主成分分析訓(xùn)練題

主成分分析在數(shù)學(xué)建模中的應(yīng)用朱 寧2012年11月13日學(xué)習(xí)目的理解主成分分析的根本思想；會(huì)用SAS軟件編寫(xiě)相關(guān)程序，對(duì)相關(guān)數(shù)據(jù)進(jìn)展主成分分析；會(huì)用SAS軟件編程結(jié)合主成分分析方法解決實(shí)際問(wèn)題。學(xué)習(xí)要求理解主成分分析的根本原理，把握主成分分析的根本步驟；會(huì)用SAS軟件編寫(xiě)相關(guān)程序，對(duì)相關(guān)數(shù)據(jù)進(jìn)展分析處理和假設(shè)檢驗(yàn)；3000字的小論文；精讀一篇優(yōu)秀論文。理論根底1析就是在這種降維的思想下產(chǎn)生的處理高維數(shù)據(jù)的方法。根本原理〔1〕.總體的主成分定義1.XXX1

,…,X2

)”為PZi

aX為X的第i主成分ii=1,2P，假設(shè)：a”a

ii當(dāng)i>1時(shí)，a” ai j

0(j1,2,…i-1);Var(Z)

max Var(a”X)i a”a1,a”aj

定理1.XX1X2,…,Xp)”是P維隨機(jī)向量，且DX)的特征值為…0，a,a,…,a

為相應(yīng)的單位正交特征向量，則Xi主成分為1 2 p 1 2 pZa”X i1,,,p).i i定義2.我們稱(chēng)k

/

為主成分Z

的奉獻(xiàn)率；又稱(chēng)

/k

為主成分ii1 k1 i1Z,…Z (m p))，diag(,…

)1 m ij 12 p

為aa

,…,a

是相應(yīng)的單位正交特征向量，記正交矩陣1 2 p 1 2 pA(a,a

,…,a

ZZ

)”Z

a”(i1,2,…,p).則總體主成分有如1 2 p 1 p i i下的性質(zhì)：1.D(Z)，即P個(gè)主成分的方差為：Var(Zi互不相關(guān)的。

)i

(i1,2,…,p，且它們是性質(zhì)2.

ii

，通常稱(chēng)i

為X的總方差〔或稱(chēng)總慣量。iii1 i1 i13.主成分Zk

Xi

的相關(guān)系數(shù)(Ziiii

X為i(Zk

,X) a i k ik

(ki1,2p )并把主成分Zk

Xi

的相關(guān)系數(shù)稱(chēng)為因子負(fù)荷量。

p2(

,X)pa2k ik

1(i1,2,…,p。k i 5.令

pi1

2(Z

,X)i

(k1,2,…,p).EXi

)，Var(Xi

)2，即iX*X

E(X

)Xi

i1,,,p)ii iiiVar(X)iiX*X*,X*,…X*)”iVar(X)ii1 2 pRR動(dòng)身求主成分，記主成分向量為Z*

Z*,…Z*)”Z*有與總體主成分相應(yīng)的性質(zhì)：1.D(Z**

diag(**,…*，其中*

1

p…*R的特征值。

1 2 p

1 2 p性質(zhì)2.pii1

p.3.主成分Z*X*的相關(guān)系數(shù)(Z*,X*為k i k k(Z*,X*) *a* (ki1,2p)k k k ik其中a*

(a*

,…,a*

)”R對(duì)應(yīng)于*的單位正交特征向量。k性質(zhì)4.

1k pk 2(Z*,X*)p*(a*)2

1 (i1,2,…,p)

k k k ik性質(zhì)5.

2(Z*,X*)

*(a*)2

* (k1,,,p)k k2.樣本的主成分

k ik k1.X,X,…,Xn次觀測(cè)數(shù)據(jù)陣X1 2 p樣本相關(guān)陣R，且1Rn11

X”X(rij

)ppR

0aa

,…,a

1 2 pZj

1 2 pa”X (j1,,,p)j類(lèi)似總體主成分，樣本主成分也具有如下的性質(zhì)：1.Z

1nn

Z

(z1

p

)”0〔由于X0，而Z”Z

0,

當(dāng)ij,i j (n1) 當(dāng)i=j.i上式說(shuō)明當(dāng)ij時(shí)，第i個(gè)主成分得分向量Zj個(gè)主成分得分向量Z是相互正交的。i jp

