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Given a set of time series data from 0 to t as $x_t$, we would like to predict time series for t+1 and, say, t+2, using trend $m_{t+1}, ...$

Now, exponential smoothing trend is defined as: $m_{t+1} = \alpha x_{t+1} + (1-\alpha)m_{t}$ (this is from Brockwell chapter 1). Where am I supposed to get the observation $x_{t+1}$ to calculate $m_{t+1}$ if $x_t$ is the last observation? The same problem goes with moving average trend estimation, but there even more points to the right from the last observation $t$ are needed depending on the window size.

Now, as for regression, it can fit pretty nice. But how about the rule that "Regression should never be used for prediction outside of the interval of observation" (from my former stats course)? In time series the observations to predict $x_{t+1}, x_{t+2}$ are always outside of the interval on which the regression was fit ($x_0, ..., x_t$)!

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There are several questions here; taking them in order,

  1. Exponential smoothing, as you have written it out, does not have a trend term. $m_t$ is the estimated level of the series at time $t$, not the "exponential smoothing trend". Since the $x_t$ are assumed to have no trend in the simple exponential smoothing formulation, the forecast for all periods $t+1, \dots, t+k$ is just equal to the estimated level at time $t$, namely, $m_t$. Note that $m_{t+1}$ is the estimated level after you have seen $x_{t+1}$; since, as you note, you haven't seen it yet at time $t$, it should be clear that you don't use it when forecasting.

  2. If you do use the trend version of exponential smoothing, at time $t$ you'll have formed estimates of the level $m_t$ and the trend, label it $b_t$. You'll predict the level at time $t+k$ by $m_t + kb_t$.

  3. The rule (of thumb) about not using regression to predict outside the interval of observation refers to the range of values of $x$, not the range of values of $t$. To see the difference, consider predicting ice cream sales as a function of time of year (let's ignore population growth and other factors for our example.) We'd probably model $x = $ "time of year" using a sine, shifted appropriately. Every year, the week with January 7th in it would have a value of $3\pi/2$ (as that is the average minimum sales week of the year for ice cream), and the fact that it is a future observation, i.e, the index $t$ on $x_t$ is different than any we've observed in the past, doesn't mean we can't expect our prediction to be reasonably accurate (given a fair amount of data.)

ETA: Double exponential smoothing is a variant of exponential smoothing that estimates trend as well. Let's define the trend estimate at time $t$ as $b_t$, and the level estimate at time $t$ as $m_t$. At each time period, we'll form a level estimate and a trend estimate; for prediction, we'll predict the level at time $t+k$ by $m_t + kb_t$.

The updating equations are:

$$m_t = \alpha x_t + (1-\alpha)(m_{t-1}+b_{t-1})$$

$$b_t = \beta(m_t - m_{t-1}) + (1-\beta)b_{t-1}$$

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  • $\begingroup$ nice discussion on the rule of thumb .......+1 $\endgroup$ – IrishStat Apr 21 '18 at 19:53
  • $\begingroup$ In 1) I think you are talking about exponential smoothing as stationary model? In Brockwell this is separate from trend estimation. The model I refer is $X_t = m_t + s_t + Y_t$, where $m_t$ is trend, $s_t$ is seasonality, and {Y} is whatever stationary left. In the book, the trend is estimated using the formula I mentioned. But ofc it is estimated only over $0-t$. But if one needs a forecast, one still needs the trend to sum it up for $X_{t+1} = m_{t+1} + s_{t+1} + Y_{t+1}$. So, how is trend estimated then if we have not seen $x_{t+1}$? $\endgroup$ – SWIM S. Apr 21 '18 at 20:20
  • $\begingroup$ What you've posted in the question is the classic formula for simple exponential smoothing. It is the level, not the trend. If you look closely at it, you will realize it can't be a trend, as there's no term like $x_t - x_{t-1}$ or the like, it's just a weighted average of the last estimate and the newest data point. $\endgroup$ – jbowman Apr 21 '18 at 20:25
  • $\begingroup$ Oh, ok, I think I got your argument. So, trend is always estimated using regression then? $\endgroup$ – SWIM S. Apr 21 '18 at 20:30
  • $\begingroup$ No, there's a variant of exponential smoothing that calculates trend as well as level. I'll add it to the answer, as it's becoming clear that that is the crux of your question! $\endgroup$ – jbowman Apr 21 '18 at 20:31

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