Matlab软件包与Logistic回归

1、* *1然后，对log进行通常的线性回归。例如，Logistic型概率函数1 300e2x的图形如下:ezplot(W(1+300*exp(-2*x),0,10)Matlab软件包与 Logistic 回归在回归分析中，因变量y可能有两种情形：（1）y是一个定量的变量，这 时就用通常的regress函数对y进行回归；（2）y是一个定性的变量，比如，y0或1,这时就不能用通常的regress函数对y进行回归，而是使用所谓的Logistic回归。Logistic回归的基本思想是，不是直接对y进行回归，而是先定义一种概率 函数，令Pr Y 1|XiXi,X2X2, ,XnXn事，于是，人们改为考虑y

2、 1 的概率y 1 的概率a 0, bj0要求01。此时，如果直接对进行回归，得到的回归方程可能不满足这个条件。在现实生活中，一般有01。直接求的表达式，是比较困难的一件般的，0 k。人们经过研究发现，令Pr Y 1|X!xX2X2, ,XnXnbX1bnXn即， 是一个Logistic型的函数，效果比较理想于是,我们将其变形得到:log -bnXn* *例1企业到金融商业机构贷款，金融商业机构需要对企业进行评估。例如,Moody公司就是New York的一家专门评估企业的贷款信誉的公司。设:0,企业2年后破产y1企业2年后具备还款能力F面列出美国66家企业的具体情况:YX1X2X30-62.

3、8-89.51.703.3-3.51.10-120.8-103.22.50-18.1-28.81.10-3.8-50.60.90-61.2-56.21.70-20.3-17.41.00-194.5-25.80.5020.8-4.31.00-106.1-22.91.50-39.4-35.71.20-164.1-17.71.3* *0-308.9-65.80.807.2-22.62.00-118.3-34.21.50-185.9-280.06.70-34.6-19.43.40-27.96.31.30-48.26.81.60-49.2-17.20.30-19.2-36.70.80-18.1-6.50

4、.90-98.0-20.81.70-129.0-14.21.30-4.0-15.82.10-8.7-36.32.80-59.2-12.82.10-13.1-17.60.90-38.01.61.20-57.90.70.80-8.8-9.10.90-64.7-4.00.10-11.44.80.9143.016.41.3* *147.016.01.91-3.34.02.7135.020.81.9146.712.60.9120.812.52.4133.023.61.5126.110.42.1168.613.81.6137.333.43.5159.023.15.5149.623.81.9112.57.0

5、1.8137.334.11.5135.34.20.9149.525.12.6118.113.54.0131.415.71.9121.5-14.41.018.55.81.5140.65.81.8134.626.41.8119.926.72.3* *117.412.61.3154.714.61.7153.520.61.1135.926.42.0139.430.51.9153.17.11.9139.813.81.2159.57.02.0116.320.41.0121.7-7.81.6其中,解：在这个破产问题中,1 的次数，1y 1 的次数0,0.5y1,0.51我们讨论log -，概率0,1设 二企

6、业2年后具备还款能力的概率， 即,=企业不破产的概率。因为66个数据有33个为0,33个为1，所以，取分界值0.5，令X1未分配利润总资产X2支付利息前的利润总资产销售额X3总资产建立破产特征变量y的回归方程。* *由于我们并不知道企业在没有破产前概率的具体值，也不可能通过* *X1,X2,X3的数据把这个具体的概率值算出来，于是，为了方便做回归运算,我们取区间的中值，y 0 对应 0.25; y 1,对应0.75。数据表变为：X1X2X30.25-62.8-89.51.70.253.3-3.51.10.25-120.8-103.22.50.25-18.1-28.81.10.25-3.8-50

7、.60.90.25-61.2-56.21.70.25-20.3-17.41.00.25-194.5-25.80.50.2520.8-4.31.00.25-106.1-22.91.50.25-39.4-35.71.20.25-164.1-17.71.30.25-308.9-65.80.80.257.2-22.62.00.25-118.3-34.21.50.25-185.9-280.06.70.25-34.6-19.43.4* *0.25-27.96.31.30.25-48.26.81.6* *0.25-49.2-17.20.30.25-19.2-36.70.80.25-18.1-6.50.90.

