判别分析时，通常涉及到计算两个样本之间的距离，多元统计学理论中有多种距离计算公式。MATLAB中已有对应函数，可方便直接调用计算。距离函数有：pdist, pdist2, mahal, squareform, mdscale, cmdscale

 主要介绍pdist2 ,其它可参考matlab help

D = pdist2(X,Y)D = pdist2(X,Y,distance)D = pdist2(X,Y,'minkowski',P)D = pdist2(X,Y,'mahalanobis',C)D = pdist2(X,Y,distance,'Smallest',K)D = pdist2(X,Y,distance,'Largest',K)[D,I] = pdist2(X,Y,distance,'Smallest',K)[D,I] = pdist2(X,Y,distance,'Largest',K)

练习：

2种计算方式，一种直接利用pdist计算，另一种按公式（见最后理论）直接计算。

% distance

clc;clear;x = rand(4,3)y = rand(1,3)

for i =1:size(x,1)    for j =1:size(y,1)        a = x(i,:); b=y(j,:);       %         Euclidean distance        d1(i,j)=sqrt((a-b)*(a-b)');       %         Standardized Euclidean distance        V = diag(1./std(x).^2);        d2(i,j)=sqrt((a-b)*V*(a-b)');       %         Mahalanobis distance        C = cov(x);        d3(i,j)=sqrt((a-b)*pinv(C)*(a-b)');       %         City block metric        d4(i,j)=sum(abs(a-b));       %         Minkowski metric        p=3;        d5(i,j)=(sum(abs(a-b).^p))^(1/p);       %         Chebychev distance        d6(i,j)=max(abs(a-b));       %         Cosine distance        d7(i,j)=1-(a*b')/sqrt(a*a'*b*b');       %         Correlation distance        ac = a-mean(a); bc = b-mean(b);               d8(i,j)=1- ac*bc'/(sqrt(sum(ac.^2))*sqrt(sum(bc.^2)));

    endend

md1 = pdist2(x,y,'Euclidean');

md2 = pdist2(x,y,'seuclidean');

md3 = pdist2(x,y,'mahalanobis');

md4 = pdist2(x,y,'cityblock');

md5 = pdist2(x,y,'minkowski',p);

md6 = pdist2(x,y,'chebychev');

md7 = pdist2(x,y,'cosine');

md8 = pdist2(x,y,'correlation');

md9 = pdist2(x,y,'hamming');

md10 = pdist2(x,y,'jaccard');md11 = pdist2(x,y,'spearman');

D1=[d1,md1],D2=[d2,md2],D3=[d3,md3]

D4=[d4,md4],D5=[d5,md5],D6=[d6,md6]

D7=[d7,md7],D8=[d8,md8]

md9,md10,md11

运行结果如下：

x =

    0.5225    0.6382    0.6837    0.3972    0.5454    0.2888    0.8135    0.0440    0.0690    0.6608    0.5943    0.8384

y =

    0.5898    0.7848    0.4977

D1 =

    0.2462    0.2462    0.3716    0.3716    0.8848    0.8848    0.3967    0.3967

D2 =

    0.8355    0.8355    1.5003    1.5003    3.1915    3.1915    1.2483    1.2483

D3 =

  439.5074  439.5074  437.5606  437.5606  438.3339  438.3339  437.2702  437.2702

D4 =

    0.3999    0.3999    0.6410    0.6410    1.3934    1.3934    0.6021    0.6021

D5 =

    0.2147    0.2147    0.3107    0.3107    0.7919    0.7919    0.3603    0.3603

D6 =

    0.1860    0.1860    0.2395    0.2395    0.7409    0.7409    0.3406    0.3406

D7 =

    0.0253    0.0253    0.0022    0.0022    0.3904    0.3904    0.0531    0.0531

D8 =

    1.0731    1.0731    0.0066    0.0066    1.2308    1.2308    1.8954    1.8954

md9 =

     1     1     1     1

md10 =

     1     1     1     1

md11 =

    1.5000    0.0000    1.5000    2.0000

 基本理论公式如下：