Mahalanobis distance

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The Mahalanobis distance is a descriptive statistic that provides a relative measure of a data point's distance (residual) from a common point. It is a unitless measure introduced by P. C. Mahalanobis in 1936.[1] The Mahalanobis distance is used to identify and gauge similarity of an unknown sample set to a known one. It differs from Euclidean distance in that it takes into account the correlations of the data set and is scale-invariant. In other words, it has a multivariate effect size.

Definition[edit]

The Mahalanobis distance of an observation x = ( x_1, x_2, x_3, \dots, x_N )^T from a group of observations with mean \mu = ( \mu_1, \mu_2, \mu_3, \dots , \mu_N )^T and covariance matrix S is defined as:

D_M(x) = \sqrt{(x - \mu)^T S^{-1} (x-\mu)}.\, [2]

Mahalanobis distance (or "generalized squared interpoint distance" for its squared value[3]) can also be defined as a dissimilarity measure between two random vectors \vec{x} and \vec{y} of the same distribution with the covariance matrix S:

 d(\vec{x},\vec{y})=\sqrt{(\vec{x}-\vec{y})^T S^{-1} (\vec{x}-\vec{y})}.\,

If the covariance matrix is the identity matrix, the Mahalanobis distance reduces to the Euclidean distance. If the covariance matrix is diagonal, then the resulting distance measure is called a normalized Euclidean distance:

 d(\vec{x},\vec{y})=
\sqrt{\sum_{i=1}^N  {(x_i - y_i)^2 \over s_{i}^2}},

where si is the standard deviation of the xi and yi over the sample set.

Intuitive explanation[edit]

Consider the problem of estimating the probability that a test point in N-dimensional Euclidean space belongs to a set, where we are given sample points that definitely belong to that set. Our first step would be to find the average or center of mass of the sample points. Intuitively, the closer the point in question is to this center of mass, the more likely it is to belong to the set.

However, we also need to know if the set is spread out over a large range or a small range, so that we can decide whether a given distance from the center is noteworthy or not. The simplistic approach is to estimate the standard deviation of the distances of the sample points from the center of mass. If the distance between the test point and the center of mass is less than one standard deviation, then we might conclude that it is highly probable that the test point belongs to the set. The further away it is, the more likely that the test point should not be classified as belonging to the set.

This intuitive approach can be made quantitative by defining the normalized distance between the test point and the set to be  {x - \mu} \over \sigma . By plugging this into the normal distribution we can derive the probability of the test point belonging to the set.

The drawback of the above approach was that we assumed that the sample points are distributed about the center of mass in a spherical manner. Were the distribution to be decidedly non-spherical, for instance ellipsoidal, then we would expect the probability of the test point belonging to the set to depend not only on the distance from the center of mass, but also on the direction. In those directions where the ellipsoid has a short axis the test point must be closer, while in those where the axis is long the test point can be further away from the center.

Putting this on a mathematical basis, the ellipsoid that best represents the set's probability distribution can be estimated by building the covariance matrix of the samples. The Mahalanobis distance is simply the distance of the test point from the center of mass divided by the width of the ellipsoid in the direction of the test point.

Another intuitive description of Mahalanobis distance is that it is square root of the negative log likelihood. That is, the exponential of the negative square of the Mahalanobis distance will give you the likelihood of your data point belonging to (a presumed normal) distribution of the sample points you already have.

Discussion[edit]

In general, given a normal (Gaussian) random variable X with variance S=1 and mean \mu = 0, any other normal random variable R can be defined in terms of X by the equation R = \mu_1 + \sqrt{S_1}X. Conversely, to recover a normalized random variable from any normal random variable, one can typically solve for X = (R - \mu_1)/\sqrt{S_1} . If we square both sides, and take the square-root, we will get an equation for a metric that looks a lot like the Mahalanobis distance:

D = \sqrt{X^2} = \sqrt{(R - \mu_1)^2/S_1} = \sqrt{(R - \mu_1) S_1^{-1} (R - \mu_1) }.

The resulting magnitude is always non-negative and varies with the distance of the data from the mean, attributes that are convenient when trying to define a model for the data.

Relationship to leverage[edit]

Mahalanobis distance is closely related to the leverage statistic, h, but has a different scale:[4]

Squared Mahalanobis distance = (N − 1)(h − 1/N).

