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The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled).
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.
Excel graph of the difference between two evaluations of the smallest root of a quadratic: direct evaluation using the quadratic formula (accurate at smaller b) and an approximation for widely spaced roots (accurate for larger b). The difference reaches a minimum at the large dots, and round-off causes squiggles in the curves beyond this minimum.
with = and = (), where CDF −1 is the quantile function. When normalizing by the mean value of the measurements, the term coefficient of variation of the RMSD, CV(RMSD) may be used to avoid ambiguity. [ 5 ]
The earliest reference to a similar formula appears to be Armstrong (1985, p. 348), where it is called "adjusted MAPE" and is defined without the absolute values in the denominator. It was later discussed, modified, and re-proposed by Flores (1986).
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Thus, minimizing SURE can act as a surrogate for minimizing the MSE. Note that there is no dependence on the unknown parameter μ {\displaystyle \mu } in the expression for SURE above. Thus, it can be manipulated (e.g., to determine optimal estimation settings) without knowledge of μ {\displaystyle \mu } .
But what if we use the formula (¯) / = / = to estimate the variance? The estimate of the variance would be zero – and the estimate would be zero for any population and any sample of n = 1. The problem is that in estimating the sample mean, the process has already made our estimate of the mean close to the value we sampled—identical, for n = 1.
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