How to find the variable that minimize the root mean squared error between a known vetor and the multiplication of this variable and an another known vector?

Supposing that I have a variable , and two known vectors , where N is a known number. I want to find the a that minimizes the root mean squared error between and . Is there any neat apprach to implement it?

Answer 1

[Amal Raj]

Hi 奥 刘,

You can use linear regression to find a.

Please refer this example below:

N = 100; % Example value for N
x = linspace(0, 10, N)'; % Known vector x
y = sin(x) + 0.1 * randn(N, 1); % Known vector y
X = [ones(N, 1) x]; % Design matrix
a = X \ y; % Solve for a using least squares
y_pred = X * a; % Calculate predicted values of y
rmse = sqrt(mean((y - y_pred) .^ 2)); % Calculate RMSE

[奥 刘]

Dear Raj

Really thanks for your answer, and I will try the linear regresstion way.

[Torsten]

@Amal Raj

You mustn't include the constant term in the calculation of the regression coefficients because you are to regress

y = a*x

not

y = a*x + b

The results for "a" will differ for the two cases.

[奥 刘]

@Torsten Yes, that's right. I think @Amal Raj just gave me an example but didn't mean that I should include the constant. Anyway, thanks for reminding me!

Answer 2

[Torsten]

a = (c.'*b)/(b.'*b)

where b and c are your column vectors.

[奥 刘]

Thanks for your answer! I got this projection way.