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This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.
In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series. [1] [2] The moving-average model specifies that the output variable is cross-correlated with a non-identical to itself random-variable.
In particular, the residuals should be independent of each other and constant in mean and variance over time. (Plotting the mean and variance of residuals over time and performing a Ljung–Box test or plotting autocorrelation and partial autocorrelation of the residuals are helpful to identify misspecification.) If the estimation is inadequate ...
Dependence: autocorrelated time series might be modelled using autoregressive moving average models. Non-constant variance: in the simplest cases, weighted least squares might be used. Non-normal distribution for errors: in the simplest cases, a generalized linear model might be applicable.
The CRAN task view on Time Series contains links to most of these. Mathematica has a complete library of time series functions including ARMA. [11] MATLAB includes functions such as arma, ar and arx to estimate autoregressive, exogenous autoregressive and ARMAX models. See System Identification Toolbox and Econometrics Toolbox for details.
Volatility and Time Series Econometrics: Essays in Honor of Robert Engle (1st ed.). Oxford: Oxford University Press. pp. 137–163. ISBN 9780199549498. Enders, W. (2004). "Modelling Volatility". Applied Econometrics Time Series (Second ed.). John-Wiley & Sons. pp. 108–155. ISBN 978-0-471-45173-0. Engle, Robert F. (1982). "Autoregressive ...
Then the subset is modified by "shifting forward"; that is, excluding the first number of the series and including the next value in the subset. A moving average is commonly used with time series data to smooth out short-term fluctuations and highlight longer-term trends or cycles. The threshold between short-term and long-term depends on the ...
Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n − 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n-1 instead of n, is called ...