多元統(tǒng)計(jì)分析第八章-典型相關(guān)分析

第8章典型相關(guān)分析典型相關(guān)分析是用來描述兩組隨機(jī)變量（兩個(gè)隨機(jī)向量）間關(guān)系的統(tǒng)計(jì)分析方法。兩組隨機(jī)向量，各含有許多隨機(jī)變量，能否用少量隨機(jī)變量來描述其相關(guān)魚粉以及豬肉、牛肉、羊肉、雞肉、雞蛋、鴨肉、鴨蛋的價(jià)格，分析飼料與葷菜(由顯著性檢驗(yàn)可見)格看成一組隨機(jī)變量，肉蛋禽價(jià)格看成另一組隨機(jī)變量，找這兩組隨機(jī)變量的線性組合，使之相關(guān)系數(shù)平方最大，從而分析兩組隨機(jī)變量間的關(guān)系，判定這兩組隨機(jī)變量是否有關(guān)聯(lián)，這就是典型相關(guān)分析。8.1典型相關(guān)分析數(shù)學(xué)模型設(shè)隨機(jī)向量X(1,...xp)'與Y(1,...yp)'的方差,xxyycov(X,)。a,b為常數(shù)向量。則Ycorr(a'X,b'Y)a'b/(a'ab'b)，1/2為了計(jì)算確定性，限制D(a'X)a'xxaDb'Y)b'b1。定義8.1設(shè)1,bbaa在條件：1D(a'X)a'axxD(b'Y)b'b1yy下使cov(a'X,b'Y)大，則稱1a'X,wb'Y為第一對(duì)典型相關(guān)變量，111a'X,b'Y)稱為第一典型相關(guān)系數(shù)。由定義可見，1,wv盡可能多地反映原來p對(duì)隨機(jī)變量相關(guān)的信息。第一對(duì)1,定義8.1′若常數(shù)向量a=a，b=b在條件：22D(a'X)a'a1，D(b'Y)b'b1yy；cov(1,a'X)01bYv，cov(w,')0下使a'X,b'Y)最大，則稱va'X,wb'Y2為第二對(duì)典型相關(guān)變量，2221cov(a'X,b'Y)稱為第二典型相關(guān)系數(shù)。若常數(shù)向量a=a，b=b在條件：2233DaX，Db'Y)1；(')1,')0w；v，1,b'Y)01aXv，2,b'Y)0,')0w，2aX下使cov(a'X,b'Y)最大，則稱v3a'X,wb'Y為第三對(duì)典型相關(guān)變量，22333cov(a'X,b'Y)稱為第三典型相關(guān)系數(shù)。……33求第一對(duì)典型相關(guān)變量是在條件：D(a'X)axxa1Db'Y)b'b1',yy下使cov(a'X,b'Y)a'b最大，由Lagrange乘子法，應(yīng)當(dāng)求Lagrange函數(shù)la'xybaabb的無條件極大。1('/2('/2xx2yy對(duì)a，b求偏導(dǎo)數(shù)得：ba1ab200（8.1）假設(shè)xx,正定(否則用廣義逆處理)8.11式左乘a'得a'b；yy1xy（8.12式左乘b'得b'a；從而2xy21。當(dāng)0時(shí)（8.1）式消去a得yx1b2b0，從而，b分2xxxyyy別是yx1相對(duì)于的特征值，特征向量，或化為：xxxyyy/11/yy221/2b1/2b12yxxxxyyyyyyy0令db，則1/2yy，d是1/211/22yy的特征值，特征向量。yxxxxyyy（8.1）式消去b得1a2a0xy，從而yyyxxx2，a分別是xy1相對(duì)于的特征值，特征向量，或化為：yyyxxx11/22a1/axx2，/1/2120xyyyyxxxxxxx令ca，從而、c是1/211/21/2

