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The RMSD of predicted values ^ for times t of a regression's dependent variable, with variables observed over T times, is computed for T different predictions as the square root of the mean of the squares of the deviations:
Physical scientists often use the term root mean square as a synonym for standard deviation when it can be assumed the input signal has zero mean, that is, referring to the square root of the mean squared deviation of a signal from a given baseline or fit. [8] [9] This is useful for electrical engineers in calculating the "AC only" RMS of a signal.
If a vector of predictions is generated from a sample of data points on all variables, and is the vector of observed values of the variable being predicted, with ^ being the predicted values (e.g. as from a least-squares fit), then the within-sample MSE of the predictor is computed as
The second moment of a random variable, () is also called the mean square. The square root of a mean square is known as the root mean square (RMS or rms), and can be used as an estimate of the standard deviation of a random variable when the random variable is zero-mean.
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.
It is remarkable that the sum of squares of the residuals and the sample mean can be shown to be independent of each other, using, e.g. Basu's theorem.That fact, and the normal and chi-squared distributions given above form the basis of calculations involving the t-statistic:
Forecast errors can be evaluated using a variety of methods namely mean percentage error, root mean squared error, mean absolute percentage error, ...
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