Matlab软件包与Logistic回归

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，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| Xi Xi,X2 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1 y 1的概率k

2、y 1的概率一般的，0 k。人们经过研究发现，令1Pr Y 1|X1 X1,X2 X2, ,Xn XnbTX1一bTX；1 a ea 0, bj 0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：1logb° “为bnxn然后，对log进行通常的线性回归。例如，Logistic型概率函数11 300e 2x的图形如下：ezplot('W(1+300*exp(-2*x)',0,10)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

3、破产y1，企业2年后具备还款能力面列出美国 66 家企业的具体情况：YX1X2X30-62.8-89.51.703.3-3.51.10-120.8-103.22.50-18.1-28.81.10-3.8-50.60.90-61.2-56.21.70-20.3-17.41.00-194.5-25.80.5020.8-4.31.00-106.1-22.91.50-39.4-35.71.20-164.1-17.71.30-308.9-65.80.807.2-22.62.00-118.3-34.21.50-185.9-280.06.70-34.6-19.43.40-27.96.31.30-48.26.

4、81.60-49.2-17.20.30-19.2-36.70.80-18.1-6.50.90-98.0-20.81.70-129.0-14.21.30-4.0-15.82.10-8.7-36.32.80-59.2-12.82.10-13.1-17.60.90-38.01.61.20-57.90.70.80-8.8-9.10.90-64.7-4.00.10-11.44.80.9143.016.41.3147.016.01.91-3.34.02.7135.020.81.9146.712.60.9120.812.52.4133.023.61.5126.110.42.1168.613.81.6137.

5、333.43.5159.023.15.5149.623.81.9112.57.01.8137.334.11.5135.34.20.9149.525.12.6118.113.54.0131.415.71.9121.5-14.41.018.55.81.5140.65.81.8134.626.41.8119.926.72.3117.412.61.3154.714.61.7153.520.61.1135.926.42.0139.430.51.9153.17.11.9139.813.81.2159.57.02.0116.320.41.0121.7-7.81.6其中X1未分配利润Y支付利息前的利润总资产X

6、2总资产X3销售额总资产建立破产特征变量y的回归方程。解：在这个破产问题中，1 y 1的次数y 1的次数我们讨论log，概率 0,1。设 二企业2年后具备还款能力的概率， 即,=企业不破产的概率。因为值0.5,令66个数据有33个为0, 33个为1,所以，取分界0, 0.5y1, 0.5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

7、.25-120.8-103.22.50.25-18.1-28.81.10.25-3.8-50.60.90.25-61.2-56.21.70.25-20.3-17.41.00.25-194.5-25.80.50.2520.8-4.31.00.25-106.1-22.91.50.25-39.4-35.71.20.25-164.1-17.71.30.25-308.9-65.80.80.257.2-22.62.00.25-118.3-34.21.50.25-185.9-280.06.70.25-34.6-19.43.40.25-27.96.31.30.25-48.26.81.60.25-49.2-17

8、.20.30.25-19.2-36.70.80.25-18.1-6.50.90.25-98.0-20.81.70.25-129.0-14.21.30.25-4.0-15.82.10.25-8.7-36.32.80.25-59.2-12.82.10.25-13.1-17.60.90.25-38.01.61.20.25-57.90.70.80.25-8.8-9.10.90.25-64.7-4.00.10.25-11.44.80.90.7543.016.41.30.7547.016.01.90.75-3.34.02.70.7535.020.81.90.7546.712.60.90.7520.812.

9、52.40.7533.023.61.50.7526.110.42.10.7568.613.81.60.7537.333.43.50.7559.023.15.50.7549.623.81.90.7512.57.01.80.7537.334.11.50.7535.34.20.90.7549.525.12.60.7518.113.54.00.7531.415.71.90.7521.5-14.41.00.758.55.81.50.7540.65.81.80.7534.626.41.80.7519.926.72.30.7517.412.61.30.7554.714.61.70.7553.520.61.1

10、0.7535.926.42.00.7539.430.51.90.7553.17.11.90.7539.813.81.20.7559.57.02.00.7516.320.41.00.7521.7-7.81.6于是，在Matlab软件包中编程如下，对log -进行通常的线性回归: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,-106.