2xx的特征值和特征向量。yyxyyyyxxx可以證明：定理8．1設(shè)，c分別是1/211/22xx的最大特征值及相應(yīng)xyyyyxxx特征向量；2，d分別是1/211/2yy的最大特征值及相應(yīng)特征向yxxxxyyy量；ac，bd滿足條件D(a'X)D'Y)1，則1/21/2yyyy2va'X,wb'Y1為第一對(duì)典型相關(guān)變量，為第一典型相關(guān)系數(shù)的平方。2111更一般的，設(shè)02，2，ic分別是1/211/2xx的第i大特征值及相ixyyyyxxx應(yīng)特征向量；20，d分別是1/211/2iiyxxxxyyyyy的第i大特征值及相a，D(a'X)ic1iDb'Y)11/2應(yīng)特征向量；idi，/2ii則via'X,wb'Y為第i對(duì)典型相關(guān)變量，iii2為第i典型相關(guān)系數(shù)的平方。i(i)X(),2,..iniY是正態(tài)總體的1i1i一個(gè)樣本。X，

XYY，則()()nn1()()(ixxX，XXiXn1(X(XYYxy，i)(i)n1()()'Y(YYYi)(i)yy，n分別是xx,,的極大似然估計(jì)樣本協(xié)差陣。定理（8.1）中協(xié)差陣可用極大xyyy似然估計(jì)樣本協(xié)差陣代替。這樣做的依據(jù)是：定理8．2設(shè)20，c分別是xx1/2xyyy1yxxx1/2的第i大特征ii值及相應(yīng)特征向量；20，d分別是iiyy/yxxxxyyy的第i大特1211/2i'，di'Y的樣本方差都是1；則ci,di分別

