matlab统计功能简介[1].pdf

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1、2007-4-141 Matlab统计功能简介-兼与S_PLUS比较2007-4-142 主要内容Matlab统计功能介绍 S_PLUS对应功能2007-4-143 参考资料?statistics toolbox?Computational Statistics Handbook with MATLAB?S-PLUS 6 for Windows Guide to Statistics2007-4-144 Matlab统计功能?概率分布?描述性统计?统计作图?假设检验?线性和非线性模型?多变量统计?等等2007-4-145 概率分布?MATLAB中函数*pdf,*cdf,*inv(cdf-1),

2、*rnd?S_PLUS中函数d*,p*,q*,r*2007-4-146 连续型?Beta (beta*)(*beta)?指数分布(Exponential)(exp*)(*exp)?Gamma (gam*)(*gamma)?对数正态分布(Lognormal)(logn*)(*lnorm)?正态分布(Normal)(norm*)(*norm)?均匀分布(Uniform)(unif*)(*unif)?Weibull(wbl*)(*weibull)函数MATLABS-PLUS2007-4-147 连续型统计分布?卡方分布(Chi-square)(chi2*)(*chisq)?非中心卡方分布(Non-c

3、entral Chi-square)(ncx2*)(pchisq(3,5,ncp=1),ncx2cdp(3,5,1)?F (f*)(*f)?Non-central F (ncf*)(ncfcdf(3,5,1,2),pf(3,5,1,ncp=2)?t (t*)(*t)?Non-central t (nct*)2007-4-148 离散型?二项分布(Binomial)(bino*)(*binom)?几何分布(Geometric)(geo*)(*geom)?超几何分布(Hypergeometric)(hyge*)(*hyper)?负二项分布(Negative binomial)(nbin*)(*nb

4、inom)?泊松分布(Poisson)(poiss*)(*pois)2007-4-149 计算标准正态分布概率密度函数x=0时的值normpdf(0)ans=0.3989dnorm(0)#S_PLUS1 0.39894232007-4-1410 normcdf(2,1,3)ans=0.6306(pnorm(2,1,3)#S_PLUS1 0.6305587)计算P(2 X normcdf(4,1,3)-normcdf(2,1,3)ans=0.2108pnorm(4,1,3)-pnorm(2,1,3)#S_PLUS1 0.2107861()1,3XN设，计算()2P X 2007-4-1411 n

5、orminv(0.95,1,3)ans=5.9346qnorm(0.95,1,3)#S_PLUS1 5.9345612007-4-1412 描述性统计?均值?中位数?方差?标准差?等2007-4-1413 应用举例load sz.datap1=sz(:,1)p2=sz(:,2)mean(p1)-均值var(p1)-方差std(p1)-标准差prctile(p1,50)-分位值prctile(p1,25)2007-4-1414 应用举例skewness(p1)-偏度kurtosis(p1)-峰度min(p1)max(p1)range(p1)-范围median(p1)-中位数2007-4-1415

6、 S_PLUSsz_importData(sz.txt,type=ASCII)p1_sz,1p2_sz,2stdev(p1)-标准差1 124.8987quantile(p1,c(.25,.50,.75)-分位数25%50%75%1480.125 1669.4 1712.3252007-4-1416 作图?盒形图?分布图?散点图?等2007-4-1417 盒形图(boxplot)x=normrnd(4,1,1,100)normrnd(6,0.5,1,200);均值标准差1x100矩阵boxplot(x)2007-4-1418 234567x_append(rnorm(100,4,1),rnor

7、m(200,6,0.5)boxplot(x)2007-4-1419 直方图(hist)Matlabhist(x)2007-4-1420 hist(x,15)2007-4-1421 set.seed(100)x_append(rnorm(100,4,1),rnorm(200,6,0.5)hist(x)2345678020406080 x2007-4-1422 散点图Matlaby=88,85,88,91,92,93,93,95,96,98,97,96,98,99,100,102x=1:16plot(x,y,+)plot(x,y,.)2007-4-1423 S_PLUSy=c(88,85,88,9

8、1,92,93,93,95,96,98,97,96,98,99,100,102)x_1:16plot(x,y,pch=+)+xy510158590951002007-4-1424 线形图plot(x,y)2007-4-1425 xy51015859095100plot(x,y,type=l)2007-4-1426 分布图卡方分布2007-4-1427 f分布2007-4-1428 标准正态分布x=-5:0.01:5;y=normpdf(x,0,1);plot(x,y)2007-4-1429 t分布和标准正态分布 x=-5:0.1:5 y=tpdf(x,5);z=normpdf(x);plot(

