Mean squared error and NMSE

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Murali Krishna AG
Murali Krishna AG on 26 Oct 2021
Commented: Diwakar Diwakar on 17 Sep 2023
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
  1 Comment
Diwakar Diwakar
Diwakar Diwakar on 17 Sep 2023
Ran in:
Generate some sample data for demonstration purposes
% You should replace this with your actual data
h_true = rand(1, 100);
noise = 0.1 * randn(1, 100); % Simulated noise
he = h_true + noise; % Estimated vector with noise
% Define hyperparameters
learning_rate = 0.01;
num_iterations = 1000;
% Initialize the estimated vector with random values
he_estimate = rand(1, 100);
% Gradient descent optimization
for iter = 1:num_iterations
% Calculate the gradient of the MSE loss
gradient = 2 * (he_estimate - he) * he_estimate';
% Update the estimated vector he_estimate
he_estimate = he_estimate - learning_rate * gradient;
% Calculate the current MSE
current_mse = norm(he_estimate - he)^2;
% Display the MSE at each iteration (optional)
fprintf('Iteration %d, MSE: %.4f\n', iter, current_mse);
end
Iteration 1, MSE: 20.9528 Iteration 2, MSE: 23.5143 Iteration 3, MSE: 26.3126 Iteration 4, MSE: 29.9879 Iteration 5, MSE: 35.8007 Iteration 6, MSE: 47.2753 Iteration 7, MSE: 78.1015 Iteration 8, MSE: 214.3499 Iteration 9, MSE: 1741.9512 Iteration 10, MSE: 121250.5092 Iteration 11, MSE: 588676619.5794 Iteration 12, MSE: 13861843807810676.0000 Iteration 13, MSE: 7686028577253918231700059455488.0000 Iteration 14, MSE: 2363001411614556870655872893799869451664282048439527032225792.0000 Iteration 15, MSE: 223351026851695503741178766792080463553073827505424017635935069793609627279200652276771019157782823532296645542515048448.0000 Iteration 16, MSE: 1995427247828271751331188737846075879562766440380902641245723132818276502821813699360243035232803776681131423321736605829993707915273026187869805002558050941537293301965688079767890621165591617421801506477407407690911226051113840805937152.0000 Iteration 17, MSE: Inf Iteration 18, MSE: Inf Iteration 19, MSE: Inf Iteration 20, MSE: Inf Iteration 21, 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MSE: Inf Iteration 897, MSE: Inf Iteration 898, MSE: Inf Iteration 899, MSE: Inf Iteration 900, MSE: Inf Iteration 901, MSE: Inf Iteration 902, MSE: Inf Iteration 903, MSE: Inf Iteration 904, MSE: Inf Iteration 905, MSE: Inf Iteration 906, MSE: Inf Iteration 907, MSE: Inf Iteration 908, MSE: Inf Iteration 909, MSE: Inf Iteration 910, MSE: Inf Iteration 911, MSE: Inf Iteration 912, MSE: Inf Iteration 913, MSE: Inf Iteration 914, MSE: Inf Iteration 915, MSE: Inf Iteration 916, MSE: Inf Iteration 917, MSE: Inf Iteration 918, MSE: Inf Iteration 919, MSE: Inf Iteration 920, MSE: Inf Iteration 921, MSE: Inf Iteration 922, MSE: Inf Iteration 923, MSE: Inf Iteration 924, MSE: Inf Iteration 925, MSE: Inf Iteration 926, MSE: Inf Iteration 927, MSE: Inf Iteration 928, MSE: Inf Iteration 929, MSE: Inf Iteration 930, MSE: Inf Iteration 931, MSE: Inf Iteration 932, MSE: Inf Iteration 933, MSE: Inf Iteration 934, MSE: Inf Iteration 935, MSE: Inf Iteration 936, MSE: Inf Iteration 937, MSE: Inf Iteration 938, MSE: Inf Iteration 939, MSE: Inf Iteration 940, MSE: Inf Iteration 941, MSE: Inf Iteration 942, MSE: Inf Iteration 943, MSE: Inf Iteration 944, MSE: Inf Iteration 945, MSE: Inf Iteration 946, MSE: Inf Iteration 947, MSE: Inf Iteration 948, MSE: Inf Iteration 949, MSE: Inf Iteration 950, MSE: Inf Iteration 951, MSE: Inf Iteration 952, MSE: Inf Iteration 953, MSE: Inf Iteration 954, MSE: Inf Iteration 955, MSE: Inf Iteration 956, MSE: Inf Iteration 957, MSE: Inf Iteration 958, MSE: Inf Iteration 959, MSE: Inf Iteration 960, MSE: Inf Iteration 961, MSE: Inf Iteration 962, MSE: Inf Iteration 963, MSE: Inf Iteration 964, MSE: Inf Iteration 965, MSE: Inf Iteration 966, MSE: Inf Iteration 967, MSE: Inf Iteration 968, MSE: Inf Iteration 969, MSE: Inf Iteration 970, MSE: Inf Iteration 971, MSE: Inf Iteration 972, MSE: Inf Iteration 973, MSE: Inf Iteration 974, MSE: Inf Iteration 975, MSE: Inf Iteration 976, MSE: Inf Iteration 977, MSE: Inf Iteration 978, MSE: Inf Iteration 979, MSE: Inf Iteration 980, MSE: Inf Iteration 981, MSE: Inf Iteration 982, MSE: Inf Iteration 983, MSE: Inf Iteration 984, MSE: Inf Iteration 985, MSE: Inf Iteration 986, MSE: Inf Iteration 987, MSE: Inf Iteration 988, MSE: Inf Iteration 989, MSE: Inf Iteration 990, MSE: Inf Iteration 991, MSE: Inf Iteration 992, MSE: Inf Iteration 993, MSE: Inf Iteration 994, MSE: Inf Iteration 995, MSE: Inf Iteration 996, MSE: Inf Iteration 997, MSE: Inf Iteration 998, MSE: Inf Iteration 999, MSE: Inf Iteration 1000, MSE: Inf
% The optimized he_estimate should now be a good approximation of the original h_true

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Accepted Answer

DGM
DGM on 26 Oct 2021
Edited: DGM on 26 Oct 2021
Ran in:
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.

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