nonlinear and linear regression

I have created a script plus a function to use for nonlinear least square optimization. I have compared it with linear regression and also with built-in functions in MATLAB such as fminsearch,fminunc, lsqnonlin. The results for all regression models are surprisingly the same and I don't know why. Can anyone help me with that please?Here is my function:

function result = NonLsq(w,x,y)
    ei = -(w(1).*x+w(2))+y;
    result = sum(ei.^2);
end

And the following is my main script:

clc;  clear;  close all;

% { Linear And Nonlinear Curvefitting}


%% 1. One Dimensional data

x = [0.5 1 2 3 4];
y = [10.4 5.8 3.3 2.4 2];

xMin = min(x);
xMax = max(x);
n = 100;    % Number of data which sould be interpolated
xInterp = linspace(xMin,xMax,n);
yInterp1 = interp1(x,y,xInterp);
yInterp2 = interp1(x,y,xInterp,'spline');


%% 2. NonLinear Least Square

% Initial Guess
g = @(w,x,y) (w(1).*x+w(2))-y;

X01 = [0.15 0.55];
X02 = [0.4 0.8];
X03 = [0.7 48];

% X0 = [0.15 0.55]';

Options1 = optimset('Display','Iter','TolX',1e-5);
Options2 = optimset('Display','on');
Options3 = optimset('MaxIter',50,'TolFun',1e-4);

p1_Nonlin = fminsearch(@NonLsq,X01,Options1,x,y);
p2_Nonlin = fminunc(@NonLsq,X02,Options2,x,y);
p3_Nonlin = lsqnonlin(g,X03,[],[],Options3,x,y);
plot(x,y,'o','MarkerSize',8,'LineWidth',3,'MarkerFaceColor','k');

hold on
grid on

plot(xInterp,yInterp1,'r--','LineWidth',2)

hold on
plot(xInterp,yInterp2,'b:','LineWidth',2)
legend('Spline INterpolated')

hold on
plot(xInterp,pLinear_Interp,'k*','LineWidth',2)

plot(xInterp,P1_Nonlin_Interp,'c.','LineWidth',2,'MarkerSize',12)

hold on
plot(xInterp,P2_Nonlin_Interp,'m','LineWidth',2)


hold on
plot(xInterp,P3_Nonlin_Interp,'g','LiNEwidth',2)
legend('Original Data','Linear Interpolatn','Linear Spline','Linear Regression'...
    ,'FminSearch','FminUnc','LsqNonLinear')

Could it be related to the function I'm trying to optimize?

Answer 1

[William Rose]

@Mhmmd Sjj

You get the same results because your model g() is linear in w(1) and w(2):

g = @(w,x,y) (w(1).*x+w(2))-y;

The error function NonLsq() for the noninear case uses the same linear model. Thus linear and nonlinear fits find the same solution.

By the way, you do not need the dot-multiply in ei=-(w(1).*x+w(2))+y. Since w(1) is a scalar, you can do ei=-(w(1)*x+w(2))+y. The same is true for the deifnition of g().