Matlab软包的Logistic回归实现)

1、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，y 0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| X1 人公2 X2, ,Xn Xn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,是比较困难的一件事，于是，人们改为考虑y 1的概率ky 1的概率般的，0

2、 k。人们经过研究发现，令b1 X1 XnPr Y 1| X1 x1, X2x2, Xn xa 0, bj0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:logb。0為b.Xn然后，对log 进行通常的线性回归例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：y 0, 企业 2年后破产y 1，企业 2年后具备还款能力下面列出美国66 家企业的具体情况：YX1X2X30-62.8-89.51.703.3-3.51.10-120.8-103.22.50-18.1-28.

3、81.10-3.8-50.60.90-61.2-56.21.70-20.3-17.41.00-194.5-25.80.5020.8-4.31.00-106.1-22.91.50-39.4-35.71.20-164.1-17.71.30-308.9-65.80.807.2-22.62.00-118.3-34.21.50-185.9-280.06.70-34.6-19.43.40-27.96.31.30-48.26.81.60-49.2-17.20.30-19.2-36.70.80-18.1-6.50.90-98.0-20.81.70-129.0-14.21.30-4.0-15.82.10-8.7

4、-36.32.80-59.2-12.82.10-13.1-17.60.90-38.01.61.2-57.90.70.8-8.8-9.10.9-64.7-4.00.1-11.44.80.943.016.41.347.016.01.9-3.34.02.735.020.81.946.712.60.920.812.52.433.023.61.526.110.42.168.613.81.637.333.43.559.023.15.549.623.81.912.57.01.837.334.11.535.34.20.949.525.12.618.113.54.031.415.71.921.5-14.41.0

5、8.55.81.540.65.81.834.626.41.819.926.72.317.412.61.354.714.61.753.520.61.135.926.42.039.430.51.953.17.11.939.813.81.259.57.02.016.320.41.021.7-7.81.6未分配利润V支付利息前的利润销售额X总资产X2总资产3总资产0000111111111111111111111111111111111其中X1建立破产特征变量y的回归方程。解：在这个破产问题中,我们讨论log 1，概率=企业不破产的概率。因为 值0.5,令1 y 1的次数1y 1的次数0,1。设 二企

6、业2年后具备还款能力的概率，即,66个数据有33个为0, 33个为1,所以，取分界0.50.50,7 1,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，y0对应X1X2X30.25-62.8-89.51.70.253.3-3.51.10.25-120.8-103.22.50.25-18.1-28.81.10.25-3.8-50.60.90.25-61.2-56.21.70.25-20.3-17.41.00.25-194.5-25.80.50.2520.8-4.31.00.25-106.1-

7、22.91.50.25-39.4-35.71.20.25-164.1-17.71.30.25-308.9-65.80.80.257.2-22.62.00.25-118.3-34.21.50.25-185.9-280.06.70.25-34.6-19.43.40.25-27.96.31.30.25-48.26.81.60.25-49.2-17.20.30.25-19.2-36.70.80.25-18.1-6.50.90.25-98.0-20.81.70.25-129.0-14.21.30.25-4.0-15.82.10.25-8.7-36.32.80.25-59.2-12.82.10.25; y

8、 1,对应0.75。数据表变为:于是，在Matlab软件包中编程如下，对log 1进行通常的线性回归:0.25-13.1-17.60.90.25-38.01.61.20.25-57.90.70.80.25-8.8-9.10.90.25-64.7-4.00.10.25-11.44.80.90.7543.016.41.30.7547.016.01.90.75-3.34.02.70.7535.020.81.90.7546.712.60.90.7520.812.52.40.7533.023.61.50.7526.110.42.10.7568.613.81.60.7537.333.43.50.7559.

9、023.15.50.7549.623.81.90.7512.57.01.80.7537.334.11.50.7535.34.20.90.7549.525.12.60.7518.113.54.00.7531.415.71.90.7521.5-14.41.00.758.55.81.50.7540.65.81.80.7534.626.41.80.7519.926.72.30.7517.412.61.30.7554.714.61.70.7553.520.61.10.7535.926.42.00.7539.430.51.90.7553.17.11.90.7539.813.81.20.7559.57.02

10、.00.7516.320.41.00.7521.7-7.81.6X=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,-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

11、,-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,-38,1.6,1.2;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,

12、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;1,49.5,25.1,2.6;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

13、,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);y0=a0;a1;Y=log(1-y0)./y0);b,bint,r,rint,stats =regress(Y,X)rcoplot(r,ri

14、nt)执行后得到结果：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 -0.99371.39830.99880.96210.30720.49420.81610.39570.11411.21761.22250.86700.74680.85310.57770.

