The vector RMSE

Function for calculation of vector root-mean-square error (RMSE)

RM_STATISTICS. The vector RMSE

[Mse, Dse, stdL1, stdL2, alfa1]=RM_vec2rmse(uN, vN, uM, vM);

% INPUT
% uN is eastward component of the vector N
% vN is northward component of the vector N
% uM is eastward component of the vector M
% vM is northward component of the vector M

% OUTPUT
% Mse is module of systematic error vector (statistical bias)
% Dse is dirrection of systematic error vector (statistical bias), degrees
% (clockwise from the northward direction)
% stdL1 - the major semi-axes of random error ellipse
% stdL2 - the minor semi-axes of random error ellipse
% alfa1 - the direction of major axes of random error ellipse

The closeness of two-time series can be estimated as a root-mean-square error (RMSE). For vector processes, the root-mean-square error is a combination of % the square of the mean and variance of the discrepancy between the "verifiable" and "true" vector series.

RMSE=(mean(R))^2+((R -mrean(R))^2)/L.

The mean of vector differences mean(R)=sum(R)/L will show a systematic error (statistical bias), and the standard deviation ellipse of vector differences Std(R)=sqrt((Var(R)))=sqrt((sum(R-mean(R))^2)/L)
will describe a random error. Moreover the orientation of the vector mean(R), the orientation and the flattening of the ellipse std(R) show the features of the differences between "verifiable" and "true" vectors.

Ideally coinciding vector series have a module of the systematic error vector Mse, equal to zero, and random error ellipse with the values of the axes tending to zero (stdL1 -> 0, stdL2 -> 0).