fitting data with a combination of exponential and linear form ( a*exp(-x/b)+c*x+d )

Hello Altruists,

I have been trying to fit my data to a combination of exponential and linear form, i.e., a*exp(-x/b)+c*x+d . I am not getting a good fit (image attached). I am getting an warning as well (Warning: Start point not provided, choosing random start point). Could anyone offer some help?

I have tried with this code:

data= readmatrix('data');

x= data(:,1);
y = data(:,2); 

% getting fitting parameters
explinearfit = fittype('a*exp(-x/b)+c*x+d')
fo = fitoptions(explinearfit);
fo.normalize = 'on';
myFit = fit(x,y,explinearfit);

%plot
plot(myFit,x,y)

ylabel('Y')
xlabel('X')
legend('Data','Fit', 'location', 'best')

Answer 1

[Matt J]

If you download fminspleas, you can get a pretty good fit with a fairly naive initial guess [b,e,f]=[-1,0,0]:

[x,y]=readvars('data.csv');

flist={  @(p,x)exp(p(1)*x)  ,@(p,x) x, 1, @(p,x)tanh(p(2)*x+p(3))};

warning off
[p,coeff]=fminspleas(flist,[-1,0,0],x,y,-inf(1,3),[0,inf,inf]); warning on
p(:).'

ans = 1×3

-0.0007 0.0005 0.0075

xs=linspace(min(x),max(x));
plot(x,y,'--g',xs,  ffit(xs,p,coeff,flist));

function y=ffit(x,p,coeff,flist)

   y=0;
   for i=1:numel(flist)
       
      f=flist{i};
      if isnumeric(f)
         y=y+coeff(i)*f; 
      else
         y=y+coeff(i)*f(p,x); 
      end
   end
       
end

Answer 2

[Torsten]

Try

f(x) = a*atan(b*x)

It's too steep at the beginning and too flat at the end, but better than your model function.

data= readmatrix('data');
x = data(:,1);
y = data(:,2); 
f = @(p) p(1)*atan(p(2)*x);
fun = @(p) f(p) - y;
sol = lsqnonlin(fun,[2/pi 1])

Local minimum possible.

lsqnonlin stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

sol = 1×2

0.6070 0.0006

hold on
plot(x,y,'o')
plot(x,f(sol))
hold off

[Md Shariful Islam]

Thanks for your time answering my question. I appreciate this.

If I use 2 expoential functions (a*exp(b*x)+c*exp(d*x)), I could achieve a better fit as well. I used curve fitting tool achieve the fitting parameters. I was looking for something better; that is why I was exploring to linear+exponential form.

"

clc; clear; close all;

%%

data= readmatrix('data');

x= data(:,1);

y = data(:,2);

a = 0.7057;

b = 1.746e-05;

c = -0.6389;

d = -0.0006763;

y_f= a*exp(b*x)+c*exp(d*x);

figure;

plot(x,y, 's')

hold on

plot(x,y_f)

ylabel('X')

xlabel('Y')

legend('data', 'fit')

set(gca, 'fontname', 'times', 'fontsize', 12)

box on

"

Answer 3

[Alex Sha]

If taking fitting function as "y=a*exp(-x/b)+c*x+d", the result will be:

Sum Squared Error (SSE): 0.473516174967249
Root of Mean Square Error (RMSE): 0.0194398036424297
Correlation Coef. (R): 0.993172772220197
R-Square: 0.986392155479551

Parameter	Best Estimate      
---------	-------------      
a        	-0.626947642051749 
b        	1365.46862978889   
c        	1.57136165222552E-5
d        	0.682801486192597 

If taking fitting function as "y=a*exp(b*x)+c*exp(d*x)", the result will be:

Sum Squared Error (SSE): 0.517841961771885
Root of Mean Square Error (RMSE): 0.0203293308633968
Correlation Coef. (R): 0.992531430423344
R-Square: 0.985118640378209

Parameter	Best Estimate       
---------	-------------       
a        	-0.637793104918275  
b        	-0.00067464442828496
c        	0.705787394940032   
d        	1.74592589676531E-5 

A much better result will be obtained if taking fitting function as "y=a*exp(b*x)+c*x+d+tanh(e*x+f)*p"

Sum Squared Error (SSE): 0.154723411889559
Root of Mean Square Error (RMSE): 0.0111122622277951
Correlation Coef. (R): 0.997774312797138
R-Square: 0.995553579277802

Parameter	Best Estimate        
---------	-------------        
a        	-0.44380788683452    
b        	-0.000338851451599573
c        	1.0070723631259E-5   
d        	0.571547658395962    
e        	0.0025117355962601   
f        	-0.850904593274322   
p        	0.195041944730927

[Md Shariful Islam]

Hi Alex,

Thanks so much! I am convinced with the last one.

May I know how do I get the fitting parameters for the equation "y=a*exp(b*x)+c*x+d+tanh(e*x+f)*p" ? I have to perform similar fitting for different data sets. I tried using curve fitting tool to achieve these parameters and I am getting "Inf computed by model function, fitting cannot continue. Try using or tightening upper and lower bounds on coefficients" error.