15、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 0.0653-0.5082-1.1506-0.8882-0.5701-0.4191-0.3540-0.8289-0.4239-0.5720-0.3449-0.3153-0.4396-0.6967-0.3640-0.8616-0.8919 rint =-1.4320-0.3990-1.6975-0.7882-0.9222-1

16、.1498-0.7332-2.0696-0.3070-1.2048-0.9730-1.5626-2.9063-0.2499-1.3925-1.7217-0.0051-0.4609-0.4909-1.1505-0.9556-0.64771.42452.51131.16082.13491.92771.78562.19710.66092.57911.71541.96401.2441-0.62232.64661.5249-0.26572.80182.45852.41521.76491.94392.27991.8562-1.0648-1.3238-0.2340-0.2162-0.5911-0.7136-

17、0.6117-0.8868-0.6044-1.1944-0.4914-2.0862-1.8729-2.0558-1.9108-2.2125-1.9186-2.0271-2.0034-1.8340-1.1951-0.3186-1.7819-2.2238-1.8981-2.3643-1.5319-1.3378-1.9834-2.5850-2.3556-2.0422-1.8929-1.8195-2.2961-1.8955-2.0355-1.8178-1.7876-1.9105-2.1620-1.8335-2.32371.55212.66922.66132.32502.20732.31782.0421

18、2.31561.71202.42640.85041.07600.86711.03890.69551.02340.91900.94591.09671.63402.16811.16620.72051.04490.53421.39591.46830.96690.28390.57930.90201.05471.11160.63831.04760.89161.12801.15711.03130.76861.10550.60050.5707-2.3544 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）,F_检验值=27.3

19、8410.0000 （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:log 110.3914 0.0069xi 0.0093x2 0.3263x310.3914 0.0069x1 0.0093x2 0.3263x3以及残差图: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,-3.5,1.1;1,-120.8,-103.2,2.5;1,-18.1,-28.8,1.1;1,

20、-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,-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.

21、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;1,43,16.4,1.3;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,

22、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;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

23、,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)执行后得到结果： ans =1.0000 0.64090.6409 1.0000 ans =1.0000 0.04670.0467 1.0000ans =1.0000 -0.3501-0.3501 1.0000可见corrcoef(X1,X2) = 0.64，这说明，在做回归时，可以去

24、掉捲列，或者去掉他列。根据经济意义，我们去掉Xi列，再进行回归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.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,

25、-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,-i2.8,2.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;1,4

26、3,16.4,1.3;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;1,49.5,25.1,2.6;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,

27、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);y0=a0;a1;Y=log(1-y0)./y0);X1=X(:,2);X2=X(:,3);X3=X(:,4);E=ones(66,1);B=E

28、,X2,X3;b,bint,r,rint,stats =regress(Y,B)rcoplot(r,rint)执行后得到：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 0.8308 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

29、.7961 1.1419 1.1068 1.1949 0.5489 1.0286 0.8256 0.6992 0.4153 0.9449 -0.8603 -0.5868-0.4249-0.5020-1.1145-0.4149-0.6395-0.5923-0.76600.46881.2219-0.4490-0.7927-0.4540-1.2630-0.0987 0.3509 -0.5921 -1.5450 -0.9541 -0.8139 -0.4498 -0.2107 -0.9275 -0.7051 -0.8796 -0.3563-0.3306-0.7441-0.9530-0.6992-0.92

30、99-1.1478 rint =1.23252.50541.35602.06361.56961.85582.21731.82372.44492.35611.98822.3537-1.9280-0.7220-1.7877-1.1746-1.6382-1.3743-1.0189-1.3898-0.7833-0.8845-1.2496-0.8853-1.9330 -0.6385 -1.0852 -2.1813 0.1435 -0.4463 -0.2909 -1.3275 -1.4460 -0.8695 -0.7514 -0.8222 -0.4645 -0.4883 -0.4091 -1.0680 -

31、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 -2.4329 -2.0699 -1.82581.23352.58832.1574-0.57243.22862.76312.90591.87851.77522.35972.48432.41442.748

32、22.70202.79882.16592.63842.43642.31462.01322.55350.75431.03451.18941.11900.48651.20340.97951.02870.85312.02702.58921.17150.82681.16370.33151.51021.89951.02920.03310.66030.80521.17041.4044-2.54070.6858-2.32540.9152-2.49080.7316-1.97551.2629-1.94901.2879-2.36440.8761-2.56430.6582-2.31980.9215-2.53830.

33、6785-2.75540.4598stats =0.4716 28.11750.00000.6681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：1 2log 20.6594 0.0177x2 0.4676x32120.6594 0.0177x2 0.4676x31 e将企业的具体数据X2,X3代入 的表达式计算，再结合0,0.5y1,0.5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，一 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(1)用回归方程 11 e.3914 0.0069为 0.0093X2 0.3263x3 进行诊断。在Matlab软件包中，X=1,-62.8,-89.5,1.7;1,3.3,-3.5,1.1;