8、25-98.0-20.81.70.25-129.0-14.21.30.25-4.0-15.82.10.25-8.7-36.32.80.25-59.2-12.82.10.25-13.1-17.60.90.25-38.01.61.20.25-57.90.70.80.25-8.8-9.10.90.25-64.7-4.00.10.25-11.44.80.90.7543.016.41.30.7547.016.01.90.75-3.34.02.70.7535.020.81.90.7546.712.60.90.7520.812.52.40.7533.023.61.50.7526.110.42.1* *0.7

9、568.613.81.60.7537.333.43.50.7559.023.15.50.7549.623.81.90.7512.57.01.80.7537.334.11.50.7535.34.20.90.7549.525.12.60.7518.113.54.00.7531.415.71.90.7521.5-14.41.00.758.55.81.50.7540.65.81.80.7534.626.41.80.7519.926.72.30.7517.412.61.30.7554.714.61.70.7553.520.61.10.7535.926.42.00.7539.430.51.90.7553.

10、17.11.90.7539.813.81.2* *1,-34.6,-19.4,3.4;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,-3

11、4.2,1.5;1,-185.9,-280,6.7;0.7559.57.02.00.7516.320.41.00.7521.7-7.81.6于是，在Matlab1软件包中编程如下，对log1进行通常的线性回归:* *1,-34.6,-19.4,3.4;1,-27.9,6.3,1.3;* *1,20.8,12.5,2.4;1,-48.2,6.8,1.6;1,-492-17.2,0.3;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、,-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;1,43,16.4,1.3;1,47,16,1.9;1,33,4,2.7;1,35,20.8,1.9;1,46.7,12.6,0.9;1,33,23.6,1.5;* *1,53.1,7.1,1.9;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

13、.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;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;* *-0.00371,39.8,13.8,1.2;1,59.5,7,2;1,16.3,20.4,1;1,21.7,-7.8,1.6;a0=0.2

14、5*o nes(33,1);a1=0.75*o nes(33,1);y0=a0;a1;丫=log(1-y0)./y0);b,b in t,r,ri nt,stats =regress(Y,X)rcoplot(r,ri nt)执行后得到结果：b =0.3914-0.0069-0.0093-0.3263bint =0.00730.7755-0.0105-0.0032-0.0156-0.0030-0.5253-0.1273* *1.0561-0.26830.67330.50280.31790.7320 -0.70441.13610.25530.4955 -0.1593 -1.76431.19840.

15、0662-0.99371.39830.99880.96210.30720.49420.81610.3957* *-0.30780.11411.21761.22250.86700.74680.85310.57770.85560.25880.9675 -0.6179 -0.3984 -0.5943 -0.4360-0.7585 -0.4476 -0.5541 -0.5288 -0.36870.21940.9248* *-0.8919-0.7516-0.4266-0.9150-0.06800.0653-0.5082-1.1506-0.8882-0.5701-0.4191-0.3540-0.8289-

16、0.4239-0.5720-0.3449-0.3153-0.4396-0.6967-0.3640-0.8616 rint =* *-0.64772.2799-1.43201.4245-0.39902.5113-1.69751.1608-0.78822.1349-0.92221.9277-1.14981.7856-0.73322.1971-2.06960.6609-0.30702.5791-1.20481.7154-0.97301.9640-1.56261.2441-2.9063-0.6223-0.24992.6466-1.39251.5249-1.7217-0.2657-0.00512.801

17、8-0.46092.4585-0.49092.4152-1.15051.7649-0.95561.9439* *-0.31862.1681-1.0648-1.3238-0.2340-0.2162-0.5911-0.7136-0.6117-0.8868-0.6044-1.1944-0.4914-2.0862-1.8729-2.0558-1.9108-2.2125-1.9186-2.0271-2.0034-1.8340-1.19511.85621.55212.66922.66132.32502.20732.31782.04212.31561.71202.42640.85041.07600.8671

18、1.03890.69551.02340.91900.94591.09671.6340* *-2.35440.5707-1.7819-2.2238-1.8981-2.3643-1.5319-1.3378-1.9834-2.5850-2.3556-2.0422-1.8929-1.8195-2.2961-1.8955-2.0355-1.8178-1.7876-1.9105-2.1620-1.8335-2.32371.16620.72051.04490.53421.39591.46830.96690.28390.57930.90201.05471.11160.63831.04760.89161.128

