Mahalanobia Distance(马氏距离）的解释

马氏距离有多重定义：

1）可以表示 某一个样本与DataSet的距离。

2）可以表示两个DataSet之间的距离。

1） The Mahalanobis distance of an observation {\displaystyle {\vec {x}}=(x_{1},x_{2},x_{3},\dots ,x_{N})^{T}} from a set of observations with mean {\displaystyle {\vec {\mu }}=(\mu _{1},\mu _{2},\mu _{3},\dots ,\mu _{N})^{T}} and covariance matrix S is defined as:

Intuitive explanation

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 {\displaystyle {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(圆） 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 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.

2）Mahalanobis distance can also be defined as a dissimilarity measure between two random vectors {\displaystyle {\underline {x}}} and {\displaystyle {\underline {y}}} of the same distribution with the covariance matrix S:

{\displaystyle 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 standardized Euclidean distance:

{\displaystyle d({\vec {x}},{\vec {y}})={\sqrt {\sum _{i=1}^{N}{(x_{i}-y_{i})^{2} \over s_{i}^{2}}}},}

where s_i is the standard deviation of the x_i and y_i over the sample set.

References:

http://people.revoledu.com/kardi/tutorial/Similarity/MahalanobisDistance.html

https://en.wikipedia.org/wiki/Mahalanobis_distance