征值及相應(yīng)特征向量；滿足條件：cX為ai,b的極大似然估計(jì)，2為2的極大似然估計(jì)。iii定義8．2vc'X，wd'Y稱為第i對(duì)樣本典型相關(guān)變量，iiii2稱為i第i個(gè)樣本典型相關(guān)系數(shù)平方．冗余分析也是典型相關(guān)分析的重要內(nèi)容。設(shè)每組變量都標(biāo)準(zhǔn)化了，從第1組變量提取的典型變量為V(1,vvr)'，2從第2組變量提取的典型變量為(1,w,...wr)'W；原第1組變量為2X1,xxp，原第2組變量為(,,...)'(,...)'1yyqYy；22w與X分量的相關(guān)系數(shù)i所成向量為G(,...,)'，i1ipv與Y分量的相關(guān)系數(shù)所成向量為iHi個(gè)典型變量i'/，(,...u從第1組變量提取的方差比例為GGpi1ii則第i個(gè)典型變量v從第2組變量提取的方差比例為HHqi'/。ii令RGGpv'/u'/(i)，RHHq(i)，它們稱為冗余測(cè)度。iiiiii3冗余測(cè)度的大小表示這對(duì)典型變量能夠?qū)α硪唤M變差相互解釋程度的大小，對(duì)進(jìn)一步討論多對(duì)建模提供有用的信息。8.2典型相關(guān)過程SAS中用CANCORR型相關(guān)變量。該過程主要包括以下三個(gè)語(yǔ)句：（1）PROCCANCORR語(yǔ)句，一般形式是：PROCCANCORR選擇項(xiàng)1選擇項(xiàng)2PROCCANCORR語(yǔ)句中選項(xiàng)可以是OUT＝…或＝…，用以表明輸出數(shù)據(jù)集；還可以是ALL，用以表明輸出全部計(jì)算內(nèi)容。（2）語(yǔ)句，一般形式是變量l變量2…，用以指定第一組變量。（3WITHWITH變量12例8.1現(xiàn)有北京地區(qū)1951～1976年冬季的氣象資料見表8．，其中year：年份Dec：月份平均氣溫Jan：次年一月份平均氣溫Feb：次年二月份平均氣溫High7：7月500hpa圖上13o--14oE，o--50oN范圍內(nèi)6點(diǎn)高度距平和High44月500hpa圖上oE45oN)(100oW40oN)和(100oW50oN)3點(diǎn)高度距平和high88月500hpa圖上150oE35o--45oN100oE40o--50oN范圍內(nèi)5點(diǎn)高度距平和表8．1北京地區(qū)冬季氣溫DecJanFebHhigh7high4high819511.0-2.7-4.34-7121952-5.3-5.9-3.502151953-2.0-3.4-0.86-951954-5.7-4.7-1.1101761955-0.9-3.8-3.115111956-5.7-5.3-5.9-31-121957-2.1-5.0-1.6-1531319580.6-4.3-0.210-3041959-1.7-5.72.0-9-5-141960-3.6-3.61.311-3181961-3.0-3.1-0.85-15419620.1-3.9-1.181211963-2.6-3.0-5.2113-31964-1.4-4.9-1.7-11-871965-3.9-5.7-2.5-186-61966-4.7-4.8-3.3-9-6151967-6.0-5.6-4.940-201968-1.7-6.4-5.1-7-2-151969-3.4-5.6-2.0417-231970-3.1-4.2-2.99-16231971-3.8-4.9-3.9-135-21972-2.0-4.1-2.470101973-1.7-4.2-23.6-3.3-2.017-201975-2.7-3.70.1-1-13101976-2.4-7.6-2.259-30以Dec,Jan,Feb為第一組變量,high7,high4,high8為第二組變量作典型相關(guān)分析。解采用如下程序：datatemperat;inputyearDecJanFebhigh7high4high8;cards;19511.0-2.7-4.34-7121952-5.3-5.9-3.502151953-2.0-3.4-0.86-951954-5.7-4.7-1.1101761955-0.9-3.8-3.115111956-5.7-5.3-5.9-31-121957-2.1-5.0-1.6-1531319580.6-4.3-0.210-301959-1.7-5.72.0-9-5-141960-3.6-3.61.311-31851961-3.0-3.1-0.85-15419620.1-3.9-1.181211963-2.6-3.0-5.2113-31964-1.4-4.9-1.7-11-871965-3.9-5.7-2.5-186-61966-4.7-4.8-3.3-9-6151967-6.0-5.6-4.940-201968-1.7-6.4-5.1-7-2-151969-3.4-5.6-2.0417-231970-3.1-4.2-2.99-16231971-3.8-4.9-3.9-135-21972-2.0-4.1-2.470101973-1.7-4.2-23.6-3.3-2.017-201975-2.7-3.70.1-1-13101976-2.4-7.6-2.259-30;proccancorrall;varDecJanFeb;withhigh7high4high8;run;執(zhí)行后得到如下結(jié)果：MeansandStandardDeviations3'VAR'Variables3'WITH'Variables26ObservationsVariableMeanStdDevDEC-2.7423081.859069JAN-4.5923081.172663FEB-2.2730771.960930HIGH72.03846210.470839HIGH4-0.0384629.799922HIGH80.73076913.128771以上給出個(gè)變量的樣本均值與樣本標(biāo)準(zhǔn)差。CorrelationsAmongtheOriginalVariables6CorrelationsAmongthe'VAR'VariablesDECJANFEBDEC1.00000.32840.2652JAN0.32841.00000.1587FEB0.26520.15871.0000CorrelationsAmongthe'WITH'VariablesHIGH7HIGH4HIGH8HIGH71.0000-0.11030.1019HIGH4-0.11031.0000-0.3871HIGH80.1019-0.38711.0000以上是兩組變量的組內(nèi)樣本相關(guān)陣。CorrelationsAmongtheOriginalVariablesCorrelationsBetweenthe'VAR'Variablesandthe'WITH'VariablesHIGH7HIGH4HIGH8DEC0.1238-0.28310.1652JAN0.4378-0.44790.6645FEB0.1180-0.18120.2118以上是兩組變量的組間樣本相關(guān)陣。