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In control theory, the RMSE is used as a quality measure to evaluate the performance of a state observer. [ 10 ] In fluid dynamics , normalized root mean square deviation (NRMSD), coefficient of variation (CV), and percent RMS are used to quantify the uniformity of flow behavior such as velocity profile, temperature distribution, or gas species ...
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 ...
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 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).
Choosing N = 4 at the knee of the curve in Figure 3 yields the RMSE (square root of the MSE, which is more often used for comparison in the literature) shown in Figure 4. On the other hand, choosing N = 8 yields the second curve in Figure 4.
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.