Tree-Based MethodsTree-Based Methods(V&R 9.1)(V&R 9.1)

Demeke Kasaw, Andreas Nguyen, Mariana Alvaro

STAT 6601 Project

Overview of Tree-based Methods

What are they? How do they work? Examples…

Tree pictorials common. Simple way to depict

relationships in data Tree-based methods use

this pictorial to represent relationships between random variables.

Trees can be used for bothClassification and Regression

Presence of Surgery Complicationsvs. Patient Age and Treatment Start Date

|

Start >= 8.5 months

Start >= 14.5

Age < 12 yrs

Sex = F

Start < 8.5

Start < 14.5

Age >= 12 yrs

Sex = M

Absent

Time to Next Eruptionvs. Length of Last Eruption

|Last Eruption < 3.0 min

Last Eruption < 4 .1 min 54.49

76.83 81.18

Absent

Absent Present

Present

General Computation Issues and Unique Solutions

Over-Fitting: When do we stop splitting? Stop generating new nodes when subsequent splits only result in little improvement.

Evaluate the quality of the prediction: Prune the tree to ideally select the simplest most accurate solution.Methods:– Crossvalidation: Apply the tree computed from one set of

observations (learning sample) to another completely independent set of observations (testing sample).

– V-fold crossvalidation: Repeat the analysis with different randomly drawn samples from the data. Use the tree that shows the best average accuracy for cross-validated predicted classifications or predicted values.

Computational Details

Specify the criteria for predictive accuracy– Minimum costs: Lowest misclassification rate– Case weights

Selecting Splits– Define a measure of impurity for a node. A node is “pure” if they

contain observations of a single class. Determine when to stop splitting

– All nodes are pure or contain no more than a n cases– Until all nodes contain no more than a specified Fraction of Objects

Selecting the “right-size” tree– Test sample cross validation– V Fold cross validation– Tree selection after pruning: if there are several trees with costs

close to minimum, select the smallest-sized (least complex)

Computational Formulas

Estimation of Accuracy in Classification Trees– Resubstitution estimate

d(x) is the classifierX=1 if X(d(xn) = jn) is trueX =0 if X(d(xn) = jn) is false

Estimation of Accuracy in Regression Trees– Resubstitution estimate

))((1

)(1∑=

≠=N

inn jxdX

NdR

€

R(d) =1

N(y i − d(x i

i=1

N

∑ ))2

Computational Formulas Estimation of Node Impurity

Gini Index– Reaches zero when only one class is present at a node– P(j/t): probability of category j at node t

Entropy or Information

€

g(t) = p( j /t)p(i /t)j≠i

∑

€

pik * log pik∑

Classification Tree Example:What species are these flowers?

Sepal LengthSepal Width

Petal LengthPetal Width

Versicolor

Virginica

Setosa

tree

Iris Classification Data

Iris dataset relates species to petal and sepal dimensions reported in centimeters. Originally used by R.A. Fisher and E. Anderson for a discriminant analysis example.

Data is pre-packaged in R dataset library and is available on DASYL.

Sepal.Length Sepal.Width Petal.Length Petal.Width Species

6.7 3.0 5.0 1.7 versicolor

5.8 2.7 3.9 1.2 versicolor

7.3 2.9 6.3 1.8 virginica

5.2 4.1 1.5 0.1 setosa

4.4 3.2 1.3 0.2 setosa

Iris ClassificationMethod and Code

library(rpart) # Load tree fitting packagedata(iris) # Load iris data

# Let x = tree object fitting Species vs. all other# variables in iris with 10-fold cross validationx = rpart(Species~.,iris,xval=10)

# Plot tree diagram with uniform spacing,# diagonal branches, a 10% margin, and a titleplot(x, uniform=T, branch=0, margin=0.1, main="Classification Tree\nIris Species by Petal and Sepal Length")

# Add labels to tree with final counts,# fancy shapes, and blue text colortext(x,use.n=T,fancy=T,col="blue")

Results:

Classification TreeIris Species by Petal and Sepal Length

Petal.Length < 2 .45

Petal.Width < 1 .75

Petal.Length >= 2 .45

Petal.Width >= 1 .75

setosa 50/0/0

versicolor0/49/5

virginica 0/1/45

Tree-based approach much simpler than the alternative

Classification with Cross-validation True GroupPut into Group setosa versicolor virginicasetosa 50 0 0versicolor 0 48 1virginica 0 2 49Total N 50 50 50N correct 50 48 49Proportion 1.000 0.960 0.980N = 150 N Correct = 147

Linear Discriminant Function for Groups setosa versicolor virginicaConstant -85.21 -71.75 -103.27Sepal.Length 23.54 15.70 12.45Sepal.Width 23.59 7.07 3.69Petal.Length -16.43 5.21 12.77Petal.Width -17.40 6.43 21.08

Identify this flower…Sepal Length 6Sepal Width 3.4Petal Length 4.5Petal Width 1.6

Setosa -85+24*6+24*3.4-16*4.5-17*1.6=41

Versicolor -72+16*6+7*3.4+5*4.5+6*1.6=80

Virginica -103+12*6+4*3.4+13*4.5+21*1.6=75

Since Versicolor has highest score,we classify this flower as an Iris versicolor.