19、01.15711.03130.76861.10550.6005* *1,-18.1,-28.8,1.1;1,-3.8,-50.6,0.9;stats =0.569927.38410.00000.5526即，得到：R2值二0.5699（说明回归方程刻画原问题不是太好），FJ检验值二27.38410.0000（这个值比较好），与显著性概率0.05相关的p值=0.55260.05，说明变量Xi,X2,X3之间存在线性相关关系。回归方程为:log110.3914 0.0069xi0.0093x20.3263x310.3914 0.0069x-!0.0093x20.3263x3e以及残差图:询2030崛

20、tuHJ通过残差图看出，残差连续的出现在0的上方，或者连续地出现在0的下方，这也暗示变量X1,X2,X3之间存在线性相关。编程计算它们的相关系数：X=1,-62.8,-89.5,1.7;1,3.3,35,1.1;1,-120.8,-103.2,2.5;1,-61.2,-56.2,1.7;* *1,-59.2,-12.8,2.1;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

21、.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,-13.1,-17.6,0.9;* *1,35.3,4.2,0.9;1,49.5,25.1,2.6;1,-38,1.6,1.2;1,-57.9,0.7,0.8;1,-8.8,-9.

22、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;1,33,4,2.7;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;* *1,18.1,13.5,4;1,31.4,15.7,1.9;1,21.5,-14.4,1;1,8.5,5.8,1

23、.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;X仁X(:,2);X2=X(:,3);X3=X(:,4);corrcoef(X1,X2)corrcoef(X1,X3)corrcoef(X2,X3)执行后得到结果:* *1,-106.1,-22.9,

24、1.5;1,-39.4,-35.7,1.2;ans =ans =ans =1.0000-0.3501-0.35011.0000可见corrcoef(X1,X2)=0.64，这说明，在做回归时，可以去掉捲列，或者去掉沁列。根据经济意义，我们去掉 为列，再进行回归。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;1.00000.64090.

25、64091.00001.00000.04670.04671.0000* *1,-64.7,-4,0.1;1,-11.4,4.8,0.9;1,-164.1,-17.7,1.3;1,-308.9,-65.8,0.8;1,72-22.6,2.0;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,-1

26、5.8,2.1;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;* *1,34.6,26.4,1.8;1,43,16.4,1.3;1,47,16,1.9;1,33,4,2.7;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

27、;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;* *-0.01771,19.9,26.7,2.3;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.

28、7,-7.8,1.6;a0=0.25*o nes(33,1);a1=0.75*o nes(33,1);y0=a0;a1;丫=log(1-y0)./y0);X1= X(:,2);X2=X(:,3);X3=X(:,4);E=o nes(66,1);B=E,X2,X3;b,b in t,r,ri nt,stats =regress(Y,B)rcoplot(r,ri nt)执行后得到：b =0.6594* *-1.3769-0.4676bint =0.26721.0516-0.0226-0.0127-0.6702-0.2649r =-0.34780.8917-0.21590.4445-0.03430.

29、24080.59920.21700.83080.73580.36930.7342-0.34970.97490.5361* *-1.11451.68611.15841.30750.27550.16460.74510.86650.79611.14191.10681.19490.54891.02860.82560.69920.41530.9449-0.8603-0.5868-0.4249-0.5020* *-0.3563-0.4149-0.6395-0.5923-0.76600.46881.2219-0.4490-0.7927-0.4540-1.2630-0.09870.3509-0.5921-1.

30、5450-0.9541-0.8139-0.4498-0.2107-0.9275-0.7051-0.8796* *-1.08522.1574-0.3306-0.7441-0.9530-0.6992-0.9299-1.1478 rint =-1.9280-0.7220-1.7877-1.1746-1.6382-1.3743-1.0189-1.3898-0.7833-0.8845-1.2496-0.8853-1.9330* *-1.08522.1574-0.6385-2.1813-0.57240.14353.2286-0.4463-0.2909-1.3275-1.4460-0.8695-0.7514

31、-0.8222-0.4645-0.4883-0.4091-1.0680-0.5813-0.7851-0.9163-1.1827-0.6638-2.4750-2.2082-2.03922.76312.90591.87851.23252.50541.35602.06361.56961.85582.21731.82372.44492.35611.98822.35371.23352.5883* *-2.12301.11901.77522.35972.48432.41442.74822.70202.79882.16592.63842.43642.31462.01322.55350.75431.03451

32、.1894* *-2.49080.7316-2.7155-2.0332-2.2586-2.2133-2.3850-1.0894-0.1453-2.0695-2.4121-2.0716-2.8575-1.7076-1.1978-2.2135-3.1230-2.5686-2.4329-2.0699-1.8258-2.5407-2.32540.48651.20340.97951.02870.85312.02702.58921.17150.82681.16370.33151.51021.89951.02920.03310.66030.80521.17041.40440.68580.9152* *120

