總算知道【馬氏距離】的意義，
加上實際操作MATLAB來驗證理論，
真正瞭然於心。
話說MATLAB真是一個強大的工具呢！
Mahalanobis distance
From WiKi (http://en.wikipedia.org/wiki/Mahalanobis_distance)
In statistics, Mahalanobis distance is a distance measure introduced by P. C. Mahalanobis in 1936. It is based on correlations between variables by which different patterns can be identified and analyzed. It is a useful way of determining 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 is a multivariate effect size.
Definition
Formally, the Mahalanobis distance of a multivariate vector
from a group of values with mean
and covariance matrix S is defined as:
Test in MATLAB
#data
>> m %(covariance matrix)
m =
1.2000 0.4000
0.4000 1.8000
>> m1 %(mean vector1)
m1 =
0.1000
0.1000
>> m2 %(mean vector2)
m2 =
2.1000
1.9000
>> m3 %(mean vector3)
m3 =
1.5000
2.0000
>> av %(feature vector a)
av =
1.6000
1.5000
>> bv %(feature vector b)
bv =
1
1
>> cv %(feature vector c)
cv =
2
0
#result
>> a1=(av-m1)’*inv(m)*(av-m1)
%(compute the Mahalanobis distances from av to m1)
a1 =
2.3610
>> a2=(av-m2)’*inv(m)*(av-m2)
%(compute the Mahalanobis distances from av to m2)
a2 =
0.2410
>> a3=(av-m3)’*inv(m)*(av-m3)
%(compute the Mahalanobis distances from av to m3)
a3 =
0.1790
>> b1=(bv-m1)’*inv(m)*(bv-m1)
%(compute the Mahalanobis distances from bv to m1)
b1 =
0.8910
>> b2=(bv-m2)’*inv(m)*(bv-m2)
%(compute the Mahalanobis distances from bv to m2)
b2 =
1.1790
>> b3=(bv-m3)’*inv(m)*(bv-m3)
%(compute the Mahalanobis distances from bv to m3)
b3 =
0.6250
>> c1=(cv-m1)’*inv(m)*(cv-m1)
%(compute the Mahalanobis distances from cv to m1)
c1 =
3.3310
>> c2=(cv-m2)’*inv(m)*(cv-m2)
%(compute the Mahalanobis distances from cv to m2)
c2 =
2.0990
>> c3=(cv-m3)’*inv(m)*(cv-m3)
%(compute the Mahalanobis distances from cv to m3)
c3 =
3.0250
總結：有三個feature vector，分別為a、b、c，各去計算三個mean vector，紅色變數就是D(m)2，取結果為小者，亮紅色數字即答案（a3、b3、c2），表示a、b、c的類別分別為3、3、2。
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Comments on: "[MatLab] 馬氏距離的理論與實做 (Mahalanobis Distance)" (2)
[…] 至於Mahalanobis Distance和Discriminant Function定義可參考上一篇。 […]
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