Second order polynomial coefficients with one equation

Hi! I have the following equation: y=c*u+d*u^2

Known variables are y and u for a given timeseries. c and d are unknown constants. Observe that it don't exist a constant term in the equation.

I'm sure that one solution lies within least squares, but I've sort of given up without assistance. Is there another, less complex way to solve this pherhaps?

Anyone who could help progress with this problem?

Thanks!

[Star Strider]

Azzi’s solution will work. All you need to do is to remove the vector of ones.

[Andreas Volden]

OK. Thank you!

[Star Strider]

My pleasure!

Answer 1

[Image Analyst]

I don't see Azzi's solution here any longer so I'll just give the standard least squares solution here:

This is your model:

y = alpha1 * u + alpha2 * u^2

I don't think you can use polyfit() because there's no constant term, but you can use the standard least squares formula

alpha = inv(x' * x) * x' * y;  % Get estimate of the alphas.

Where x = an N rows by 2 columns matrix.

u(1), u(1)^2
u(2), u(2)^2
u(3), u(3)^2
u(4), u(4)^2
...
u(N), u(N)^2

Now of course alpha(1) is what you called c and alpha(2) is what you called d. This is pretty much just straight out of the least squares derivation in a textbook of mine.

[Star Strider]

Azzi’s solution is in Fitting vector of n length.

[Andreas Volden]

Thank you!

Answer 2

[MariapL]

hi , I am new here, looking for a solution for the same problem. I am trying to use what you said in matlab, but its not working. Could you please take a look ? Maybe you will know what I am doing wrong. So I have the same equation: y=aN^2+bN , with coefficient c already set as 0. I am typing in matlab ( my date is a time series)

x=[0 1 2 3 4 5]
y=[100 250 680 150 200 221]
y = alpha1 * x + alpha2 * x^2
alpha = inv(x' * x) * x' * y

But I will need alpha1 and alpha2 to be known in matlab in order for this to work. I dont get what should I do here to get this two coefficient.

[Star Strider]

See my Answer to your original Question: Estimating two coefficients out of three in quadratic function (link).

[MariapL]

thanks :) !

[Star Strider]

As always, my pleasure!

[Image Analyst]

No, you don't put X into the alpha equation, you put the matrix built from X, like this:

x=[0 1 2 3 4 5]
y=[100 250 680 150 200 221]
plot(x, y, 'b*');
A = [x', x'.^2]
alpha = inv(A' * A) * A' * y'
% yFit = alpha1 * x + alpha2 * x^2
yFit = alpha(1) * x + alpha(2) * x.^2
hold on;
plot(x, yFit, 'rd-');
grid on;
legend('Training data', 'Fitted Data');