CanonicalCorrelationAnalysisAdjustedApproxSquaredCanonicalCanonicalStandardCanonicalCorrelationCorrelationErrorCorrelation10.7935620.7616860.0740520.62974120.190066-.0073680.1927750.03612530.022657.0.1998970.0005130.7935620.1900660.022657；0.6297410.0361250.000513型相關(guān)系數(shù)0.793562遠(yuǎn)大于兩組變量間單個(gè)相關(guān)系數(shù)。EigenvaluesofINV(E)*H=CanRsq/(1-CanRsq)EigenvalueDifferenceProportionCumulative11.70081.66330.97820.978220.03750.03700.02160.999730.0005.0.00031.0000CanonicalCorrelationAnalysis7TestofH0:ThecanonicalcorrelationsinthecurrentrowandallthatfollowarezeroLikelihoodRatioApproxFNumDFDenDFPr>F10.356700322.8612948.825350.008620.963380070.19774420.938230.999486660.01131220.9163似然比檢驗(yàn)表明第對(duì)典型相關(guān)是高度顯著的（0.00860.01）；第，3對(duì)典型相關(guān)是不顯著的（概率0.9382，0.9163遠(yuǎn)大于0.05）。CanonicalCorrelationAnalysisMultivariateStatisticsandFApproximationsS=3M=-0.5N=9StatisticValueFNumDFDenDFPr>FWilks'Lambda0.356700322.861948.825350.0086Pillai'sTrace0.666379292.0949660.0424Hotelling-LawleyTrace1.738803463.60649560.0013Roy'sGreatestRoot1.7008107912.47263220.0001NOTE:FStatisticforRoy'sGreatestRootisanupperbound.多種檢驗(yàn)表明兩組變量存在相關(guān)性。CanonicalCorrelationAnalysisRawCanonicalCoefficientsforthe'VAR'VariablesV1V2V3DEC-0.032779661-0.568666035-0.13313535JAN0.83397895730.2818830288-0.212608817FEB0.0889953418-0.0028848890.5230182828CanonicalCoefficientsforthe'WITH'VariablesW1W2W3HIGH70.0435982890.0116551032-0.085060488HIGH4-0.0249253530.1078948423-0.007697455HIGH80.05420836620.04778505480.0403880758上表給出原始變量典型相關(guān)變量的系數(shù)，第對(duì)典型變量是v1=-0.032779661Dec+0.8339789578Jan+0.0889953418w1=0.043598289high7-0.024925353high4+0.0542083662hign8第對(duì)典型變量是v2=-0.568666035Dec+0.2818830288Jan-0.002884889Feb8w2=0.0116551032high7+0.1078948423high7+0.0477850548high8第對(duì)典型變量讀者自己找一找。CanonicalCorrelationAnalysisStandardizedCanonicalCoefficientsforthe'VAR'VariablesV1V2V3DEC-0.0609-1.0572-0.2475JAN0.97800.3306-0.2493FEB0.1745-0.00571.0256StandardizedCanonicalCoefficientsforthe'WITH'VariablesW1W2W3HIGH70.45650.1220-0.8907HIGH4-0.24431.0574-0.0754HIGH80.71170.62740.5302上表給出標(biāo)準(zhǔn)化變量典型相關(guān)變量的系數(shù)，第對(duì)典型變量是v1=-0.0609Dec+0.9780Jan+0.1754Febw1=0.4565hign7-0.2443huigh4+0.7117high8第，對(duì)典型變量讀者自己找一找。CanonicalStructureCorrelationsBetweenthe'VAR'VariablesandTheirCanonicalVariablesV1V2V3DEC0.3065-0.9501-0.0574JAN0.9857-0.0175-0.1678FEB0.3136-0.23360.9204以上給出第組變量與自己典型變量間的相關(guān)系數(shù)，即冗余分析的0.3065.95010G0.9857，G0.0175，G1230.31360.23360.05740.16780.9204CorrelationsBetweenthe'WITH'VariablesandTheirCanonicalVariablesW1W2W3HIGH70.55600.0693-0.8283HIGH4-0.57010.8010-0.1825HIGH80.85280.23040.4687以上給出第組變量與自己典型變量間的相關(guān)系數(shù)，即冗余分析的90.5560H0.5701，10.85280.0693H0.8010，H230.2304.828300.18250.4687CanonicalStructureCorrelationsBetweenthe'VAR'VariablesandtheCanonicalVariablesofthe'WITH'VariablesW1W2W3DEC0.2432-0.1806-0.0013JAN0.7822-0.0033-0.0038FEB0.2488-0.04440.0209CanonicalStructureCorrelationsBetweenthe'WITH'VariablesandtheCanonicalVariablesofthe'VAR'VariablesV1V2V3HIGH70.44120.0132-0.0188HIGH4-0.45240.1522-0.0041HIGH80.67670.04380.0106以上是典型變量與對(duì)方變量間的相關(guān)系數(shù)。