9、x,y,-,x,z,-.)2007-4-1430 S_PLUS标准正态分布-3-2-101230.00.10.20.30.4#标准正态分布x=seq(-3,3,0.01)plot(x,dnorm(x),type=l,xlab=,ylab=,col=8,main=标准正态分布)2007-4-1431 t分布-3-2-101230.10.20.3自由度为5的t分布自由度为1的t分布x=seq(-3,3,0.01)plot(x,dt(x,5),type=n,xlab=,ylab=,main=t分布)lines(x,dt(x,5),lwd=3,col=8)lines(x,dt(x,1),lty=3,l

10、wd=3,col=6)legend(-1.2,0.1,c(自由度为5的t分布,自由度为1的t分布),lty=c(1,3),col=c(8,6)2007-4-1432 卡方分布 02468100.00.050.100.15自由度为4的卡方分布自由度为7的卡方分布x=seq(0,10,0.1)plot(x,dchisq(x,4),xlab=,ylab=,type=n,main=卡方分布)lines(x,dchisq(x,4),lwd=3,col=10)lines(x,dchisq(x,7),lty=4,lwd=3,col=12)legend(5,0.15,c(自由度为4的卡方分布,自由度为7的卡方

11、分布),lty=c(1,4),lwd=3,col=c(10,12)2007-4-1433?x=seq(0,10,0.1)?plot(x,df(x,1,3),xlab=,ylab=,type=n,main=f分布)?lines(x,df(x,1,3),lwd=3,col=6)?lines(x,df(x,5,9),lty=4,lwd=3,col=8)?legend(4,0.8,c(自由度为(1,3)的f分布,自由度为(5,9)的f分布),lty=c(1,4),lwd=3,col=c(6,8)f分布 02468100.00.20.40.60.81.0自由度为(1,3)的f分布自由度为(5,9)的f分

12、布2007-4-1434 假设检验?零假设(Null hypothesis)?备择假设(Alternative hypotheses)?置信水平(Significance level)?P值(p-value)?置信区间(Confidence intervals)2007-4-1435 假设检验(S_PLUS)x_c(850,740,900,1070,930,850,950,980,980,880,1000,980,930,650,760,810,1000,1000,960,960)t.test(x,mu=990)One-sample t-Testdata:x t=-3.4524,df=19,p

13、-value=0.0027 alternative hypothesis:mean is not equal to 990 95 percent confidence interval:859.8931 958.1069 sample estimates:mean of x 909单样本t检验2007-4-1436 Matlabx=850,740,900,1070,930,850,950,980,980,880,1000,980,930,650,760,810,1000,1000,960,960 h,pvalue,ci=ttest(x,990)h=1pvalue=0.0027ci=859.89

14、31 958.10692007-4-1437?t.test(x,conf.level=.9,mu=990)#S?One-sample t-Test?data:x?t=-3.4524,df=19,p-value=0.0027?alternative hypothesis:mean is not equal to 990?90 percent confidence interval:?868.4308 949.5692?sample estimates:?mean of x?909置信水平0.92007-4-1438 h,pvalue,ci=ttest(x,990,0.1)#Mh=1pvalue=

15、0.0027ci=868.4308 949.5692显著性水平0.12007-4-1439?var.test(x,y)?F test for variance equality?data:x and y?F=1.0755,num df=11,denom df=6,p-value=0.9788?alternative hypothesis:ratio of variances is not equal to 1?95 percent confidence interval:?0.198811 4.173718?sample estimates:?variance of x variance of

16、 y?457.4545 425.3333检验方差是否相等，结果表明方差相等不容拒绝。2007-4-1440 x_c(134,146,104,119,124,161,107,83,113,129,97,123)y_c(70,118,101,85,107,132,94)t.test(x,y)Standard Two-Sample t-Testdata:x and y t=1.8914,df=17,p-value=0.0757 alternative hypothesis:difference in means is not equal to 0 95 percent confidence inte