…+

i1

p。稱(chēng)k為樣本主成分Zi p

的奉獻(xiàn)率；又稱(chēng)1p

m為樣本主成分Z,…,Z1

(mp的累計(jì)奉獻(xiàn)率。性質(zhì)3.樣本主成分具有是殘差平方和最小的優(yōu)良性。根本步驟xx⑴數(shù)據(jù)標(biāo)準(zhǔn)化ij

ij S

, (i1,2,n;j1,2pS為第j列的方j(luò)j差；R；IR0求出按大小排列特征值

以及相應(yīng)的特征向量e ；j j⑷選擇前幾個(gè)特征向量，確定主成分；根本命令主成分分析，其主要語(yǔ)句格式如下：Procprincomp <選項(xiàng)列表>；Var變量列表；Run；1〕Procprincomp功能如下：數(shù)據(jù)集。②out=2：命名一個(gè)輸出SAS數(shù)據(jù)集，其中包含原始數(shù)據(jù)以及各主成分得分〔即各主成分的觀測(cè)值。SAS數(shù)據(jù)集。④covariance(cov)：要求從協(xié)方差矩陣動(dòng)身作主成分分析，假設(shè)省略此選項(xiàng)，式的標(biāo)準(zhǔn)化，否則不宜使用此選項(xiàng)，應(yīng)從相關(guān)矩陣動(dòng)身作主成分分析。⑤N=n：指定要計(jì)算的主成分的個(gè)數(shù)，其默認(rèn)值為參與分析的變量的個(gè)數(shù)。⑥prefix=nameSAS系統(tǒng)自動(dòng)賜予各主成分名稱(chēng)分別為prin1,prin2,??；假設(shè)“name=A1，A28個(gè)字符?！?〕VARvariables;此句中的“variables”局部列出數(shù)據(jù)集中參與主成分分析的變量名稱(chēng)。假設(shè)省略此句，則被分析數(shù)據(jù)集中全部數(shù)值變量均參與分析。舉例1】(中學(xué)生身體四項(xiàng)指標(biāo)的主成分分析)在某中學(xué)隨機(jī)抽取某年級(jí)30生，測(cè)量其身高〔X1、體重〔X2、胸圍〔X3〕和坐高〔X4〔數(shù)據(jù)見(jiàn)教材P2777.4。試對(duì)這30datad721;cards;inputnumberx1-x4@@;cards;1148417278213934717631604977864149366779515945808661423166767153437683815043777991514277801013931687414029647412161477884158497883141403367771373166731615235737914947827918145357077160477487201564478851514273822214738737815739688024147306575157488088261513674801443668762814130677613932687330148387078;varx1-x4;procprincompdata=d721prefix=zout=o721;varx1-x4;runrun;procplotdata=o721;optionsps=32procplotdata=o721;plotz2*z1$number=”*”/href=-1href=2vref=0;runrun;procprocsortdata=o721;bybyz1;runrun;varnumberz1z2varnumberz1z2x1-x4;runrun;quitquit;過(guò)程由相關(guān)陣動(dòng)身進(jìn)展主成分分析。由下面的相關(guān)陣來(lái)看，〔也就是說(shuō)用較96.36%，因此只需用兩個(gè)主成分就能很好地概括這組數(shù)據(jù)。另由第四個(gè)特征值近似為此只需用兩個(gè)主成分就能很好地概括這組數(shù)據(jù)。另由第四個(gè)特征值近似為0，可Z0.496966X*Z0.496966X*0.514571X*0.480901X*0.506928X*1 1 2 3 4Z 0.543213X*0.210246X*0.724621X*0.368294X*2 1 2 3 40.5四周，而且都是正值，它反映學(xué)生的魁梧程度。44從其次主成分得分對(duì)第一主成分得分的散布圖看，很直觀地看出，按學(xué)生30名學(xué)生大約分三組〔以第一主成分得分值為-12為從其次主成分得分對(duì)第一主成分得分的散布圖看，很直觀地看出，按學(xué)生30名學(xué)生大約分三組〔以第一主成分得分值為-12為分界點(diǎn)〕。每一組包括哪幾名學(xué)生可由每一給散點(diǎn)旁邊的序號(hào)可以得知。訓(xùn)練題1138表1 某市工業(yè)部門(mén)13個(gè)行業(yè)8項(xiàng)指標(biāo)的數(shù)據(jù)全員勞動(dòng)百元固定能源利用年末固定工業(yè)總產(chǎn)資金利稅標(biāo)準(zhǔn)燃料職工人數(shù)產(chǎn)率原資產(chǎn)值效果資產(chǎn)凈值值率消費(fèi)量〔人〕〔元/人實(shí)現(xiàn)產(chǎn)值〔萬(wàn)元/〔萬(wàn)元〕〔萬(wàn)元〕〔％〕〔噸〕年〕〔元〕噸〕1(冶金)90342524551010911927282.00016.1001974350.1722(電力)4903197320351031334.2007.1005920770.0033(煤炭)6735211393767178036.1008.2007263960.0034(化學(xué))4945436241815572250498.10025.9003482260.9855(機(jī)械)1391902035052158981060993.20012.6001395720.6286(建材)122151621910351638262.5008.7001458180.0667(森工)23726572810312329184.40022.200209210.1528(食品)11062230785493523804370.40041.000654860.2639(紡織)17111239075210821796221.50021.500638060.27610(縫紉)12063930612615586330.40029.50018400.43711(皮革)21505704620010870184.20012.00089130.27412(造紙)525161551038316875146.40027.500787960.15113(文教藝術(shù)用1434113203193961469194.60017.80063541.574品)〔綜合變量15％，應(yīng)取幾個(gè)主成分；并對(duì)這幾個(gè)主成分進(jìn)展解釋?zhuān)焕弥鞒煞值梅謱?duì)13個(gè)行業(yè)進(jìn)展排序和分類(lèi)。