Multi Variable Non-linear Curve Fitting in MATLAB

Dear all,

I am trying to curve fit my objective variable "tau" which is called Ignition delay (unit, milli-sec). tau is modeled as a function of three variables as listed below

1. Temperature, T (K),
2. Pressure, P (atm), and
3. Equivalence ration, phi (unitless)

The proposed curve-fitting equation has the following form. Here, a, b, c, d, e are the curve-fitting constant. And, here i = 3. Therefore, in total there is 15 curve-fitting constants.

tau(i) = ai * (p^bi) * (T^ci) * EXP(di / T) * (phi^ei)

Can anyone please suggest which MATLAB curve-fitting/regression tool should I use? Thanks in advance. I have attached a test data file if anyone is interested. Thanks.

****

Ultimate expression for the tau (if anyone is interested why i is set to be 3)

tau = (tau-1 + tau-2 + tau-3) / { (tau-1+tau-2) * tau-3 }

Answer 1

[Star Strider]

I would do this:

[D,S] = xlsread('Data_Source.xlsx');
EPT = D(:,1:3);
tauv = D(:,4);
% tau(i) = ai * (p^bi) * (T^ci)  * EXP(di / T) * (phi^ei)
taufcn = @(b,x) b(1) * x(:,2).^b(2) .* x(:,3).^b(3) .* exp(b(4)./x(:,3)) .* x(:,1).^b(5);

[B,R,J,CovB] = nlinfit(EPT,tauv,taufcn,rand(5,1));
BCI = nlparci(B,R,'covar',CovB);

You can have a matrix independent variable in all curve-fitting and optimization applications in MATLAB. You simply have to refer to the individual columns of the matrix as the different independent variables. (Here, they are ‘EPT’=[‘Equivalence Ratio’, ‘Pressure’, ‘Temperature’]), so ‘x(:,1)’ is ‘Equiv. Ratio’, and so for the others), with ‘tau’ as the single dependent variable.

I also calculate the parameter confidence intervals in this code, finding that ‘b(1)’ is not significantly different from zero, and so is not needed in the model. The others are all significant.

Experiment to get the result you want.

[M Fahd]

Dear Star Strider, Noted with thanks for your response and suggestion.

[Star Strider]

As always, my pleasure.

[Miraboreasu]

@Star Strider Hello, sorry to bother you, I am using nlinfit, I have 2 variable, so I cannot plot 2D to see if the fitting is fine, so should I sum up the R? or how to evaluate the my fitting?

Answer 2

[Stephan]

Hi,

i suggest to model this as a system of 4 equations with the 15 unknownse and use . 3 equations to calculate the tau(i) values and the 4th equation to calculate tau from the results of equations 1-3. Then subtract the measured value for tau from the result of equation 4. This is a job for fsolve.

Best regards

Stephan

[M Fahd]

Dear Stephan, thanks for the reply. I will try to implement your suggestion along with what Star Strider suggested below. Thank you.

Answer 3

[Alex Sha]

Hi, Fahd , refer to the results below：

Root of Mean Square Error (RMSE): 1.09231944926872

Sum of Squared Residual: 1355.43178122881

Correlation Coef. (R): 0.999883881381431

R-Square: 0.999767776246396

Adjusted R-Square: 0.999766954942228

Determination Coef. (DC): 0.999758349281268

Chi-Square: 190.539760828559

F-Statistic: 331766.054300988

Parameter Best Estimate

---------- -------------

a1 0.509520813430713

b1 -0.410333298909599

c1 -1.88156624504009

d1 11326.3070797332

e1 -0.0582321791543802

a2 3.53682438415215E-6

b2 0.378224086620891

c2 -2.32994413523028

d2 19151.2160516163

e2 -0.0238944122916122

a3 2.16069394348543

b3 -0.87187579659814

c3 -0.258436168897444

d3 3550.15220868462

e3 -2.03331105345266

[Star Strider]

Please post the complete code that produced those parameter estimates.

[harsh Brar]

Hey Alex sha... Can you please share the code?

[Pedro Machado]

Hi @Alex Sha, can you please share the full code you used? Thanks!!

Answer 4

[S0852306]

Try this: Multivariable surface fitting

The fitted surface matches your data quite well:

root mean square error : 0.1482

sum of square error : 24.9477

mean absolute error : 0.0971

code: (to run this script, download the toolbox at file exchange : https://tinyurl.com/wre9r5uk)

clear; clc; close all;
T=readtable("Data_Source.xlsx");
%%
data(1,:)=T.Equiv_Ratio_Phi';
data(2,:)=T.Pressure_P_atm_';
data(3,:)=T.Temperature_T_K_';
label=T.IDT_Tau_milli_sec_';
%%
NN.InputAutoScaling='on'; NN.LabelAutoScaling='on';
InSize=3; OutSize=1;
LayerStruct=[InSize,10,10,10,OutSize];
NN=Initialization(LayerStruct,NN);
%%
option.MaxIteration=1500;
NN=OptimizationSolver(data,label,NN,option);

Stats

R=FittingReport(data,label,NN);
%%
figure
p=NN.Evaluate(data); % use this function to evaluate fitted model
plot(label,'.')
hold on
plot(p)
legend('target value','fitted n-d surface')
% SSE=sum(R.ErrorVector.^2); % MSE=mean(R.ErrorVector.^2); % RMSE=sqrt(MSE)

The model I used is a neural net, it's a universal approximator (able to approximate any continuous function given

enough parameters), although it has a fancy name, its mathematical model is actually quite simple.

Detail math info can be found at file exchange website.