Improving fit of custom function

Hi everyone,

I would like to fit a dataset to a custom equation. Equation:

I used cftool and chose custom equation and specified the function: A+m*x+(B+n*x)/(1+10^-[(x-C)/D]) :

How can I improve the fit? In paricular I would like the beginning of the curve to follow the data points much better. I am sure there are better functions to fit this dataset to, but I require to use the equation given to analyze and extract parameters C and D in particular.

I have attached the data in a .txt file.

Appreciate any help!

Answer 1

[Rajendra Gupta]

Alex Shaw's parameter determination for this problem is very different from what 'mz86' obtained from the Matlab curve fitting tool. Why is it so? I too find problems in fitting my custom equations that involve integrals. Also, Alex Shaw only provides the parameters, not the 95% confidence bounds for each parameters. What is the difference in the two approaches that yield drastically different parameters for fitting the same data using the same equation? How can I replicate the fits and parameter results obtained by Alex so that I could try to implement the same approach to my curve fitting problem?

Your early response will be gratefully appreciated.

Thank you.

Here is one of my custom equation to fit 1048 data points with errors. Excel data file is attached.

y=VPCexp(x,Om0,299840/H0,a)+5*log10(1+x)+25

function [y] = VPCexp(z,Om0,R0,a)

n = length(z);

y=zeros(size(z));

for i=1:n

zl=z(i);

fun = @(x) (Om0.*((1+x).^3.*exp(a./3.*(1-(1+x).^(-3)))-exp(2.*a./3.*(1-(1+x).^(-3))))+(1+x).^2).^-0.5;

yl = integral(fun,0,zl);

dP=R0*sinh(yl/exp(-a/3*(1-(1+zl)^(-3))));

y(i)= 5*log10(dP);

end

The fit I get is as follows:

General model:

f(x) = VPCexp(x,Om0,299840/H0,a)+5*log10(1+x)+25

Coefficients (with 95% confidence bounds):

H0 = 71.24 (70.47, 72.01)

Om0 = 0.5749 (0.3704, 0.7793)

a = 0.8509 (0.6572, 1.045)

Goodness of fit:

SSE: 1034

R-square: 0.997

Adjusted R-square: 0.997

RMSE: 0.9948

Answer 2

[Alex Sha]

Hi, what is the function you want to fit?

y= A+m*x+(B+n*x)/(1+10^(-((x-C)/D)))

or

y= A+m*x+(B+n*x)/(1+exp(-((x-C)/D)))

if the former:

Root of Mean Square Error (RMSE): 172.220715864347

Sum of Squared Residual: 5783695.11970149

Correlation Coef. (R): 0.99832342215494

R-Square: 0.996649655223151

Adjusted R-Square: 0.996614755798392

Determination Coef. (DC): 0.996649655223151

Chi-Square: 926.211553161795

F-Statistic: 11244.6209348302

Parameter Best Estimate

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

a 11846.2854391282

m 24.3940304694188

b -1.64416859185817E18

n -8.09054974984671E17

c -83.4293845593878

d -5.89360891288855

if the later:

Root of Mean Square Error (RMSE): 172.220715864346

Sum of Squared Residual: 5783695.11970144

Correlation Coef. (R): 0.998323422154939

R-Square: 0.996649655223149

Adjusted R-Square: 0.99661475579839

Determination Coef. (DC): 0.996649655223151

Chi-Square: 926.211451996129

F-Statistic: 11244.6209212544

Parameter Best Estimate

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

a 11846.2854211205

m 24.3940307991782

b -6.12857667267666E17

n -3.01572212177201E17

c -80.9034614293268

d -2.55956180987299