83對(duì)所選取的主成分作出解釋。變量企業(yè)號(hào)凈產(chǎn)值利潤(rùn)率固定資產(chǎn)利潤(rùn)變量企業(yè)號(hào)凈產(chǎn)值利潤(rùn)率固定資產(chǎn)利潤(rùn)總產(chǎn)值利潤(rùn)率銷(xiāo)售收入利潤(rùn)產(chǎn)品成本利潤(rùn)物耗利潤(rùn)率人均利潤(rùn)率流淌資金利潤(rùn)率率率率140.424.77.26.18.38.72.44220.0225.012.711.211.012.920.23.5429.1313.23.33.94.34.45.50.5783.6422.36.75.63.76.07.40.1767.3534.311.87.17.18.08.91.72627.5635.612.516.416.722.829.33.01726.6722.07.89.910.212.617.60.84710.6848.413.410.99.910.913.91.77217.8940.619.119.819.029.739.62.44935.81024.88.09.88.911.916.20.78913.71112.59.74.24.24.66.50.8743.9121.80.60.70.70.81.10.0561.01332.313.99.48.39.813.32.12617.11438.59.111.39.512.216.41.32711.6【模型一】美國(guó)各種類(lèi)型犯罪的主成分模型〔1數(shù)據(jù)進(jìn)展主成分分析，并分析犯罪特征。1：美國(guó)各個(gè)州的各種類(lèi)型犯罪率數(shù)據(jù)MurderRapeRobberyAssaultBurglaryLarcenyAutoALABAMA14.225.296.8278.31135.51881.9280.7ALASKA10.851.696.8284.01331.73369.8753.3ARIZONA9.534.2138.2312.32346.14467.4439.5ARKANSAS8.827.683.2203.4972.61862.1183.4CALIFORNIA11.549.4287.0358.02139.43499.8663.5COLORADO6.342.0170.7292.91935.23903.2477.1CONNECTICUT4.216.8129.5131.81346.02620.7593.2DELAWARE6.024.9157.0194.21682.63678.4467.0FLORIDA10.239.6187.9449.11859.93840.5351.4GEORGIA11.731.1140.5256.51351.12170.2297.9HAWAII7.225.5128.064.11911.53920.4489.4IDAHO5.519.439.6172.51050.82599.6237.6ILLINOIS9.921.8211.3209.01085.02828.5528.6INDIANA7.426.5123.2153.51086.22498.7377.4IOWA2.310.641.289.8812.52685.1219.9KANSAS6.622.0100.7180.51270.42739.3244.3KENTUCKY10.119.181.1123.3872.21662.1245.4LOUISIANA15.530.9142.9335.51165.52469.9337.7MAINE2.413.538.7170.01253.12350.7246.9MARYLAND8.034.8292.1358.91400.03177.7428.5MASSACHUSETTS3.120.8169.1231.61532.22311.31140.1MICHIGAN9.338.9261.9274.61522.73159.0545.5MINNESOTA2.719.585.985.81134.72559.3343.1MISSISSIPPI14.319.665.7189.1915.61239.9144.4MISSOURI9.628.3189.0233.51318.32424.2378.4MONTANA5.416.739.2156.8804.92773.2309.2NEBRASKA3.918.164.7112.7760.02316.1249.1NEVADA15.849.1323.1355.02453.14212.6559.2NEWHAMPSHIRE3.210.723.276.01041.72343.9293.4NEWJERSEY5.621.0180.4185.11435.82774.5511.5NEWMEXICO8.839.1109.6343.41418.73008.6259.5NEWYORK10.729.4472.6319.11728.02782.0745.8NORTHAROLINA10.617.061.3318.31154.12037.8192.1NORTHDAKOTA0.99.013.343.8446.11843.0144.7OHIO7.827.3190.5181.11216.02696.8400.4OKLAHOMA8.629.273.8205.01288.22228.1326.8OREGON4.939.9124.1286.91636.43506.1388.9PENNSYLVANIA5.619.0130.3128.0877.51624.1333.2RHODEISLAND3.610.586.5201.01489.52844.1791.4SOUTHCAROLINA11.933.0105.9485.31613.62342.4245.1SOUTHDAKOTA2.013.517.9155.7570.51704.4147.5TENNESSEE10.129.7145.8203.91259.71776.5314.0TEXAS13.333.8152.4208.21603.12988.7397.6UTAH3.520.368.8147.31171.63004.6334.5VERMONT1.415.930.8101.21348.22201.0265.2VIRGINIA9.023.392.1165.7986.22521.2226.7WASHINGTON4.339.6106.2224.81605.63386.9360.3WESTVIRG