Question

Describe different types of Distance metrics for measuring similarity between data points.

Note: The subject is Advanced Data Mining

Answer 1

Measures of Similarity and Dissimilarity:

Distance or similarity measures are essential in solving many pattern recognition problems such as classification and clustering. Various distance/similarity measures are available in the literature to compare two data distributions. As the names suggest, a similarity measures how close two distributions are. For multivariate data complex summary methods are developed to answer this question

Similarity Measure

Numerical measure of how alike two data objects often fall between 0 (no similarity) and 1 (complete similarity)

Dissimilarity Measure

Numerical measure of how different two data objects are range from 0 (objects are alike) to ∞ (objects are different)

Proximity

refers to a similarity or dissimilarity

Similarity/Dissimilarity for Simple Attributes Here, p and q are the attribute values for two data objects. Attribute Type Similarity if p = 9 Nominal S Dissimilarity _ So ifp=2 " 1 if p 79 $90 if p 79 Ordinal d - lp - 911 S=1_llp-911 n - 1 n-1 (values mapped to integer 0 to n-1, where n is the number of values) s = 1 - ||p-9||.s= 1+ le-all Interval or Ratio d=lp - all Distance, such as the Euclidean distance, is a dissimilarity measure and has some well-known properties: Common Properties of Dissimilarity Measures 1. d(p, q) 2 0 for all p and q, and d(p, q) = 0 if and only if p = 9, 2. d(p, q) = d(q.p) for all p and q, 3. d(p, r) sd(p, q) + d(q, r) for all p, q, and r, where d(p, q) is the distance (dissimilarity) between points (data objects), p and q. A distance that satisfies these properties is called a metric. Following is a list of several common distance measures to compare multivariate data. We will assume that the attributes are all continuous.

Here, some of the different types of Distance Metrics for Measuring Similarity are:

Euclidean Distance Assume that we have measurements Xik, i = 1, ...,N, on variables k = 1, ...,p (also called attributes). The Euclidean distance between the ith and jth objects is deli,j) = (EX=1 (xix – xjx)) for every pair (i, j) of observations. The weighted Euclidean distance is: dụEG.J) =(E2 , w; (x - 1)^) If scales of the attributes differ substantially, standardization is necessary.

Minkowski Distance The Minkowski distance is a generalization of the Euclidean distance. With the measurement, Xik, i = 1, ...,N, k = 1,...,p, the Minkowski distance is duli,j) = (X=1 prik – xjx[") where 2 > 1. It is also called the L, metric. • i=1: L1 metric, Manhattan or City-block distance. • 1 = 2: L2 metric, Euclidean distance. .iņ00: Lo metric, Supremum distance. lim 2 * 00 = (ZE=1 frik – xje!")* = max (fxin – xj1]...., Þrip – xjpl) Note that and p are two different parameters. Dimension of the data matrix remains finite.

Mahalanobis Distance Let X be a N * p matrix. Then the ith row of X is x} = (Xi1 , . . . , Xip) The Mahalanobis distance is duu li.) = ((*– x;)?p-1(x; – x;)) where is the pxp sample covariance matrix.