Matlab統(tǒng)計(jì)工具箱

1、Matlab統(tǒng)計(jì)工具箱Getting StartedToolbox functions and GUIsOrganizing DataData arrays and groupsDescriptive StatisticsData summariesStatistical VisualizationData patterns and trendsProbability DistributionsModeling data frequencyRandom Number GenerationTopics in random number generationHypothesis TestsInfe

2、rences from dataAnalysis of VarianceModeling data varianceRegression AnalysisContinuous data modelsMultivariate MethodsVisualization and reductionCluster AnalysisIdentifying data categoriesClassificationCategorical data modelsMarkov ModelsStochastic data modelsDesign of ExperimentsSystematic data co

3、llectionStatistical Process ControlProduction monitoring方差分析ANOVA Operationsanova1One-way analysis of varianceanova2Two-way analysis of varianceanovanN-way analysis of varianceaoctoolInteractive analysis of covariancedummyvarCreate dummy variablesfriedmanFriedman's testkruskalwallisKruskal-Walli

4、s testmanova1One-way multivariate analysis of variancemanovaclusterDendrogram of group mean clusters following MANOVAmultcompareMultiple comparison test方差分析模型方差分析是分析試驗(yàn)（或觀測）數(shù)據(jù)的一種統(tǒng)計(jì)方法。在工農(nóng)業(yè)生產(chǎn)和科學(xué)研究中，經(jīng)常要分析各種因素及因素之間的交互作用對(duì)研究對(duì)象某些指標(biāo)值的影響。這時(shí)需要用到方差分析。利用方差分析，我們能推斷哪些因素對(duì)所考察指標(biāo)的影響是顯著的，哪些是不顯著的。方差分析包括了：單因素方差分析、雙因素方差分析

5、和多因素方差分析。單因素方差分析單因素方差分析問題：某一因素（A）對(duì)結(jié)果（）的影響分析。一般情況下，假設(shè)因素A有個(gè)不同的水平。單因素方差分析模型如下：其中表示在第組（水平）第個(gè)觀察值，為總的（的 ）均值，為第組所的均值且滿足，表示第組第個(gè)樣本的誤差，為第組的數(shù)據(jù)個(gè)數(shù)且。方差分析的目的是要基于數(shù)據(jù)分析確定這組數(shù)據(jù)的均值有沒有顯著性差異，即假設(shè)檢驗(yàn)問題： v.s. 不全相等。 （1）原假設(shè)表明因素A對(duì)的值沒有影響。格式p = anova1(X)p = anova1(X,group)p = anova1(X,group,displayopt)p,table = anova1(.)p,table,st

6、ats = anova1(.)實(shí)例 1Create X with columns that are constants plus random normal disturbances with mean zero and standard deviation one:X = meshgrid(1:5)X = 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5X = X + normrnd(0,1,5,5)X = 1.3550 2.0662 2.4688 5.9447 5.4897 2.0693 1.7611 1.4864 4.8826 6.322

7、2 2.1919 0.7276 3.1905 4.8768 4.6841 2.7620 1.8179 3.9506 4.4678 4.9291 -0.3626 1.1685 3.5742 2.1945 5.9465Perform one-way ANOVA:p = anova1(X)p = 7.9370e-006the very small p value indicates that differences between column means are highly significant. The probability of this outcome under the null h

8、ypothesis (that samples drawn from the same population would have means differing by the amounts seen in X) is equal to the p value.實(shí)例 2The following example is from a study of the strength of structural beams in Hogg. The vector strength measures deflections of beams in thousandths of an inch under

9、 3,000 pounds of force. The vector alloy identifies each beam as steel (鐵'st'), alloy 1 (合金1'al1'), or alloy 2 (合金2'al2'). (Although alloy is sorted in this example, grouping variables do not need to be sorted.) The null hypothesis is that steel beams are equal in strength to

10、 beams made of the two more expensive alloys.strength = 82 86 79 83 84 85 86 87.74 82 78 75 76 77 .79 79 77 78 82 79;alloy = 'st','st','st','st','st','st','st','st',. 'al1','al1','al1','al1','al1'