17、rval:-2.193679 40.193679 sample estimates:mean of x mean of y 120 101双样本t检验2007-4-1441 h,pvalue,ci=ttest2(x,y)h=0pvalue=0.0757ci=-2.1937 40.19372007-4-1442 线性和非线性模型?多元线性回归?非线性回归?逐步回归?广义线性模型?稳健性和非参数方法2007-4-1443 回归分析回归分析1多元线性回归在Matlab统计工具箱中使用命令regress()实现多元线性回归，调用格式为b=regress(y，x)或b，bint，r，rint，stats

18、=regess(y，x，alpha)2007-4-1444 其中因变量数据向量y和自变量数据矩阵x按以下排列方式输入1111221111kknknxxxxxx=X?12nyyy=Y?2007-4-1445 2R对一元线性回归，取k=1即可。alpha为显著性水平(缺省时设定为0.05)，输出向量b，bint为回归系数估计值和它们的置信区间，r，rint为残差及其置信区间，stats是用于检验回归模型的统计量，有三个数值，第一个是，其中R是相关系数，第二个是F统计量值，第三个是与统计量F对应的概率P，当P z=1,25,1300;yhat=b*zyhat=24.33522007-4-1452 S

19、_PLUS命令y_c(33.3,35.5,27.6,30.4,31.9,53.1,35.6,29.0,35.1,34.5)x1_c(32.4,29.1,26.3,31.2,29.2,40.7,29.8,23.0,28.2,26.9)x2_c(1250,1650,1450,1310,1310,1580,1490,1520,1620,1570)lm.fit_lm(yx1+x2)summary(lm.fit)data_data.frame(25,1300)dimnames(data)_list(1,c(x1,x2)predict(lm.fit,newdata=data,se=T,ci.fit=T,p

20、i.fit=T)#预测2007-4-1453 2非线性回归非线性回归可由命令nlinfit来实现，调用格式为beta,r,j=nlinfit(x,y,model,beta0)其中，输人数据x，y分别为nm矩阵和n维列向量，对一元非线性回归，x为n维列向量model是事先用 m-文件定义的非线性函数，beta0是回归系数的初值，beta是估计出的回归系数，r是残差，j是Jacobian矩阵，它们是估计预测误差需要的数据。2007-4-1454 预测和预测误差估计用命令y,delta=nlpredci(model,x,beta,r,j)2007-4-1455 【例例例例】以下是19002000奥林

21、匹克男子1500米获胜者所需时间数据，和建议拟合下列模型：其中123-最终极限时间-第一纪录超过最终极限时间值-记录序列向最终极限时间下降的固定比率2007-4-1456 load Olympic.m;Year=Olympic(:,1);Time=Olympic(:,2);plot(Year,Timeplot(Year,Time,*);,*);beta0=200,40,-0.01;beta,r,j=nlinfit(Year,Time,model1,beta0);2007-4-1457 betabeta=206.088241.8638-0.0173因此得到：()0.01731900206.088

22、241.8638*YearTimee=+2007-4-1458 Time1,delta=nlpredci(model1,Year,beta,r,j);plot(Year,Time,k+,Year,Time1,r);2007-4-1459 利用S_PLUS中函数nls来进行模型参数的估计source(“/olympic.dat)Olympic.fit step(lm.fit,trace=F)Call:lm(formula=y x1+x2)Coefficients:(Intercept)x1 x2 52.57735 1.468306 0.6622505Degrees of freedom:13 t

23、otal;10 residualResidual standard error(on weighted scale):2.406335 2007-4-1475 多变量统计?主成分分析Principal components analysis?因子分析Factor analysis?方差分析MANOVA?聚类分析Cluster analysis2007-4-1476 主成份分析主成份分析主要求解特征值和特征向量，使用命令 eig()，调用格式为V，D=eig(R)其中R为X的相关系数矩阵，D为R的特征值矩阵，V为特征向量矩阵2007-4-1477 【例例例例】已知某湖八年来湖水中COD浓度实测值

24、(y)与影响因素湖区工业产值(x1)、总人口数(x2)、捕鱼量(x3)、降水量(x4)资料，建立污染物y的水质分析模型。数据x1=1.376,1.375,1.387,1.401,1.412,1.428,1.445,1.477;x2=0.450,0.475,0.485,0.500,0.535,0.545,0.550,0.575;x3=2.170,2.554,2.676,2.713,2.823,3.088,3.122,3.262;x4=0.8922,1.1610,0.5346,0.9589,1.0239,1.0499,1.1065,1.1387y=5.19,5.30,5.60,5.82,6.00,

