How do I make the fit better so that there are no negative y-values for the fit?

I am attempting to fit a cubic spline to rheological data which can be found in the attached excel file. The data is best visualized on a log-log plot, however, the cubic spline being fit between some points has negative y-values. How do I improve the fit between these points? The code is below:

clc;
clear all;

%% --Variable Assignment--
osc_strain = xlsread("ag_gel_data.xlsx",1,'A1:A25');
stor_mod = xlsread("ag_gel_data.xlsx",1,'K1:K25');
loss_mod = xlsread("ag_gel_data.xlsx",1,'L1:L25');

osc_strain_t = xlsread("ag_gel_data.xlsx",1,'A1:A21');
stor_mod_t = xlsread("ag_gel_data.xlsx",1,'K1:K21');

tb1 = table(osc_strain,stor_mod,loss_mod);


%% --Fit: Cubic Spline Interpolant--
[xData, yData] = prepareCurveData( osc_strain, stor_mod );

% Set up fittype and options.
ft = 'cubicinterp';
excludedPoints = yData < 0;
opts = fitoptions( 'Method', 'CubicSplineInterpolant' );
opts.ExtrapolationMethod = 'none';
opts.Exclude = excludedPoints;

% Fit model to data.
[fitresult, gof] = fit( xData, yData, ft, opts );


%% --Plotting--
lg1 = loglog(tb1,"osc_strain","stor_mod",'LineWidth',1);
hold on;
lg2 = loglog(tb1,"osc_strain","loss_mod",'LineWidth',1);

% Plot fit with data on a log-log scale.
h = plot(fitresult);


%% --Figure Stylization--
lg1.LineStyle = "-";
lg1.Color = "magenta";
lg1.Marker = ".";
lg1.MarkerSize = 16;

lg2.LineStyle = "-";
lg2.Color = "magenta";
lg2.Marker = "o";
lg2.MarkerSize = 4;

h.LineStyle = "--";
h.Color = "black";

xlabel('Oscillation Stress, \gamma (%)')
ylabel("G',G'' (Pa)")
legend('Storage Modulus','Loss Modulus','Cubic Spline','Location','northwest')

Below is the figure that is produced. The splines between the last 6 data points are the ones I would like to smooth more. Any ideas?

Answer 1

[Matt J]

Why use fit() if you just want to interpolate? Why not use interp1 with the 'pchip' method instead?

clc;
clear all;

%% --Variable Assignment--
osc_strain = xlsread("ag_gel_data.xlsx",1,'A1:A25');
stor_mod = xlsread("ag_gel_data.xlsx",1,'K1:K25');
loss_mod = xlsread("ag_gel_data.xlsx",1,'L1:L25');

osc_strain_t = xlsread("ag_gel_data.xlsx",1,'A1:A21');
stor_mod_t = xlsread("ag_gel_data.xlsx",1,'K1:K21');

tb1 = table(osc_strain,stor_mod,loss_mod);


%% --Fit: Cubic Spline Interpolant--
[xData, yData] = prepareCurveData( osc_strain, stor_mod );

%% --Plotting--
lg1 = loglog(tb1,"osc_strain","stor_mod",'LineWidth',1);
hold on;
lg2 = loglog(tb1,"osc_strain","loss_mod",'LineWidth',1);

% Plot fit with data on a log-log scale.
xup=linspace(xData(1) , xData(end),1e4);
h = plot(xup, interp1(xData,yData,xup,'pchip'),'--');


%% --Figure Stylization--
lg1.LineStyle = "-";
lg1.Color = "magenta";
lg1.Marker = ".";
lg1.MarkerSize = 16;

lg2.LineStyle = "-";
lg2.Color = "magenta";
lg2.Marker = "o";
lg2.MarkerSize = 4;

h.LineStyle = "--";
h.Color = "black";

xlabel('Oscillation Stress, \gamma (%)')
ylabel("G',G'' (Pa)")
legend('Storage Modulus','Loss Modulus','Cubic Spline','Location','northwest')

[Elias Kerstein]

I used the pre-generated code from the MATLAB Curve Fitting tool box which has a UI. After selecting Interpolation>Cubic for my data, it fit this curve. I exported the code (below) and implemented it into my script.

function [fitresult, gof] = createFit1(osc_strain, stor_mod)
%CREATEFIT1(OSC_STRAIN,STOR_MOD)
%  Create a fit.
%
%  Data for 'untitled fit 1' fit:
%      X Input: osc_strain
%      Y Output: stor_mod
%  Output:
%      fitresult : a fit object representing the fit.
%      gof : structure with goodness-of fit info.
%
%  See also FIT, CFIT, SFIT.

