# RMSE - Root mean square Error

5,049 views (last 30 days)
Joe on 27 Mar 2011
[EDIT: 20110610 00:17 CDT - reformat - WDR]
So i was looking online how to check the RMSE of a line. found many option, but I am stumble about something,
there is the formula to create the RMSE: http://en.wikipedia.org/wiki/Root_mean_square_deviation
Dates - a Vector
Scores - a Vector
is this formula is the same as RMSE=sqrt(sum(Dates-Scores).^2)./Dates
or did I messed up with something?

John D'Errico on 10 Jun 2011
Yes, it is different. The Root Mean Squared Error is exactly what it says.
(y - yhat) % Errors
(y - yhat).^2 % Squared Error
mean((y - yhat).^2) % Mean Squared Error
RMSE = sqrt(mean((y - yhat).^2)); % Root Mean Squared Error
What you have written is different, in that you have divided by dates, effectively normalizing the result. Also, there is no mean, only a sum. The difference is that a mean divides by the number of elements. It is an average.
sqrt(sum(Dates-Scores).^2)./Dates
Thus, you have written what could be described as a "normalized sum of the squared errors", but it is NOT an RMSE. Perhaps a Normalized SSE.

imo88 on 23 Feb 2017
Dear John, your answer has helped many of us! I'm also struggling with RMSE and I want to calculate the minimum and maximum RMSE for each row of data. based on this example from Joe, would it make sense to use these functions for the calculation of the minimum and maximum value to have an idea about the rmse range?
RMSE_min_range=RMSE./abs(min(y,[],yhat))
RMSE_max_range=RMSE./abs(max(y,[],yhat))
Image Analyst on 23 Feb 2017
To compute the range of an array (of any dimension), simply do this:
RMSE_min = min(RMSE(:));
RMSE_max = max(RMSE(:));
RMSE_range = RMSE_max - RMSE_min;
imo88 on 23 Feb 2017
Dear image analyst, Thank you very much for your reply and help! You really helped me a lot!

Image Analyst on 9 Jan 2016
If you have the Image Processing Toolbox, you can use immse():
rmse = sqrt(immse(scores, dates));

Lilya on 25 Jul 2016
Dear Analyst, could you please re-write this command for the matrix? I need to calculate the RMSE between every point. thank you
Image Analyst on 23 Feb 2017
It will work with matrixed, no problem. Just pass in your two matrices:
err = immse(X,Y) calculates the mean-squared error (MSE) between the arrays X and Y. X and Y can be arrays of any dimension, but must be of the same size and class.
arun kumar on 26 Jul 2017
Thank you. Even i was having same doubt

ziad zaid on 4 Jun 2017
How to apply RMSE formula to measure differences between filters to remove noisy pictures such a median , mean and weiner fiters ? how can i get the result or how to apply it . Rgards .

#### 1 Comment

Image Analyst on 4 Jun 2017
Just do it like my code says. Compare each of your results with the original noisy image. Whichever had the higher RMSE had the most noise smoothing because it's most different from the noisy original..

Siddhant Gupta on 3 Jul 2018
if true
% code
end
y=[1 2 3]
yhat=[4 5 6]
(y - yhat)
(y - yhat).^2
mean((y - yhat).^2)
RMSE = sqrt(mean((y - yhat).^2));
RMSE

Amin Mohammed on 29 Jul 2019
What is the benefit of the first three lines?
Image Analyst on 29 Jul 2019
No benefit. This was with the old web site editor where the person clicked the CODE button before inserting the code instead of after highlighting already inserted code. It does not happen anymore with the new reply text editor.

Sadiq Akbar on 22 Oct 2019
If I have 100 vectors of error and each error vector has got four elements, then how can we we find its MSE, RMSE and any other performance metric? e.g. If I have my desired vector as u=[0.5 1 0.6981 0.7854] and I have estimated vectors like as: Est1=[0.499 0.99 0.689 0.779], Est2=[0.500 1.002 0.699 0.77], Est3=[0.489 0.989 0.698 0.787],---Est100=[---],
Then Error1=u-Est1; Error2=u-Est2 and so on up to Error100=u-Est100. Now how can we find the MSE, RMSE and tell me others as well that are used to indicate the perofrmance of the algorithm. please tell me in the form of easy code.
Regards,

Yella on 10 Jun 2011
Root mean square error is difference of squares of output an input. Let say x is a 1xN input and y is a 1xN output. square error is like (y(i) - x(i))^2. Mean square error is 1/N(square error). and its obvious RMSE=sqrt(MSE).
ur code is right. But how r dates and scores related?

#### 1 Comment

Enne Hekma on 9 Jan 2016
RMSE= sqrt(MSE) = sqrt( 1/length(y)* sum( (y-yhat).^2 )) = sqrt( mean(y-yhat).^2 )
However, he divided after the square root.