《非参数统计》与MATLAB编程 第八章 相关分析

《《非参数统计》与MATLAB编程 第八章 相关分析》由会员分享，可在线阅读，更多相关《《非参数统计》与MATLAB编程 第八章 相关分析（27页珍藏版）》请在装配图网上搜索。

1、第八章 相关分析协方差命令：C = cov(X)当X为行或列向量时，它等于 var(X) 样本标准差。X=1:15;cov(X)ans = 20 var(X)ans =20当X为矩阵时，此时X的每行为一次观察值，每列为一个变量。cov（X）为协方差矩阵,它是对称矩阵。例：x=rand(100,3);c= cov(x)c= 0.089672 -0.012641 -0.0055434-0.012641 0.07928 0.012326-0.00554340.012326 0.082203c的对角线为：diag(c)ans = 0.0897 0.0793 0.0822它等于：var(x)ans =

2、0.0897 0.0793 0.0822sqrt(diag(cov(x)ans = 0.2995 0.2816 0.2867它等于：std(x)ans = 0.2995 0.2816 0.2867命令：c = cov(x,y) 其中x和y是等长度的列向量（不是行向量），它等于cov(x y)或cov(x,y)例：x=1;4;9;y=5;8;6; c=cov(x,y)c = 16.3333 1.1667 1.1667 2.3333 cov(x,y)ans = 16.3333 1.16671.1667 2.3333COV(X)、 COV(X,0)两者相等 或COV(X,Y)、COV(X,Y,0)

3、两者相等，它们都是除以n-1，而COV(X,1) or COV(X,Y,1)是除以nx=1;4;9;y=5;8;6; cov(x,y,1)ans = 10.8889 0.7778 0.7778 1.5556它的对角线与var(x y,1) 相等ans = 10.8889 1.5556协差阵的代数计算：n,p = size(X);X = X - ones(n,1) * mean(X);Y = X*X/(n-1); Y为X的协差阵相关系数（一）命令：r=corrcoef(x)x为矩阵，此时x的每行为一次观察值，每列为一个变量。r为相关系数矩阵。它称为Pearson相关系数例：x=rand(18,3

4、);r=corrcoef(x)r = 1.0000 0.1509 -0.2008 0.1509 1.0000 0.1142 -0.2008 0.1142 1.0000r为对称矩阵，主对角阵为1命令：r=corrcoef(x,y)其中x和y是等长度的列向量（不是行向量），它等于cov(x y)或cov(x,y)，或x和y是等长度的行向量，r=corrcoef(x,y)它则等于r=corrcoef(x,y), r=corrcoef(x,y)例：x=1;4;9;y=5;8;6; corrcoef(x,y)ans = 1.0000 0.1890 0.1890 1.0000corrcoef(x,y)an

5、s = 1.0000 0.1890 0.1890 1.0000C = COV(X)Rij =C(i,j)/SQRT(C(i,i)*C(j,j)如：X=1 2 7 4 ;5 12 7 8;9 17 11 17;cov(X)ans = 16.0000 30.0000 8.0000 26.0000 30.0000 58.3333 13.3333 46.6667 8.0000 13.3333 5.3333 14.6667 26.0000 46.6667 14.6667 44.3333corrcoef(X)ans = 1.0000 0.9820 0.8660 0.9762 0.9820 1.0000 0

6、.7559 0.9177 0.8660 0.7559 1.0000 0.9538 0.9762 0.9177 0.9538 1.0000则有：30/sqrt(16*58.3333)ans = 0.9820命令：r,p=corrcoef.它还将返回p值，原假设是变量之间不相关。例：销售量x430335520490470210195270400480气温y3021354237208173525x = 430 335 520 490 470 210 195 270 400 480;y=30 21 35 42 37 20 8 17 35 25;r,p=corrcoef(x,y)r = 1.0000 0

7、.8594 0.8594 1.0000p = 1.0000 0.0014 0.0014 1.0000P矩阵主对角矩阵全为1，当总体变量X和Y都服从正态分布，并且总体相关系数等于0时，有：P矩阵的计算，即上例中0.0014的算法。 (1-tcdf(4.7540,8)*2 得：0.0014在显著性水平0.05下，0.0014小于0.05，拒绝两总体不相关的原假设，即销售量与气温相关。命令：r,p,rlo,rup=corrcoef(.)rlo与rup是与r矩阵大小相同的矩阵，rlo为相关系数r的下限，rup为相关系数r的上限。在缺失情况下，置信度为95%。例：x = 430 335 520 490

