MATLAB实例：多元函数拟合（线性与⾮线性）
MATLAB实例：多元函数拟合(线性与⾮线性)
作者：凯鲁嘎吉 - 博客园
更多请看：
之前写过⼀篇博⽂，是。
现在⽤拟合多元函数，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
function [beta, r]=fit_nonlinear_data(X, Y, choose)
% Input: X ⾃变量数据(N, D)， Y 因变量(N, 1)，choose 1-regress, 2-nlinfit 3-lsqcurvefit
if choose==1
X1=[ones(length(X(:, 1)), 1), X];
[beta, bint, r, rint, states]=regress(Y, X1)
% 多元线性回归
% y=beta(1)+beta(2)*x1+beta(3)*x2+beta(4)*x3+...
% beta—系数估计
% bint—系数估计的上下置信界
% r—残差
% rint—诊断异常值的区间
% states—模型统计信息
rcoplot(r, rint)
saveas(gcf,sprintf('线性曲线拟合_残差图.jpg'),'bmp');
elseif choose==2
beta0=ones(7, 1);
% 初始值的选取可能会导致结果具有较⼤的误差。
[beta, r, J]=nlinfit(X, Y, @myfun, beta0)
% ⾮线性回归
% beta—系数估计
% r—残差
matlab拟合数据% J—雅可⽐矩阵
[Ypred,delta]=nlpredci(@myfun, X, beta, r, 'Jacobian', J)
% ⾮线性回归预测置信区间
% Ypred—预测响应
% delta—置信区间半⾓
plot(X(:, 1), Y, 'k.', X(:, 1), Ypred, 'r');
saveas(gcf,sprintf('⾮线性曲线拟合_1.jpg'),'bmp');
elseif choose==3
beta0=ones(7, 1);
% 初始值的选取可能会导致结果具有较⼤的误差。
[beta,resnorm,r, ~, ~, ~, J]=lsqcurvefit(@myfun,beta0,X,Y)
% 在最⼩⼆乘意义上解决⾮线性曲线拟合（数据拟合）问题
% beta—系数估计
% resnorm—残差的平⽅范数 sum((fun(x,xdata)-ydata).^2)
% r—残差 r=fun(x,xdata)-ydata
% J—雅可⽐矩阵
[Ypred,delta]=nlpredci(@myfun, X, beta, r, 'Jacobian', J)
plot(X(:, 1), Y, 'k.', X(:, 1), Ypred, 'r');
saveas(gcf,sprintf('⾮线性曲线拟合_2.jpg'),'bmp');
end
end
function yy=myfun(beta,x) %⾃定义拟合函数
yy=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);
end
demo.m
clear
clc
X=[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];
Y=[0.330; 0.336; 0.294; 0.476; 0.209; 0.451; 0.482; 0.5];
choose=1;
fit_nonlinear_data(X, Y, choose)
2. 结果
(1)多元线性拟合(regress)
choose=1:
>> demo
beta =
0.200908829282537
0.044949392540298
-0.003878606875016
0.070813489681112
bint =
-0.026479907290565 0.428297565855639
-0.057656451966002 0.147555237046598
-0.017251051845827 0.009493838095795
0.000201918738160 0.141425*********
r =
0.028343433030705
-
0.066584917256987
0.038333946339215
0.037954851676187
-0.082126284727611
0.058945364984698
-0.010982985302994
-0.003883408743214
rint =
-0.151352966773048 0.208039832834458
-0.188622801533810 0.055452967019837
-0.090283529625345 0.166951422303776
-
0.090266067743345 0.166175771095720
-0.108068661106325 -0.056183908348897
-0.130409602930181 0.248300332899576
-0.206254481234707 0.184288510628719
-0.184329400080620 0.176562582594191
states =
0.768591079367914 4.428472778943478 0.092289917768436 0.004625488283939
(2)多元⾮线性拟合(nlinfit)
choose=2:
>> demo
beta =
0.312525876099987
0.015300533415459
-0.036942272680920
0.299760796634952
0.009412595106141
0.000976411370591
-0.062931846673372
r =
1.0e-03 *
-0.047521336834000
0.127597019984715
-0.092883949615763
-0.040370056416994
0.031209476614974
0.211856736183458
-0.727835090583939
0.537947200592082
J =
1.0e+02 *
