Mahalanobis distance is also called quadratic distance. It measures the separation of two groups of objects. Suppose we have two groups with means and , Mahalanobis distance is given by the following
The data of the two groups must have the same number of variables (the same number of columns) but not necessarily to have the same number of data (each group may have different number of rows).
In Matlab, the code is as follow:
function d=MahalanobisDistance(A, B)
% Return mahalanobis distance of two data matrices
% A and B (row = object, column = feature)
% @author: Kardi Teknomo
disp('number of columns of A and B must be the same')
xDiff=mean(A)-mean(B); % mean diff row vector
pC=n1/n*cA+n2/n*cB; % pooled covariance matrix
d=sqrt(xDiff*inv(pC)*xDiff'); % mahalanobis distance
The code above requires computation of Covariance matrix, which code is given below
% Return covariance given data matrix X (row = object, column = feature)
% @author: Kardi Teknomo
Xc=X-repmat(mean(X),n,1); % centered data
C=Xc'*Xc/n; % covariance
For example: Suppose we have two groups of data, each of group consists of two variables (x, y). The scattered plot of data is shown below.
First, we center the data on the arithmetic mean of each variable.
Covariance matrix of group is computed using centered data matrix
It produces covariance matrices for group 1 and 2 as follow
The pooled covariance matrix of the two groups is computed as weighted average of the covariance matrices. The weighted average takes this form
The pooled covariance is computed using weighted average (10/15)*Covariance group 1 + (5/15)*Covariance group 2 yields
The Mahalanobis distance is simply quadratic multiplication of mean difference and inverse of pooled covariance matrix.
To perform the quadratic multiplication, check again the formula of Mahalanobis distance above. When you get mean difference, transpose it, and multiply it by inverse pooled covariance. After that, multiply the result with the mean difference again and you take the square root. The final result of Mahalanobis distance is
Spreadsheet example (MS Excel) of this Mahalanobis computation can be downloaded here.
How to use the program:
Input are two matrices name matrix A and matrix B that represent features coordinates of two objects. The columns indicate the features, and the rows are the observations. The number of features of the two objects must be equal (i.e. columns of matrix A = columns of matrix B). Each matrix should have at least 2 rows and 1 column.
A matrix is sequence of numbers in a tabular format, inputted using the following format:
- each number in a row is separated by a comma or a space
- each row is separated by semicolon ;
Validate your input before running the program. The initial input values are the example.
Refresh your browser to get back the example.
This program is presented by © Kardi Teknomo
Samples of Applications of Mahalanobis Distance
- Mahalanobis distances in habitat selection studies
- Mahalanobis distance for skin color range for face detection
- Mahalanobis distance for observational epidemiology, health promotion and social determinants
- Mahalanobis Distance for Leadership and Education Science
- Mahalanobis Distance for fuzzy classifier
- Mahalanobis Distance for Classifiers
- Mahalanobis distance for classifying species
- Mahalanobis distance for computer vision
- Mahalanobis distance in robotic
This tutorial is copyrighted.
Preferable reference for this tutorial is
Teknomo, Kardi (2015) Similarity Measurement. http:\people.revoledu.comkardi tutorialSimilarity