非参数检验及matlab实现

1、6 / 6非参数检验及matlab实现Kolmogorov-Smirnovtest检验两个样本是否有相同分布KstestTeststatistics:maxF(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(continuousdistributiondefinedbycumulativedistributionfunction)CDF：被检验的样本cumulativedistributionfunction,缺省值为N(0,1)Alpha：显著性水平，缺省时为0.05Type：字符输入。unequal'(缺省

2、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CDFhh=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:criticalvalue/cutoffvalue,determiningifksstatissignificant.Kstest2:h,p,ks2stat=kstest2(x1,x2,alpha,type)详见ketestLillieforstest检验两个样本是否有相同分布Teststatis

3、tics:maxF(x)G(x)x2-sidedgoodness-of-fittestlillietesth,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-Beratest检验样本是否来自均值和方差未知的正态分布two-sidedgoodness-of-fittest假设：x为正态分布2teststatistic：jb-s2k3,wherenisthesamplesize,sis64thesampleskewness,andkisthesamplekurtosis.

4、jbtesth,p,jbstat,critval=jbtest(x,alpha,mctol)各个参数意义详见lillietest。Wilcoxon-Mann-WhitneyRanksTest检验两个样本是否来自于同一分布。以下functionf=Wilcoxon_Rank_Test(x,y,alpha)直接是秩和检验方法，另有functionf=Pre_Wilcoxon(x,y,alpha)是改进的秩和检验方法。区别在于样本的处理方式。functionf=Wilcoxon_Rank_Test(x,y,alpha);%x,yarevectors%x,ycanbeatdifferentlength

5、Z=x;y;%z1forrankingfromsmalltolarge%z2forplacez1,z2=sort(Z,'ascend');fori=1:size(x,1)forj=i:size(z1,1)ifz1(j)=x(i)X(i,1)=j;endendend%Xistherankofxinvectorm=size(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);ifabs(beta-Ex)>cdisp('Reje

6、ctNullHypothesis');f=1;elsedisp('FailtoRejectNullHypothesis');f=0;endfunctionf=Pre_Wilcoxon(x,y,alpha);%tofindwhetherdivisionisneededmemo=zeros(size(x,1),size(y,1);fori=1:size(x,1)forj=1:size(y,1)ifx(i,1)=y(j,1)memo(i,j)=1;endendend%memocontainstheinformationofdivision;%memoisasymmetricm

7、atrix;ifmemo=zeros(size(memo)disp('Noneedfordivision');disp('GoonRankTeststraightly');disp('ReaultasFollowing','significantlevel',num2str(alpha);Wilcoxon_Rank_Test(x,y,alpha);elsedisp('Divisionisneeded');cx,cy=find(memo>0);if(cx(1,1)=1)&&(cy(1,1)=1)

8、disp('group1');gamma=Wilcoxon_Rank_Test(x(1:cx(1),y(1:cy(1),alpha);ifgamma=1;returnendfork=1:size(cx,1)disp('group',num2str(k+1);gamma=Wilcoxon_Rank_Test(x(cx(k):cx(k+1),y(cy(k):cy(k+1),alpha);ifgamma=1;breakendendelseif(cx(1)=1)&&(cy(1)=1)|(cx(1)=1)&&(cy(1)=1)gamma=0

9、;disp('RejectNullHypothesis');end%(cx(1)=1)&&(cy(1)=1)fork=1:size(cx,1)disp('group',num2str(k+1);gamma=Wilcoxon_Rank_Test(x(cx(k):cx(k+1),y(cy(k):cy(k+1),alpha);ifgamma=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=s

10、igntest(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-squaregoodness-of-fittest离散态的检验样本是否来自于已知分布。2NOiEi2Teststatistics:,whereOiaretheobservedcountsi1EiandEiaretheexpectedcounts.Thestatistichasanapproximatechisquaredistributionwhenthecountsaresufficientlylarge.chi2gofh=chi2gof(x)H0:x

11、(列向量)来自正态分布(期望和方差来源于x)(defaultsignificancelevel:0.05)h=0failtorejectH0h=1rejectH0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functionf=symmetric_test(x,alpha)%xisavector%alphaisthesignificantlevelx1,x2=sort(x,'ascend');%x1isxfrommintomax%x2istheplacen=size(x,1);x_ME=median(x,1);%medianincolumnx_IQR=iqr(x);%