Talk:Exponential smoothing

Is $x_{t}$ right?

Should it be

${\begin{aligned}s_{t}&=\alpha x_{t}+(1-\alpha )s_{t-1}\,\end{aligned}}$

or

${\begin{aligned}s_{t}&=\alpha x_{t-1}+(1-\alpha )s_{t-1}\,\end{aligned}}$?

This indicates the first one, but this PowerPoint presentation implies the second one.

If the formula is used for forecasting purposes, then it looks to me like the second one is the only usable one (and also looks more natural somehow). Or am I missing something? (I'm very, very rusty on this now!). Hope someone can clear this up.--A bit iffy 11:18, 3 March 2007 (UTC)

The choice is a matter of convention (for strictly periodic data), but I think that the first one (which we use here) is more natural, since people would compute the smoothed value as soon as possible and would naturally want to label it with the time that they computed it. JRSpriggs 11:27, 4 March 2007 (UTC)

Agreed to this, when there is a large amount of terms, is does not matter whether x_t or x_{t-1} is used. For few terms the first one should be used. 137.56.63.144 (talk) 15:33, 21 October 2022 (UTC)

Agree: using x[t] and s[t] to mean the signal (or filtered signal) using all information up to time t (inclusive) keeps everything simple. If instead s[t] is lagged then at time t users will actually be much more interested in s[t+1], and then things get even more confusing with second order filters etc, where the user, at time t, will be considering s'[t+2] and so on - best avoided by just using t to mean "all information up to time t".

The textbooks that I use for teaching time series use the second one (with the lagged value of the series). See Moore, McCabe, Duckworth and Sclove, The Practice of Business Statistics. In Minitab, on the other hand, the smoothed value at time t is defined using the first formula, while the fitted value at time t is the smoothed value at time (t-1). EconProf86 16:20, 28 May 2007 (UTC)

In my view it should be X_t and not X_{t-1}. The simple exponential filter is analageous to a single pole low pass IIR filter. In both cases you are saying that the current out put is formed from the sum of the new input plus a fraction of the old output — Preceding unsigned comment added by 212.77.61.18 (talk) 08:09, 9 September 2011 (UTC)

Both are correct. As said above, X_t is used when the focus is smoothing a series and X_{t-1} is used when prediction is the focus. As the article mostly follows the NIST handbook we use X_{t-1}.--Muhandes (talk) 16:55, 10 September 2011 (UTC)

Another related point: the following paragraph states "In the limiting case with α = 1 the output series is just the same as the original series". This will only be the case with X_t. Using X_{t-1} means adding a lag in the time series.Eric thiebaut (talk) 17:28, 24 December 2011 (UTC)

Corrected. --Muhandes (talk) 11:45, 25 December 2011 (UTC)

looks like it is back to t-1. I like it better as t. That is how you would program it. — Preceding unsigned comment added by 128.114.150.27 (talk) 22:59, 17 March 2014 (UTC)

Since there is only one $x$ term in the equation, the results will, in all cases, be the same except for one time step delay. Gah4 (talk) 08:45, 11 April 2018 (UTC)

Now the formula is x(t-1), but the text says: "the smoothed statistic s(t) is a simple weighted average of the current observation x(t) and the previous smoothed statistic s(t−1)" --85.212.105.213 (talk) —Preceding undated comment added 19:20, 29 September 2019 (UTC)

Corrected an Error

I removed an innacuracy that stated that simple exponential smoothing was the same as Brown exponential smoothing. This is not the case; Brown's method is double exponential smoothing. JLT 1451, 16 Dec 1009 (CST) —Preceding unsigned comment added by 131.10.254.62 (talk)

Division by 0 in triple exponential smoothing

How does the method cope with scenarios in which either c(t) is 0 or s(t) is 0? The problem occurs in the next time iteration:

• if c(t) is 0 then s(t+1) = ... x(t) / 0 ...
• if s(t) is 0 then c(t+1) = ... x(t) / 0 ...

Merging Exponential moving average here

same concept. fgnievinski (talk) 05:47, 19 June 2021 (UTC)

fgnievinski, Exponential moving averageExponential moving average is a redirect, not an article. – wbm1058 (talk) 02:36, 24 October 2021 (UTC)

@Wbm1058: I meant merging here the redirect's target, section Moving average#Exponential. fgnievinski (talk) 03:37, 24 October 2021 (UTC)

No need to merge; just delete all but the first two sentences of Moving average#Exponential moving average, plus perhaps the paragraph citing Hunter (1986). That should remove the WP:OR problem and leave the section as a summary of the main article (see WP:SUMMARY). Klbrain (talk) 16:28, 24 October 2022 (UTC)

I've done the hard pruning and removed the merge templates. The other page should now be a WP:SUMMARY. Klbrain (talk) 09:51, 26 November 2022 (UTC)

Resolved

Conflation with exponential window function

The article mentions the exponential window function, but that's a two-sided function, unlike the impulse response of the EMA, which is a one-sided exponential. It would be better to specifically contrast these than to leave them confused. The window function convolution can be realized using forward and backward one-sided EMA (e.g. by using the Matlab or Numpy filtfilt function), resulting in a non-causal smoothing with zero delay. Dicklyon (talk) 00:52, 5 August 2025 (UTC)