# Mean squared error and NMSE

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Murali Krishna AG on 26 Oct 2021
Edited: DGM on 26 Oct 2021
Let he is estimated vector and h is original vector which is unknown. The Mean squared error (MSE) is defined as norm(he-h)^2. Now , how to get h original vector using he and MSE? ..pls help me

DGM on 26 Oct 2021
Edited: DGM on 26 Oct 2021
The title is about NMSE, but there's no mention of it. I'll refer to both.
How to back calculate the reference from an estimate/observation and the error? You don't have enough information to do that, just the same as you don't have enough information to unambiguously back-calculate the reference from the mean absolute error either.
Consider the simple demonstration:
observation = [0 0];
mse = @(x,y) norm(x(:)-y(:),2)^2 / numel(x);
nmse = @(x,y) mse(x,y)/norm(x(:))^2;
% regular MSE
mse([-1 0],observation)
ans = 0.5000
mse([ 1 0],observation)
ans = 0.5000
mse([0 -1],observation)
ans = 0.5000
mse([0 1],observation)
ans = 0.5000
mse(1./sqrt([2 2]),observation) % and so on
ans = 0.5000
% normalized MSE
nmse([-1 0],observation)
ans = 0.5000
nmse([ 1 0],observation)
ans = 0.5000
nmse([0 -1],observation)
ans = 0.5000
nmse([0 1],observation)
ans = 0.5000
nmse(1./sqrt([2 2]),observation) % and so on
ans = 0.5000
Observe that the solution to any back-calculation must be ambiguous. Compared to multiple references, the observations have the same error. This is true for NMSE, just as it is true for MSE, RMS, or MAE.