8、470 210 195 270 400 480;y=30 21 35 42 37 20 8 17 35 25;r,p,rlo,rup=corrcoef(x,y)r = 1.0000 0.8594 0.8594 1.0000p = 1.0000 0.0014 0.0014 1.0000rlo = 1.0000 0.5006 0.5006 1.0000rup = 1.0000 0.9662 0.9662 1.0000因此销售量与气温相关系数95%的置信区间为0.5006，0.9662如果要求99%的销售量与气温相关系数的置信区间：r,p,rlo,rup=corrcoef(x,y,alpha,0.0

9、1)注意不是：r,p,rlo,rup=corrcoef(x,y,0.01)r = 1.0000 0.8594 0.8594 1.0000p = 1.0000 0.0014 0.0014 1.0000rlo = 1.0000 0.3071 0.3071 1.0000rup = 1.0000 0.9786 0.9786 1.0000得销售量与气温相关系数99%的置信区间为0.3071，0.9786比95%的置信区间0.5006，0.9662更宽。 rlo和rup具体怎么算出来的还没弄明白。帮助文件是这样说的：The confidence bounds are based on an asympto

10、tic normal distribution of 0.5*log(1+R)/(1-R), with an approximate variance equal to 1/(n-3).命令：.=corrcoef(.,param1,val1,param2,val2,.)例：.=corrcoef(.,alpha,0.01) 是求在置信度为99%，求r置信区间。.=corrcoef(.,rows, all) 只计算两个变量（两列）所有观察值（所有行）都存在的两个变量。缺失情况下，就是用这种方法计算。例：x = 3 9 2 34 5 NaN 6 66 NaN 4 9 7 8 7 33 9 9 14

11、11 22r,p,rlo,rup=corrcoef(x,rows, all)r = NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 1.0000 -0.5296 NaN NaN -0.5296 1.0000p = NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 1.0000 0.3587 NaN NaN 0.3587 1.0000rlo = NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 1.0000 -0.9623 NaN NaN -0.9623 1.0000rup = NaN NaN NaN N

12、aN NaN NaN NaN NaN NaN NaN 1.0000 0.6619 NaN NaN 0.6619 1.0000说明：NaN表示不存在数据，在x中，只有第四列和每五列每行的数据都存在，r,p,rlo,rup=corrcoef(x,rows, all)只计算第三列和第四列的相关系数及相应的区间。即：r,p,rlo,rup=corrcoef(x(:,3), x(:,4)r = 1.0000 -0.5296 -0.5296 1.0000p = 1.0000 0.3587 0.3587 1.0000rlo = 1.0000 -0.9623 -0.9623 1.0000rup = 1.000

13、0 0.66190.6619 1.0000命令：r,p,rlo,rup=corrcoef(x,rows, complete)如果某行含有NaN，则去掉所有含NaN的行，再计算相关系数和区间。例：x = 3 9 2 34 5 NaN 6 66 NaN 4 9 7 8 7 33 9 9 14 11 22r,p,rlo,rup=corrcoef(x,rows, complete)r = 1.0000 0.3883 0.6080 -0.7630 0.3883 1.0000 -0.4956 0.2995 0.6080 -0.4956 1.0000 -0.9771 -0.7630 0.2995 -0.97

14、71 1.0000p = 1.0000 0.7462 0.5840 0.4475 0.7462 1.0000 0.6699 0.8064 0.5840 0.6699 1.0000 0.1365 0.4475 0.8064 0.1365 1.0000rlo = 1 NaN NaN NaN NaN 1 NaN NaN NaN NaN 1 NaN NaN NaN NaN 1rup = 1 NaN NaN NaN NaN 1 NaN NaN NaN NaN 1 NaN NaN NaN NaN 1r,p,rlo,rup=corrcoef(x,rows, complete)与去掉二、三行所得矩阵再求 co

