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In some disciplines, the RMSD is used to compare differences between two things that may vary, neither of which is accepted as the "standard". For example, when measuring the average difference between two time series x 1 , t {\displaystyle x_{1,t}} and x 2 , t {\displaystyle x_{2,t}} , the formula becomes
When the model has been estimated over all available data with none held back, the MSPE of the model over the entire population of mostly unobserved data can be estimated as follows.
The MSE could be a function of unknown parameters, in which case any estimator of the MSE based on estimates of these parameters would be a function of the data (and thus a random variable). If the estimator θ ^ {\displaystyle {\hat {\theta }}} is derived as a sample statistic and is used to estimate some population parameter, then the ...
The standard deviation of the observed field () is side a, the standard deviation of the test field () is side b, the centered RMS difference (centered RMS difference is the mean-removed RMS difference, and is equivalent to the standard deviation of the model errors [17]) between the two fields (E′) is side c, and the cosine of the angle ...
For color images with three RGB values per pixel, the definition of PSNR is the same except that the MSE is the sum over all squared value differences (now for each color, i.e. three times as many differences as in a monochrome image) divided by image size and by three.
Note that the Brier score, in its most common formulation, takes on a value between zero and one, since this is the square of the largest possible difference between a predicted probability (which must be between zero and one) and the actual outcome (which can take on values of only 0 or 1).
Standard method like Gauss elimination can be used to solve the matrix equation for .A more numerically stable method is provided by QR decomposition method. Since the matrix is a symmetric positive definite matrix, can be solved twice as fast with the Cholesky decomposition, while for large sparse systems conjugate gradient method is more effective.
A consequence of the above discussion is the following counterintuitive result: When three or more unrelated parameters are measured, their total MSE can be reduced by using a combined estimator such as the James–Stein estimator; whereas when each parameter is estimated separately, the least squares (LS) estimator is admissible. A quirky ...