DM825 - Introduction to Machine Learning
Sheet 14, Spring 2013 [pdf format]

Exercise 1

Do exercises 1, 4, 5 from Exam 2010.

Exercise 2 – Tree based methods

Consider a data set comprising 400 data points from class C₁ and 400 data points from class C₂. Suppose that a tree model A splits these into (300,100) assigned to the first leaf node (predicting C₁ and (100,300) assigned to the second leaf node (predicting C₂, where (n,m) denotes that n points come from class C₁ and m points come from class C₂. Similarly, suppose that a second tree model B splits them into (200,400) and (200,0), respectively. Evaluate the misclassification rates for the two trees and show that they are equal. Similarly, evaluate the pruning criterion for the cross-entropy case for the two trees.

Exercise 3 – Tree based methods

You are given the following data points: Negative: (-1, -1) (2, 1) (2, -1); Positive: (-2, 1) (-1, 1) (1,-1). The points are depicted in Figure 1.

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Figure 1: The data points for classification.

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1. Construct a decision tree using the greedy recursive bi-partitioning algorithm based on information gain described in class. Use both criteria the Gini index and the entropy. In the search for the split threshold θ discretize the continue scale of the two features and consider only values in {−1.5,0,1.5} for f₁ and {0} for f₂. Represent graphically the tree constructed and draw the decision boundaries in the Figure 1. Table 1 might be useful for some computations

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   x	y	−(x/y) · log(x/y)	x	y	−(x/y) · log(x/y)
   1	2	0.50	1	5	0.46
   1	3	0.53	2	5	0.53
   2	3	0.39	3	5	0.44
   1	4	0.50	4	5	0.26
   3	4	0.31

   Table 1: Numerical values for the computation of information gains.

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2. Use the tree to predict the outcome for the new point (1,1).

Exercise 4 – Nearest Neighbor

1. Draw the decision boundaries for 1-Nearest Neighbor on the Figure 1. Make it accurate enough so that it is possible to tell whether the integer-valued coordinate points in the diagram are on the boundary or, if not, which region they are in.
2. What class does 1-NN predict for the new point: (1, 1).
3. What class does 3-NN predict for the new point: (1, 0).

Exercise 5 – Practical

Analyze by means of classification tree the data on spam email from the UCI repository. Use rpart from the rpart package and the ctree from the party package.

Exercise 6 – PCA

Using the iris data readily available in R use principle component analysis to identify two components and plot the data in these components. Can you classify the data at this stage?

Exercise 7 – Probability and Independence

A joint probability table for the binary variables A, B, and C is given below.

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A / B	b1	b2
a1	(0.006, 0.054)	(0.048, 0.432)
a2	(0.014, 0.126)	(0.032, 0.288)

Table 2: Joint probability distribution P (A, B, C)

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• Calculate P (B, C) and P (B).
• Are A and C independent given B? (Remember to report the justification of your answer.)