非参数检验及matlab实现

1/6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：max()()FxGx[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)2/6详见ketestLillieforstest：检验两个样本是否有相同分布Teststatistics：max()()xFxGx2-sidedgoodness-of-fittestlillietest[h,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teststatistic：22364knJBs，wherenisthesamplesize,sisthesampleskewness,andkisthesamplekurtosis.jbtest[h,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。3/6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,ycanbeatdifferentlengthZ=[x;y];%z1forrankingfromsmalltolarge%z2forplace[z1,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('RejectNullHypothesis');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;4/6endendend%memocontainstheinformationofdivision;%memoisasymmetricmatrix;ifmemo==zeros(size(memo))disp('Noneedfordivision');disp('GoonRankTeststraightly');disp({'ReaultasFollowing','significantlevel',num2str(alpha)});Wilcoxon_Rank_Test(x,y,alpha);elsedisp('Divisionisneeded');[cx,cy]=find(memo0);if(cx(1,1)~=1)&&(cy(1,1)~=1)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;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;breakend5/6endendend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-squaregoodness-of-fittest离散态的检验样本是否来自于已知分布。Teststatistics：221NiiiiOEE,whereOiaretheobservedcountsandEiaretheexpectedcounts.Thestatistichasanapproximatechi-squaredistributionwhenthecountsaresufficientlylarge.chi2gofh=chi2gof(x)H0:x(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)})6/6关于样本是否来自对称的分布functionf=symmetric_test(x,alpha)%xisavector%alphaisthesignificantlevel[x1,x2]=sort(x,'ascend');%x1isxfrommintomax%x2istheplacen=size(x,1);x_ME=median(x,1);%medianincolumnx_IQR=iqr(x);%IQRincolumnx3=abs(x-x_ME);x4=var(x3);x5=norminv(1-alpha/2);ifabs(x3-x_IQR/2)x5*sqrt(x4)disp('AsymmetricDistribution');elsedisp('SymmetricDistribution');end