11、,'al1',. 'al2','al2','al2','al2','al2','al2'p = anova1(strength,alloy)p = 1.5264e-004雙因素方差分析模型雙因素方差分析問題。一般地，不妨假設(shè)因素A有個(gè)水平，因素B有個(gè)水平。在雙因素（多因素方差）分析中，不僅要考慮每個(gè)因素的影響，有時(shí)還要考慮因素之間的交互作用。下面是考慮有交互作用的雙因素方差分析的模型如下：其中表示A因素第組B因素第組第個(gè)樣本的觀察值，為總的均值，為A因素第組的均值，為B因素第組的均值

12、，表示A因素第組B因素第組交互作用的均值，滿足。方差分析的目的是要基于數(shù)據(jù)分析確定兩因素是否存在交互作用，各因素不同組的均值有沒有顯著性差異，也就是假設(shè)檢驗(yàn)問題： v.s. 不全相等， （2） v.s. 不全相等， （3） v.s. （4）格式p = anova2(X,reps)p = anova2(X,reps,displayopt)p,table = anova2(.)p,table,stats = anova2(.)實(shí)例The data below come from a study of popcorn brands and popper type (Hogg 1987). The c

13、olumns of the matrix popcorn are brands (Gourmet, National, and Generic). The rows are popper type (Oil and Air.) The study popped a batch of each brand three times with each popper. The values are the yield in cups of popped popcorn. load popcornpopcornpopcorn = 5.5000 4.5000 3.5000 5.5000 4.5000 4

14、.0000 6.0000 4.0000 3.0000 6.5000 5.0000 4.0000 7.0000 5.5000 5.0000 7.0000 5.0000 4.5000p = anova2(popcorn,3)p = 0.0000 0.0001 0.7462回歸分析Linear RegressioncoxphfitCox proportional hazards regressiondummyvarCreate dummy variablesglmfitGeneralized linear model regressionglmvalGeneralized linear model

15、valuesinvpredInverse predictionleverageLeveragemnrfitMultinomial logistic regressionmnrvalMultinomial logistic regression valuesmvregressMultivariate linear regressionmvregresslikeNegative log-likelihood for multivariate regressionplsregressPartial least-squares regressionpolyconfPolynomial confiden

16、ce intervalspolytoolInteractive polynomial fittingregressMultiple linear regressionregstatsRegression diagnosticsridgeRidge regressionrobustdemoInteractive robust regressionrobustfitRobust regressionrsmdemoInteractive response surface demonstrationrstoolInteractive response surface modelingstepwiseI

17、nteractive stepwise regressionstepwisefitStepwise regressionx2fxConvert predictor matrix to design matrix回歸分析模型模型假設(shè)和相關(guān)問題設(shè)變量與多個(gè)變量 (p>1)有線性相關(guān)關(guān)系，相應(yīng)的觀察數(shù)據(jù)為，。我們討論的多元線性回歸模型為：其中和是未知參數(shù)。格式b = regress(y,X)b,bint = regress(y,X)b,bint,r = regress(y,X)b,bint,r,rint = regress(y,X)b,bint,r,rint,stats = regress(y

18、,X). = regress(y,X,alpha)實(shí)例導(dǎo)入數(shù)據(jù)cars; 確定weight 和 horsepower 作為自變量（預(yù)測變量）, mileage 為響應(yīng)變量:load carsmallx1 = Weight;x2 = Horsepower; % Contains NaN datay = MPG;計(jì)算有交互作用項(xiàng)的線性回歸模型的回歸系數(shù):X = ones(size(x1) x1 x2 x1.*x2;b = regress(y,X) % Removes NaN datab = 60.7104 -0.0102 -0.1882 0.0000Plot the data and the mo

19、del:scatter3(x1,x2,y,'filled')hold onx1fit = min(x1):100:max(x1);x2fit = min(x2):10:max(x2);X1FIT,X2FIT = meshgrid(x1fit,x2fit);YFIT = b(1) + b(2)*X1FIT + b(3)*X2FIT + b(4)*X1FIT.*X2FIT;mesh(X1FIT,X2FIT,YFIT)xlabel('Weight')ylabel('Horsepower')zlabel('MPG')view(50,10)

