Non-linear curve fitting to a model with multiple observational variables in MATLAB (codes included)

Utpal Kumar   3 minute read      

How to fit data to non-linear model

Non-linear model is the one in which observational data is modeled by a non-linear combination of one or more model parameters and observational variables.

The non-linear equation is of the form:

\[y = f(x1,x2)\]

In this case, we use the MATLAB function lsqcurvefit:

Description

%% Fit Model
% - Utpal Kumar

clear; close all; clc
% generate some random data
Kdp = 0:40;
Zdr = 100:100+length(Kdp)-1;

xdata = [Kdp; Zdr]; %define independent variable

noise = 0.1*randn(size(xdata(1,:)));
ydata = 26.778*xdata(1,:).^0.946 .*xdata(2,:).^-1.249 + noise; %define dependent, ydata



% define optimization options
options = optimset('Display','iter','FunValCheck','on', ...
                   'MaxFunEvals',Inf,'MaxIter',Inf, ...
                   'TolFun',1e-6,'TolX',1e-6);
paramslb = [-Inf -Inf -Inf];  % lower bound
paramsub = [ Inf Inf Inf];  % upper bound

% define the initial seed
params0 = [20,0.9,-1.2];

% define model function
modelfun = @(pp,xdata) pp(1)*xdata(1,:).^pp(2).*xdata(2,:).^pp(3);

[params,resnorm,residual,exitflag,output] = lsqcurvefit(modelfun,params0,xdata,ydata,paramslb,paramsub,options);
params

% compute model fit
modelfit = modelfun(params,xdata);

% check squared error (the aim is to minimize squared error)
squarederror = sum((ydata(:)-modelfit(:)).^2)


% visualize the data and results
figure;
scatter3(xdata(1,:),xdata(2,:),ydata,'k') %scatter plot of data
hold on
[X,Y] = meshgrid(xdata(1,:),xdata(2,:));
Z = params(1)*X.^params(2) .*Y.^params(3);
s = surf(X,Y,Z,'FaceColor','interp','FaceAlpha',0.7); %surface plot of the results with the estimated parameters
s.EdgeColor = 'none';
colorbar
title(sprintf('squared error = %.1f; params: [%.2f, %.2f, %.2f]',squarederror,params(1),params(2),params(3)));

OUTPUT:

                                        Norm of      First-order
Iteration  Func-count     f(x)          step          optimality
    0          4         3.00771                          45.8
    1          8         1.01784             10           18.5
    2         12         1.01784             20           18.5
    3         16        0.749083              5           3.19
    4         20        0.749083             10           3.19
    5         24        0.739709            2.5          0.941
    6         28        0.736382              5           2.32
    7         32        0.730617              5           2.06
    8         36        0.725309              5           1.68
    9         40        0.720793              5           1.39
  10         44        0.716863              5           1.17
  11         48        0.716863             10           1.17
  12         52        0.714394            2.5          0.363
  13         56        0.702756              5           1.61
  14         60        0.702475             10           2.95
  15         64         0.69606            2.5          0.302
  16         68        0.692996              5          0.649
  17         72        0.692936             10           2.11
  18         76        0.689562            2.5          0.248
  19         80        0.685872              5           1.03
  20         84        0.684458             10           1.45
  21         88        0.682476             10           1.15
  22         92        0.680904             10          0.983
  23         96        0.679528             10          0.849
  24        100        0.678304             10           0.74
  25        104        0.677204             10          0.649
  26        108        0.677204             20          0.649
  27        112        0.676476              5          0.188
  28        116        0.675021             10          0.279
  29        120        0.675021             20          0.279
  30        124        0.674572              5          0.461
  31        128        0.673816             10          0.439
  32        132        0.673816             20          0.439
  33        136        0.673347              5          0.165
  34        140        0.666859             10           1.04
  35        144        0.666859             20           1.04
  36        148         0.66599              5           0.11
  37        152        0.665912             10          0.313
  38        156        0.665785             10          0.301
  39        160        0.665663             10          0.278
  40        164        0.665549             10          0.257
  41        168        0.665443             10          0.238
  42        172        0.665344             10          0.221
  43        176        0.665251             10          0.206
  44        180        0.665164             10          0.192
  45        184        0.665082             10           0.18
  46        188        0.665082             20           0.18
  47        192        0.665025              5         0.0544
  48        196         0.66496             10          0.135
  49        200         0.66496             20          0.135
  50        204        0.664915              5         0.0478
  51        208        0.664852             10          0.117
  52        212        0.664852             20          0.117
  53        216        0.664813              5          0.043
  54        220        0.664754             10          0.101
  55        224        0.664754             20          0.101
  56        228        0.664719              5         0.0385
  57        232        0.664664             10         0.0868
  58        236        0.664664             20         0.0868
  59        240        0.664633              5         0.0342
  60        244        0.664581             10         0.0722
  61        248        0.664581             20         0.0722
  62        252        0.664553              5         0.0296
  63        256        0.664503             10         0.0543
  64        260        0.664503             20         0.0543
  65        264        0.664479              5         0.0227
  66        268        0.664421             10         0.0334
  67        272        0.664421             20         0.0334
  68        276        0.664401              5         0.0725
  69        280        0.664365             10         0.0896
  70        284        0.664364             20          0.321
  71        288         0.66428              5         0.0308
  72        292         0.66425             10         0.0596
  73        296        0.664243             20          0.273
  74        300        0.664181              5          0.027
  75        304        0.664156             10         0.0495
  76        308        0.664146             20          0.235
  77        312          0.6641              5         0.0237
  78        316        0.664078             10         0.0421
  79        320        0.664068             20          0.205
  80        324        0.664032              5         0.0208
  81        328        0.664015             10         0.0369
  82        332        0.664006             20          0.181
  83        336        0.663979             20          0.174
  84        340        0.663953             20          0.163
  85        344        0.663931             20          0.152
  86        348        0.663912             20          0.143
  87        352        0.663895             20          0.134
  88        356        0.663881             20          0.126
  89        360         0.66387             20          0.119
  90        364        0.663861             20          0.112
  91        368        0.663854             20          0.105
  92        372         0.66385             20         0.0991
  93        376        0.663847        19.0329         0.0845
  94        380        0.663842       0.413735       2.64e-05
  95        384        0.663842      0.0136481       3.46e-08

Local minimum found.

Optimization completed because the size of the gradient is less than
the selected value of the optimality tolerance.

<stopping criteria details>


params =

804.4592    0.9611   -1.9579


squarederror =

  0.6638
Description Description

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