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)));

end

end

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：