15、rrcoef相同。即与：r,p,rlo,rup=corrcoef(x(1,:);x(4,:);x(5,:)r = 1.0000 0.3883 0.6080 -0.7630 0.3883 1.0000 -0.4956 0.2995 0.6080 -0.4956 1.0000 -0.9771 -0.7630 0.2995 -0.9771 1.0000p = 1.0000 0.7462 0.5840 0.4475 0.7462 1.0000 0.6699 0.8064 0.5840 0.6699 1.0000 0.1365 0.4475 0.8064 0.1365 1.0000rlo = 1 NaN

16、 NaN NaN NaN 1 NaN NaN NaN NaN 1 NaN NaN NaN NaN 1rup = 1 NaN NaN NaN NaN 1 NaN NaN NaN NaN 1 NaN NaN NaN NaN 1这里rlo,rup中显示NaN，是因为求区间要用到n-3,这里n-3=0命令：r,p,rlo,rup=corrcoef(x,rows, pairwise)r(i,j)的计算是把X中i或j列中含有NaN的行去掉，再计算i与j列的相关系数等指标。例：x = 3 9 2 34 5 NaN 6 66 NaN 4 9 7 8 7 33 9 9 14 11 22r,p,rlo,rup=c

17、orrcoef(x,rows, pairwise)r = 1.0000 0.3883 0.6476 -0.5791 0.3883 1.0000 -0.1687 0.5868 0.6476 -0.1687 1.0000 -0.5296 -0.5791 0.5868 -0.5296 1.0000p = 1.0000 0.7462 0.3524 0.4209 0.7462 1.0000 0.8313 0.4132 0.3524 0.8313 1.0000 0.3587 0.4209 0.4132 0.3587 1.0000rlo = 1.0000 NaN -0.8302 -0.9895 NaN 1.

18、0000 -0.9722 -0.8584 -0.8302 -0.9722 1.0000 -0.9623 -0.9895 -0.8584 -0.9623 1.0000rup = 1.0000 NaN 0.9915 0.8614 NaN 1.0000 0.9457 0.9897 0.9915 0.9457 1.0000 0.66190.8614 0.9897 0.6619 1.0000r矩阵中0.3883只按以下求得：r,p,rlo,rup=corrcoef(3;8;9;,9;7;14)r = 1.0000 0.3883 0.3883 1.0000p = 1.0000 0.7462 0.7462

19、1.0000rlo = 1 NaN NaN 1rup = 1 NaN NaN 1r矩阵中0.6476只按以下求得：r=corrcoef(3;5;8;9;,2;6;33;11)r = 1.0000 0.6476 0.6476 1.0000相关系数（二）corrceof只能计算pearson相关系数。corr则还可计算其它相关系数。不加说明时都是指pearson相关系数。 corr不能做区间估计。命令：rho=corr(x) x为np矩阵，返回pp的相关系数矩阵。它与corcoef(x)计算结果相同。例：x=1 2 7 4 ;5 12 7 8;9 17 11 17;rho=corr(x)rho =

20、 1.0000 0.9820 0.8660 0.9762 0.9820 1.0000 0.7559 0.9177 0.8660 0.7559 1.0000 0.9538 0.9762 0.9177 0.9538 1.0000与corcoef(x)相同corrcoef(x)ans = 1.0000 0.9820 0.8660 0.9762 0.9820 1.0000 0.7559 0.9177 0.8660 0.7559 1.0000 0.9538 0.9762 0.9177 0.9538 1.0000命令：rho=corr(x,y) x为np1矩阵, y为np2矩阵,rho为p1p2相关矩阵。

21、它不是对称矩阵，rho(i,j)为x的第i列（变量）与y的第j列（变量）的相关系数。注意：corrcoef(x,y)只适合大小相同的列或行向量。例：x=1 2 7 4 ;5 12 7 8;9 17 11 17;y=4 9;23 12;-2 -7;rho=corr(x,y)rho = -0.2299 -0.7832 -0.0418 -0.6516 -0.6857 -0.9892 -0.4354 -0.8994如-0.6857为x第3列与y的第1列的相关系数。它等于：rho=corr(x(:,3),y(:,1)rho = -0.6857命令：rho,pval=corr()Pval是返回原假设为两变

22、量相关系数为0时，返回P值。例：x=1 2 7 4 ;5 12 7 8;9 17 11 17;y=4 9;23 12;-2 -7;rho,pval=corr(x,y)rho = -0.2299 -0.7832 -0.0418 -0.6516 -0.6857 -0.9892 -0.4354 -0.8994pval = 0.8523 0.4272 0.9734 0.5482 0.5190 0.0938 0.7132 0.2881我们看pval值计算，如0.0938的计算：即x的第三列与y的第二列相关系数所对应的值的计算。r,p=corrcoef(7;7;11, 9 12 -7)r = 1.0000

