MATLAB: Hyperbolic Least Squares Interpolation

hyperbolic functionleast squares interpolation

Hello Everybody,

I have got 4 datapoints from trials. They seem to be aligned in a hyperbolic manner. So what i want to do is to find the least squares regression of those values with a kind of a/(bx+c)-Function, where the c-value is equal to zero.

Does matlab provide a sort of standard-function like polyfit for such a problem? Or is it possible to modify the data in a way (coordinate-transformation) to apply polyfit?

Thanks for your help! Georg

Best Answer

You can use core MATLAB functions to do the regression:

x = ...;                                                    % Independent Variable
y = ...;                                                    % Dependent Variable
fcn1 = @(b,x) b(1)./(b(2).*x + b(3));                       % Objective Function #1
fcn2 = @(b,x) b(1)./(b(2).*x);                              % Objective Function #2
SSECF = @(b) sum((y - fcn2(b,x)).^2);                       % Sum-Squared-Error Cost Function (Use ‘fcn2’ Here)
B0 = [1; 1];                                                % Initial Parameter Estimates
[B,SSE] = fminsearch(SSECF, [1; 1]);                        % Estimate Parameters
xv = linspace(min(x), max(x));
figure(1)
plot(x, y, 'bp')
hold on
plot(xv, fcn2(B,xv), '-r')
hold off
grid

I tested this with random vectors and it ran without error.

EDIT — Note that the two-parameter model you want requires only one parameter. A simple ratio (or product) of parameters will not uniquely identify either of them, only the ratio (or product). The three-parameter model actually makes sense.

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MATLAB: How to fit inverse proportional function

Your function could be difficult to fit. You can use the fminsearch function to estimate the parameters:

% Define ‘x’ and ‘y’ here
% Parameter Vector: b(1) = a,  b(2) = b,  b(3) = c
yfit = @(b,x) b(1)./(x + b(2)) + b(3);  % Objective Function
CF = @(b) sum((y-yfit(b,x)).^2);        % Cost Function
b0 = rand(1,3)*10;                      % Initial Parameter Estimates
[B, fv] = fminsearch(CF, b0);           % Estimate Parameters

Note that ‘x’ and ‘y’ have to be defined in your workspace

MATLAB: Non linear Least square fitting

The ‘trick’ here is to create a matrix of your ‘x’ and ‘y’ data vectors and give them to your objective function as a single argument. The objective function can then refer to the appropriate columns of that matrix to use ‘x’ and ‘y’ correctly in your equation. I created random ‘x’, ‘y’, and ‘z’ vectors to test my code, so substitute your data for them. My code would otherwise not change. The estimated parameters are the ‘B’ output of the fminsearch call, the second ‘resnorm’ output being the norm of the residuals. (The ‘resnorm’ output is not necessary, but is helpful in assessing how good the fit is.)

The Code —

% % % z = A*(y^B)*exp(C*x)                            % Original Equation
% % % MAPPING: b(1) = A,  b(2) = B,  b(3) = C
z_fit = @(b, xy) b(1) .* xy(:,2).^b(2) .* exp(b(3) .* xy(:,1));
x = randi(99, 1, 10);                               % Create Data


y = randi(99, 1, 10);                               % Create Data
z = randi(99, 1, 10);                               % Create Data
xy = [x(:), y(:)];                                  % Create Column Vectors & Concatenate In One Matrix To Form Independent Variable Argument
NCF = @(b) norm(z - z_fit(b,xy));                   % Norm Cost Function
B0 = [1; 2; 3];                                     % Initial Parameter Estimates
[B,resnorm] = fminsearch(NCF, B0);                  % Estimate Parameters

You can plot the result using scatter3 or stem3 for your data, and plot3 for the objective function fit. Use the hold function to plot them on the same axes, and grid on to show the grid lines.

Best Answer

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