11、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,-18.1,-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,-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

13、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

14、.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,rint)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

15、.19840.0662-0.99371.39830.99880.96210.30720.49420.81610.39570.11411.21761.22250.86700.74680.85310.57770.85560.25880.9675 -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 -0.06800.0653 -0.5082 -1.1506 -0.8882 -0.5701 -0.4191 -0.3540 -

16、0.8289 -0.4239 -0.57201.42452.51131.16082.13491.92771.78562.19710.66092.57911.71541.96401.2441-0.62232.64661.5249-0.26572.80182.45852.41521.76491.94392.27991.85621.55212.66922.66132.32502.20732.31782.04212.31561.71202.42640.85041.07600.8671-0.3449-0.3153-0.4396-0.6967-0.3640-0.8616-0.8919 rint =-1.4

17、320-0.3990-1.6975-0.7882-0.9222-1.1498-0.7332-2.0696-0.3070-1.2048-0.9730-1.5626-2.9063-0.2499-1.3925-1.7217-0.0051-0.4609-0.4909-1.1505-0.9556-0.6477-1.0648-1.3238-0.2340-0.2162-0.5911-0.7136-0.6117-0.8868-0.6044-1.1944-0.4914-2.0862-1.8729-2.0558-1.91081.0389-2.21250.6955-1.91861.0234-2.02710.9190

18、-2.00340.9459-1.83401.0967-1.19511.6340-0.31862.1681-1.78191.1662-2.22380.7205-1.89811.0449-2.36430.5342-1.53191.3959-1.33781.4683-1.98340.9669-2.58500.2839-2.35560.5793-2.04220.9020-1.89291.0547-1.81951.1116-2.29610.6383-1.89551.0476-2.03550.8916-1.81781.1280-1.78761.1571-1.91051.0313-2.16200.7686-

19、1.83351.1055-2.32370.6005-2.35440.5707s =0.569927.38410.00000.5526R2值二0.5699 （说明回归方程刻画原问题不是太好）,F_检验值=0.05相关的p值=说明变量x1, x2, x3之间存在线性相关关系。回归方程为：即，得到：27.3841>0.0000 （这个值比较好）,与显著性概率0.5526>0.05 ,log110.3914 0.0069% 0.0093x2 0.3263x31 e'0.3914 0.0069x., 0.0093x2 0.3263x3以及残差图:Fee JlmJ士r =lEt1D3D

20、3D<160CaE> Mdrrwr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, 也暗示变量Xi,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,-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

21、.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,-38,1.6,1.2;1,-57.9

22、,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;1,49.5,25.1,2.6;1,18.1,

23、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;X1=X(:,2);X2=X(:,3);X3=X(:,4);corrcoef(X1,X2)corrco

24、ef(X1,X3)corrcoef(X2,X3) 执行后得到结果： ans =1.0000 0.64090.64091.0000ans =1.00000.0467ans =1.0000-0.35010.04671.0000-0.35011.0000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x列。根据经济意义，我们去掉x1 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.

25、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,-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,

26、-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;1,26.1,10.4,2.1;1,68.6,13.8,1.6;1,37.3,33

27、.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;1,39.4,30.5,1.9;1,53.1,7.1,1.9;1,39

28、.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,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 =1.0516-0.0127-0.26490.2672-0.0226-0.6702r

29、 =-0.34780.8917-0.21590.4445-0.03430.24080.59920.21700.83080.73580.36930.7342-0.34970.97490.5361-1.37691.68611.15841.30750.27550.16460.74510.86650.79611.14191.10681.19490.54891.02860.82560.69920.41530.9449-0.8603-0.5868-0.4249-0.5020-1.1145-0.4149-0.6395-0.5923-0.76600.46881.2219-0.4490-0.7927-0.454

30、0-1.2630-0.09870.3509-0.5921-1.5450-0.9541-0.8139-0.4498-0.2107-0.9275-0.7051-0.8796-0.3563-0.3306-0.7441-0.9530-0.6992-0.9299-1.1478rint =-1.9280 1.2325-0.7220 2.5054-1.7877 1.3560-1.1746 2.0636-1.6382 1.5696-1.3743 1.8558-1.0189 2.2173-1.3898 1.8237-0.7833 2.4449-0.8845 2.3561-1.2496 1.9882-0.8853