20、主成分分析Principal Component AnalysisbarttestBartlett's testparetoPareto chartpcacovPrincipal component analysis on covariance matrixpcaresResiduals from principal component analysisprincompPrincipal component analysis on data模型在研究某個(gè)問題時(shí)涉及到個(gè)變量（指標(biāo)）（記為列向量，且有均值向量和方差協(xié)方差矩陣）。通常較大且這些指標(biāo)之間存在一定程度的相關(guān)性。因此我們考慮用以下

21、線性變換將這個(gè)變量轉(zhuǎn)換成新的個(gè)兩兩不相關(guān)的變量：記，且限定，則上述的線性變換中的變換矩陣的轉(zhuǎn)置。由于這組新的變量中任意兩個(gè)不相關(guān)，即對(duì)任意有。而且變量中每一個(gè)變量的方差依次是余下所有變量中最大的，因此有。方差越大，包含的信息就越多。我們稱為的第主成分。理論上可以證明第個(gè)主成分對(duì)應(yīng)于的方差協(xié)方差矩陣的第個(gè)特征值，與特征值相應(yīng)的標(biāo)準(zhǔn)化特征向量記為。且有的特征根是主成分的方差，即 ，這里有。因此主成分的方差協(xié)方差矩陣為對(duì)角矩陣diag()。格式COEFF,SCORE = princomp(X)COEFF,SCORE,latent = princomp(X)COEFF,SCORE,latent,tsq

22、uare = princomp(X). = princomp(X,'econ')實(shí)例Compute principal components for the ingredients data in the Hald data set, and the variance accounted for by each component.load hald;pc,score,latent,tsquare = princomp(ingredients);pc,latentpc = 0.0678 -0.6460 0.5673 -0.5062 0.6785 -0.0200 -0.5440

23、-0.4933 -0.0290 0.7553 0.4036 -0.5156 -0.7309 -0.1085 -0.4684 -0.4844latent = 517.7969 67.4964 12.4054 0.2372The following command and plot show that two components account for 98% of the variance: cumsum(latent)./sum(latent)ans = 0.86597 0.97886 0.9996 1biplot(pc(:,1:2),'Scores',score(:,1:2

24、),'VarLabels',.'X1' 'X2' 'X3' 'X4')因子分析Factor AnalysisfactoranFactor analysis模型假設(shè)類似于在主成分分析中的情形，在研究某個(gè)問題時(shí)涉及到個(gè)容易觀測的變量（記為列向量）。其均值向量，方差協(xié)方差矩陣）。我們假定存在個(gè)潛在相互獨(dú)立的的公共因子（變量）（記為列向量）和個(gè)相互獨(dú)立的特殊因子變量（記為列向量）滿足以下的模型：其中，記為。相互獨(dú)立。模型用矩陣向量形式可表示為其中被稱為因子載荷矩陣。格式lambda = factoran(X,m)lambd

25、a,psi = factoran(X,m)lambda,psi,T = factoran(X,m)lambda,psi,T,stats = factoran(X,m)lambda,psi,T,stats,F = factoran(X,m). = factoran(.,param1,val1,param2,val2,.)實(shí)例1導(dǎo)入carbig數(shù)據(jù), 用兩因子擬合默認(rèn)的模型.load carbigX = Acceleration Displacement Horsepower MPG Weight; X = X(all(isnan(X),2),:);Lambda,Psi,T,stats,F = f

26、actoran(X,2,. 'scores','regression');inv(T'*T) % Estimated correlation matrix of F, = eye(2)Lambda*Lambda'+diag(Psi) % Estimated correlation matrixLambda*inv(T) % Unrotate the loadingsF*T' % Unrotate the factor scoresbiplot(Lambda,. % Create biplot of two factors 'Lin

27、eWidth',2,. 'MarkerSize',20)實(shí)例 2Although the estimates are the same, the use of a covariance matrix rather than raw data doesn't let you request scores or significance level:Lambda,Psi,T = factoran(cov(X),2,'xtype','cov')Lambda,Psi,T = factoran(corrcoef(X),2,'xtyp

