Matlab與Logistic回歸

1、Matlab軟件包與Logistic回歸在回歸分析中，因變量y可能有兩種情形：（1） y是一個(gè)定量的 變量，這時(shí)就用通常的regress函數(shù)對(duì)y進(jìn)行回歸；（2） y是一個(gè)定 性的變量，比如，y = 0或1,這時(shí)就不能用通常的regress函數(shù)對(duì)y進(jìn) 行回歸，而是使用所謂的Logistic回歸。Logistic回歸的基本思想是，不是直接對(duì)y進(jìn)行回歸，而是先定義 一種概率函數(shù)二，令二二 Pr Y = 1| X二片，X2 = X2， , Xn 二 Xn要求0乞二叮。此時(shí)，如果直接對(duì)二進(jìn)行回歸，得到的回歸方程可能不 滿足這個(gè)條件。在現(xiàn)實(shí)生活中，一般有 0 ：二：1。直接求二的表達(dá)式， 是比較困難的一件

2、事，于是，人們改為考慮1 -二 y = 1的概率k兀 y =1的概率一般的，0 ：k。人們經(jīng)過研究發(fā)現(xiàn)，令二 Xn 二二 Pr Y =1| X,a 0,bj 一0即，二是一個(gè)Logistic型的函數(shù)，效果比較理想。于是，我們將其變形得到：(1nlog二 6 - J/ 一 - bnXn12然后，對(duì)log 進(jìn)行通常的線性回歸#例1 企業(yè)到金融商業(yè)機(jī)構(gòu)貸款，金融商業(yè)機(jī)構(gòu)需要對(duì)企業(yè)進(jìn)行評(píng)估。例如，Moody公司就是New York的一家專門評(píng)估企業(yè)的貸款信譽(yù)的公司。設(shè)：“°，企業(yè)2年后破產(chǎn)一1,企業(yè)2年后具備還款能力F面列出美國(guó)66家企業(yè)的具體情況:YX1X2X30-62.8-89.51.7

3、03.3-3.51.10-120.8-103.22.50-18.1-28.81.10-3.8-50.60.90-61.2-56.21.70-20.3-17.41.00-194.5-25.80.5020.8-4.31.00-106.1-22.91.50-39.4-35.71.20-164.1-17.71.30-308.9-65.80.807.2-22.62.00-118.3-34.21.50-185.9-280.06.70-34.6-19.43.40-27.96.31.30-48.26.81.60-49.2-17.20.30-19.2-36.70.80-18.1-6.50.90-98.0-20.

4、81.70-129.0-14.21.30-4.0-15.82.10-8.7-36.32.80-59.2-12.82.10-13.1-17.60.90-38.01.61.20-57.90.70.80-8.8-9.10.90-64.7-4.00.10-11.44.80.9143.016.41.3147.016.01.91-3.34.02.7135.020.81.9146.712.60.9120.812.52.4133.023.61.5126.110.42.1168.613.81.6137.333.43.5159.023.15.5149.623.81.9112.57.01.8137.334.11.5

5、135.34.20.9149.525.12.6118.113.54.0131.415.71.9121.5-14.41.018.55.81.5140.65.81.8134.626.41.8119.926.72.3117.412.61.3154.714.61.7153.520.61.1135.926.42.0139.430.51.9153.17.11.9139.813.81.2159.57.02.0116.320.41.0121.7-7.81.6其中，未分配利潤(rùn)支付利息前的利潤(rùn)銷售額XXX ：1 總資產(chǎn)2總資產(chǎn)3總資產(chǎn)建立破產(chǎn)特征變量y的回歸方程。解：在這個(gè)破產(chǎn)問題中,1 一二y - 1的次數(shù)1n

6、y = 1的次數(shù)我們討論log ，概率二=0,1。設(shè)二二企業(yè)2年后具備還款能力的概率，即,I只丿二二企業(yè)不破產(chǎn)的概率。因?yàn)?6個(gè)數(shù)據(jù)有33個(gè)為0, 33個(gè)為1,所以，取分界 值0.5,令丄0,愿玄0.5y =1,二 0.5由于我們并不知道企業(yè)在沒有破產(chǎn)前概率二的具體值，也不可能通過X1,X2,X3的數(shù)據(jù)把這個(gè)具體的概率值算出來，于是，為了方便做回歸運(yùn)算，我們?nèi)^(qū)間的中值，y=0對(duì)應(yīng)二=0.25;y =1,對(duì)應(yīng)二=0.75。數(shù)據(jù)表變?yōu)?兀X1X2X30.25-62.8-89.51.70.253.3-3.51.10.25-120.8-103.22.50.25-18.1-28.81.10.25-3.

