Curve fitting with Levenberg-Marquardt algorithm

I want to fit a custom function to a dataset with the following values:

%This data is collected from the data below. It comes from a 10x10 matrix
%and I need the rows [1 4 5] for specific purposes. That is why the codes
%looks too big, but is in fact a restribution of data.

x_data = [1.68995000000000	1.67356950000000	1.69012730000000	1.69841930000000	1.70660157500000	1.72105803000000	1.73417499500000	1.74831049750000	1.77170521600000	1.78799623600000 ; ...
    0.845360000000000	0.751521500000000	0.761509900000000	0.756846750000000	0.755428325000000	0.752491960000001	0.752734425000001	0.756773460000004	0.761931829999993	0.766921715000000 ; ...
    2.60905500000000	2.52328050000000	2.53137270000000	2.54665280000000	2.57327365000000	2.61874219000000	2.65582985000000	2.68521748250000	2.71079400200000	2.69824300800000 ]

y_data = [0.219970548869443	0.815706518602178	1.39509549501506	1.71961499263378	2.11330719376825	2.92533458286614	3.87232815557023	5.26474561602075	7.86182243535394	10.5374746473838 ; ...
    0.0642339602535873	0.169476385473391	0.333488789915357	0.386605265203827	0.440998338154743	0.538448868958661	0.620497982025440	0.725751134956693	0.922329295454408	1.20273021731183 ; ...
    0.721637615151627	2.37225924529110	4.21538558708732	5.33208295040392	6.79868950885182	9.65661324356717	12.5494793161649	16.2135119764150	22.1717696376411	27.3348431675366 ]

eave_data = [0.511623450000000	0.510701460000000	0.509541980000000	0.508871890500000	0.508029996750000	0.506469197900000	0.504673369800000	0.502044141175002	0.496770950180000	0.490995012415000 ; ...
    0.508641150000000	0.508465500000000	0.508151158000000	0.507993901000000	0.507816865000000	0.507554680200001	0.507337630400000	0.507106704825006	0.506744231299999	0.506337948470000 ; ...
    0.502123600000000	0.500216710000000	0.497755699000000	0.496243499500000	0.494246570250000	0.490432166300000	0.486290536300001	0.480809998450000	0.471233465220000	0.462093520630000]

Ce_data = [0.485782500000000	0.484491450000000	0.483157650000000	0.482326400000000	0.481256700000000	0.479023250000000	0.476403150000000	0.472544250000000	0.465113350000000	0.457141900000000 ; ...
    0.483133900000000	0.482860300000000	0.482498350000000	0.482327350000000	0.482136400000000	0.481847600000000	0.481619600000000	0.481382100000000	0.480901400000000	0.480256350000000 ; ...
    0.476538050000000	0.473938850000000	0.470942550000000	0.468958000000000	0.466260950000000	0.460949500000000	0.455317900000000	0.448167250000000	0.436138350000000	0.425210500000000]

%Original dataset, which came from an even bigger one. But these are
%important for the this curve fitting.