28、e','cov')實(shí)例 3斜交旋轉(zhuǎn)Use promax rotation:Lambda,Psi,T,stats,F = factoran(X,2,'rotate','promax',. 'powerpm',4);inv(T'*T) % Est'd corr of F, % no longer eye(2)Lambda*inv(T'*T)*Lambda'+diag(Psi) % Est'd corr of XPlot the unrotated variables with obliq

29、ue axes superimposed.invT = inv(T)Lambda0 = Lambda*invTline(-invT(1,1) invT(1,1) NaN -invT(2,1) invT(2,1), . -invT(1,2) invT(1,2) NaN -invT(2,2) invT(2,2), . 'Color','r','linewidth',2)hold onbiplot(Lambda0,. 'LineWidth',2,. 'MarkerSize',20)xlabel('Loadings

30、 for unrotated Factor 1')ylabel('Loadings for unrotated Factor 2')聚類分析Hierarchical ClusteringclusterConstruct agglomerative clusters from linkagesclusterdataConstruct agglomerative clusters from datacophenetCophenetic correlation coefficientinconsistentInconsistency coefficientlinkageCre

31、ate agglomerative hierarchical cluster treepdistPairwise distance between pairs of objectspdist2Pairwise distance between two sets of observationssquareformFormat distance matrix系統(tǒng)聚類法是聚類分析中應(yīng)用最為廣泛的一種方法.它的基本原理是：首先將一定數(shù)量的樣品(或指標(biāo))各自看成一類，然后根據(jù)樣品(或指標(biāo))的親疏程度，將親疏程度最高的兩類合并，然后重復(fù)進(jìn)行，直到所有的樣品都合成一類.衡量親疏程度的指標(biāo)有兩類：距離、相似系

32、數(shù).1.常用距離1）歐氏距離假設(shè)有兩個(gè)維樣本和，則它們的歐氏距離為2）標(biāo)準(zhǔn)化歐氏距離假設(shè)有兩個(gè)維樣本和，則它們的標(biāo)準(zhǔn)化歐氏距離為其中：表示個(gè)樣本的方差矩陣，表示第列的方差.3）馬氏距離假設(shè)共有個(gè)指標(biāo)，第個(gè)指標(biāo)共測得個(gè)數(shù)據(jù)(要求)：， 于是，我們得到階的數(shù)據(jù)矩陣，每一行是一個(gè)樣本數(shù)據(jù).階數(shù)據(jù)矩陣的階協(xié)方差矩陣記做.兩個(gè)維樣本和的馬氏距離如下：馬氏距離考慮了各個(gè)指標(biāo)量綱的標(biāo)準(zhǔn)化，是對(duì)其它幾種距離的改進(jìn).馬氏距離不僅排除了量綱的影響，而且合理考慮了指標(biāo)的相關(guān)性.4）布洛克距離兩個(gè)維樣本和的布洛克距離如下：5）閔可夫斯基距離兩個(gè)維樣本和的閔可夫斯基距離如下：注：時(shí)是布洛克距離；時(shí)是歐氏距離.6）余弦距

33、離這是受相似性幾何原理啟發(fā)而產(chǎn)生的一種標(biāo)準(zhǔn)，在識(shí)別圖像和文字時(shí)，常用夾角余弦為標(biāo)準(zhǔn).7）相似距離2.MATLAB中常用的計(jì)算距離的函數(shù)假設(shè)我們有階數(shù)據(jù)矩陣，每一行是一個(gè)樣本數(shù)據(jù). 在MATLAB中計(jì)算樣本點(diǎn)之間距離的內(nèi)部函數(shù)為y=pdist(x) 計(jì)算樣本點(diǎn)之間的歐氏距離y=pdist(x,'seuclid') 計(jì)算樣本點(diǎn)之間的標(biāo)準(zhǔn)化歐氏距離y=pdist(x,'mahal') 計(jì)算樣本點(diǎn)之間的馬氏距離y=pdist(x,'cityblock') 計(jì)算樣本點(diǎn)之間的布洛克距離y=pdist(x,'minkowski') 計(jì)算樣本點(diǎn)之