CanonicalRedundancyAnalysisRawVarianceofthe'VAR'VariablesExplainedbyTheirOwnTheOppositeCanonicalVariablesCanonicalVariablesCumulativeCanonicalCumulativeProportionProportionR-SquaredProportionProportion10.23500.23500.62970.14800.148020.38380.61880.03610.01390.161830.38121.00000.00050.00020.1620CanonicalRedundancyAnalysisRawVarianceofthe'WITH'VariablesExplainedbyTheirOwnTheOppositeCanonicalVariablesCanonicalVariablesCumulativeCanonicalCumulativeProportionProportionR-SquaredProportionProportion1010.50380.50380.62970.31730.317320.18860.69240.03610.00680.324130.30761.00000.00050.00020.3242以上給出每組原始變量用它們自己的典型變量和用對(duì)方的典型變量解釋的方差的累計(jì)比例。CanonicalRedundancyAnalysisStandardizedVarianceofthe'VAR'VariablesExplainedbyTheirOwnTheOppositeCanonicalVariablesCanonicalVariablesCumulativeCanonicalCumulativeProportionProportionR-SquaredProportionProportion10.38790.38790.62970.24430.244320.31920.70710.03610.01150.255830.29291.00000.00050.00020.2560以上給出第組變量，2，個(gè)典型變量從標(biāo)準(zhǔn)化的第組變量提取的比例2G/3分別是0.38790.39120.2929R(i)分別是0.24430.2558，iv0.2560。CanonicalRedundancyAnalysisStandardizedVarianceofthe'WITH'VariablesExplainedbyTheirOwnTheOppositeCanonicalVariablesCanonicalVariablesCumulativeCanonicalCumulativeProportionProportionR-SquaredProportionProportion10.45380.45380.62970.28580.285820.23320.68700.03610.00840.294230.31301.00000.00050.00020.2944以上給出第組變量，2，個(gè)典型變量從標(biāo)準(zhǔn)化的第組變量提取的比例2H/3分別是0.45380.23320.3130R(i)分別是0.28580.2942，iw0.2944。CanonicalRedundancyAnalysisSquaredMultipleCorrelationsBetweenthe'VAR'VariablesandtheFirst'M'CanonicalVariablesofthe'WITH'VariablesM123DEC0.05920.09180.0918JAN0.61180.61180.6118FEB0.06190.06390.0643以上給出第一組變量與第二組，2，個(gè)典型變量相關(guān)系數(shù)累計(jì)平方和，即0.0592=0.24322，0.0918=0.0592+（-0.1806）2，0.0918=0.0918+（-0.0013）2……0.0619=0.24882，0.0639=0.0619+（-0.0419）2，0.0643=0.0639+0.02092。CanonicalRedundancyAnalysisSquaredMultipleCorrelationsBetweenthe'WITH'VariablesandtheFirst'M'CanonicalVariablesofthe'VAR'VariablesM123HIGH70.19470.19480.1952HIGH40.20470.22790.2279HIGH80.45800.45990.4600以上給出第二組變量與第一組，2，個(gè)典型變量相關(guān)系數(shù)累計(jì)平方和，即0.1947=0.44122,0.1948=0.1947+0.01322,0.1952=0.1948+（-0.0188）2；……0.4580=0.67672，0.4599=0.4580+0.04382，0.4600=4599+0.01062。例8.2利用武漢市2005年五月份的每天的各監(jiān)測(cè)站平均的SO2（so2NO（no2PM10pm10windtemp三小時(shí)降水（rain）作典型相關(guān)分析。數(shù)據(jù)見表8.2：表8.2Dateso2no2pm10windtemprain200505065.71123.018.0122.4325.4300200505045.5726.1494.71220.40.0020200505070.1426.8679.43122.30.0030200505047.1427.1476.86222.97.0040200505042.2927.0069.00123.00.105012200505034.5718.8653.71117.80.0060200505046.1429.7163.14120.00.0070200505027.8621.1460.43121.60.0080200505045.5724.43109.71121.90.0090200505153.5727.0098.00124.20.0000200505143.2928.4381.71224.10.0010200505154.7134.8693.71225.20.0020200505149.2925.0076.86023.80.7030200505126.5722.4368.43220.70.0040200505120.7123.5750.43220.00.0050200505126.1429.2975.43122.013.0600200505118.2923.4363.14220.318.0700200505130.2922.2953.14220.20.0080200505133.0024.7153.00220.40.2090200505246.1439.43116.86020.90.0000200505224.1428.57113.57119.25.0010200505226.1431.8694.29121.10.002013200505242.7138.29134.86023.30.3030200505245.0034.4392.14022.30.0040200505248.2930.7185.14224.10.0050200505252.8632.4386.00223.80.0060200505252.2942.86109.00024.30.0070200505233.4341.86125.14125.30.0080200505244.1427.71104.00224.30.0090200505354.8637.8694.43227.60.0000200505330.7128.4397.43125.90.0010解采用以下程序：datawuhan;inputdateso2no2pm10windtemprain;cards;2005050122.4325.4365.71123.0018.00

