335 23 3 7 (0.043 0.043 0 0.043 0 0.87) *

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Tree plot

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plot(object, uniform=FALSE, branch=1, compress=FALSE, nspace, margin=0, minbranch=.3, args)

> plot(rpart.model,compress=TRUE)> text(rpart.model, use.n=TRUE)

And if you are bravesummary(rpart.model)

… pages….

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Remember to LOOK at the data

> names(Glass)

[1] "RI" "Na" "Mg" "Al" "Si" "K" "Ca" "Ba" "Fe" "Type"

> head(Glass)

RI Na Mg Al Si K Ca Ba Fe Type

1 1.52101 13.64 4.49 1.10 71.78 0.06 8.75 0 0.00 1

2 1.51761 13.89 3.60 1.36 72.73 0.48 7.83 0 0.00 1

3 1.51618 13.53 3.55 1.54 72.99 0.39 7.78 0 0.00 1

4 1.51766 13.21 3.69 1.29 72.61 0.57 8.22 0 0.00 1

5 1.51742 13.27 3.62 1.24 73.08 0.55 8.07 0 0.00 1

6 1.51596 12.79 3.61 1.62 72.97 0.64 8.07 0 0.26 1

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rpart.pred> rpart.pred

91 163 138 135 172 20 1 148 169 206 126 157 107 39 150 203 151 110 73 104 85 93 144 160 145 89 204 7 92 51

1 1 2 1 5 2 1 1 5 7 2 1 7 1 1 7 2 2 2 1 2 2 2 2 1 2 7 1 2 1

186 14 190 56 184 82 125 12 168 175 159 36 117 114 154 62 139 5 18 98 27 183 42 66 155 180 71 83 123 11

7 1 7 2 2 2 1 1 5 5 2 1 1 1 1 7 2 1 1 1 1 5 1 1 1 5 2 1 2 2

195 101 136 45 130 6 72 87 173 121 3

7 2 1 1 5 2 1 2 5 1 2

Levels: 1 2 3 5 6 7

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plot(rpart.pred)

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Cross-validation• Cross-validation is a model validation

technique for assessing how the results of a statistical analysis will generalize to an independent data set.

• It is mainly used in settings where the goal is prediction, and one wants to estimate how accurately a predictive model will perform in practice.

• I.e. predictive and prescriptive analytics…35

Cross-validationIn a prediction problem, a model is usually given a dataset of known data on which training is run (training dataset), and a dataset of unknown data (or first seen data) against which the model is tested (testing dataset).

Sound familiar?

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Cross-validationThe goal of cross validation is to define a dataset to "test" the model in the training phase (i.e., the validation dataset), in order to limit problems like overfitting

And, give an insight on how the model will generalize to an independent data set (i.e., an unknown dataset, for instance from a real problem), etc.

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Common type of x-validation• K-fold• 2-fold (do you know this one?)• Rep-random-subsample• Leave out-subsample

• Lab in a few weeks … to try these out

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K-fold• Original sample is randomly partitioned into k

equal size subsamples. • Of the k subsamples, a single subsample is

retained as the validation data for testing the model, and the remaining k − 1 subsamples are used as training data.

• Repeat cross-validation process k times (folds), with each of the k subsamples used exactly once as the validation data. – The k results from the folds can then be

averaged (usually) to produce a single estimation.

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Leave out subsample• As the name suggests, leave-one-out cross-

validation (LOOCV) involves using a single observation from the original sample as the validation data, and the remaining observations as the training data.

• i.e. K=n-fold cross-validation

• Leave out > 1 = bootstraping and jackknifing

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boot(strapping)• Generate replicates of a statistic applied to

data (parametric and nonparametric).– nonparametric bootstrap, possible methods:

ordinary bootstrap, the balanced bootstrap, antithetic resampling, and permutation.

• For nonparametric multi-sample problems stratified resampling is used: – this is specified by including a vector of strata in

the call to boot. – importance resampling weights may be specified.

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Jackknifing• Systematically recompute the statistic

estimate, leaving out one or more observations at a time from the sample set

• From this new set of replicates of the statistic, an estimate for the bias and an estimate for the variance of the statistic can be calculated.

• Often use log(variance) [instead of variance] especially for non-normal distributions

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Repeat-random-subsample• Random split of the dataset into training and

validation data. – For each such split, the model is fit to the training

data, and predictive accuracy is assessed using the validation data.

• Results are then averaged over the splits.