7、8-50.60.90.25-61.2-56.21.70.25-20.3-17.41.00.25-194.5-25.80.50.2520.8-4.31.00.25-106.1-22.91.50.25-39.4-35.71.20.25-164.1-17.71.30.25-308.9-65.80.80.257.2-22.62.00.25-118.3-34.21.50.25-185.9-280.06.70.25-34.6-19.43.40.25-27.96.31.30.25-48.26.81.60.25-49.2-17.20.30.25-19.2-36.70.80.25-18.1-6.50.90.25

8、-98.0-20.81.70.25-129.0-14.21.30.25-4.0-15.82.10.25-8.7-36.32.80.25-59.2-12.82.10.25-13.1-17.60.90.25-38.01.61.20.25-57.90.70.80.25-8.8-9.10.90.25-64.7-4.00.10.25-11.44.80.90.7543.016.41.30.7547.016.01.90.75-3.34.02.70.7535.020.81.90.7546.712.60.90.7520.812.52.40.7533.023.61.50.7526.110.42.10.7568.6

9、13.81.60.7537.333.43.50.7559.023.15.50.7549.623.81.90.7512.57.01.80.7537.334.11.50.7535.34.20.90.7549.525.12.60.7518.113.54.00.7531.415.71.90.7521.5-14.41.00.758.55.81.50.7540.65.81.80.7534.626.41.80.7519.926.72.30.7517.412.61.30.7554.714.61.70.7553.520.61.10.7535.926.42.00.7539.430.51.90.7553.17.11

10、.90.7539.813.81.20.7559.57.02.00.7516.320.41.00.7521.7-7.81.6于是'在Matlab軟件包中編程如下'對(duì)log U進(jìn)行通常的線性回歸:X=1,-62.8,-89.5,1.7;1,3.3,-3.5,1.1;1,-120.8,-103.2,2.5;1,-18.1,-28.8,1.1;1,-3.8,-50.6,0.9;1,-61.2,-56.2,1.7;1,-20.3,-17.4,1;1,-194.5,-25.8,0.5;1,20.8,-4.3,1;1,-106.1,-22.9,1.5;1,-39.4,-35.7,1.2;1,

11、-164.1,-17.7,1.3;1,-308.9,-65.8,0.8;1,7.2,-22.6,2.0;1,-118.3,-34.2,1.5;1,-185.9,-280,6.7;1,-34.6,-19.4,3.4;1,-27.9,6.3,1.3;1,-48.2,6.8,1.6;1,-49.2,-17.2,0.3;1,-19.2,-36.7,0.8;1,-18.1,-6.5,0.9;1,-98,-20.8,1.7;1,-129,-14.2,1.3;1,-4,-15.8,2.1;1,-8.7,-36.3,2.8;1,-59.2,-12.8,2.1;1,-13.1,-17.6,0.9;1,-38,1

12、.6,1.2;1,-57.9,0.7,0.8;1,-8.8,-9.1,0.9;1,-64.7,-4,0.1;1,-11.4,4.8,0.9;1,43,16.4,1.3;1,47,16,1.9;1,-3.3,4,2.7;1,35,20.8,1.9;1,46.7,12.6,0.9;1,20.8,12.5,2.4;1,33,23.6,1.5;1,26.1,10.4,2.1;1,68.6,13.8,1.6;1,37.3,33.4,3.5;1,59,23.1,5.5;1,49.6,23.8,1.9;1,12.5,7,1.8;1,37.3,34.1,1.5;1,35.3,4.2,0.9;1,49.5,25

13、.1,2.6;1,18.1,13.5,4;1,31.4,15.7,1.9;1,21.5,-14.4,1;1,8.5,5.8,1.5;1,40.6,5.8,1.8;1,34.6,26.4,1.8;1,19.9,26.7,2.3;1,17.4,12.6,1.3;1,54.7,14.6,1.7;1,53.5,20.6,1.1;1,35.9,26.4,2;1,39.4,30.5,1.9;1,53.1,7.1,1.9;1,39.8,13.8,1.2;1,59.5,7,2;1,16.3,20.4,1;1,21.7,-7.8,1.6;a0=0.25*ones(33,1);a1=0.75*ones(33,1)

14、;y0=a0;a1;Y=log(1-y0)./y0);b,bint,r,rint,stats =regress(Y,X)rcoplot(r,rint)執(zhí)行后得到結(jié)果：b =0.3914-0.0069-0.0093-0.3263bint =0.0073 0.7755-0.0105 -0.0032-0.0156 -0.0030-0.5253 -0.1273r =-0.00371.0561-0.26830.67330.50280.31790.7320 -0.70441.13610.25530.4955 -0.1593 -1.76431.19840.0662 -0.99371.39830.99880.

