Matlab软件包与Logistic回归.docx

Matlab软件包与Logistic回归.docx

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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：

（１）

是一个定量的变量，这时就用通常的regress函数对

进行回归；（２）

是一个定性的变量，比如，

０或１，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NewYork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：

YX1X2X3

0-62.8-89.51.7

03.3-3.51.1

0-120.8-103.22.5

0-18.1-28.81.1

0-3.8-50.60.9

0-61.2-56.21.7

0-20.3-17.41.0

0-194.5-25.80.5

020.8-4.31.0

0-106.1-22.91.5

0-39.4-35.71.2

0-164.1-17.71.3

0-308.9-65.80.8

07.2-22.62.0

0-118.3-34.21.5

0-185.9-280.06.7

0-34.6-19.43.4

0-27.96.31.3

0-48.26.81.6

0-49.2-17.20.3

0-19.2-36.70.8

0-18.1-6.50.9

0-98.0-20.81.7

0-129.0-14.21.3

0-4.0-15.82.1

0-8.7-36.32.8

0-59.2-12.82.1

0-13.1-17.60.9

0-38.01.61.2

0-57.90.70.8

0-8.8-9.10.9

0-64.7-4.00.1

0-11.44.80.9

143.016.41.3

147.016.01.9

1-3.34.02.7

135.020.81.9

146.712.60.9

120.812.52.4

133.023.61.5

126.110.42.1

168.613.81.6

137.333.43.5

159.023.15.5

149.623.81.9

112.57.01.8

137.334.11.5

135.34.20.9

149.525.12.6

118.113.54.0

131.415.71.9

121.5-14.41.0

18.55.81.5

140.65.81.8

134.626.41.8

119.926.72.3

117.412.61.3

154.714.61.7

153.520.61.1

135.926.42.0

139.430.51.9

153.17.11.9

139.813.81.2

159.57.02.0

116.320.41.0

121.7-7.81.6

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

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：

X1X2X3

0.25-62.8-89.51.7

0.253.3-3.51.1

0.25-120.8-103.22.5

0.25-18.1-28.81.1

0.25-3.8-50.60.9

0.25-61.2-56.21.7

0.25-20.3-17.41.0

0.25-194.5-25.80.5

0.2520.8-4.31.0

0.25-106.1-22.91.5

0.25-39.4-35.71.2

0.25-164.1-17.71.3

0.25-308.9-65.80.8

0.257.2-22.62.0

0.25-118.3-34.21.5

0.25-185.9-280.06.7

0.25-34.6-19.43.4

0.25-27.96.31.3

0.25-48.26.81.6

0.25-49.2-17.20.3

0.25-19.2-36.70.8

0.25-18.1-6.50.9

0.25-98.0-20.81.7

0.25-129.0-14.21.3

0.25-4.0-15.82.1

0.25-8.7-36.32.8

0.25-59.2-12.82.1

0.25-13.1-17.60.9

0.25-38.01.61.2

0.25-57.90.70.8

0.25-8.8-9.10.9

0.25-64.7-4.00.1

0.25-11.44.80.9

0.7543.016.41.3

0.7547.016.01.9

0.75-3.34.02.7

0.7535.020.81.9

0.7546.712.60.9

0.7520.812.52.4

0.7533.023.61.5

0.7526.110.42.1

0.7568.613.81.6

0.7537.333.43.5

0.7559.023.15.5

0.7549.623.81.9

0.7512.57.01.8

0.7537.334.11.5

0.7535.34.20.9

0.7549.525.12.6

0.7518.113.54.0

0.7531.415.71.9

0.7521.5-14.41.0

0.758.55.81.5

0.7540.65.81.8

0.7534.626.41.8

0.7519.926.72.3

0.7517.412.61.3

0.7554.714.61.7

0.7553.520.61.1

0.7535.926.42.0

0.7539.430.51.9

0.7553.17.11.9

0.7539.813.81.2

0.7559.57.02.0

0.7516.320.41.0

0.7521.7-7.81.6

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

-0.3263

bint=

0.00730.7755

-0.0105-0.0032

-0.0156-0.0030

-0.5253-0.1273

r=

-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

-0.8919

rint=

-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

-0.00512.8018

-0.46092.4585

-0.49092.4152

-1.15051.7649

-0.95561.9439

-0.64772.2799

-1.06481.8562

-1.32381.5521

-0.23402.6692

-0.21622.6613

-0.59112.3250

-0.71362.2073

-0.61172.3178

-0.88682.0421

-0.60442.3156

-1.19441.7120

-0.49142.4264

-2.08620.8504

-1.87291.0760

-2.05580.8671

-1.91081.0389

-2.21250.6955

-1.91861.0234

-2.02710.9190

-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

-1.83351.1055

-2.32370.6005

-2.35440.5707

stats=

0.569927.38410.00000.5526

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＝27.3841>0.0000（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０的上方，或者连续地出现在０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=

1.00000.6409

0.64091.0000

ans=

1.00000.0467

0.04671.0000

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

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