fitting data to a decaying exponential distribution

Hello there,

I have this data (attached) and i dont know how to fit it to a decaying exponential distribution, if it possible I want to have the mean and the decaying constant

[Star Strider]

What are we suppolsed to do with these?

Data = readmatrix('https://www.mathworks.com/matlabcentral/answers/uploaded_files/1306265/data.txt')

Data = 1070×2

0 0.2635 2.3400 0.0471 0 0.0387 0 0.0379 0 0.0245 0 0.0355 0.4000 0.0178 0 0.0293 0.3000 0.0178 0.7000 0.0178

% Data = sortrows(Data,2)

figure
yyaxis left
plot(Data(:,1))
yyaxis right
plot(Data(:,2))
grid

figure
plot(Data(:,1), Data(:,2))
grid

Sorting by either column does not help.

.

[Hozifa]

Thank you,

I want something close to figure 2 you made, where a fitting decaying line is shown (the curve has mean and decay constant)

Answer 1

[Image Analyst]

See attached demo of fitnlm

[Hozifa]

Thank you very much

I already has this file, from a previous question, and you gave it to me.

The thing is I use it now but it does not work,

Can you help me please

[Image Analyst]

OK but from your data that @Star Strider plotted, it looks virtually nothing like an exponential decay, unless you're talking about fitting only the local maxima. Are you sure you attached the right data? Can you plot it and draw a red line on it where you expect the fit would be?

[Hozifa]

Thank you @Image Analyst,

May I know something, in the figure uploaded, what are the mean of the distribution and the time decay constant?

[Image Analyst]

If you want to use fitnlm to fit the histogram of that data to an exponential decay, see the code below:

% Uses fitnlm() to fit a non-linear model (an exponential decay curve, Y = a * exp(-b*x) + c) through noisy data.
% Requires the Statistics and Machine Learning Toolbox, which is where fitnlm() is contained.
% Initialization steps.
clc;    % Clear the command window.
close all;  % Close all figures (except those of imtool.)
clear;  % Erase all existing variables. Or clearvars if you want.
workspace;  % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 20;

%------------------------------------------------------------------------------------------------------
% Read in the data samples drawn from an exponential decay distribution.
data = importdata('data.txt')
[rows, columns] = size(data)

%------------------------------------------------------------------------------------------------------
% Plot the histograms of each column.
% I don't know what each column represents though!!!
subplot(1, 2, 1);
histObject1 = histogram(data(:, 1))
grid on;
title('Histogram of Column 1', 'FontSize',fontSize);
subplot(1, 2, 2);
histObject2 = histogram(data(:, 2))
title('Histogram of Column 2', 'FontSize',fontSize);
grid on

%------------------------------------------------------------------------------------------------------
% Create the Y coordinates from the first histogram object, 
% which is for data column 1. 
% (You can repeat later to fit column #2).
X = histObject1.BinEdges(1:end-1);
Y = histObject1.Values;

%--------------------------------------------------------------------------------------------------------------------------------------
% Now we have noisy training data that we can send to fitnlm().
% Plot the noisy initial data.
figure;
plot(X, Y, 'b*', 'LineWidth', 2, 'MarkerSize', 15);
grid on;
% Convert X and Y into a table, which is the form fitnlm() likes the input data to be in.
% Note: it doesn't matter if X and Y are row vectors or column vectors since we use (:) to get them into column vectors for the table.
tbl = table(X(:), Y(:));
% Define the model as Y = a * exp(-b*x) + c
% 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);  

%------------------------------------------------------------------------------------------------------
% Guess values to start with.  Just make your best guess.
aGuessed = 500 % Arbitrary sample values I picked.
bGuessed = 1
cGuessed = 10
beta0 = [aGuessed, bGuessed, cGuessed]; % 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:
% First get a lot more X values so the curve will look smooth and
% not be evaluated only at the training points.
xFitted = linspace(min(X), max(X), 1000);
yFitted = coefficients(1) * exp(-coefficients(2)*xFitted) + coefficients(3);

%------------------------------------------------------------------------------------------------------
% Now we're done and we can plot the smooth model as a red line going through the noisy blue markers.
hold on;
plot(xFitted, 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', 'north');
legendHandle.FontSize = 30;

%------------------------------------------------------------------------------------------------------
% Place formula text roughly in the middle of the plot.
formulaString = sprintf('Y = %.3f * exp(-%.3f * X) + %.3f', coefficients(1), coefficients(2), coefficients(3))
xl = xlim;
yl = ylim;
xt = xl(1) + abs(xl(2)-xl(1)) * 0.325;
yt = yl(1) + abs(yl(2)-yl(1)) * 0.59;
text(xt, yt, formulaString, 'FontSize', 25, 'FontWeight', 'bold');

%------------------------------------------------------------------------------------------------------
% Set up figure properties:
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
% Get rid of tool bar and pulldown menus that are along top of figure.
% set(gcf, 'Toolbar', 'none', 'Menu', 'none');
% Give a name to the title bar.
set(gcf, 'Name', 'Demo by ImageAnalyst', 'NumberTitle', 'Off') 

Note: the fit above is only for the first column of your data (the left histogram). You'd need to repeat it on the second histogram to get the other equation.