Classification TreeIris Species by Petal and Sepal Length

PetalLength< 2 .45

PetalWidth< 1 .75

PetalLength>= 2 .45

PetalWidth

>= 1 .75

setosa 50/0/0

versicolor0/49/5

virginica 0/1/45

Regression Tree Example

Software used : R, rpart package

Goal:– Applying the regression tree method on CPU data,

and predicting the response variable, ‘performance’.

CPU Data

CPU performance of 209 different processors.

name syct mmin mmax cach chmin chmax perf

1 ADVISOR 32/60 125 256 6000 256 16 128 198

2 AMDAHL 470V/7 29 8000 32000 32 8 32 269

3 AMDAHL 470/7A 29 8000 32000 32 8 32 220

4 AMDAHL 470V/7B 29 8000 32000 32 8 32 172

5 AMDAHL 470V/7C 29 8000 16000 32 8 16 132

6 AMDAHL 470V/8 26 8000 32000 64 8 32 318

...

Mem

ory (

kb)

System

Spe

ed

(mhz

)

Cache

(kb)

Chann

els

Perfor

mance

Bench

mark

R Code

library(MASS); library(rpart); data(cpus); attach(cpus)

# Fit regression tree to datacpus.rp <-rpart(log(perf)~.,cpus[,2:8],cp=0.001)

# Print and plot complexity Parameter (cp) tableprintcp(cpus.rp); plotcp(cpus.rp)

# Prune and display tree cpus.rp<-prune(cpus.rp,cp=0.0055)plot(cpus.rp,uniform=T,main="Regression Tree")text(cpus.rp,digits=3)

# Plot residual vs. predictedplot(predict(cpus.rp),resid(cpus.rp)); abline(h=0)

Determine the Best Complexity Parameter (cp) Value for the Model

CP nsplit rel error xerror xstd1 0.5492697 0 1.00000 1.00864 0.0968382 0.0893390 1 0.45073 0.47473 0.0482293 0.0876332 2 0.36139 0.46518 0.0467584 0.0328159 3 0.27376 0.33734 0.0328765 0.0269220 4 0.24094 0.32043 0.0315606 0.0185561 5 0.21402 0.30858 0.0301807 0.0167992 6 0.19546 0.28526 0.0280318 0.0157908 7 0.17866 0.27781 0.0276089 0.0094604 9 0.14708 0.27231 0.02878810 0.0054766 10 0.13762 0.25849 0.02697011 0.0052307 11 0.13215 0.24654 0.02629812 0.0043985 12 0.12692 0.24298 0.02717313 0.0022883 13 0.12252 0.24396 0.02702314 0.0022704 14 0.12023 0.24256 0.02706215 0.0014131 15 0.11796 0.24351 0.02724616 0.0010000 16 0.11655 0.24040 0.026926

1 – R2

Cross-Validated

Error

cp

X-v

al

Re

lati

ve

Err

or

0.2

0.4

0.6

0.8

1.0

1.2

Inf 0.03 0.0072 0.0012

1 3 5 7 11 14 17

size of tree

# SplitsComplexityParameter

Cross-Validated Error SD

Regression Tree

Regression TreeBefore Pruning

|cach< 27

mmax< 6100

mmax< 1750

mmax< 2500

chmax< 4.5

syct< 110

syct>=360

chmin< 5.5

cach< 0.5

chmin>=1.5

mmax< 1.4e+04

mmax< 2.8e+04

cach< 96.5

mmax< 1.124e+04

chmax< 14

cach< 56

2.51

3.05

3.12

3.263.54

2.95

3.52

3.89

4.044.31

4.554.21

4.695.14

5.355.226.14

Regression TreeAfter Pruning

|cach< 27

mmax< 6100

mmax< 1750 syct>=360

chmin< 5.5

cach< 0.5

mmax< 2.8e+04

cach< 96.5

mmax< 1.1e+04

cach< 56

2.51 3.292.95

3.52 4.03

4.55

4.21 4.92

5.35

5.22 6.14

How well does it fit?