epsilonampl = [1.68995000000000	1.67356950000000	1.69012730000000	1.69841930000000	1.70660157500000	1.72105803000000	1.73417499500000	1.74831049750000	1.77170521600000	1.78799623600000 ; 
1.80913500000000	1.77612850000000	1.77461040000000	1.77382280000000	1.77118812500000	1.76756418000000	1.76984687500000	1.77848557500001	1.80125089100001	1.80059083500000 ; 
1.42231000000000	1.41365750000000	1.44327620000000	1.45454830000000	1.46592077500000	1.49942036000000	1.52881787500000	1.56116331500000	1.60480011299999	1.63390681050000 ; 
0.845360000000000	0.751521500000000	0.761509900000000	0.756846750000000	0.755428325000000	0.752491960000001	0.752734425000001	0.756773460000004	0.761931829999993	0.766921715000000 ; 
2.60905500000000	2.52328050000000	2.53137270000000	2.54665280000000	2.57327365000000	2.61874219000000	2.65582985000000	2.68521748250000	2.71079400200000	2.69824300800000 ; 
1.25974500000000	1.15777800000000	1.18113270000000	1.17785195000000	1.17870765000000	1.19160369000000	1.21128859000000	1.23377914250000	1.26779698600000	1.29254151900000 ; 
2.06843000000000	1.96272550000000	1.98881510000000	2.00038080000000	2.01178252500000	2.03431910000000	2.06737115500000	2.10785441250000	2.16685295800000	2.19641107850000 ;
1.59727500000000	1.56965400000000	1.61065480000000	1.61401480000000	1.62046692500000	1.63481720000000	1.64983897000000	1.67536094750000	1.71700520600000	1.75151178850000 ; 
1.65214500000000	1.64498350000000	1.65268200000000	1.66672140000000	1.68123452500000	1.70255762000000	1.72431821000000	1.75237155500000	1.78956005300000	1.81168614150000 ; 
1.31446500000000	1.34041950000000	1.36544320000000	1.37398980000000	1.38366432500000	1.39477109000000	1.40866851000000	1.43014082750000	1.46985591700000	1.50364010100000]

eliminierung_campl = [0.219970548869443	0.815706518602178	1.39509549501506	1.71961499263378	2.11330719376825	2.92533458286614	3.87232815557023	5.26474561602075	7.86182243535394	10.5374746473838 ; 
0.178212538524405	0.610891612212584	1.05873240345686	1.32057490932360	1.66272908653406	2.35017757341005	3.19217848022701	4.39544605838876	6.71506858407999	8.97745161731723 ; 
0.317374045573957	1.31533503453183	2.27416029843252	2.77867550745613	3.34090395849249	4.39157279755229	5.59916994226081	7.39821805023363	11.0712000028568	14.9433243509966 ; 
0.0642339602535873	0.169476385473391	0.333488789915357	0.386605265203827	0.440998338154743	0.538448868958661	0.620497982025440	0.725751134956693	0.922329295454408	1.20273021731183 ; 
0.721637615151627	2.37225924529110	4.21538558708732	5.33208295040392	6.79868950885182	9.65661324356717	12.5494793161649	16.2135119764150	22.1717696376411	27.3348431675366 ; 
0.277387135970351	0.816199784223801	1.62796170470247	1.79429210358029	2.03003583146033	2.61988161812730	3.42571414878310	4.66549357115017	7.24569141698775	10.0756212106590 ; 
0.433664540530333	1.04389545903620	1.70366206417345	2.02840711616856	2.44663548568462	3.26670408509225	4.29006082136354	5.75289862637344	8.54605544601133	11.4817886043978 ; 
0.296289880737664	0.825906844633911	1.47891262699295	1.73375721380982	2.07105831778929	2.79606909732602	3.67809507974306	5.01795852014047	7.74596871737049	10.6915272467139 ; 
0.228770613022142	0.718261535758381	1.20754669762319	1.50052534533639	1.90133688008586	2.72047090926408	3.70024032220998	5.11856065180731	7.81320857170644	10.4566681534346 ; 
0.234908809894484	0.636085998286533	1.00600951455756	1.22645349625657	1.48142020124878	1.94632942267008	2.41887607337860	3.17720915635447	4.91836945589996	7.00981003502703]