2005050245.5726.1494.71220.400.00

2005050370.1426.8679.43122.300.00

2005050447.1427.1476.86222.907.00

2005050542.2927.0069.00123.000.10

2005050634.5718.8653.71117.800.00

2005050746.1429.7163.14120.000.00

2005050827.8621.1460.43121.600.00

2005050945.5724.43109.71121.900.00

2005051053.5727.0098.00124.200.00

2005051143.2928.4381.71224.100.00

2005051254.7134.8693.71225.200.00

2005051349.2925.0076.86023.800.70

2005051426.5722.4368.43220.700.00

2005051520.7123.5750.43220.000.00

2005051626.1429.2975.43122.0013.00

2005051718.2923.4363.14220.3018.00

2005051830.2922.2953.14220.200.00

2005051933.0024.7153.00220.400.20

2005052046.1439.43116.86020.900.00142005052124.1428.57113.57119.205.002005052226.1431.8694.29121.100.002005052342.7138.29134.86023.300.302005052445.0034.4392.14022.300.002005052548.2930.7185.14224.100.002005052652.8632.4386.00223.800.002005052752.2942.86109.00024.300.002005052833.4341.86125.14125.300.002005052944.1427.71104.00224.300.002005053054.8637.8694.43227.600.002005053130.7128.4397.43125.900.00;proccancorrall;varso2no2pm10;withwindtemprain;run;輸出結(jié)果如下：TheCANCORRProcedureVARVariables3WITHVariables3Observations31MeansandStandardDeviationsStandardVariableMeanDeviationso239.94451612.480598no229.1019356.126653pm1085.14225822.463019wind1.2580650.728823temp22.4483872.202101rain2.0096775.054460CorrelationsAmongtheOriginalVariablesCorrelationsAmongtheVARVariablesso2no2pm10so21.00000.38290.3309no20.38291.00000.7413pm100.33090.74131.0000CorrelationsAmongtheWITHVariableswindtemprainwind1.0000-0.00390.0735temp-0.00391.0000-0.1402rain0.0735-0.14021.0000CorrelationsBetweentheVARVariablesandtheWITHVariableswindtemprainso2-0.12920.4704-0.4792no2-0.38160.5514-0.1974pm10-0.41970.4517-0.2310以上分別給出兩組變量的組內(nèi)、組間樣本相關(guān)陣。CanonicalCorrelationAnalysisAdjustedApproximateSquaredCanonicalCanonicalStandardCanonical15CorrelationCorrelationErrorCorrelation10.7663490.7331210.0753500.58729120.3735390.2996650.1570990.13953230.148090.0.1785700.021931TestofH0:canonicalcorrelationsintheEigenvaluesofInv(E)*Hcurrentrowandallthatfollowarezero=CanRsq/(1-CanRsq)LikelihoodApproximateEigenvalueDifferenceProportionCumulativeRatioFValueNumDFDenDFPr>F11.42301.26090.88520.88520.347335303.69960.9940.001020.16220.13970.10090.98610.841597741.174520.334530.02240.01391.00000.978069340.611270.4433MultivariateStatisticsandFApproximationsS=3M=-0.5N=11.5StatisticValueFValueNumDFDenDFPr>FWilks'Lambda0.347335303.69960.9940.0010Pillai'sTrace0.748752872.999810.0040Hotelling-LawleyTrace1.607592534.35936.2330.0007Roy'sGreatestRoot1.4230123812.81327<.0001NOTE:FStatisticforRoy'sGreatestRootisanupperbound.0.766349，0.373539，0.148090相關(guān)系數(shù)0.766349大于兩組變量的單個(gè)相關(guān)系數(shù)。根據(jù)似然比檢驗(yàn)結(jié)果得，第一典型相關(guān)顯著，第二、三典型相關(guān)不太顯著，其它多種相關(guān)性檢驗(yàn)得出兩組變量是存在相關(guān)性的。