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Tags

#m249 #mathematics #open-university #statistics #time-series

Question

Answer

α(1 − α)^{i}

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

simple exponential smoothing: the term exponential refers to the fact that the weights α(1 − α) i lie on an exponential curve.

Tags

#m249 #mathematics #open-university #statistics #time-series

Question

what does the expanded **m-times **(i.e. non recursive) simple exponential smoothing formula looks like?

fully recursive is:

\(\hat{x}_{n+1}\)= αx_{n} + (1 − α)\(\hat{x}_n\)

fully recursive is:

\(\hat{x}_{n+1}\)= αx

Answer

expanded m-times is

\(\large \hat{x}_{n+1} = \sum_{i=0}^m\alpha(1-\alpha)^ix_{n-i}+(1-\alpha)^{m+1}\hat{x}_{n-m}\)

\(\large \hat{x}_{n+1} = \sum_{i=0}^m\alpha(1-\alpha)^ix_{n-i}+(1-\alpha)^{m+1}\hat{x}_{n-m}\)

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

If a time series X t is described by an additive model with constant level and no seasonality, 1-step ahead forecasts may be obtained by simple exponential smoothing using the formula \(\hat{x}_{n+1}\)= αx n + (1 − α)\(\hat{x}_n\) where: x n is the observed value at time n, \(\hat{x}_n\)and \(\hat{x}_{n+1}\)are the 1-step ahead forecasts of X n and X n+1 , and α is a smoothing parameter, 0 ≤ α ≤ 1.