[Torsten]

Try if you can reproduce the results if you use the fitting parameters as initial guesses in MATLAB.

@Alex Sha has his "special tool" to do the fitting (not part of MATLAB), and I think he invests quite a long time to adequate initial guesses for the parameters.

[Alex Sha]

I'm ashamed that it's not my product, although hopefully it's a commercial optimization solver package, I just like to use it because it's simple to use but works well.

Answer 4

[Sam Chak]

Hi @Md Shariful Islam

The data seems to exhibit the pattern of an nth-root function (a form of the power function), given by

where is a negative exponent function that varies with x. Because the data is bounded by 1, we can assume that . Since the exponential function-based models yield some good results in @Alex Sha's fitting, I attempted with the following model:

Fit model #1

% Data Sets
[x, y] = readvars('data.csv');

% Curve-fitting
fo = fitoptions('Method', 'NonlinearLeastSquares', ...
               'Lower', [-0.9, -0.006, -0.07, -0.0002, -1.2, -0.04, -0.2, -0.0008], ...
               'Upper', [0, 0, 0, 0, 0, 0, 0, 0], ...
               'StartPoint', [1 1 1 1 1 1 1 1]);
ft = fittype('x^(a*exp(b*x) + c*exp(d*x) + e*exp(f*x) + g*exp(h*x))', 'options', fo);
[yfit, gof] = fit(x, y, ft)

yfit = 
     General model:
     yfit(x) = x^(a*exp(b*x) + c*exp(d*x) + e*exp(f*x) + g*exp(h*x))
     Coefficients (with 95% confidence bounds):
       a =      -0.289  (-0.5166, -0.06147)
       b =   -0.003427  (-0.004609, -0.002245)
       c =    -0.06304  (-0.06519, -0.06088)
       d =  -0.0001293  (-0.0001322, -0.0001264)
       e =      -0.975  (-1.186, -0.7636)
       f =   -0.008562  (-0.01062, -0.006502)
       g =     -0.1089  (-0.1211, -0.09673)
       h =  -0.0007063  (-0.0007685, -0.0006441)

gof = struct with fields:
           sse: 0.1447
       rsquare: 0.9958
           dfe: 1245
    adjrsquare: 0.9958
          rmse: 0.0108

figure
plot(x, y, '.', 'color', '#A7B7F7'), hold on
plot(yfit, 'r'), hold off, grid on
legend('Data', 'Fitted curve', 'location', 'best')

Note that the coefficients are not unique. In my machine, I got different results"

Fit model #2

General model:

f(x) = x^(a*exp(b*x) + c*exp(d*x) + e*exp(f*x) + g*exp(h*x))

Coefficients (with 95% confidence bounds):

a = -0.1227 (-0.1285, -0.1169)

b = -0.0007697 (-0.0008078, -0.0007315)

c = -0.06455 (-0.06617, -0.06292)

d = -0.0001311 (-0.0001335, -0.0001288)

e = -1.12 (-1.968, -0.2716)

f = -0.03423 (-0.0573, -0.01116)

g = -0.8431 (-0.929, -0.7572)

h = -0.005286 (-0.005609, -0.004962)

Goodness of fit:

SSE: 0.1438

R-square: 0.9959

Adjusted R-square: 0.9958

RMSE: 0.01075

Fit model #3

General model:

f(x) = x^(a*exp(b*x) + c*exp(d*x) + e*exp(f*x) + g*exp(h*x))

Coefficients (with 95% confidence bounds):

a = -0.8429 (-0.9291, -0.7566)

b = -0.005285 (-0.00561, -0.00496)

c = -0.06455 (-0.06617, -0.06292)

d = -0.0001311 (-0.0001335, -0.0001288)

e = -1.107 (-1.94, -0.2753)

f = -0.03402 (-0.05694, -0.01109)

g = -0.1227 (-0.1285, -0.1169)

h = -0.0007697 (-0.0008078, -0.0007315)

Goodness of fit:

SSE: 0.1438

R-square: 0.9959

Adjusted R-square: 0.9958

RMSE: 0.01075

[Sam Chak]

Hi @Md Shariful Islam

You can also use the Curve-fitting App. But you definitely need to tighten the lower and upper bounds of the coefficients. The upper bounds can be set to zero. 7 out of 8 coefficients have lower bounds in the range , except for the fifth coefficient e ().

[Alex Sha]

For fitting function provided by @Sam Chak, the results are actually unique, only the order is different

Sum Squared Error (SSE): 0.143801140894944
Root of Mean Square Error (RMSE): 0.0107128649564239
Correlation Coef. (R): 0.997931669471303
R-Square: 0.995867616933781

Parameter	Best Estimate        
---------	-------------        
a        	-0.842982011054431   
b        	-0.00528502909652476 
c        	-0.122676258206711   
d        	-0.000769613241785951
e        	-0.0645440331501187  
f        	-0.000131143071549622
g        	-1.11996380236003    
h        	-0.0342317204260728

[Sam Chak]

Thanks @Alex Sha for the clarification on the coefficients in Fit models #2 and #3.