Plot of residuals3456predict(cpus.rp)

Summary

Advantages of C & RT Simplicity of results:

– The interpretation of results summarized in a tree is very simple.

– This simplicity is useful for purposes of rapid classification of new observations

– It is much easier to evaluate just one or two logical conditions.

Tree methods are nonparametric and nonlinear – There is no implicit assumption that the underlying

relationships between the predictor variables and the dependent variable are linear, follow some specific non-linear link function

References

Venables, Ripley (2002), Modern Applied Statistics with S,251-266. StatSoft (2003) “Classification and Regression Trees”, Electronic

Textbook, StatSoft, 2003, retrieved on 11/8/2004 from http://www.statsoft.com/textbook/stcart.html

Fisher, R. A. (1936) “The use of multiple measurements in taxonomic problems”. Annals of Eugenics, 7, Part II, 179-188.

Using Trees in R (the 30 second version)1)Load the rpart librarylibrary(rpart)

2)For classification trees, make sure the response is of the type factor. If you don’t know how to do this lookup help(as.factor)or consult a general R reference.y=as.factor(y)

3)Fit the tree modelf=rpart(y~x1+x2+…,data=…,cp=0.001)If using an unattached dataframe, you must specify data.If using global variables, then data= can be omitted.A good starting point for cp, which controls the complexity of the tree, is given.

4)Plot and check the modelplot(f,uniform=T,margin=0.1); text(f,use.n=T)plotcp(f); printcp(f)Look at the xerrors in the summary and choose the smallest number of splits that achieve the smallest xerror. Consider the tradeoff between model fit and complexity (ie overfitting). Based on your judgement, repeat step 3 with the cp value of your choice.

5)Predict resultspredict(f,newdata,type=“class”)where newdata is a dataframe with the independent variables.

Using Trees in R (the 30 second version)1)Load the rpart librarylibrary(rpart)

2)For classification trees, make sure the response is of the type factor. If you don’t know how to do this lookup help(as.factor)or consult a general R reference.y=as.factor(y)

3)Fit the tree modelf=rpart(y~x1+x2+…,data=…,cp=0.001)If using an unattached dataframe, you must specify data.If using global variables, then data= can be omitted.A good starting point for cp, which controls the complexity of the tree, is given.

4)Plot and check the modelplot(f,uniform=T,margin=0.1); text(f,use.n=T)plotcp(f); printcp(f)Look at the xerrors in the summary and choose the smallest number of splits that achieve the smallest xerror. Consider the tradeoff between model fit and complexity (ie overfitting). Based on your judgement, repeat step 3 with the cp value of your choice.

5)Predict resultspredict(f,newdata,type=“class”)where newdata is a dataframe with the independent variables.

Using Trees in R (the 30 second version)1)Load the rpart librarylibrary(rpart)

2)For classification trees, make sure the response is of the type factor. If you don’t know how to do this lookup help(as.factor)or consult a general R reference.y=as.factor(y)

3)Fit the tree modelf=rpart(y~x1+x2+…,data=…,cp=0.001)If using an unattached dataframe, you must specify data.If using global variables, then data= can be omitted.A good starting point for cp, which controls the complexity of the tree, is given.

4)Plot and check the modelplot(f,uniform=T,margin=0.1); text(f,use.n=T)plotcp(f); printcp(f)Look at the xerrors in the summary and choose the smallest number of splits that achieve the smallest xerror. Consider the tradeoff between model fit and complexity (ie overfitting). Based on your judgement, repeat step 3 with the cp value of your choice.

5)Predict resultspredict(f,newdata,type=“class”)where newdata is a dataframe with the independent variables.

Using Trees in R (the 30 second version)1)Load the rpart librarylibrary(rpart)

2)For classification trees, make sure the response is of the type factor. If you don’t know how to do this lookup help(as.factor)or consult a general R reference.y=as.factor(y)

3)Fit the tree modelf=rpart(y~x1+x2+…,data=…,cp=0.001)If using an unattached dataframe, you must specify data.If using global variables, then data= can be omitted.A good starting point for cp, which controls the complexity of the tree, is given.

4)Plot and check the modelplot(f,uniform=T,margin=0.1); text(f,use.n=T)plotcp(f); printcp(f)Look at the xerrors in the summary and choose the smallest number of splits that achieve the smallest xerror. Consider the tradeoff between model fit and complexity (ie overfitting). Based on your judgement, repeat step 3 with the cp value of your choice.

5)Predict resultspredict(f,newdata,type=“class”)where newdata is a dataframe with the independent variables.