How can I evaluate the custom curve fitting result

Hi everyone!

As the title above? How can I figure out an index such as R-square to evaluate the custom curve fitting result? my solution: X=1:21; Y=[]; Custom function : Y=1./(1+exp(a*X+b)) After simplified: y=LN(1./Y-1)=a*X+b; The problem is transformed into a linear regression problem;

if true
  p=polyfit(X,y,1);
  x1=linspace(min(X),max(X));
  py=1./(1+exp(a*x+b));
  n=size(Y);

    xba=mean(X);
    yba=mean(py);
    fengzi=0;
    fengmuX=0;
    fengmuY=0;
    for i=1:n
        fengzi=fengzi+((x(i)-xba)*(y(i)-yba));
        fengmuX=fengmuX+(x(i)-xba)^2;
        fengmuY=fengmuY+(y(i)-yba)^2;
    end
    fengmu=(fengmuX*fengmuY)^0.5;
    R=fengzi/fengmu;
    R-square=R*R;
  end

Is that right?

Answer 1

[Star Strider]

It is incorrectly transformed into a linear regression problem. Taking the log of Y converts additive errors into multiplicative errors, violating the assumptions of least-squares parameter estimation. Only in the complete absence of noise will the estimated parameters be accurate.

It is relatively easy to use fminsearch to do a nonlinear curve fit to your exponential equation, and if your model is appropriate for your data, the estimated parameters will be reasonably accurate.

For example:

X=1:21;
Y=[something];

Yest = @(b,x) 1./(1+exp(b(1).*x+b(2)));     % Objective function

OLS = @(b) sum((Y - Yest(b,X)).^2);         % Cost function

b = fminsearch(OLS, [1; 1])                 % Estimate parameters

Yfit = Yest(b,X);                           % Estimated Y-values

figure(1)                                   % Plot data & fitted curve
plot(X, Y, '*r')
hold on
plot(X, Yfit, '-b')
hold off
grid

There are several ways to evaluate the fit. Probably the easiest is to create a (21 x 2) matrix of [Y; Yfit]' then call corrcoef:

[R, P, RLO, RUP] = corrcoef([Y; Yfit]')