I need to fits the attached data as in image

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I need to fit the attached data as in the image below:
  2 Comments
Mushtaq Al-Jubbori
Mushtaq Al-Jubbori on 5 Jun 2021
Yes, I used tanh(x) but I need another function?

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Accepted Answer

Alex Sha
Alex Sha on 6 Jun 2021
Hi, the fitting function "y = p1/(1+Exp(p2+p3*x)+p4*x)^p5" is pretty good for all data set, where y=L, x=T
1: T1-L1:
Root of Mean Square Error (RMSE): 0.0481024350848356
Sum of Squared Residual: 0.0277661311330899
Correlation Coef. (R): 0.999085841980414
R-Square: 0.998172519645713
Parameter Best Estimate
---------- -------------
p1 6.26675447680147
p2 109.105635295479
p3 -67.1792841156261
p4 404935160.81852
p5 0.0161845071634169
2: T2-L2:
Root of Mean Square Error (RMSE): 0.0728301819569392
Sum of Squared Residual: 0.111388943481498
Correlation Coef. (R): 0.999334306206347
R-Square: 0.998669055560921
Parameter Best Estimate
---------- -------------
p1 7.42506477062961
p2 19.1556613661477
p3 -9.26234046573527
p4 -0.0219401101448161
p5 0.101288835010354
3: T3-L3:
Root of Mean Square Error (RMSE): 0.134092463526771
Sum of Squared Residual: 0.413558141817605
Correlation Coef. (R): 0.999276494987953
R-Square: 0.998553513435409
Parameter Best Estimate
---------- -------------
p1 11.4516493147292
p2 17.1095166703901
p3 -4.66159981573592
p4 0.0192821696183541
p5 0.129232672490336
4: T4-L4:
Root of Mean Square Error (RMSE): 0.174646688455142
Sum of Squared Residual: 1.06755130259216
Correlation Coef. (R): 0.999289186413636
R-Square: 0.998578878083226
Parameter Best Estimate
---------- -------------
p1 15.0727185425809
p2 61.2119358895463
p3 -10.7637556382665
p4 0.236072827848106
p5 0.037477078210382
5: T5-L5:
Root of Mean Square Error (RMSE): 0.154312409495508
Sum of Squared Residual: 0.500058714210495
Correlation Coef. (R): 0.999654251549009
R-Square: 0.999308622640009
Parameter Best Estimate
---------- -------------
p1 17.5989211228526
p2 138.340503536049
p3 -17.2818282854456
p4 -0.0357446437398955
p5 0.0179266207350469
  11 Comments
Mushtaq Al-Jubbori
Mushtaq Al-Jubbori on 10 Jun 2021
Thank you, but when run the cod appear.. Undefined function or variable 'FitCoth'

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More Answers (5)

Sulaymon Eshkabilov
Sulaymon Eshkabilov on 6 Jun 2021
You can start using curve fitting toolbox, cftool that is quite straightforward and does not require any addional coding.
An alternative way is the least squares method for that you have you data ready. You'd need to generate Vandermonde matrix and then compute the fit model coefficients.
  4 Comments
Sulaymon Eshkabilov
Sulaymon Eshkabilov on 8 Jun 2021
@John D'Errico This polynomial model is shown here mean to be an example not the proposed model. For V matrix it is viable to insert exp(), sin(), cos() tan() and polynonials in combination.

