非参数检验及matlab实现

1、非参数检验及 matlab实现Kolmogorov-Smirnov test检验两个样本是否有相同分布KstestTest Statistics : max ( F (x) G (x)h,p,ksstat,cv = kstest(x,CDF,alpha,type)x：被测试的数据样本，以列向量输入(continuous distributiondefined by cumulative distribution function )CDF:被检验的样本 cumulative distribution function ,缺省值为 N(0,1)Alpha：显著性水平，缺省时为 0.05Type

2、:字符输入。'unequal'(缺省值)检验两者分布是否相同'larger'检验x的CDF大于给定的CDF'smaller'检验x的CDF小于给定的CDFh h=0不拒绝原假设，即两个分布相同h=1拒绝原假设，即两个分布不同p：拒绝原假设的最小显著性水平ksstat ：假设为真时，满足 student分布cv: critical value/cutoff value , determining if ksstat is significant.Kstest2 :h,p,ks2stat = kstest2(x1,x2,alpha,type)详见ke

3、testLilliefors test检验两个样本是否有相同分布Test statistics : max(|f(x)-g(x)| )2-sided goodness-of-fit testlillietesth,p,kstat,critval = lillietest (x,alpha,distr,mctol)各参数参见kstest,特别的，mctol为使用蒙特卡洛方法计算p值Jarque-Bera test检验样本是否来自均值和方差未知的正态分布two-sided goodness-of-fit test假设：x为正态分布1 2、test statistic : jb =- s2 +(k

4、-3), where n is the sample size, s is61 4 ；the sample skewness, and k is the sample kurtosis.jbtesth,p,jbstat,critval = jbtest(x,alpha,mctol)各个参数意义详见lillietest 。Wilcoxon-Mann-Whitney Ranks Test检验两个样本是否来自于同一分布。以下 function f=Wilcoxon_Rank_Test(x,y,alpha) 直接是秩和检验方法， 另有function f=Pre_Wilcoxon(x,y,alpha)

5、是改进的秩和检验方法。区别 在于样本的处理方式。function f=Wilcoxon_Rank_Test(x,y,alpha);%x,y are vectors%x,y can be at different lengthZ=x;y;%z1 for ranking from small to large%z2 for placez1,z2=sort(Z, 'ascend' );for i=1:size(x,1)for j=i:size(z1,1)if z1(j)=x(i)X(i,1)=j;endendend%X is the rank of x in vectorm=size

6、(x,1);n=size(y,1);Ex=m*(m+n+1)/2;Varx=m*n*(m+n+1)/12;c=sqrt(Varx)*norminv(1-alpha/2);beta=sum(X,1);if abs(beta-Ex)>cdisp( 'Reject Null Hypothesis' );f=1;elsedisp('Fail to Reject Null Hypothesis' );f=0;end function f=Pre_Wilcoxon(x,y,alpha);%to find whether division is neededmemo=z

7、eros(size(x,1),size(y,1);for i=1:size(x,1)for j=1:size(y,1)if x(i,1)=y(j,1)memo(i,j)=1;endendend%memo contains the information of division;%memo is a symmetric matrix;if memo=zeros(size(memo)disp( 'No need for division' );disp('Go on Rank Test straightly' );disp('Reault as Follow

8、ing' ,'significant level' ,num2str(alpha);Wilcoxon_Rank_Test(x,y,alpha);elsedisp('Division is needed' );cx,cy=find(memo>0);if (cx(1,1)=1)&&(cy(1,1)=1)disp('group 1' );gamma=Wilcoxon_Rank_Test(x(1:cx(1),y(1:cy(1),alpha);if gamma=1;returnendfor k=1:size(cx,1)disp

9、('group ' ,num2str(k+1);gamma=Wilcoxon_Rank_Test(x(cx(k):cx(k+1),y(cy(k):cy(k+1),alpha );if gamma=1;breakendendelseif (cx(1)=1)&&(cy(1)=1)|(cx(1)=1)&&(cy(1)=1)gamma=0;disp( 'Reject Null Hypothesis' );end%(cx(1)=1)&&(cy(1)=1)for k=1:size(cx,1)disp('group &#

10、39; ,num2str(k+1);gamma=Wilcoxon_Rank_Test(x(cx(k):cx(k+1),y(cy(k):cy(k+1),alpha );if gamma=1;breakendendendend另外matlab中也有相关的自带的秩和检验函数:signrank/signtestp = signrank(x,y)为双侧检验，检验x, y是否来自中位数为0的整体;相似的，有 p = signtest(x,y)ranksump = ranksum(x,y)为双侧检验，检验x, y之间是否相互独立，要求x, y之间有相同的中位数Chi-square goodness-of-f

11、it test离散态的检验样本是否来自于已知分布。N2Test statistics : 片=£(Oi -Ei), where O i are the observed counts i aEiand E i are the expected counts. The statistic has an approximate chisquare distribution when the counts are sufficiently large.chi2gofh = chi2gof(x)H0 :x(列向量)来自正态分布(期望和方差来源于x) (defaultsignificance

12、level: 0.05 )h=0 fail to reject H 0h=1 rejectH 0或者h,p = chi2gof(x,'cdf',normcdf,mean(x),std(x)关于样本是否来自对称的分布function f=symmetric_test(x,alpha) %x is a vector%alpha is the significant level x1,x2=sort(x, 'ascend' );%x1 is x from min to max%x2 is the placen=size(x,1);x_ME=median(x,1); %median in column x_IQR=iqr