15、96210.30720.49420.81610.39570.11411.21761.22250.86700.74680.85310.57770.85560.25880.9675 -0.6179 -0.3984 -0.5943 -0.4360 -0.7585 -0.4476 -0.5541 -0.5288 -0.36870.21940.9248 -0.3078 -0.7516-0.4266-0.9150-0.0680 0.0653-0.5082-1.1506-0.8882-0.5701-0.4191-0.3540-0.8289-0.4239-0.5720-0.3449-0.3153-0.4396

16、-0.6967-0.3640-0.8616-0.8919 rint =-1.4320-0.3990-1.6975-0.7882-0.9222-1.1498-0.7332-2.0696-0.3070-1.2048-0.9730-1.5626-2.9063-0.2499-1.3925-1.7217-0.0051-0.4609-0.4909-1.1505-0.9556-0.64771.42452.51131.16082.13491.92771.78562.19710.66092.57911.71541.96401.2441-0.62232.64661.5249-0.26572.80182.45852

17、.41521.76491.94392.27991.8562-1.0648-1.3238-0.2340-0.2162-0.5911-0.7136-0.6117-0.8868-0.6044-1.1944-0.4914-2.0862-1.8729-2.0558-1.9108-2.2125-1.9186-2.0271-2.0034-1.8340-1.1951-0.3186-1.7819-2.2238-1.8981-2.3643-1.5319-1.3378-1.9834-2.5850-2.3556-2.0422-1.8929-1.8195-2.2961-1.8955-2.0355-1.8178-1.78

18、76-1.9105-2.1620-1.8335-2.32371.55212.66922.66132.32502.20732.31782.04212.31561.71202.42640.85041.07600.86711.03890.69551.02340.91900.94591.09671.63402.16811.16620.72051.04490.53421.39591.46830.96690.28390.57930.90201.05471.11160.63831.04760.89161.12801.15711.03130.76861.10550.60050.5707-2.3544 stat

19、s =0.569927.38410.00000.5526即，得到：R2值二0.5699 （說明回歸方程刻畫原問題不是太好），F(xiàn)_檢驗(yàn)值二27.3841>0.0000 （這個(gè)值比較好），與顯著性概率a =0.05相關(guān)的p值=0.5526>，-0.05，說明變量xi, X2, X3之間存在線性相關(guān)關(guān)系?；貧w方程為：'1 一匚log 1 =0.3914-0.0069-0.0093x2 -0.3263x3I 5丿1-0.39140.0069為-0.0093X20.3263X31 e以及殘差圖: 通過殘差圖看出，殘差連續(xù)的出現(xiàn)在0的上方，或者連續(xù)地出現(xiàn)在0的下方，這 也暗示變量X1，

20、X2，X3之間存在線性相關(guān)。編程計(jì)算它們的相關(guān)系數(shù)：X=1,-62.8,-89.5,1.7;1,3.3,-3.5,1.1;1,-120.8,-103.2,2.5;1,-18.1,-28.8,1.1;1,-3.8,-50.6,0.9;1,-61.2,-56.2,1.7;1,-20.3,-17.4,1;1,-194.5,-25.8,0.5;1,20.8,43,1;1,-106.1,-22.9,1.5;1,-39.4,-35.7,1.2;1,-164.1,-17.7,1.3;1,-308.9,-65.8,0.8;1,7.2,-22.6,2.0;1,-118.3,-34.2,1.5;1,-185.9,

21、-280,6.7;1,-34.6,-19.4,3.4;1,-27.9,6.3,1.3;1,-48.2,6.8,1.6;1,-49.2,-17.2,0.3;1,-19.2,-36.7,0.8;1,-18.1,-6.5,0.9;1,-98,-20.8,1.7;1,-129,-14.2,1.3;1,-4,-15.8,2.1;1,-8.7,-36.3,2.8;1,-59.2,-12.8,2.1;1,-13.1,-17.6,0.9;1,-38,1.6,1.2;1,-57.9,0.7,0.8;1,-8.8,-9.1,0.9;1,-64.7,-4,0.1;1,-11.4,4.8,0.9;1,43,16.4,

