MATLAB实例：多元函数拟合(线性与非线性)

作者：凯鲁嘎吉 - 博客园 http://www.cnblogs.com/kailugaji/

更多请看：随笔分类 - MATLAB作图

之前写过一篇博文，是关于一元非线性曲线拟合，自定义曲线函数。

现在用最小二乘法拟合多元函数，实现线性拟合与非线性拟合，其中非线性拟合要求自定义拟合函数。

下面给出三种拟合方式，第一种是多元线性拟合(回归)，第二三种是多元非线性拟合，实际中第二三种方法是一个意思，任选一种即可，推荐第二种拟合方法。

1. MATLAB程序

fit_nonlinear_data.m

1function [beta, r]=fit_nonlinear_data(X, Y, choose)
2% Input: X 自变量数据(N, D)， Y 因变量(N, 1)，choose 1-regress, 2-nlinfit 3-lsqcurvefit
3if choose==1
4    X1=[ones(length(X(:, 1)), 1), X];
5    [beta, bint, r, rint, states]=regress(Y, X1)
6    % 多元线性回归
7    % y=beta(1)+beta(2)*x1+beta(3)*x2+beta(4)*x3+...
8    % beta—系数估计
9    % bint—系数估计的上下置信界
10    % r—残差
11    % rint—诊断异常值的区间
12    % states—模型统计信息
13    rcoplot(r, rint)
14    saveas(gcf,sprintf('线性曲线拟合_残差图.jpg'),'bmp');
15elseif choose==2
16    beta0=ones(7, 1);
17    % 初始值的选取可能会导致结果具有较大的误差。
18    [beta, r, J]=nlinfit(X, Y, @myfun, beta0)
19    % 非线性回归
20    % beta—系数估计
21    % r—残差
22    % J—雅可比矩阵
23    [Ypred,delta]=nlpredci(@myfun, X, beta, r, 'Jacobian', J)
24    % 非线性回归预测置信区间
25    % Ypred—预测响应
26    % delta—置信区间半角
27    plot(X(:, 1), Y, 'k.', X(:, 1), Ypred, 'r');
28    saveas(gcf,sprintf('非线性曲线拟合_1.jpg'),'bmp');
29elseif choose==3
30    beta0=ones(7, 1);
31    % 初始值的选取可能会导致结果具有较大的误差。
32    [beta,resnorm,r, ~, ~, ~, J]=lsqcurvefit(@myfun,beta0,X,Y)
33    % 在最小二乘意义上解决非线性曲线拟合（数据拟合）问题
34    % beta—系数估计
35    % resnorm—残差的平方范数 sum((fun(x,xdata)-ydata).^2)
36    % r—残差 r=fun(x,xdata)-ydata
37    % J—雅可比矩阵
38    [Ypred,delta]=nlpredci(@myfun, X, beta, r, 'Jacobian', J)
39    plot(X(:, 1), Y, 'k.', X(:, 1), Ypred, 'r');
40    saveas(gcf,sprintf('非线性曲线拟合_2.jpg'),'bmp');
41end
42end
43
44
45function yy=myfun(beta,x) %自定义拟合函数
46yy=beta(1)+beta(2)*x(:, 1)+beta(3)*x(:, 2)+beta(4)*x(:, 3)+beta(5)*(x(:, 1).^2)+beta(6)*(x(:, 2).^2)+beta(7)*(x(:, 3).^2);
47end

demo.m

1clear
2clc
3X=[1 13 1.5; 1.4 19 3; 1.8 25 1; 2.2 10 2.5;2.6 16 0.5; 3 22 2; 3.4 28 3.5; 3.5 30 3.7];
4Y=[0.330; 0.336; 0.294; 0.476; 0.209; 0.451; 0.482; 0.5];
5choose=1;
6fit_nonlinear_data(X, Y, choose)

2. 结果

(1)多元线性拟合(regress)

choose=1:

1>> demo
2
3beta =
4
5   0.200908829282537
6   0.044949392540298
7  -0.003878606875016
8   0.070813489681112
9
10
11bint =
12
13  -0.026479907290565   0.428297565855639
14  -0.057656451966002   0.147555237046598
15  -0.017251051845827   0.009493838095795
16   0.000201918738160   0.141425060624065
17
18
19r =
20
21   0.028343433030705
22  -0.066584917256987
23   0.038333946339215
24   0.037954851676187
25  -0.082126284727611
26   0.058945364984698
27  -0.010982985302994
28  -0.003883408743214
29
30
31rint =
32
33  -0.151352966773048   0.208039832834458
34  -0.188622801533810   0.055452967019837
35  -0.090283529625345   0.166951422303776
36  -0.090266067743345   0.166175771095720
37  -0.108068661106325  -0.056183908348897
38  -0.130409602930181   0.248300332899576
39  -0.206254481234707   0.184288510628719
40  -0.184329400080620   0.176562582594191
41
42
43states =
44
45   0.768591079367914   4.428472778943478   0.092289917768436   0.004625488283939

(2)多元非线性拟合(nlinfit)

choose=2:

1>> demo
2
3beta =
4
5   0.312525876099987
6   0.015300533415459
7  -0.036942272680920
8   0.299760796634952
9   0.009412595106141
10   0.000976411370591
11  -0.062931846673372
12
13
14r =
15
16   1.0e-03 *
17
18  -0.047521336834000
19   0.127597019984715
20  -0.092883949615763
21  -0.040370056416994
22   0.031209476614974
23   0.211856736183458
24  -0.727835090583939
25   0.537947200592082
26
27
28J =
29
30   1.0e+02 *
31
32   0.010000000000266   0.010000000001236   0.129999999998477   0.014999999999641   0.010000000007909   1.689999999969476   0.022499999999756
33   0.010000000000266   0.014000000006524   0.189999999999301   0.029999999999283   0.019600000006932   3.609999999769248   0.089999999999024
34   0.010000000000266   0.018000000011811   0.249999999990199   0.009999999999965   0.032399999999135   6.250000000033778   0.010000000000377
35   0.009999999999679   0.022000000005116   0.099999999998065   0.025000000000218   0.048400000003999   1.000000000103046   0.062500000001264
36   0.009999999999972   0.025999999998421   0.159999999998889   0.004999999999982   0.067599999997174   2.559999999999039   0.002499999999730
37   0.009999999999679   0.029999999991726   0.219999999999713   0.019999999999930   0.089999999993269   4.839999999890361   0.040000000000052
38   0.009999999999092   0.033999999985031   0.279999999990611   0.034999999998348   0.115599999997155   7.839999999636182   0.122500000000614
39   0.010000000000266   0.034999999992344   0.299999999994194   0.037000000000420   0.122499999994626   8.999999999988553   0.136899999999292
40
41
42Ypred =
43
44   0.330047521336834
45   0.335872402980015
46   0.294092883949616
47   0.476040370056417
48   0.208968790523385
49   0.450788143263817
50   0.482727835090584
51   0.499462052799408
52
53
54delta =
55
56   0.011997285626178
57   0.011902559677366
58   0.011954353934643
59   0.012001513980794
60   0.012005923574387
61   0.011706970437467
62   0.007666390995581
63   0.009878186927507

(3)多元非线性拟合(lsqcurvefit)

choose=3:

1>> demo
2
3beta =
4
5   0.312525876070457
6   0.015300533464733
7  -0.036942272680581
8   0.299760796608728
9   0.009412595094407
10   0.000976411370579
11  -0.062931846666179
12
13
14resnorm =
15
16     8.937848643213721e-07
17
18
19r =
20
21   1.0e-03 *
22
23   0.047521324135769
24  -0.127597015215197
25   0.092883952947764
26   0.040370060121864
27  -0.031209466218374
28  -0.211856745335304
29   0.727835089662676
30  -0.537947200236699
31
32
33J =
34
35   1.0e+02 *
36
37   (1,1)      0.010000000000000
38   (2,1)      0.010000000000000
39   (3,1)      0.010000000000000
40   (4,1)      0.010000000000000
41   (5,1)      0.010000000000000
42   (6,1)      0.010000000000000
43   (7,1)      0.010000000000000
44   (8,1)      0.010000000000000
45   (1,2)      0.010000000000000
46   (2,2)      0.014000000059605
47   (3,2)      0.017999999970198
48   (4,2)      0.022000000029802
49   (5,2)      0.026000000014901
50   (6,2)      0.030000000000000
51   (7,2)      0.034000000059605
52   (8,2)      0.035000000000000
53   (1,3)      0.130000000000000
54   (2,3)      0.190000000000000
55   (3,3)      0.250000000000000
56   (4,3)      0.100000000000000
57   (5,3)      0.160000000000000
58   (6,3)      0.220000000000000
59   (7,3)      0.280000000000000
60   (8,3)      0.300000000000000
61   (1,4)      0.015000000000000
62   (2,4)      0.030000000000000
63   (3,4)      0.010000000000000
64   (4,4)      0.025000000000000
65   (5,4)      0.005000000000000
66   (6,4)      0.020000000000000
67   (7,4)      0.035000000000000
68   (8,4)      0.036999999880791
69   (1,5)      0.010000000000000
70   (2,5)      0.019599999934435
71   (3,5)      0.032399999983609
72   (4,5)      0.048400000035763
73   (5,5)      0.067599999997765
74   (6,5)      0.090000000000000
75   (7,5)      0.115600000023842
76   (8,5)      0.122500000000000
77   (1,6)      1.690000000000000
78   (2,6)      3.610000000000000
79   (3,6)      6.250000000000000
80   (4,6)      1.000000000000000
81   (5,6)      2.560000000000000
82   (6,6)      4.840000000000000
83   (7,6)      7.840000000000000
84   (8,6)      9.000000000000000
85   (1,7)      0.022500000000000
86   (2,7)      0.090000000000000
87   (3,7)      0.010000000000000
88   (4,7)      0.062500000000000
89   (5,7)      0.002500000000000
90   (6,7)      0.040000000000000
91   (7,7)      0.122500000000000
92   (8,7)      0.136899999976158
93
94
95Ypred =
96
97   0.330047521324136
98   0.335872402984785
99   0.294092883952948
100   0.476040370060122
101   0.208968790533782
102   0.450788143254665
103   0.482727835089663
104   0.499462052799763
105
106
107delta =
108
109   0.011997285618724
110   0.011902559623756
111   0.011954353977139
112   0.012001513949620
113   0.012005923574975
114   0.011706970418735
115   0.007666391016173
116   0.009878186931566

注意：多元非线性函数拟合中参数的初始值需要提前设置，有些情况下，参数的初始选取对函数拟合结果影响极大，需要谨慎处理。第二三种方法中，由于数据是多维的，因此只展示了第一个维度的拟合函数图。如有需要，可自行修改。