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Image Analyst
Image Analyst on 6 Jun 2021
@Mushtaq Al-Jubbori. What I did was to find where the flat part starts and fit the data to the left of that to an exponential growth curve. As an output you have the location where the flat part starts and the coefficients of the exponential fitted equation. Try this (click the copy icon in the upper right of the code block below and then paste into MATLAB):
clc; % Clear command window.
fprintf('Running %s.m ...\n', mfilename);
clear; % Delete all variables.
close all; % Close all figure windows except those created by imtool.
workspace; % Make sure the workspace panel is showing.
fontSize = 14;
L1=[1.4 1.9 2.4 3.132 4.2 4.532 4.532 4.4 4.532 4.532 4.4 4.5];%1.5Mev
L2=[1.65 2.2 2.8 3.4 4.266 5.6 6.6 7.4 7.5 7.534 7.5 7.5 7.4 7.4 7.4 7.5 7.6 7.532 7.4 7.6 7.6];%2.26MeV
L3=[1.8 2.2 2.8 3 3.8 4.8 5.8 6.4 7.8 8.6 9.8 10.6 11.2 11.4 11.2 11.4 11.2 11.4 11 11.2 11.4 11.2 11.4 ];%3.2MeV
L4=[2.05 2.4 2.8 3.4 3.7 4 4.5 5 6.3 7.2 8 8.6 9.8 10.4 11.2 12.2 13.8 14.6 14.6 14.532 14.532 14.6 14.6 14.532 14.532 14.4 14.4 14.4 14.4 14.532 14.5 14.2 14.4 14.4 14.6];%4MeV
L5=[3 3.4 3.8 4.4 5.2 6 6.8 8 9.2 10.8 12.8 15 16.4 17.4 17.8 17.6 17.8 17.8 17.6 17.8 17.8];%4.85MeV
T1=0.25:0.25:3;
T2=[0.5:0.25:2.25 2.75:0.25:5 5.5 5.75 6];
T3=[0.75:0.25:1.75 2.25:0.25:4 4.5:0.25:5 5.5 6 6.25 6.75:0.25:7.5 ];
T4=[1 1.25 1.75:0.25:3 3.5:0.25:9.25 9.75 10.25 10.75];
T5=[2:0.5:7.5 7.7 8:0.5:10.5 11.5 12];
subplot(5, 2, 1);
plot(T1,L1)
grid on;
title('L1 vs. T1', 'FontSize', fontSize);
kneeIndex = FindKneeIndex(L1, T1);
xline(T1(kneeIndex), 'Color', 'r', 'LineWidth', 2);
% Now fit the left portion of the data that follows an exponential to an exponential formula.
coefficients1 = FitExponentialGrowth(T1(1:kneeIndex), L1(1:kneeIndex), 2)
subplot(5, 2, 3);
plot(T2,L2)
grid on;
title('L2 vs. T2', 'FontSize', fontSize);
kneeIndex = FindKneeIndex(L2, T2);
xline(T2(kneeIndex), 'Color', 'r', 'LineWidth', 2);
% Now fit the left portion of the data that follows an exponential to an exponential formula.
coefficients2 = FitExponentialGrowth(T2(1:kneeIndex), L2(1:kneeIndex), 4)
subplot(5, 2, 5);
plot(T3,L3)
grid on;
title('L3 vs. T3', 'FontSize', fontSize);
kneeIndex = FindKneeIndex(L3, T3);
xline(T3(kneeIndex), 'Color', 'r', 'LineWidth', 2);
% Now fit the left portion of the data that follows an exponential to an exponential formula.
coefficients3 = FitExponentialGrowth(T3(1:kneeIndex), L3(1:kneeIndex), 6)
subplot(5, 2, 7);
plot(T4,L4)
grid on;
title('L4 vs. T4', 'FontSize', fontSize);
kneeIndex = FindKneeIndex(L4, T4);
xline(T4(kneeIndex), 'Color', 'r', 'LineWidth', 2);
% Now fit the left portion of the data that follows an exponential to an exponential formula.
coefficients4 = FitExponentialGrowth(T4(1:kneeIndex), L4(1:kneeIndex), 8)
subplot(5, 2, 9);
plot(T5,L5)
grid on;
title('L5 vs. T5', 'FontSize', fontSize);
kneeIndex = FindKneeIndex(L5, T5);
xline(T5(kneeIndex), 'Color', 'r', 'LineWidth', 2);
% Now fit the left portion of the data that follows an exponential to an exponential formula.
coefficients5 = FitExponentialGrowth(T5(1:kneeIndex), L5(1:kneeIndex), 10)
% Maximize figure window.
g = gcf;
g.WindowState = 'maximized';
fprintf('Done running %s.m\n', mfilename);
function kneeIndex = FindKneeIndex(L, T)
slopes = (L(end) - L) ./ (T(end) - T);
kneeIndex = find(slopes < 0.15, 1, 'first');
% figure;
% plot(T, slopes, 'b.-');
% grid on;
end
function coefficients = FitExponentialGrowth(X, Y, plotNumber)
fontSize = 14;
% Convert X and Y into a table, which is the form fitnlm() likes the input data to be in.
tbl = table(X(:), Y(:));
% Define the model as Y = a + exp(-b*x)
% Note how this "x" of modelfun is related to big X and big Y.
% x((:, 1) is actually X and x(:, 2) is actually Y - the first and second columns of the table.
modelfun = @(b,x) b(1) * exp(b(2)*x(:, 1)) + b(3);
beta0 = [11, .5, 0]; % Guess values to start with. Just make your best guess.
% Now the next line is where the actual model computation is done.
mdl = fitnlm(tbl, modelfun, beta0);
% Now the model creation is done and the coefficients have been determined.
% YAY!!!!
% Extract the coefficient values from the the model object.
% The actual coefficients are in the "Estimate" column of the "Coefficients" table that's part of the mode.
coefficients = mdl.Coefficients{:, 'Estimate'}
% Create smoothed/regressed data using the model:
yFitted = coefficients(1) * exp(coefficients(2)*X) + coefficients(3);
% Now we're done and we can plot the smooth model as a red line going through the noisy blue markers.
subplot(5, 2, plotNumber);
plot(X, Y, 'b.', 'MarkerSize', 20);
hold on;
plot(X, yFitted, 'r-', 'LineWidth', 2);
grid on;
title('Exponential Regression with fitnlm()', 'FontSize', fontSize);
xlabel('X', 'FontSize', fontSize);
ylabel('Y', 'FontSize', fontSize);
legendHandle = legend('Noisy Y', 'Fitted Y', 'Location', 'northwest');
legendHandle.FontSize = 12;
formulaString = sprintf('Y = %.3f * exp(%.3f * X) + %.3f', coefficients(1), coefficients(2), coefficients(3))
text(5, 17000, formulaString, 'FontSize', 12, 'FontWeight', 'bold');
end
  12 Comments
Mushtaq Al-Jubbori
Mushtaq Al-Jubbori on 8 Jun 2021
Did we can develobment the function by + or - any parameters?