22、1.3;1,47,16,1.9;1,-3.3,4,2.7;1,35,20.8,1.9;1,46.7,12.6,0.9;1,20.8,12.5,2.4;1,33,23.6,1.5;1,26.1,10.4,2.1;1,68.6,13.8,1.6;1,37.3,33.4,3.5;1,59,23.1,5.5;1,49.6,23.8,1.9;1,12.5,7,1.8;1,37.3,34.1,1.5;1,35.3,4.2,0.9;1,49.5,25.1,2.6;1,18.1,13.5,4;1,31.4,15.7,1.9;1,21.5,-14.4,1;1,8.5,5.8,1.5;1,40.6,5.8,1.8

23、;1,34.6,26.4,1.8;1,19.9,26.7,2.3;1,17.4,12.6,1.3;1,54.7,14.6,1.7;1,53.5,20.6,1.1;1,35.9,26.4,2;1,39.4,30.5,1.9;1,53.1,7.1,1.9;1,39.8,13.8,1.2;1,59.5,7,2;1,16.3,20.4,1;1,21.7,-7.8,1.6;X1=X(:,2);X2=X(:,3);X3=X(:,4);corrcoef(X1,X2)corrcoef(X1,X3)corrcoef(X2,X3)15執(zhí)行后得到結(jié)果： ans =1.0000 0.64090.6409 1.0000

24、 ans =1.0000 0.04670.0467 1.0000 ans =1.0000 -0.3501-0.3501 1.0000可見corrcoef(X1,X2) = 0.64，這說明，在做回歸時(shí)，可以去掉為列，或者去掉X2列。根據(jù)經(jīng)濟(jì)意義，我們?nèi)サ鬤i列，再進(jìn)行回歸X=1,-62.8,-89.5,1.7;i,3.3,-3.5,i.i;i,-i20.8,-i03.2,2.5;i,-i8.i,-28.8,i.i;i,-3.8,-50.6,0.9;i,-6i.2,-56.2,i.7;i,-20.3,-i7.4,i;i,-i94.5,-25.8,0.5;i,20.8,-4.3,i;i,-i06.

25、i,-22.9,i.5;i,-39.4,-35.7,i.2;i,-i64.i,-i7.7,i.3;i,-308.9,-65.8,0.8;i,7.2,-22.6,2.0;i,-ii8.3,-34.2,i.5;i,-i85.9,-280,6.7;i,-34.6,-i9.4,3.4;i,-27.9,6.3,i.3;i,-48.2,6.8,i.6;i,-49.2,-i7.2,0.3;i,-i9.2,-36.7,0.8;i,-i8.i,-6.5,0.9;i,-98,-20.8,i.7;i,-i29,-i4.2,i.3;i,-4,-i5.8,2.i;i,-8.7,-36.3,2.8;i,-59.2,-i2

26、.8,2.i;i,-i3.i,-i7.6,0.9;i,-38,i.6,i.2;i,-57.9,0.7,0.8;1,-8.8,-9.1,0.9;1,-64.7,-4,0.1;1,-11.4,4.8,0.9;1,43,16.4,1.3;1,47,16,1.9;1,-3.3,4,2.7;1,35,20.8,1.9;1,46.7,12.6,0.9;1,20.8,12.5,2.4;1,33,23.6,1.5;1,26.1,10.4,2.1;1,68.6,13.8,1.6;1,37.3,33.4,3.5;1,59,23.1,5.5;1,49.6,23.8,1.9;1,12.5,7,1.8;1,37.3,3

27、4.1,1.5;1,35.3,4.2,0.9;1,49.5,25.1,2.6;1,18.1,13.5,4;1,31.4,15.7,1.9;1,21.5,-14.4,1;1,8.5,5.8,1.5;1,40.6,5.8,1.8;1,34.6,26.4,1.8;1,19.9,26.7,2.3;1,17.4,12.6,1.3;1,54.7,14.6,1.7;1,53.5,20.6,1.1;1,35.9,26.4,2;1,39.4,30.5,1.9;1,53.1,7.1,1.9;1,39.8,13.8,1.2;1,59.5,7,2;1,16.3,20.4,1;1,21.7,-7.8,1.6;a0=0.

