Matlab软件包与Logistic回归

Matlab软件包与Logistic回归在回归分析中，因变量可能有两种情形：（）是一个定量的变量，这时就用通常的regress函数对进行回归；（）是一个定性的变量，比如，或，这时就不能用通常的regress函数对进行回归，而是使用所谓的Logistic回归。Logistic回归的基本思想是，不是直接对进行回归，而是先定义一种概率函数，令要求。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。直接求的表达式，是比较困难的一件事，于是，人们改为考虑一般的，。人们经过研究发现，令即，是一个Logistic型的函数，效果比较理想。于是，我们将其变形得到：然后，对进行通常的线性回归。例如，Logistic型概率函数的图形如下：ezplot(1/(1+300*exp(-2*x),0,10)例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：下面列出美国66家企业的具体情况：Y X1 X2 X3 0 -62.8 -89.5 1.7 0 3.3 -3.5 1.1 0 -120.8 -103.2 2.5 0 -18.1 -28.8 1.1 0 -3.8 -50.6 0.9 0 -61.2 -56.2 1.7 0 -20.3 -17.4 1.0 0 -194.5 -25.8 0.5 0 20.8 -4.3 1.0 0 -106.1 -22.9 1.5 0 -39.4 -35.7 1.2 0 -164.1 -17.7 1.3 0 -308.9 -65.8 0.8 0 7.2 -22.6 2.0 0 -118.3 -34.2 1.5 0 -185.9 -280.0 6.7 0 -34.6 -19.4 3.4 0 -27.9 6.3 1.3 0 -48.2 6.8 1.6 0 -49.2 -17.2 0.3 0 -19.2 -36.7 0.8 0 -18.1 -6.5 0.9 0 -98.0 -20.8 1.7 0 -129.0 -14.2 1.3 0 -4.0 -15.8 2.1 0 -8.7 -36.3 2.8 0 -59.2 -12.8 2.1 0 -13.1 -17.6 0.9 0 -38.0 1.6 1.2 0 -57.9 0.7 0.8 0 -8.8 -9.1 0.9 0 -64.7 -4.0 0.1 0 -11.4 4.8 0.9 1 43.0 16.4 1.3 1 47.0 16.0 1.9 1 -3.3 4.0 2.7 1 35.0 20.8 1.9 1 46.7 12.6 0.9 1 20.8 12.5 2.4 1 33.0 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 59.0 23.1 5.5 1 49.6 23.8 1.9 1 12.5 7.0 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 4.0 1 31.4 15.7 1.9 1 21.5 -14.4 1.0 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 2.0 1 39.4 30.5 1.9 1 53.1 7.1 1.9 1 39.8 13.8 1.2 1 59.5 7.0 2.0 1 16.3 20.4 1.0 1 21.7 -7.8 1.6 其中，建立破产特征变量的回归方程。解：在这个破产问题中，我们讨论，概率。设企业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，。数据表变为： X1 X2 X3 0.25 -62.8 -89.5 1.7 0.25 3.3 -3.5 1.1 0.25-120.8 -103.2 2.5 0.25 -18.1 -28.8 1.1 0.25 -3.8 -50.6 0.9 0.25 -61.2 -56.2 1.7 0.25 -20.3 -17.4 1.0 0.25 -194.5 -25.8 0.5 0.25 20.8 -4.3 1.0 0.25 -106.1 -22.9 1.5 0.25 -39.4 -35.7 1.2 0.25 -164.1 -17.7 1.3 0.25 -308.9 -65.8 0.8 0.25 7.2 -22.6 2.0 0.25 -118.3 -34.2 1.5 0.25 -185.9 -280.0 6.7 0.25 -34.6 -19.4 3.4 0.25 -27.9 6.3 1.3 0.25 -48.2 6.8 1.6 0.25 -49.2 -17.2 0.3 0.25 -19.2 -36.7 0.8 0.25 -18.1 -6.5 0.9 0.25 -98.0 -20.8 1.7 0.25 -129.0 -14.2 1.3 0.25 -4.0 -15.8 2.1 0.25 -8.7 -36.3 2.8 0.25 -59.2 -12.8 2.1 0.25 -13.1 -17.6 0.9 0.25 -38.0 1.6 1.2 0.25 -57.9 0.7 0.8 0.25 -8.8 -9.1 0.9 0.25 -64.7 -4.0 0.1 0.25 -11.4 4.8 0.9 0.75 43.0 16.4 1.3 0.7547.0 16.0 1.9 0.75-3.3 4.0 2.7 0.7535.0 20.8 1.9 0.7546.7 12.6 0.9 0.7520.8 12.5 2.4 0.7533.0 23.6 1.5 0.7526.1 10.4 2.1 0.7568.6 13.8 1.6 0.7537.3 33.4 3.5 0.7559.0 23.1 5.5 0.7549.6 23.8 1.9 0.7512.5 7.0 1.8 0.7537.3 34.1 1.5 0.7535.3 4.2 0.9 0.7549.5 25.1 2.6 0.7518.1 13.5 4.0 0.7531.4 15.7 1.9 0.7521.5 -14.4 1.0 0.758.5 5.8 1.5 0.7540.6 5.8 1.8 0.7534.6 26.4 1.8 0.7519.9 26.7 2.3 0.7517.4 12.6 1.3 0.7554.7 14.6 1.7 0.7553.5 20.6 1.1 0.7535.9 26.4 2.0 0.7539.4 30.5 1.9 0.7553.1 7.1 1.9 0.7539.8 13.8 1.2 0.7559.5 7.0 2.0 0.7516.3 20.4 1.0 0.7521.7 -7.8 1.6 于是，在Matlab软件包中编程如下，对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,-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,-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; 1,37.3,33.4,3.5; 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; 1,39.8,13.8,1.2; 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,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 = -0.0037 1.0561 -0.2683 0.6733 