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Scott MacKenzie
Scott MacKenzie on 6 Jun 2021
Edited: Scott MacKenzie on 6 Jun 2021
Here's what I put together. This is only for the T5-L5 data. You can repeat this for the rest of the data and build up the plot with the remaining points and lines.
L5=[3 3.4 3.8 4.4 5.2 6 6.8 8 9.2 10.8 12.8 15 16.4 17.4 17.8 17.6 17.8 17.8 17.6 17.8 17.8];%4.85MeV
T5=[2:0.5:7.5 7.7 8:0.5:10.5 11.5 12];
threshold = 0.2; % adjust accordingly
% find elbow (index where curve transitions to flat line)
elbow = find(diff(L5) < threshold, 1);
% plot markers, set axis limits
plot(T5, L5, 'or', 'markerfacecolor', 'r');
ax = gca;
ax.YLim = [0 20];
ax.XLim = [0 15];
% plot flat line
x1 = T5(elbow); x2 = T5(end);
y1 = mean(L5(elbow):L5(end)); y2 = y1;
line([x1 x2], [y1 y2], 'color', 'r', 'linewidth', 1.5);
hold on;
% plot curved line using polynomial fit
x = T5(1:elbow);
ySample = L5(1:elbow);
pf = polyfit(x,ySample,5); % order 5 looks pretty good
y = polyval(pf, x)
y(end) = L5(elbow); % ensure curved line meets flat line
plot(x, y, 'r', 'linewidth', 1.5);

Sulaymon Eshkabilov
Sulaymon Eshkabilov on 6 Jun 2021
This nonlinear least squares fit gives quite nice fit models:
...
x = T2;
y = L2;
p = [1, 3]; % Guess: Fit model parameters
q = [-1, -3]; % Guess: Fit model parameters
plot(x,y,'r*')
xlabel 't'
ylabel 'Response'
p = optimvar('p',2);
q = optimvar('q',2);
fun = p(1)*(1+exp(q(1)+q(2)*x) + p(2)*x).^-0.65;
obj = sum((fun - y).^2);
LSQ_Prob = optimproblem("Objective",obj);
x0.p = [1/2, 5/2];
x0.q = [-1/2,-5/2];
show(LSQ_Prob) % Display the problem formulation and parameters
[Sol,fval] = solve(LSQ_Prob,x0) % Show the results
figure
FM_val = evaluate(fun,Sol);
plot(x,y,'r*', x, FM_val,'b-')
legend('Original Data','Fitted Curve', 'location', 'southeast')
xlabel('T')
ylabel('L & Fit Model')
title("Data vs. Fitted Model Response")
SStot = sum((y-mean(y)).^2);
SSres = sum((y-FM_val).^2);
Rsq = 1 - SSres/SStot;
gtext(['R^2 = ' num2str(Rsq)])
gtext(['Fit model: ' num2str(Sol.p(1)) '*(1+exp(' num2str(Sol.q(1)) ' + ' num2str(Sol.q(2)) '*x) + ' num2str(Sol.p(2)) '*x).^{-0.65}'])
fprintf('Found fit model is: %f *(1+exp(%f+%f*x) + %f*x)^-0.65 \n', [Sol.p(1), Sol.q(1), Sol.q(2), Sol.p(2)])
  7 Comments
Image Analyst
Image Analyst on 8 Jun 2021
OK, fine. Then use the piecewise function I gave you, or use ad hoc function mysteriously found by that third party software.
but you may need to upgrade your MATLAB, as you said.

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Mushtaq Al-Jubbori
Mushtaq Al-Jubbori on 6 Jun 2021
Thanks
Sulaymon Eshkabilov
But the cod dont run in Matlab-2016, the function optimvar undefine

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