31、 2.3537-1.9330 1.2335-0.6385 2.5883-1.0852 2.1574 -2.1813 -0.5724 0.1435 3.2286 -0.4463 2.7631 -0.2909 2.9059 -1.3275 1.8785 -1.4460 1.7752 -0.8695 2.3597 -0.7514 2.4843 -0.8222 2.4144 -0.4645 2.7482 -0.4883 2.7020 -0.4091 2.7988 -1.0680 2.1659 -0.5813 2.6384 -0.7851 2.4364 -0.9163 2.3146 -1.1827 2.

32、0132 -0.6638 2.5535 -2.4750 0.7543 -2.2082 1.0345 -2.0392 1.1894 -2.1230 1.1190 -2.7155 0.4865 -2.0332 1.2034 -2.2586 0.9795 -2.2133 1.0287 -2.3850 0.8531 -1.0894 2.0270 -0.1453 2.5892 -2.0695 1.1715 -2.4121 0.8268 -2.0716 1.1637 -2.8575 0.3315 -1.7076 1.5102 -1.1978 1.8995 -2.2135 1.0292 -3.1230 0.

33、0331 -2.5686 0.6603 -2.4329 0.8052 -2.0699 1.1704 -1.8258 1.4044 -2.5407 0.6858 -2.3254 0.9152 -2.4908 0.7316-1.97551.2629-1.94901.2879-2.36440.8761-2.56430.6582-2.31980.9215-2.53830.6785-2.75540.4598stats =0.4716 28.11750.00000.6681以及残差图：工OT-/I1JCE10303CJOacase Nurmei3:irf； nini cnnft"們 pin3 S

34、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：log 1-0.6594 0.0177x2 0.4676x3212,0.6594 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0,0.51,0.5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

35、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 10.3914 0.0069为 0.0093x2 0.3263x3 进行诊断。1 e在Matlab软件包中，编程诊断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

36、.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,-18.1,-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,-1

37、7.6,0.9;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;1,26.1,10.4,2.1;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.

38、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;for j=1:66;f=1/(1+exp(

39、0.3914-0.0069*X(j,2)-0.0093*X(j,3)-0.3263*X(j,4);if f<=0.5;jy=0else jy=1 endendj,在Mathematica软件包中编程如下:62.8,3.3,120.8,18.1,3.8,61.2,20.3,194.5,20.8,106.1,39.4,164.1,308.9,7.2,118.3,185.9,34.6,27.9,48.2,49.2,19.2,18.1,98,129,4,8.7,59.2,13.1,38,57.9,8.8,64.7,11.4,43, 47,3.3,35,46.7,20.8,33, 26.1,68.

40、6,37.3,59, 49.6,12.5,37.3,35.3,49.5,18.1,31.4,21.5,8.5,40.6,34.6,19.9,17.4,54.7,53.5,35.9,39.4,53.1,39.8,59.5,16.3,21.7189.5,3.5,103.2,28.8,50.6,56.2,17.4,25.8,4.3,22.9,35.7,17.7,65.8,22.6,34.2,280,19.4,6.3,6.8,17.2,3676.5,20.8,14.2,15.8,36.3,12.8,17.6,1.6,0.7,9.1,1, 4.8,16.4,16, 4, 20.8,12.6,12.5,2

41、3.6,10.4,13.8,33.4,23.1,23.8,7, 34.1,4.2,25.1,13.5,15.7,14.4,5.8,5.8,26.4,26.7,12.6,14.6,20.6,26.4,30.5,7.1,13.8,7,20.4,7.8;1.7,1.1,2.5,1.1,0.9,1.7,1, 0.5,1,1.5,1.2,1.3,0.8,2, 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,

42、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, 1.9,1.9,f1.2, 2, 1, 1.6 ;11E°.39140.0069 x1j0.0093 x2 j0.3263 x3 j'Iff 0.5,0, 1 ; Print "", j, ",", y,IIII1x1x2x3Doy66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134235336437538639740841901421043114412451346140147154816491