0.5028 0.3179 0.7320 -0.7044 1.1361 0.2553 0.4955 -0.1593 -1.7643 1.1984 0.0662 -0.9937 1.3983 0.9988 0.9621 0.3072 0.4942 0.8161 0.3957 0.1141 1.2176 1.2225 0.8670 0.7468 0.8531 0.5777 0.8556 0.2588 0.9675 -0.6179 -0.3984 -0.5943 -0.4360 -0.7585 -0.4476 -0.5541 -0.5288 -0.3687 0.2194 0.9248 -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 -0.8919rint = -1.4320 1.4245 -0.3990 2.5113 -1.6975 1.1608 -0.7882 2.1349 -0.9222 1.9277 -1.1498 1.7856 -0.7332 2.1971 -2.0696 0.6609 -0.3070 2.5791 -1.2048 1.7154 -0.9730 1.9640 -1.5626 1.2441 -2.9063 -0.6223 -0.2499 2.6466 -1.3925 1.5249 -1.7217 -0.2657 -0.0051 2.8018 -0.4609 2.4585 -0.4909 2.4152 -1.1505 1.7649 -0.9556 1.9439 -0.6477 2.2799 -1.0648 1.8562 -1.3238 1.5521 -0.2340 2.6692 -0.2162 2.6613 -0.5911 2.3250 -0.7136 2.2073 -0.6117 2.3178 -0.8868 2.0421 -0.6044 2.3156 -1.1944 1.7120 -0.4914 2.4264 -2.0862 0.8504 -1.8729 1.0760 -2.0558 0.8671 -1.9108 1.0389 -2.2125 0.6955 -1.9186 1.0234 -2.0271 0.9190 -2.0034 0.9459 -1.8340 1.0967 -1.1951 1.6340 -0.3186 2.1681 -1.7819 1.1662 -2.2238 0.7205 -1.8981 1.0449 -2.3643 0.5342 -1.5319 1.3959 -1.3378 1.4683 -1.9834 0.9669 -2.5850 0.2839 -2.3556 0.5793 -2.0422 0.9020 -1.8929 1.0547 -1.8195 1.1116 -2.2961 0.6383 -1.8955 1.0476 -2.0355 0.8916 -1.8178 1.1280 -1.7876 1.1571 -1.9105 1.0313 -2.1620 0.7686 -1.8335 1.1055 -2.3237 0.6005 -2.3544 0.5707stats = 0.5699 27.3841 0.0000 0.5526即，得到：值0.5699（说明回归方程刻画原问题不是太好），F_检验值27.38410.0000（这个值比较好），与显著性概率相关的p值0.5526，说明变量之间存在线性相关关系。回归方程为：以及残差图：通过残差图看出，残差连续的出现在的上方，或者连续地出现在的下方，这也暗示变量之间存在线性相关。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,-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,-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; 1,37.3,33.4,3.5; 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; 1,39.8,13.8,1.2; 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 0.6409 0.6409 1.0000ans = 1.0000 0.0467 0.0467 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，可以去掉列。根据经济意义，我们去掉列，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,-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,-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; 1,37.3,33.4,3.5; 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; 1,39.8,13.8,1.2; 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 = 0.2672 1.0516 -0.0226 -0.0127 -0.6702 -0.2649r = -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 0.7961 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 -0.7660 0.4688 1.2219 -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 -0.9299 -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 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 2.0132 -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 0.0331 -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 -1.9755 1.2629 -1.9490 1.2879 -2.3644 0.8761 -2.5643 0.6582 -2.3198 0.9215 -2.5383 0.6785 -2.7554 0.4598stats = 0.4716 28.1175 0.0000 0.6681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，个指标没有选好。最后得到：将企业的具体数据代入的表达式计算，再结合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 所谓的回归诊断，就是将的原始数据代入求得的回归方程中，计算值，看看有多少个由回归方程计算所得的值与原始的值不同，因而判断回归方程的好坏。（1）用回归方程进行诊断。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,-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,-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;