Applications[edit]

Mahalanobis's discovery was prompted by the problem of identifying the similarities of skulls based on measurements in 1927.[5]

Mahalanobis distance is widely used in cluster analysis and classification techniques. It is closely related to Hotelling's T-square distribution used for multivariate statistical testing and Fisher's Linear Discriminant Analysis that is used for supervised classification.[6]

In order to use the Mahalanobis distance to classify a test point as belonging to one of N classes, one first estimates the covariance matrix of each class, usually based on samples known to belong to each class. Then, given a test sample, one computes the Mahalanobis distance to each class, and classifies the test point as belonging to that class for which the Mahalanobis distance is minimal.

Mahalanobis distance and leverage are often used to detect outliers, especially in the development of linear regression models. A point that has a greater Mahalanobis distance from the rest of the sample population of points is said to have higher leverage since it has a greater influence on the slope or coefficients of the regression equation. Mahalanobis distance is also used to determine multivariate outliers. Regression techniques can be used to determine if a specific case within a sample population is an outlier via the combination of two or more variable scores. A point can be a multivariate outlier even if it is not a univariate outlier on any variable (consider a probability density similar to a hollow cube in three dimensions, for example).

Mahalanobis distance was also widely used in biology, such as predicting protein structural class, [7] predicting membrane protein type,[8] predicting protein subcellular localization,[9]as well as predicting many other attributes of proteins through their pseudo amino acid composition [10] or Chou's PseAAC,[11] based on Chou's invariance theorem, as done in the papers. [12] [13]

See also[edit]

References[edit]

  1. ^ Mahalanobis, Prasanta Chandra (1936). "On the generalised distance in statistics". Proceedings of the National Institute of Sciences of India 2 (1): 49–55. Retrieved 2012-05-03. 
  2. ^ De Maesschalck, Roy; Jouan-Rimbaud, Delphine; and Massart, Désiré L. (2000); The Mahalanobis distance, Chemometrics and Intelligent Laboratory Systems 50:1–18
  3. ^ Gnanadesikan, Ramanathan; and Kettenring, John R. (1972); Robust estimates, residuals, and outlier detection with multiresponse data, Biometrics 28:81–124
  4. ^ Schinka, John A.; Velicer, Wayne F.; and Weiner, Irving B. (2003); Handbook of psychology: Research methods in psychology, John Wiley and Sons
  5. ^ Mahalanobis, Prasanta Chandra (1927); Analysis of race mixture in Bengal, Journal and Proceedings of the Asiatic Society of Bengal, 23:301–333
  6. ^ McLachlan, Geoffrey J. (1992); Discriminant Analysis and Statistical Pattern Recognition, Wiley Interscience, p. 12. ISBN 0-471-69115-1
  7. ^ Chou, Kuo-Chen (April 1995). "A novel approach to predicting protein structural classes in a (20-1)-D amino acid composition space". Proteins 21 (4): 319–44. doi:10.1002/prot.340210406. PMID 7567954. 
  8. ^ Chou, Kuo-Chen; and Elrod, David W. (1999); Prediction of membrane protein types and subcellular locations, Proteins: Structure, Function, and Genetics, 34, 137–153
  9. ^ Chou, Kuo-Chen; and Elrod, David W. (1999); Protein subcellular location prediction, Protein Engineering, 12, 107–118
  10. ^ Chou, Kuo-Chen (May 2001). "Prediction of protein cellular attributes using pseudo-amino acid composition". Proteins 43 (3): 246–55. doi:10.1002/prot.1035. PMID 11288174. 
  11. ^ Lin, Sheng-Xiang; Lapointe, Jacques (2013). "Theoretical and experimental biology in one —A symposium in honour of Professor Kuo-Chen Chou’s 50th anniversary and Professor Richard Giegé’s 40th anniversary of their scientific careers". JBiSE 6: 435–442. doi:10.4236/jbise.2013.64054. 
  12. ^ Pan YX, Zhang ZZ, Guo ZM, Feng GY, Huang ZD, He L (May 2003). "Application of pseudo amino acid composition for predicting protein subcellular location: stochastic signal processing approach". J. Protein Chem. 22 (4): 395–402. doi:10.1023/A:1025350409648. PMID 13678304. 
  13. ^ Zhou, Guo-Ping & Doctor, Kutbuddin (January 2003). "Subcellular location prediction of apoptosis proteins". Proteins 50 (1): 44–8. doi:10.1002/prot.10251. PMID 12471598. 

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