0.010000000000266 0.010000000001236 0.129999999998477 0.014999999999641 0.010000000007909 1.689999999969476 0.022499999999756 0.010000000000266 0.014000000006524 0.189999999999301 0.029999999999283 0.019600000006932 3.609999999769248 0.089999999999024 0.010000000000266 0.018000000011811 0.249999999990199 0.009999999999965 0.032399999999135 6.250000000033778 0.010000000000377 0.009999999999679 0.022000000005116 0.099999999998065 0.025000000000218 0.048400000003999 1.000000000103046 0.062500000001264 0.009999999999972 0.025999999998421 0.159999999998889 0.004999999999982 0.067599999997174 2.559999999999039 0.002499999999730 0.009999999999679 0.029999999991726 0.219999999999713 0.019999999999930 0.089999999993269 4.839999999890361 0.040000000000052 0.009999999999092 0.033999999985031 0.279999999990611 0.034999999998348 0.115599999997155 7.839999999636182 0.122500000000614
0.010000000000266 0.034999999992344 0.299999999994194 0.037000000000420 0.122499999994626 8.999999999988553 0.136899999999292
Ypred =
0.330047521336834
0.335872402980015
0.294092883949616
0.476040370056417
0.208968790523385
0.450788143263817
0.482727835090584
0.499462052799408
delta =
0.011997285626178
0.011902559677366
0.011954353934643
0.012001513980794
0.012005923574387
0.011706970437467
0.007666390995581
0.009878186927507
(3)多元⾮线性拟合(lsqcurvefit) choose=3:
>> demo
beta =
0.312525876070457
0.015300533464733
-0.036942272680581
0.299760796608728
0.009412595094407
0.000976411370579
-0.062931846666179
resnorm =
8.937848643213721e-07
r =
1.0e-03 *
0.047521324135769
-0.127597015215197
0.092883952947764
0.040370060121864
-0.031209466218374
-0.211856745335304
0.727835089662676
-0.537947200236699
J =
1.0e+02 *
(1,1) 0.010000000000000 (2,1) 0.010000000000000 (3,1) 0.010000000000000 (4,1) 0.010000000000000 (5,1) 0.010000000000000 (6,1) 0.010000000000000 (7,1) 0.010000000000000 (8,1) 0.010000000000000 (1,2) 0.010000000000000 (2,2) 0.014000000059605 (3,2) 0.017999999970198 (4,2) 0.022000000029802 (5,2) 0.026000000014901 (6,2) 0.030000000000000 (7,2) 0.034000000059605 (8,2) 0.035000000000000 (1,3) 0.130000000000000 (2,3) 0.190000000000000 (3,3) 0.250000000000000 (4,3) 0.100000000000000 (5,3) 0.160000000000000 (6,3) 0.220000000000000 (7,3) 0.280000000000000 (8,3) 0.300000000000000 (1,4) 0.015000000000000 (2,4) 0.030000000000000 (3,4) 0.010000000000000 (4,4) 0.025000000000000 (5,4) 0.005000000000000 (6,4) 0.020000000000000 (7,4) 0.035000000000000 (8,4) 0.036999999880791 (1,5) 0.010000000000000 (2,5) 0.019599999934435 (3,5) 0.032399999983609 (4,5) 0.048400000035763 (5,5) 0.067599999997765 (6,5) 0.090000000000000 (7,5) 0.115600000023842 (8,5) 0.122500000000000 (1,6) 1.690000000000000 (2,6) 3.610000000000000 (3,6) 6.2500000
00000000 (4,6) 1.000000000000000 (5,6) 2.560000000000000 (6,6) 4.840000000000000 (7,6) 7.840000000000000 (8,6) 9.000000000000000 (1,7) 0.022500000000000 (2,7) 0.090000000000000 (3,7) 0.010000000000000 (4,7) 0.062500000000000 (5,7) 0.002500000000000 (6,7) 0.040000000000000 (7,7) 0.122500000000000 (8,7) 0.136899999976158 Ypred =
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