Diffrence between RMSE selfcalculated and RMSE calculated with Statistics Toolbox

Hi all, i calculated the RMSE of these Data:

Y_hat=[
 9.774614325191857
 9.453084986417043
 9.502166049524247
 7.817755496590051
 7.031233831915310
 8.392026077578970
 6.881255539731626
 6.488927374899896
 6.779374282307657
 6.474790314047517
13.842988631876649
13.113764172190285
14.244292841981128
12.470726075747763]

Y=[
 8.900000000000000
 8.600000000000000
 9.167000000000000
 7.000000000000000
 7.030000000000000
 7.270000000000000
 7.430000000000000
 7.270000000000000
 7.370000000000000
 7.030000000000000
15.029999999999999
13.170000000000000
13.369999999999999
13.630000000000001]

my calculation is based on the Formula: RMSE= sqrt(mean((Y_hat-Y)^2)). with the calculation i got RMSE=0.7894.

But with the Statistics Toolbox of matlab I got RMSE=0.885 which is the sqrt of my calculated Value!!!. Who is wrong: I or the Toolbox??

Thank you!

[Brendan Hamm]

Using the Statistic Toolbox how? Presumably these data values were not hand entered as when I calculate the RMSE I get:

>> sqrt(mean((Y-Y_hat).^2))
ans =
    0.7847

[the cyclist]

Just adding a bit more to Brendan's comment ...

Your Y_hat and RMSE are presumably the output of a MATLAB fitting function. Which one, and what specific output do you associate with RMSE? Ideally, you could post all of your data and code, and we could replicate completely.

[Rica]

Hi, Thanks for the comments. I made a multivariate regression wit these Parameters X1 and X2. the function fitlm calculates the regression coeffitionts, r^2 and rmse.

% X1=1.0e+02 *[  

   4.794100000000000
   4.830800000000000
   5.043100000000000
   4.059800000000000
   3.179700000000000
   4.608300000000000
   3.795500000000000
   3.299600000000000
   3.431000000000000
   3.635300000000000
   8.896799999999999
   8.344199999999999
   8.839100000000000
   5.600200000000000
]
 %
X2=[33.979999999999997
  32.450000000000003
  32.310000000000002
  26.309999999999999
  24.230000000000000
  27.989999999999998
  22.489999999999998
  21.550000000000001
  22.649999999999999
  20.910000000000000
  45.509999999999998
  43.130000000000003
  47.439999999999998
  44.899999999999999]

the Result is Y_hat= 0.5262+0.003757*X1+0.21916*X2. the Code is:

%
X_f=[ones(size(X1)) X1 X2];
X_f_lm=X_f(:,2:end);
mdl=fitlm(X_f_lm,Y,'linear').

I got this:

Answer 1

[the cyclist]

You calculated the RMSE incorrectly -- and then had a remarkable numerical coincidence.

You calculated

RMSE = sqrt(mean(((Y-Y_hat).^2)))

which is equivalent to

RMSE = sqrt(sum(((Y-Y_hat).^2)/N_obs))

where N_obs is the number of observations. (N_obs = 14 in your case.) You got the value RMSE = 0.7847.

But the correct calculation of RMSE divides by the number of degrees of freedom, not the number of observations. The correct RMSE calculation is

RMSE = sqrt(sum(((Y-Y_hat).^2)/(N_obs-rankX)))

where rankX = 3 in your case.

So,

RMSE = sqrt(sum(((Y-Y_hat).^2)/11))

and is equal to 0.8853 (as MATLAB got).

The numerical coincidence, and complete red herring, is that this is very nearly equal to the square root of your incorrect value.

You can see where (the latest version of) MATLAB does the calculation of MSE around lines 1436-1440 of the file LinearModel.

[Rica]

Thank you for the helpfull explanation..

Is the explanation in Wikipidea Wrong?..

[the cyclist]

I looked at a couple articles, including this one and this one, and they both reference two different ways of calculating the MSE (specifically mentioning regression).

[Sairam Seshapalli]

how your going to find rank