• Note: for this method can the results will vary if the analysis is repeated with different random splits.

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Advantage?• The advantage of K-fold over repeated

random sub-sampling is that all observations are used for both training and validation, and each observation is used for validation exactly once. – 10-fold cross-validation is commonly used

• The advantage of rep-random over k-fold cross validation is that the proportion of the training/validation split is not dependent on the number of iterations (folds). 44

Disadvantage• The disadvantage of rep-random is that some

observations may never be selected in the validation subsample, whereas others may be selected more than once. – i.e., validation subsets may overlap.

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New dataset to work with treesfitK <- rpart(Kyphosis ~ Age + Number + Start, method="class", data=kyphosis)

printcp(fitK) # display the results

plotcp(fitK) # visualize cross-validation results

summary(fitK) # detailed summary of splits

# plot tree

plot(fitK, uniform=TRUE, main="Classification Tree for Kyphosis")

text(fitK, use.n=TRUE, all=TRUE, cex=.8)

# create attractive postscript plot of tree

post(fitK, file = “kyphosistree.ps", title = "Classification Tree for Kyphosis") # might need to convert to PDF (distill)

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> pfitK<- prune(fitK, cp= fitK$cptable[which.min(fitK$cptable[,"xerror"]),"CP"])> plot(pfitK, uniform=TRUE, main="Pruned Classification Tree for Kyphosis")> text(pfitK, use.n=TRUE, all=TRUE, cex=.8)> post(pfitK, file = “ptree.ps", title = "Pruned Classification Tree for Kyphosis”)

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> fitK <- ctree(Kyphosis ~ Age + Number + Start, data=kyphosis)> plot(fitK, main="Conditional Inference Tree for Kyphosis”)

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> plot(fitK, main="Conditional Inference Tree for Kyphosis",type="simple")

Trees for the Titanicdata(Titanic)

rpart, ctree, hclust, etc. for Survived ~ .

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randomForest> require(randomForest)

> fitKF <- randomForest(Kyphosis ~ Age + Number + Start, data=kyphosis)

> print(fitKF) # view results

Call:

randomForest(formula = Kyphosis ~ Age + Number + Start, data = kyphosis)

Type of random forest: classification

Number of trees: 500

No. of variables tried at each split: 1

OOB estimate of error rate: 20.99%

Confusion matrix:

absent present class.error

absent 59 5 0.0781250

present 12 5 0.7058824

> importance(fitKF) # importance of each predictor

MeanDecreaseGini

Age 8.654112

Number 5.584019

Start 10.168591 52

Random forests improve predictive accuracy by generating a large number of bootstrapped trees (based on random samples of variables), classifying a case using each tree in this new "forest", and deciding a final predicted outcome by combining the results across all of the trees (an average in regression, a majority vote in classification).

More on another dataset.# Regression Tree Example

library(rpart)

# build the tree

fitM <- rpart(Mileage~Price + Country + Reliability + Type, method="anova", data=cu.summary)

printcp(fitM) # display the results….

Root node error: 1354.6/60 = 22.576

n=60 (57 observations deleted due to missingness)

CP nsplit rel error xerror xstd

1 0.622885 0 1.00000 1.03165 0.176920

2 0.132061 1 0.37711 0.51693 0.102454

3 0.025441 2 0.24505 0.36063 0.079819

4 0.011604 3 0.21961 0.34878 0.080273

5 0.010000 4 0.20801 0.36392 0.075650 53

Mileage…plotcp(fitM) # visualize cross-validation results

summary(fitM) # detailed summary of splits

<we will look more at this in a future lab> 54

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par(mfrow=c(1,2)) rsq.rpart(fitM) # visualize cross-validation results

# plot tree

plot(fitM, uniform=TRUE, main="Regression Tree for Mileage ")

text(fitM, use.n=TRUE, all=TRUE, cex=.8)

# prune the tree

pfitM<- prune(fitM, cp=0.01160389) # from cptable

# plot the pruned tree

plot(pfitM, uniform=TRUE, main="Pruned Regression Tree for Mileage")

text(pfitM, use.n=TRUE, all=TRUE, cex=.8)

post(pfitM, file = ”ptree2.ps", title = "Pruned Regression Tree for Mileage”)

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# Conditional Inference Tree for Mileage

fit2M <- ctree(Mileage~Price + Country + Reliability + Type, data=na.omit(cu.summary))

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Enough of trees!

