Find minimum sse for logistic transformation

Hi, I have a little issue with a logistic regression transformation.

I need to find the values of a and b that give the minimum Sum of squared residuals, categorised as:

Where yhat is a logistic transformation of my known variable, categorised as:

Thanks for your time and patience.

The function that I wrote, run with fmincon, gives always an SSE = number of observation;

function [SSE,y]=logit2(Par,x)
if nargin<5;
    ir=0;  %default ir=0
end;

if rows(Par)<cols(Par); % put parameters in column
    Par=Par';
end;
if rows(Par)~=3; % check for the the right number of parameters as inputs
error('Dimension of Par not compatible');
end;
%% Assign the Parameters to the Elements in the Vector
a=Par(1);
b=Par(end-1);
SSE=Par(end)
%%
T=rows(x);
y=zeros(T,1);
for i=1:T
    y(i)=1/(1+exp(-(a+b*x(i))));
end

SSE=-sum((y - ones(size(y))).^2)

end

Answer 1

[Nipun]

Hi Andrea,

I understand that you are trying to find the values of a and b that minimize the sum of squared residuals (SSE) for a logistic regression transformation. Your current function seems to have issues since it always gives an SSE equal to the number of observations.

Here is a corrected version of your function that should properly calculate the SSE:

function SSE = logit2(Par, x, y)
    if nargin < 3
        error('Not enough input arguments.');
    end
    
    % Ensure parameters are column vector
    if isrow(Par)
        Par = Par';
    end
    
    % Check if Par has the correct number of parameters
    if length(Par) ~= 2
        error('Parameter vector must have exactly 2 elements.');
    end
    
    % Assign the parameters to the elements in the vector
    a = Par(1);
    b = Par(2);
    
    % Calculate predicted values using logistic transformation
    yhat = 1 ./ (1 + exp(-(a + b * x)));
    
    % Calculate residuals
    resid = yhat - y;
    
    % Calculate SSE
    SSE = sum(resid.^2);
end

To use the logit2 function with fmincon, you can follow this example:

% Example data
x = [0.5, 1.5, 2.5, 3.5, 4.5]'; % Replace with your x data
y = [0.2, 0.4, 0.6, 0.8, 1.0]'; % Replace with your y data

% Initial guess for parameters [a, b]
initial_guess = [0, 0];

% Define the objective function for fmincon
objective_function = @(Par) logit2(Par, x, y);

% Set options for fmincon
options = optimoptions('fmincon', 'Display', 'iter', 'Algorithm', 'sqp');

% Run fmincon to minimize SSE
[optimal_params, fval] = fmincon(objective_function, initial_guess, [], [], [], [], [], [], [], options);

% Display the results
fprintf('Optimal parameters: a = %.4f, b = %.4f\n', optimal_params(1), optimal_params(2));
fprintf('Minimum SSE: %.4f\n', fval);

Explanation:

1. The logit2 function now correctly calculates the residuals by subtracting the actual 𝑦y values from the predicted 𝑦^y^​ values.
2. The fmincon function is used to find the values of 𝑎a and 𝑏b that minimize the SSE.
3. The initial_guess array contains the starting values for 𝑎a and 𝑏b.
4. The objective_function handles the minimization of the SSE.

Make sure to replace the example data x and y with your actual data points. This code should provide you with the optimal parameters for your logistic regression model.

Hope this helps.

Regards,

Nipun