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Distance correlation ranges from 0 to 1, where 0 implies independence between X & Y and 1 implies that the linear subspaces of X & Y are equal.

Correlation distance is a popular way of measuring the distance between two random variables with finite variances¹. If the correlation² between two random variables is r, then their correlation distance is defined as d=1-r.

Correlation-based distance considers two objects to be similar if their features are highly correlated, even though the observed values may be far apart in terms of Euclidean distance. The distance between two objects is 0 when they are perfectly correlated. Pearson's correlation is quite sensitive to outliers.

Whereas euclidean distance was the sum of squared differences, correlation is basically the average product. There is a further relationship between the two. If we expand the formula for euclidean distance, we get this: But if X and Y are standardized, the sums Σx2 and Σy2 are both equal to n.

Whereas euclidean distance was the sum of squared differences, correlation is basically the average product. There is a further relationship between the two. If we expand the formula for euclidean distance, we get this: But if X and Y are standardized, the sums Σx2 and Σy2 are both equal to n.

Euclidean distance is the shortest path between source and destination which is a straight line as shown in Figure 1.3. but Manhattan distance is sum of all the real distances between source(s) and destination(d) and each distance are always the straight lines as shown in Figure 1.4.

Euclidean distance is the most widely used distance metric in KNN classifications, however, only few studies examined the effect of different distance metrics on the performance of KNN, these used a small number of distances, a small number of datasets, or both.

That is, let (4) dCov α 2 ( X , Y ) = E ( | X − X ′ | α | Y − Y ′ | α ) + E ( | X − X ′ | α ) E ( | Y − Y ′ | α ) − 2 E ( | X − X ′ | α | Y − Y ′ ′ | α ) . By this definition, the -distance correlation coefficient can be expressed in the usual way, as dCor α ( X , Y ) = dCov α ( X , Y ) dCov α ( X , X ) dCov α ( Y , Y ...