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

metaPCA (in R)• Singular Value Decomposition• Feature selection, reduction• Built into a lot of 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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Simple example

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More dimensions

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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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On your own…library(EDR) # effective dimension reduction

library(dr)

library(clustrd)

install.packages("edrGraphicalTools")

library(edrGraphicalTools)

demo(edr_ex1)

demo(edr_ex2)

demo(edr_ex3)

demo(edr_ex4)64

Some examples• Lab8b_dr1_2015.R• Lab8b_dr2_2015.R• Lab8b_dr3_2015.R• Lab8b_dr4_2015.R

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Multidimensional Scaling• Visual representation ~ 2-D plot - patterns of

proximity in a lower dimensional space• "Similar" to PCA/DR but uses dissimilarity as input -

> dissimilarity matrix– An MDS algorithm aims to place each object in N-

dimensional space such that the between-object distances are preserved as well as possible.

– Each object is then assigned coordinates in each of the N dimensions.

– The number of dimensions of an MDS plot N can exceed 2 and is specified a priori.

– Choosing N=2 optimizes the object locations for a two-dimensional scatterplot 66

Four types of MDS• Classical multidimensional scaling

– Also known as Principal Coordinates Analysis, Torgerson Scaling or Torgerson–Gower scaling. Takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain.

• Metric multidimensional scaling– A superset of classical MDS that generalizes the

optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress, which is often minimized using a procedure called stress majorization.

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Four types of MDS ctd• Non-metric multidimensional scaling

– In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. The relationship is typically found using isotonic regression.

• Generalized multidimensional scaling– An extension of metric multidimensional scaling, in which

the target space is an arbitrary smooth non-Euclidean space. In cases where the dissimilarities are distances on a surface and the target space is another surface, GMDS allows finding the minimum-distortion embedding of one surface into another. 68

In Rfunction (library)• cmdscale() (stats)• smacofSym() (smacof)• wcmdscale() (vegan)• pco() (ecodist)• pco() (labdsv)• pcoa() (ape)

• Only stats is loaded by default, and the rest are not installed by default 69

cmdscale()cmdscale(d, k = 2, eig = FALSE, add = FALSE, x.ret = FALSE)

d - a distance structure such as that returned by dist or a full symmetric matrix containing the dissimilarities.

k - the maximum dimension of the space which the data are to be represented in; must be in {1, 2, …, n-1}.

eig - indicates whether eigenvalues should be returned.

add - logical indicating if an additive constant c* should be computed, and added to the non-diagonal dissimilarities such that the modified dissimilarities are Euclidean.

x.ret - indicates whether the doubly centred symmetric distance matrix should be returned.

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Distances between Australian cities

row.names(dist.au) <- dist.au[, 1]

dist.au <- dist.au[, -1]

dist.au

## A AS B D H M P S

## A 0 1328 1600 2616 1161 653 2130 1161

## AS 1328 0 1962 1289 2463 1889 1991 2026

## B 1600 1962 0 2846 1788 1374 3604 732

## D 2616 1289 2846 0 3734 3146 2652 3146

## H 1161 2463 1788 3734 0 598 3008 1057

## M 653 1889 1374 3146 598 0 2720 713

## P 2130 1991 3604 2652 3008 2720 0 3288

## S 1161 2026 732 3146 1057 713 3288 071

Distances between Australian cities

fit <- cmdscale(dist.au, eig = TRUE, k = 2)

x <- fit$points[, 1]

y <- fit$points[, 2]

plot(x, y, pch = 19, xlim = range(x) + c(0, 600))

city.names <- c("Adelaide", "Alice Springs", "Brisbane", "Darwin", "Hobart",

"Melbourne", "Perth", "Sydney")

text(x, y, pos = 4, labels = city.names)

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R – many ways (of course)library(igraph)

g <- graph.full(nrow(dist.au))

V(g)$label <- city.names

layout <- layout.mds(g, dist = as.matrix(dist.au))

plot(g, layout = layout, vertex.size = 3)

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On your own• Lab8b_mds1_2015.R• Lab8b_mds2_2015.R• Lab8b_mds3_2015.R

• http://www.statmethods.net/advstats/mds.html

• http://gastonsanchez.com/blog/how-to/2013/01/23/MDS-in-R.html

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Next week…• A5 – oral presentations on Tues (maybe Frid)• Rest of Friday – lab

• Next week - Spring break

• March 31 – SVM• April 3 – SVM and lab

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