23、 -0.9892 -0.9892 1.0000p = 1.0000 0.0938 0.0938 1.00000.0938小于显著性水平0.1，即拒绝x的第三列与y第二列不相关的原假设。命令：. = corr(.,param1, val1, param2, val2,.)ParameterValuestype1、Pearson (the default) computes Pearsons linear correlation coefficient; 2、Kendall computes Kendalls tau; 3、Spearman computes Spearmans rho rows1

24、、all (the default) uses all rows regardless of missing values (NaNs) 2、complete uses only rows with no missing values 3、 pairwisecomputes RHO(i,j) using rows with no missing values in column i or j tail - The alternative hypothesis against which to compute p-values for testing the hypothesis of no c

25、orrelation1、ne - Correlation is not zero (the default) 2、gt - Correlation is greater than zero 3、lt - Correlation is less than zero 例：统计学原理黄良文 p226页复习时间（小时）考试成绩（分）原始数据排队等级原始数据排队等级xiyi33863044874011410228520559361869151108958.50.25979470119958.50.25131096100合 计55826552.5Spearman等级相关系数r,p=corr(x,y,typ

26、e,spearman)r = 0.9848p = 2.2888e-007t=r*sqrt(8)/sqrt(1-r2)t = 16.0404p=2*(1-tcdf(t,8) 得： p=2.2888e-007Kendall等级相关系数（适合于打结的情况）r,p=corr(x,y,type,Kendall)r = 0.9439p = 2.3149e-004其中r的计算，请参见SPSS for Windows 统计分析（第2版） 主编 卢纹岱 p208页。相应的p值不知道怎么算出的。单侧、双侧检验所对应的p值右侧检验的p值：r,p=corr(x,y,type,spearman,tail,gt)r =

27、0.9848p = 1.1444e-007t=r*sqrt(8)/sqrt(1-r2)t = 16.0404p=1-tcdf(t,8)得： p= 1.1444e-007p*2 得：2.2888e-007，它等于上面双侧检验对应的p值。左侧检验的p值：r,p=corr(x,y,type,spearman,tail,lt)r = 0.9848p = 1.0000p值的计算：t=r*sqrt(8)/sqrt(1-r2)t = 16.0404p=tcdf(t,8)得： p= 1.0000rows的用法与corrcoef的rows用法相同，这里不再论述。8.1x=1:9; y=1.5 2 17 3 4

28、5 17.5 6 18 矩相关系数：corr(x,y,type,pearson)ans =0.5209或corr(x,y)ans =0.5209秩相关系数：corr(x,y,type,spearman)ans =0.8000Kendall相关系数corr(x,y,type,kendall)ans =0.7222 习题 八1x=45 52 54 63 62 68 75 76 92 88;y=100 110 120 130 140 150 160 170 180 190;corr(x,y,type,spearman)ans =0.9758当两者变量独立时，在显著性水平面0.01下，n=10,临界值

29、为0.745, 0.9758大于0.745，正相关。corr(x,y,type,kendall)ans = 0.9111当两者变量独立时，在显著性水平面0.01下，n=10,临界值为0.6, 0.9111大于0.6，正相关。5x=79 87 81;83 96 81;60 61 65;80 83 84;73 68 78;69 55 72;57 71 77;74 80 80;78 79 68;76 76 80;77 78 78;82 90 87;75 70 74;72 63 81;71 74 73;54 73 67x = 79 83 60 80 73 69 57 74 78 76 77 82 75

30、 72 71 54 87 96 61 83 68 55 71 80 79 76 78 90 70 63 74 73 81 81 65 84 78 72 77 80 68 80 78 87 74 81 73 67p,table,stats=friedman(x,1)p = 0.0015table = Source SS df MS Chi-sq ProbChi-sq Columns 826.0000 15 55.0667 36.5487 0.0015 Error 191.0000 30 6.3667 Total 1017 47 stats = source: friedman n: 3 meanranks: 1x16 double sigma: 4.7539p值0.0015，小于0.01，所以一致性成立。协和系数36.5487/32ans = 1.1421