how to find satistics of the fitted parameter of an equation [closed] - matlab

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For instance: a, b, c and n are the three constants, which are required to be calculated by using data fitting method in a particular equation.
How can I calculate the statistics (mean, standard deviation, variance, skewness value and student t-test value) of the parameters as of a custom equation, for example the quadratic plateau equation?
Example:
x=[0,40,80,100,120,150,170,200],
y=[1865,2855,3608,4057,4343,4389,4415,4478]
y=a*(x+n)^2+b*(x+n)+c, x < xc(Ymax) ....(1) y=yp, x >= xc(Ymax) ....(2)
I have fitted this equation by given code:
yf = #(b,x) b(1).*(x+n).^2+b(2)*(x+n)+b(3); B0 = [0.006; 21; 1878];
[Bm,normresm] = fminsearch(#(b) norm(y - yf(b,x)), B0); a=Bm(1);
b=Bm(2); c=Bm(3); xc=(-b/(2*a))-n; p=p=a*(xc+n)^2+b*(xc+n)+c;
if (x < xc)
yfit = a.*(x+n).^2+ b*(x+n)+c;
else
yfit = p;
end
plot(x,yfit,'*')
hold on; plot(x,y); hold off
Note: I have already used the polyfit command, it was helpful and provided me the results. However, I really don’t find it suitable, as there is no option to customize the equation. Can I find these statistics by any code?

Answer 1

Questions 1, 2 and 4)
Good practice is to set initial values close to the final result if you have previous knowledge about the equation system:
What you have is an overdetermined system of linear equations.
y(1) = a*x(1)^2 + b*x(1) + c
y(2) = a*x(2)^2 + b*x(2) + c
y(3) = a*x(3)^2 + b*x(3) + c
…
y(n) = a*x(n)^2 + b*x(n) + c
or in general:
y = A*X, where
A = [a; b; c]
X = [x(1)^2 x(1) 1;
x(2)^2 x(2) 1;
x(3)^2 x(2) 1;
...
x(n)^2 x(n) 1]
One of the common practices to fit the overdetermined system (since it has no solution) is "least square fit" (mldivide,\ (link))
x=[0; 40; 80; 100; 120; 150; 170; 200];
y=[1865; 2855; 3608; 4057; 4343; 4389; 4415; 4478];
X = [x.^2 x ones(numel(x),1)];
A = y\X;
a0=A(1); %- initial value for a
b0=A(2); %- initial value for b
c0=A(3); %- initial value for c
You can customize equation, when you customize your X and A
but you also can set initial values to ones, it should have neglectable small impact on the result. More related to Question 4
a0=1;
b0=1;
c0=1;
or to random values
rng(10);
A = rand(3,1);
a0=A(1);
b0=A(2);
c0=A(3);
Question 3 - Statistics
If you need more control on monitoring of optimization process, use more general form of writing anonymous function (in code below> myfun) to save all intermediate values of parameters (a_iter, b_iter, c_iter)
function Fiting_ex()
global a_iter b_iter c_iter
a_iter = 0;
b_iter = 0;
c_iter = 0;
x=[0; 40; 80; 100; 120; 150; 170; 200];
y=[1865; 2855; 3608; 4057; 4343; 4389; 4415; 4478];
X = [x.^2 x ones(numel(x),1)];
A = y\X;
a0=A(1);
b0=A(2);
c0=A(3);
B0 = [a0; b0; c0];
[Bm,normresm] = fminsearch(#(b) myfun(b,x,y),B0);
a=Bm(1);
b=Bm(2);
c=Bm(3);
xc=-b/(2*a);
p=c-(b^2/(4*a));
yfit = zeros(numel(x),1);
for i=1:numel(x)
if (x(i) < xc)
yfit(i) = a.*x(i).^2+ b*x(i)+c;
else
yfit(i) = p;
end
end
plot(x,yfit,'*')
hold on;
plot(x,y);
hold off
% Statistic on optimization process
a_mean = mean(a_iter(2:end)); % mean value
a_var = var(a_iter(2:end)); % variance
a_std = std(a_iter(2:end)); % standard deviation
function f = myfun(Bm, x, y)
global a_iter b_iter c_iter
a_iter = [a_iter Bm(1)];
b_iter = [b_iter Bm(2)];
c_iter = [c_iter Bm(3)];
yf = Bm(1)*(x).^2+Bm(2)*(x)+Bm(3);
a=Bm(1);
b=Bm(2);
c=Bm(3);
xc=-b/(2*a);
p=c-(b^2/(4*a));
yfit = zeros(numel(x),1);
for i=1:numel(x)
if (x(i) < xc)
yfit(i) = a.*x(i).^2+ b*x(i)+c;
else
yfit(i) = p;
end
end
f = norm(y - yfit);