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two data points x and y whose attributes are all numeric, the Euclidean distance kx − yk is a popular

measurement. To be consistent with the textbook, we assume a vector x by default is a row vector.

However, Euclidean distance is not effective if

• the ranges (scales) of attributes are different, or

• there exist correlations among attributes.

Mahalanobis distance has been introduced to address the above problems. Given a dataset whose

attributes are all numeric, we first construct the the covariance matrix Σ = (si,j )n×n, where an element

si,j is the covariance of i-th and j-th attributes, and n is the number of attributes. The Mahalanobis

distance of x and y is then defined as

M ahalanobis(x, y) = (x − y)Σ−(x − y)



where Σ

− is the inverse of the matrix Σ, and (x − y)


is the transpose of the row vector x − y.

In the part, you will first implement the above two distance measurements, and then give an experimental

evaluation of these measurements. Usually, evaluation is performed in a specific data mining


tasks, such as classification, clustering. Since we currently have not yet got into these topics, we will

simply check the consistency between two measurements based on random datasets. Specifically, you

will need to

1. generate a random dataset of two-attribute instances in Gaussian distribution

2. build the covariance matrix Σ

3. set counter = 0

4. for each instance,

(a) Compute the nearest neighbor using Euclidean distance

(b) Compute the nearest neighbor using Mahalanobis distance

(c) if the nearest neighbors are same, count++

5. output the consistency ratio (count / number-of-instances)

You will need to read the documentation of the packages [url removed, login to view] and [url removed, login to view], and

typically the class Matrix to complete this part. To generate data in Gaussian distribution, you may

refer to the standard Java class Random, and use the Java method nextGaussian().

Skills: Data Mining, Java

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