RawCanonicalCoefficientsfortheVARVariablesV1V2V3so20.0403074949-0.0770730250.0012825474no20.07054769910.1015801810.216293083pm100.01367767580.0117879452-0.064027397RawCanonicalCoefficientsfortheWITHVariablesW1W2W3wind-0.600101907-0.9048102190.845039713temp0.33861069210.06284965520.3029060086rain-0.0755459570.16013607470.0937780851CanonicalCorrelationAnalysisStandardizedCanonicalCoefficientsfortheVARVariablesV1V2V3so20.5031-0.96190.0160no20.43220.62231.3252pm100.30720.2648-1.4382StandardizedCanonicalCoefficientsfortheWITHVariablesW1W2W3wind-0.4374-0.65940.6159temp0.74570.13840.6670rain-0.38180.80940.4740以上給出的是原始變量和標(biāo)準(zhǔn)化變量典型相關(guān)變量的系數(shù)。CanonicalStructure16CorrelationsBetweentheVARVariablesandTheirCanonicalVariablesV1V2V3so20.7702-0.63600.0475no20.85260.45030.2650pm100.79420.4078-0.4506CorrelationsBetweentheWITHVariablesandTheirCanonicalVariablesW1W2W3wind-0.4683-0.60050.6481temp0.80090.02750.5982rain-0.51850.74150.4258CorrelationsBetweentheVARVariablesandtheCanonicalVariablesoftheWITHVariablesW1W2W3so20.5903-0.23760.0070no20.65340.16820.0393pm100.60860.1523-0.0667CorrelationsBetweentheWITHVariablesandtheCanonicalVariablesoftheVARVariablesV1V2V3wind-0.3589-0.22430.0960temp0.61380.01030.0886rain-0.39740.27700.0631以上給出兩組變量與自己典型變量間的相關(guān)系數(shù)以及與對(duì)方典型變量間的相關(guān)系數(shù)。CanonicalRedundancyAnalysisRawVarianceoftheVARVariablesExplainedbyTheirOwnTheOppositeCanonicalVariablesCanonicalVariablesCanonicalVariableCumulativeCanonicalCumulativeNumberProportionProportionR-SquareProportionProportion10.62750.62750.58730.36850.368520.22140.84890.13950.03090.399430.15111.00000.02190.00330.4027RawVarianceoftheWITHVariablesExplainedbyTheirOwnTheOppositeCanonicalVariablesCanonicalVariablesCanonicalVariableCumulativeCanonicalCumulativeNumberProportionProportionR-SquareProportionProportion10.32640.32640.58730.19170.191720.46050.78690.13950.06430.256030.21311.00000.02190.00470.2606以上給出每組原始變量用它們自己的典型變量和用對(duì)方的典型變量解釋的方差的累計(jì)比例。CanonicalRedundancyAnalysisStandardizedVarianceoftheVARVariablesExplainedbyTheirOwnTheOppositeCanonicalVariablesCanonicalVariables17CanonicalVariableCumulativeCanonicalCumulativeNumberProportionProportionR-SquareProportionProportion10.65030.65030.58730.38190.381920.25790.90820.13950.03600.417930.09181.00000.02190.00200.4199以上給出第組變量，2，個(gè)典型變量從標(biāo)準(zhǔn)化的第組變量提取的比例2G/3分別是0.65030.25790.0918R(i)分別是0.38190.41790.4199。ivStandardizedVarianceoftheWITHVariablesExplainedbyTheirOwnTheOppositeCanonicalVariablesCanonicalVariablesCanonicalVariableCumulativeCanonicalCumulativeNumberProportionProportionR-SquareProportionProportion10.37650.37650.58730.22110.221120.30370.68030.13950.04240.263530.31971.00000.02190.00700.2705以上給出第組變量，2，個(gè)典型變量從標(biāo)準(zhǔn)化的第組變量提取的比例2H/3分別是0.37650.30370.3197R(i)分別是0.22110.26350.2705。iwCanonicalRedundancyAnalysisSquaredMultipleCorrelationsBetweentheVARVariablesandtheFirstMCanonicalVariablesoftheWITHVariablesM123so20.34840.40490.4049no20.42690.45520.4568pm100.37040.39360.3980以上給出第一組變量與第二組，2，個(gè)典型變量相關(guān)系數(shù)累計(jì)平方和。