fitting data to a decaying exponential distribution

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Hozifa
Hozifa 2023-2-24
评论： Image Analyst 2023-2-28
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
  2 个评论
Star Strider
Star Strider 2023-2-24
Ran in:
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
Hozifa 2023-2-25
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)

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Image Analyst
Image Analyst 2023-2-25
See attached demo of fitnlm
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Hozifa
Hozifa 2023-2-27
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
Image Analyst 2023-2-28
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.

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