33、.6594 0.0177X20.4676x32-1.97551.2629-1.94901.2879-2.36440.8761-2.56430.6582-2.31980.9215-2.53830.6785-2.75540.4598stats =0.471628.1175 0.00000.6681以及残差图：残差图仍然显示变量之间的相关性， 这说明，最开始调查数据时，3个指标没有选好。最后得到：logm 驴i dmiCHFR匚HRFnr3 3SD* *1,-61.2,-56.2,1.7;121 e6594 0.0177X20.4676 x3将企业的具体数据X2,X3代入 的表达式计算，再结合0,0

34、.5y1,0.5金融机构就可以知道，是否应该贷款给这家企业。注：一个通常的Regress回归，可以用R2, R2, F test等参数评价回归结果的好 坏，但对Logistic回归来说，不存在这样简单而令人满意的评价参数，所以， 般应该进行回归诊断。Logistic 回归的诊断所谓的Logistic回归诊断，就是将Xi的原始数据代入求得的回归方程中，计 算y值，看看有多少个由回归方程计算所得的y值与原始的y值不同，因而判断回归方程的好坏。（1）用回归方程111 e.3914 0.0069X!0.0093x20.3263x3进行诊断。在Matlab软件包中，编程诊断X=1,-62.8,-89.5

35、,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,-13.1,-17.6,0.9;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,-280,6.7;1,-34.6,-19.4,3.4;1,-27.9,6.3

36、,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,-38,1.6,1.2;1,-57.9,0.7,0.8;* *1,18.1,13.5,4;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;1,33,4,2.7;1,35,20.8,1.

37、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;1,31.4,15.7,1.9;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;1,5

38、4.7,14.6,1.7;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;for j=1:66;f=1心+exp(0.3914-0.0069*X(j,2)-0.0093*XQ,3)-0.3263*XQ,4);if f=0.5;jy=0else jy=1endend在Mathematica软件包中编程如下:* *x162.8,3.3,120.8,18.1,3.8,61.2,20.3,194.5,20.8,106

39、.1,39.4,164.1,308.9,7.2,118.3,185.9,34.6,27.9,48.2,49.2,19.2,18.1,98,129,4,8.7,59.2,13.1,38,57.9,8.8,64.7,11.4,43, 47,3.3,35,37.3,35.3,49.5,18.1,31.4,21.5,8.5,40.6,34.6,19.9,17.4,54.7,53.5,35.9,39.4,53.1,39.8,59.5,16.3,21.7;89.5,3.5,103.2,28.8,50.6,56.2,17.4,25.8,4.3,22.9,35.7,17.7,65.8,22.6,34.2,28

40、0,19.4,6.3,6.8,17.2,36.76.5,20.8,14.2,15.8,36.3,12.8,17.6,1.6,0.7,9.1,1, 4.8,16.4,16,4, 20.8,12.6,12.5,23.6,10.4,13.8,33.4,23.1,23.8,7, 34.1,4.2,25.1,13.5,15.7,14.4,5.8, 5.8,26.4,26.7,12.6,14.6,20.6,26.4,30.5,7.1,13.8,7,20.4,7.8J1.7,1.1,2.5,1.1,0.9,1.7,1, 0.5,1, 1.5,x2x346.7,20.8,33, 26.1,68.6,37.3,

41、59, 49.6,12.5,1.2,1.3,0.8,2, 1.5,6.7,3.4,1.3,1.6,0.3,0.8,0.9,1.7,1.3,2.1,2.8,2.1,0.9,1.2,0.8,0.9,0.1,0.9,1.3,1.9,2.7,1.9,0.9,2.4,1.5,2.1,1.6,3.5,5.5,1.9,1.8,1.5,0.9,2.6,4,1.9,1,1.5,1.8,1.8,2.3,1.3,1.7,1.1, 2,1.91.9,1.2,2, 1,1.6 ;1f1E0.39140.0069 x1 j0.0093 x2 j0.3263 x3jIff0.5,0, 1 ; PrintIIII:II IIJJJJ Jy,IIDo y66j,* *0.3914 0.0069为0.0093x20.3263x3执行后得到结果（只列出不相同的）：序号y的原始值Logistic回归值序号y的原始值Logistic回归值13423533643753863974084190142104311441245134614014715