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Exponential smoothing was first suggested in the statistical literature without citation to previous work by Robert Goodell Brown in 1956, [3] and then expanded by Charles C. Holt in 1957. [4] The formulation below, which is the one commonly used, is attributed to Brown and is known as "Brown’s simple exponential smoothing". [5]
Smoothing may be distinguished from the related and partially overlapping concept of curve fitting in the following ways: . curve fitting often involves the use of an explicit function form for the result, whereas the immediate results from smoothing are the "smoothed" values with no later use made of a functional form if there is one;
The tracking signal is then used as the value of the smoothing constant for the next forecast. The idea is that when the tracking signal is large, it suggests that the time series has undergone a shift; a larger value of the smoothing constant should be more responsive to a sudden shift in the underlying signal.
If the smoothing or fitting procedure has projection matrix (i.e., hat matrix) L, which maps the observed values vector to predicted values vector ^ =, then PE and MSPE are formulated as: P E i = g ( x i ) − g ^ ( x i ) , {\displaystyle \operatorname {PE_{i}} =g(x_{i})-{\widehat {g}}(x_{i}),}
The default Expert Modeler feature evaluates a range of seasonal and non-seasonal autoregressive (p), integrated (d), and moving average (q) settings and seven exponential smoothing models. The Expert Modeler can also transform the target time-series data into its square root or natural log.
The formula for a given N-Day period and for a given data series is: [2] [3] = = + (()) = (,) The idea is do a regular exponential moving average (EMA) calculation but on a de-lagged data instead of doing it on the regular data.
Extension packages contain related and extended functionality: package tseries includes the function arma(), documented in "Fit ARMA Models to Time Series"; packagefracdiff contains fracdiff() for fractionally integrated ARMA processes; and package forecast includes auto.arima for selecting a parsimonious set of p, q.
Exponentially weighted moving average (EWMA) is an alternative model in a separate class of exponential smoothing models. As an alternative to GARCH modelling it has some attractive properties such as a greater weight upon more recent observations, but also drawbacks such as an arbitrary decay factor that introduces subjectivity into the ...