28、25*ones(33,1);a1=0.75*ones(33,1);y0=a0;a1;Y=log(1-y0)./y0);X1=X(:,2);X2=X(:,3);X3=X(:,4);E=ones(66,1);B=E,X2,X3;b,bint,r,rint,stats =regress(Y,B)rcoplot(r,rint)執(zhí)行后得到：b =0.6594-0.0177-0.4676bint =0.2672 1.0516-0.0226 -0.0127-0.6702 -0.2649-0.3478 0.8917 -0.2159 0.4445 -0.0343 0.2408 0.5992 0.2170 0.8

29、308 0.7358 0.3693 0.7342 -0.3497 0.9749 0.5361 -1.3769 1.6861 1.1584 1.3075 0.2755 0.1646 0.7451 0.8665 0.7961 1.1419 1.1068 1.1949 0.5489 1.0286 0.8256 0.6992 0.4153 0.9449 -0.8603 -0.5868-0.4249-0.5020-1.1145-0.4149-0.6395-0.5923-0.76600.46881.2219-0.4490-0.7927-0.4540-1.2630-0.09870.3509-0.5921-1

30、.5450-0.9541-0.8139-0.4498-0.2107-0.9275-0.7051-0.8796-0.3563-0.3306-0.7441-0.9530-0.6992-0.9299-1.1478 rint =1.23252.50541.35602.06361.56961.85582.21731.82372.44492.35611.98822.3537-1.9280-0.7220-1.7877-1.1746-1.6382-1.3743-1.0189-1.3898-0.7833-0.8845-1.2496-0.885318-1.9330 -0.6385 -1.0852 -2.1813

31、0.1435 -0.4463 -0.2909 -1.3275 -1.4460 -0.8695 -0.7514 -0.8222 -0.4645 -0.4883 -0.4091 -1.0680 -0.5813 -0.7851 -0.9163 -1.1827 -0.6638 -2.4750 -2.2082 -2.0392 -2.1230 -2.7155 -2.0332 -2.2586 -2.2133 -2.3850 -1.0894 -0.1453 -2.0695 -2.4121 -2.0716 -2.8575 -1.7076 -1.1978 -2.2135 -3.1230 -2.5686 -2.43

32、29 -2.0699 -1.82581.23352.58832.1574-0.57243.22862.76312.90591.87851.77522.35972.48432.41442.74822.70202.79882.16592.63842.43642.31462.01322.55350.75431.03451.18941.11900.48651.20340.97951.02870.85312.02702.58921.17150.82681.16370.33151.51021.89951.02920.03310.66030.80521.17041.404419-2.54070.6858-2

33、.32540.9152-2.49080.7316-1.97551.2629-1.94901.2879-2.36440.8761-2.56430.6582-2.31980.9215-2.53830.6785-2.75540.4598stats =0.4716 28.11750.00000.6681以及殘差圖：殘差圖仍然顯示變量之間的相關(guān)性，這說明，最開始調(diào)查數(shù)據(jù)時(shí)，3個(gè)指標(biāo)沒有選好。最后得到：1 -兀 2 Jlog -=0.6594-0.0177x2 -0.4676x3I心丿_ 12 _0.6594-0.0177x2 -0.4676x31 e將企業(yè)的具體數(shù)據(jù)X2, X3代入二的表達(dá)式計(jì)算，再結(jié)合

34、0 <0.510.5金融機(jī)構(gòu)就可以知道，是否應(yīng)該貸款給這家企業(yè) 注：一個(gè)通常的Regress回歸，可以用R2, R2-., F-test等參數(shù)評(píng)價(jià)回歸結(jié)果的好壞，但對(duì)Logistic回歸來說，不存在這樣簡(jiǎn)單而令人滿意的評(píng)價(jià)參數(shù)，所以，一 般應(yīng)該進(jìn)行回歸診斷。Logistic回歸的診斷所謂的Logistic回歸診斷，就是將Xi的原始數(shù)據(jù)代入求得的回歸方程中，計(jì)算y值，看看有多少個(gè)由回歸方程計(jì)算所得的 y值與原始的y值不同，因而判斷回歸方程的好壞。1(1) 用回歸方程 10.3914 _0.0069為 _0.0093x2 _0.3263x3 進(jìn)行診斷。1 + e在Matlab軟件包中，編程診斷X=1,-62.8,-89.5,1.7;1,3.