eave = [0.511623450000000	0.510701460000000	0.509541980000000	0.508871890500000	0.508029996750000	0.506469197900000	0.504673369800000	0.502044141175002	0.496770950180000	0.490995012415000 ; 
0.510243700000000	0.509318530000000	0.508121999000000	0.507421250500000	0.506506232000000	0.504705086500001	0.502566729650001	0.499390536075001	0.493113976700001	0.486402081405000 ; 
0.493188850000000	0.492474695000000	0.491534898000000	0.491014827500000	0.490374352750000	0.489270044900000	0.488091355500000	0.486388426150000	0.482883968699999	0.478827431860000 ; 
0.508641150000000	0.508465500000000	0.508151158000000	0.507993901000000	0.507816865000000	0.507554680200001	0.507337630400000	0.507106704825006	0.506744231299999	0.506337948470000 ; 
0.502123600000000	0.500216710000000	0.497755699000000	0.496243499500000	0.494246570250000	0.490432166300000	0.486290536300001	0.480809998450000	0.471233465220000	0.462093520630000 ; 
0.500206750000000	0.499500520000000	0.498217855000000	0.497682240500000	0.497130307250000	0.496112222700000	0.494860150099999	0.492905573300000	0.488731526790002	0.483871039955000 ; 
0.497748500000000	0.496921155000000	0.495837435000000	0.495266439500000	0.494578131500000	0.493339576300000	0.491938604200000	0.489877810925001	0.485673815829998	0.480839703225000 ; 
0.481535600000000	0.480925235000000	0.480061888000000	0.479630788000000	0.479123975250000	0.478177226500000	0.477058980900000	0.475361809474999	0.471749224670001	0.467555096565000 ; 
0.531027100000000	0.530124655000000	0.528954710000000	0.528237271000000	0.527313733250000	0.525530434900000	0.523419833000000	0.520284491225000	0.513938645160002	0.507106289950000 ; 
0.437652600000000	0.437408155000000	0.437121344000000	0.436952670000000	0.436748043750000	0.436399650500000	0.436042361900000	0.435518252075002	0.434288406240004	0.432692610370000 ]

Ce_0 = [ 0.485782500000000	0.484491450000000	0.483157650000000	0.482326400000000	0.481256700000000	0.479023250000000	0.476403150000000	0.472544250000000	0.465113350000000	0.457141900000000 ; 
0.484469600000000	0.483170000000000	0.481786800000000	0.480892850000000	0.479655000000000	0.477003550000000	0.473775450000000	0.469086250000000	0.460372850000000	0.451540700000000 ; 
0.468333850000000	0.467309750000000	0.466249550000000	0.465606400000000	0.464813150000000	0.463397650000000	0.461735150000000	0.459228100000000	0.454050600000000	0.448238500000000 ; 
0.483133900000000	0.482860300000000	0.482498350000000	0.482327350000000	0.482136400000000	0.481847600000000	0.481619600000000	0.481382100000000	0.480901400000000	0.480256350000000 ; 
0.476538050000000	0.473938850000000	0.470942550000000	0.468958000000000	0.466260950000000	0.460949500000000	0.455317900000000	0.448167250000000	0.436138350000000	0.425210500000000 ; 
0.475024700000000	0.473995850000000	0.472436900000000	0.472030300000000	0.471371950000000	0.469909900000000	0.467929150000000	0.464934750000000	0.458702750000000	0.451766800000000 ; 
0.472596500000000	0.471450800000000	0.470242400000000	0.469574550000000	0.468711000000000	0.467009550000000	0.464953750000000	0.461921350000000	0.455826150000000	0.448861700000000 ; 
0.457238800000000	0.456438900000000	0.455411950000000	0.454946450000000	0.454328950000000	0.452952400000000	0.451262350000000	0.448671700000000	0.443308000000000	0.437285950000000 ;
0.504230550000000	0.502981300000000	0.501583850000000	0.500667100000000	0.499423550000000	0.496834800000000	0.493668450000000	0.489073300000000	0.480056850000000	0.470997650000000 ; 
0.415623100000000	0.415323850000000	0.414980900000000	0.414779500000000	0.414529650000000	0.414098350000000	0.413593900000000	0.412770250000000	0.410770500000000	0.408280550000000 ]

Here ist the code of my plot:

%A for loop in order to do the fitting for each cycle.
i_vals = 1:10
j_vals = [1 4 5]

x_data = zeros(length(j_vals), length(i_vals))
y_data = zeros(length(j_vals), length(i_vals))
eave_data = zeros(length(j_vals), length(i_vals))
Ce_data = zeros(length(j_vals), length(i_vals))

figure

for i = 1:length(i_vals)
    for j = 1:length(j_vals)

x_data (j,i) = epsilonampl(j_vals(j), i)
y_data (j,i) = eliminierung_campl(j_vals(j), i)
eave_data (j,i) = eave(j_vals(j), i)
Ce_data (j,i) = Ce_0(j_vals(j), i)

fun = @(x,x_data)x(1)*exp(x(2)*x_data);

x0 = [100,-1];
x = lsqcurvefit(fun,x0,x_data,y_data)



times = linspace(x_data(1),x_data(end));
plot(x_data,y_data,'ko',times,fun(x,times),'b-')

legend('Data','Fitted exponential')
title('Data and Fitted Curve')

hold on

    end
end

This code creates the plot which is seen. I want to use the following function to fit the curve:

%fun = k .* (Ce_data - eave_data) .^2 ./ (1 + eave_data)
%k is a factor for the function so that the curve fitting is adapted to the
%function. Therwise the function is the same and the curve cannot be fitted
%to the data correctly.

It should look like that:

I would appreciate the help!

Answer 1

[Infinite_king]

Hi,

I understand that you want to fit a subset of the dataset using the selected non-linear function. The 'lsqcurvefit' function is not fitting the curve properly, possibly due to not including all the data points or, in some cases, including data points from other datasets.

To resolve this problem, you can try the following fixes,

• Copy all the datapoints before calling “lsqcurvefit"

num_columns = 10; % number of columns of dataset
selected_rowns = j_vals;

for j = 1:num_columns

x_data (:,j) = epsilonampl(selected_rowns, j); %copy j'th column of selected rows
y_data (:,j) = eliminierung_campl(selected_rowns, j); %copy j'th column of selected rows

end

• Fit the curve by passing the set of datapoints

for j = 1:num_columns

x = lsqcurvefit(fun,x0,x_data(:,j),y_data(:,j)); % fit the curve for only j'th column

times = linspace(min(x_data(:,j)),max(x_data(:,j))); % find the lowest and highest to calculate the range of plot
plot(x_data(:,j),y_data(:,j),'ko',times,fun(x,times),'b-')

legend('Data','Fitted exponential')
title('Data and Fitted Curve')
hold on
end

Please find the complete code and its output below,

i_vals = 1:10;
j_vals = [1 4 5];

x_data = zeros(length(j_vals), length(i_vals));
y_data = zeros(length(j_vals), length(i_vals));
eave_data = zeros(length(j_vals), length(i_vals));
Ce_data = zeros(length(j_vals), length(i_vals));

figure

% ------------- Modified ------------- 
num_columns = 10;
selected_rowns = j_vals;

for j = 1:num_columns

x_data (:,j) = epsilonampl(selected_rowns, j); %copy j'th column of selected rows
y_data (:,j) = eliminierung_campl(selected_rowns, j); %copy j'th column of selected rows

end

fun = @(x,x_data)x(1)*exp(x(2)*x_data);
x0 = [100,-1];

% ------------- Modified ------------- 
for j = 1:num_columns

x = lsqcurvefit(fun,x0,x_data(:,j),y_data(:,j)); % fit the curve for only j'th column

times = linspace(min(x_data(:,j)),max(x_data(:,j))); % find the lowest and highest to calculate the range of plot
plot(x_data(:,j),y_data(:,j),'ko',times,fun(x,times),'b-')

legend('Data','Fitted exponential')
title('Data and Fitted Curve')
hold on
end

For more information, please refer the following resources -

• https://in.mathworks.com/help/optim/ug/lsqcurvefit.html
• https://in.mathworks.com/help/matlab/ref/max.html
• https://in.mathworks.com/help/matlab/ref/min.html

Hope this is helpful.