%  Auto-generated by MATLAB on 31-Jan-2024 14:13:01


%% Fit: 'untitled fit 1'.
[xData, yData] = prepareCurveData( osc_strain, stor_mod );

% Set up fittype and options.
ft = 'cubicinterp';
excludedPoints = yData < 0;
opts = fitoptions( 'Method', 'CubicSplineInterpolant' );
opts.ExtrapolationMethod = 'none';
opts.Exclude = excludedPoints;

% Fit model to data.
[fitresult, gof] = fit( xData, yData, ft, opts );

% Plot fit with data.
figure( 'Name', 'untitled fit 1' );
h = plot( fitresult, xData, yData, excludedPoints );
legend( h, 'stor_mod vs. osc_strain', 'Excluded stor_mod vs. osc_strain', 'untitled fit 1', 'Location', 'NorthEast', 'Interpreter', 'none' );
% Label axes
xlabel( 'osc_strain', 'Interpreter', 'none' );
ylabel( 'stor_mod', 'Interpreter', 'none' );
grid on

[Elias Kerstein]

Would I be able to extract formulas for each interpolation from your solution? The end goal is to find the first derivative of the curve over the entire range of x.

[Torsten]

The end goal is to find the first derivative of the curve over the entire range of x.

Then you'll better use the pchip interpolation function directly. Using Breaks and Coefs of the output, you should be able to build the derivative of the obtained piecewise cubic function.

https://uk.mathworks.com/help/matlab/ref/pchip.html

[Elias Kerstein]

Thank you both so much!

[Matt J]

Would I be able to extract formulas for each interpolation from your solution? The end goal is to find the first derivative of the curve over the entire range of x.

Easiest would be to take a numerical derivative,

clc;
clear all;

%% --Variable Assignment--
osc_strain = xlsread("ag_gel_data.xlsx",1,'A1:A25');
stor_mod = xlsread("ag_gel_data.xlsx",1,'K1:K25');
loss_mod = xlsread("ag_gel_data.xlsx",1,'L1:L25');

osc_strain_t = xlsread("ag_gel_data.xlsx",1,'A1:A21');
stor_mod_t = xlsread("ag_gel_data.xlsx",1,'K1:K21');

tb1 = table(osc_strain,stor_mod,loss_mod);


%% --Fit: Cubic Spline Interpolant--
[xData, yData] = prepareCurveData( osc_strain, stor_mod );

%% --Plotting--

% Plot derivative
xup=linspace(xData(1) , xData(end),1e4);
yup=interp1(xData,yData,xup,'pchip');

plot( xup,gradient(yup,xup),'-.')
xlabel('Oscillation Stress, \gamma (%)')
ylabel("Derivative (Pa)")

[Elias Kerstein]

Oh, this is almost what I need! Is there a way to generate a table from the derivative curve containing the values of the gradient over the range of x?

[Matt J]

Perhaps as follows?

T = table( xup, yup, gradient(yup,xup),'Var',{'x','y', 'dy/dx'} )

[Elias Kerstein]

I ended up making a matrix and transposing it so I could paste the data into a graphing software. Because I need to put the data on a log-log plot though, much of the data is irrelavant as they contain negative y-values. I am trying to truncate the table where the y-values become negative to decrease file size. Do you have an idea on how to do this.

Below is the code I used to generate the matrix:

K = gradient(yup,xup);
Ktable0 = [xup; K];
Ktable1 = Ktable0.';

[Matt J]

I am trying to truncate the table where the y-values become negative to decrease file size. Do you have an idea on how to do this.

But they don't become negative. That was what was achieved by using pchip interpolation instead of cubic spline interpolation.

[Elias Kerstein]

Many of the values of the gradient are negative, I apologize if that was unclear.

[Matt J]

Perhaps as follows?

K = gradient(yup,xup);
Ktable0 = [xup; K].';
Ktable1=Ktable0(K>=0,:);

You realize I hope that the region where the gradient is >=0 may not be contiguous. Basically, you are keeping only the regions where stor_mod is increasing and discarding wherever it is decreasing.

[Elias Kerstein]

Thank you so much! Your solution worked. Yes, I am aware, however the culture surrounding how rheological data is plotted demands it to be on a log-log plot, which eliminates the region in which yData decreases. I greatly appreciate your assistance!