34、間的閔可夫斯基距離y=pdist(x,'minkowski',p) 計(jì)算樣本點(diǎn)之間的參數(shù)為p的閔可夫斯基距離y=pdist(x,'cosine') 計(jì)算樣本點(diǎn)之間的余弦距離y=pdist(x,'correlation') 計(jì)算樣本點(diǎn)之間的相似距離另外，內(nèi)部函數(shù)yy=squareform(y)表示將樣本點(diǎn)之間的距離用矩陣的形式輸出.3.常用的聚類方法常用的聚類方法主要有以下幾種：最短距離法、最長距離法、中間距離法、重心法、平方和遞增法等等.4. 創(chuàng)建系統(tǒng)聚類樹假設(shè)已經(jīng)得到樣本點(diǎn)之間的距離y，可以用linkage函數(shù)創(chuàng)建系統(tǒng)聚類樹，格式為z=link

35、age(y).其中：z為一個(gè)包含聚類樹信息的(m-1)×3的矩陣.例如：z=2.000 5.000 0.23.000 4.000 1.28則z的第一行表示第2、第5樣本點(diǎn)連接為一個(gè)類，它們距離為0.2；則z的第二行表示第3、第4樣本點(diǎn)連接為一個(gè)類，它們距離為1.28.在MATLAB中創(chuàng)建系統(tǒng)聚類樹的函數(shù)為z=linkage(y) 表示用最短距離法創(chuàng)建系統(tǒng)聚類樹z=linkage(y,'complete') 表示用最長距離法創(chuàng)建系統(tǒng)聚類樹z=linkage(y,'average') 表示用平均距離法創(chuàng)建系統(tǒng)聚類樹z=linkage(y,'cent

36、roid') 表示用重心距離法創(chuàng)建系統(tǒng)聚類樹z=linkage(y,'ward') 表示用平方和遞增法創(chuàng)建系統(tǒng)聚類樹格式T = cluster(Z,'cutoff',c)T = cluster(Z,'cutoff',c,'depth',d)T = cluster(Z,'cutoff',c,'criterion',criterion)T = cluster(Z,'maxclust',n)實(shí)例比較Fisher iris數(shù)據(jù)的聚類結(jié)果和類別數(shù)據(jù):load fisheririsd =

37、 pdist(meas);Z = linkage(d);c = cluster(Z,'maxclust',3:5);crosstab(c(:,1),species)ans = 0 0 2 0 50 48 50 0 0crosstab(c(:,2),species)ans = 0 0 1 0 50 47 0 0 2 50 0 0crosstab(c(:,3),species)ans = 0 4 0 0 46 47 0 0 1 0 0 2 50 0 0判別分析Discriminant AnalysisclassifyDiscriminant analysismahalMahalan

38、obis distance判別分析概說在已知研究對(duì)象分成若干類型，并已取得各種類型的一批已知樣品的觀測數(shù)據(jù)，在此基礎(chǔ)上根據(jù)某些準(zhǔn)則建立判別式，然后對(duì)未知類型的樣品進(jìn)行判別分類。距離判別法首先根據(jù)已知分類的數(shù)據(jù)，分別計(jì)算各類的重心，計(jì)算新個(gè)體到每類的距離，確定最短的距離（歐氏距離、馬氏距離）Fisher判別法利用已知類別個(gè)體的指標(biāo)構(gòu)造判別式（同類差別較小、不同類差別較大），按照判別式的值判斷新個(gè)體的類別Bayes判別法計(jì)算新給樣品屬于各總體的條件概率，比較概率的大小，然后將新樣品判歸為來自概率最大的總體 判別分析是利用原有的分類信息，得到體現(xiàn)這種分類的函數(shù)關(guān)系式（稱之為判別函數(shù)，一般是與分類相關(guān)的若干個(gè)指標(biāo)的線性關(guān)系式），然后利用該函數(shù)去判斷未知樣品屬于哪一類。Matlab中對(duì)于給定的數(shù)據(jù)，用c