SquaredMultipleCorrelationsBetweentheWITHVariablesandtheFirstMCanonicalVariablesoftheVARVariablesM123wind0.12880.17910.1883temp0.37670.37680.3846rain0.15790.23460.2386以上給出第二組變量與第一組，2，個(gè)典型變量相關(guān)系數(shù)累計(jì)平方和。例8.3有1952~1980年長(zhǎng)江下游五站上海,南京蕪湖,武漢九江)平均的6,7,8月降水量(x1,x2,x3)與9月次年4月的副熱帶高壓面積指數(shù)資料(x4~x11),對(duì)此進(jìn)行典型相關(guān)分析.表8.3長(zhǎng)江下游月降水量與副熱帶高壓面積指數(shù)YearX1X2X3X4X5X6X7X8X9X10X11195279.5123.7163.91414181412145101953293.7106.3104.7161434107071954389.2448.795.41227172020371955298.9126.5102.214108161671956274.397.3157.2247101410181957172.7257.5132.4831040036195855.559.6218.4191314141917181959221.873.557.02010161010128196073.016211515129381961191.957.884.42817171610310196256.5243.4246.21751472775196386.3145.1224.4291812925341964305.558.737.12971713710741965137.4138.7147.023120020631966137.5102.427.915161215104101967130.479.637.120181063100196871.6105.480.610143500101969139.4507.9238.081200519211970197.2215.584.522131491014191971218.072.677.821149700001972182.1103.8105.014179500061973159.8127.845.513222615101918211974155.3259.596.02679600001975278.4140.9190.41219940021976243.364.599.9160200041977143.4166.2160.91019138715148197840.326.91924121915161979238.2165.150.91919121914810171980219.1257.4289.5232012104919解采用以下程序：Datarainfall;Inputyearx1x2x3x4x5x6x7x8x9x10x11;Card;195279.5123.7163.91414181412145101953293.7106.3104.7143410707……1980219.1257.4289.52012104919;proccancorrall;19varx1x2x3;withx4x5x6x7x8x9x10run;運(yùn)行結(jié)果如下：MeansandStandardDeviations3'VAR'Variables8'WITH'Variables29ObservationsVariableMeanStdDevX1186.34137983.087267X2153.858621110.325834X3119.11034572.106629X417.9310345.891465X514.3793105.796636X611.3103455.542714X79.1034485.820805X85.8965525.524910X96.0000005.994045X106.6551726.142800X118.6206906.586858CorrelationsAmongtheOriginalVariablesCorrelationsAmongthe'VAR'VariablesX1X2X3X11.00000.1234-0.2587X20.12341.00000.4282X3-0.25870.42821.0000CorrelationsAmongtheOriginalVariablesCorrelationsAmongthe'WITH'VariablesX4X5X6X7X41.0000-0.0776-0.00700.1377X5-0.07761.00000.29300.5026X6-0.00700.29301.00000.7052X70.13770.50260.70521.0000X80.04590.24330.46640.6556X9-0.02120.27750.59120.515920X100.00430.09810.47320.3237X11-0.07060.20880.46900.3560CorrelationsAmongtheOriginalVariablesCorrelationsAmongthe'WITH'VariablesX8X9X10X11X40.0459-0.02120.0043-0.0706X50.24330.27750.09810.2088X60.46640.59120.47320.4690X70.65560.51590.32370.3560X81.00000.70530.47040.5171X90.70531.00000.69260.6386X100.47040.69261.00000.8387X110.51710.63860.83871.0000CorrelationsAmongtheOriginalVariablesCorrelationsBetweenthe'VAR'Variablesandthe'WITH'VariablesX4X5X6X7X10.02860.1711-0.07040.1044X2-0.3078-0.06320.1480-0.0617X3-0.0274-0.1535-0.0661-0.3010X8X9X10X11X1-0.1204-0.2913-0.2510-0.0822X2-0.3690-0.17950.17650.2447X3-0.2524-0.09800.15360.1661以上給出的是原始變量的樣本均值、樣本標(biāo)準(zhǔn)差，以及兩組變量的組內(nèi)、組間樣本相關(guān)陣。CanonicalCorrelationAnalysisAdjustedAppro