25、6.06,6.45,6.95;2007-4-1478 x=x1 x2 x3 x4计算相关系数矩阵R=corrcoef(x)R=1.0000 0.9483 0.9119 0.46130.9483 1.0000 0.9683 0.47470.9119 0.9683 1.0000 0.40360.4613 0.4747 0.4036 1.0000求特征根、特征向量V,D=eig(R)结果：R*V=V*D.2007-4-1479 V=-0.2531 0.7854 0.1634 0.54070.8022 -0.1560 0.1660 0.5519-0.5387 -0.5968 0.2534 0.5380

26、-0.0477 -0.0519 -0.9389 0.3369D=0.0223 0 0 00 0.0882 0 00 0 0.7269 00 0 0 3.16262007-4-1480 按特征根由大到小写出各主成份第一主成份f1=0.5407x1+0.5519x2+0.5380 x3+0.3369x4方差贡献率为3.1626/4=79.06第二主成份f2=0.1634x1+0.1660 x2+0.2534x3-0.9389x4方差贡献率为0.7269/4=18.17第三主成份f3=0.7854x1-0.1560 x2-0.5968x3-0.0519x4方差贡献率为 0.0882/4=2.21%注

27、：这里xi表示标准化向量iiixx2007-4-1481 coefs,scores,variances,t2=princomp(x)coefs=0.0887 -0.0069 -0.8649 -0.49400.1133 -0.0127 -0.4839 0.86770.9456 -0.2928 0.1306 -0.05490.2918 0.9561 0.0274 -0.0088variances=0.13770.03210.00020.0001或：V1 D1=eig(cov(x)2007-4-1482 variances(1)/sum(variances)ans=0.8096 variances/

28、sum(variances)ans=0.80960.18860.00150.00032007-4-1483 x1=c(1.376,1.375,1.387,1.401,1.412,1.428,1.445,1.477)x2=c(0.450,0.475,0.485,0.500,0.535,0.545,0.550,0.575)x3=c(2.170,2.554,2.676,2.713,2.823,3.088,3.122,3.262)x4=c(0.8922,1.1610,0.5346,0.9589,1.0239,1.0499,1.1065,1.1387)y=c(5.19,5.30,5.60,5.82,6.

29、00,6.06,6.45,6.95)x_cbind(x1,x2,x3,x4)R_cor(x)eig_eigen(R)2007-4-1484 eig$values1 3.16264069 0.72686908 0.08819302 0.02229720 eig$vectors,1 ,2 ,3 ,4 1,0.5407396-0.1634458-0.78538098 0.253106352,0.5519479-0.1659517 0.15604114-0.802162563,0.5380120-0.2533985 0.59676807 0.538702314,0.3368936 0.9389008

30、0.05192049 0.047667782007-4-1485 x.prc0_princomp(x)summary(x.prc0,loadings=T)Importance of components:Comp.1 Comp.2 Comp.3 Standard deviation 0.3470673 0.1675116 0.014723495Proportion of Variance 0.8096374 0.1886049 0.001457086Cumulative Proportion 0.8096374 0.9982423 0.999699374Comp.4 Standard devi

31、ation 0.0066877760Proportion of Variance 0.0003006261Cumulative Proportion 1.0000000000Loadings:Comp.1 Comp.2 Comp.3 Comp.4 x1 -0.865 0.494x2 0.113 -0.484-0.868x3 0.946-0.293 0.131 x4 0.292 0.956 2007-4-1486 x.prc_princomp(x,cor=T)summary(x.prc,loadings=T)Importance of components:Comp.1 Comp.2 Comp.

32、3 Standard deviation 1.7783815 0.8525662 0.29697309Proportion of Variance 0.7906602 0.1817173 0.02204825Cumulative Proportion 0.7906602 0.9723774 0.99442570Comp.4 Standard deviation 0.149322485Proportion of Variance 0.005574301Cumulative Proportion 1.000000000Loadings:Comp.1 Comp.2 Comp.3 Comp.4 x1 0.541-0.163-0.785 0.253x2 0.552-0.166 0.156-0.802x3 0.538-0.253 0.597 0.539x4 0.337 0.939 2007-4-1487 Comp.1Comp.2Comp.3Comp.40.00.020.040.060.080.100.12xVariances0.810.99811Comp.1Comp.2Comp.3Comp.40.00.51.01.52.02.53.0 xVariances0.7910.9720.9941