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What's the difference between smoothed and moving average?

Are they the same?

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    $\begingroup$ What do you mean by "smoothed" average? Can you write the mathematical expressions down for both smoothed average and moving average? $\endgroup$
    – jbowman
    Dec 28, 2017 at 16:06

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There's no official definition, but in univariate time series analysis, a "smoother" uses data to the left (t-1,2..) and to the right (t+1,2,..) of the current observation to calculate its current (t) value. In other words: it is "not causal","acausal", or "time symmetric". A Moving average is a filter that uses only past (and current) values to calculate the current value of the filter: no future data is used.

If you use a smoother on an online system, e.g. a live stock chart (so the graph is moving all the time), you will see that the past calculated values of the smoother can change as new data arrives (not so with the moving average: past mov avg values stay the same as new stock prices are created). A good way to detect a non-causal smoother is to look at sudden jumps: if your input data is in a decreasing segment, after which there is a sudden jump, and your smoothed line is already increasing in the declining segment (or vice versa), you intuitively know the used method is not causal: a causal filter can not "know" so many observations ahead of time it has to jump, only a non-causal smoother can. Causal methods can't "see into the future", while a smoother will sometimes appear to be able to do so.

And no, trying to make a smoother "one-sided" does not solve this problem.

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    $\begingroup$ Many fields don't make this distinction. For example, in climatology a moving average often -- in my reading almost always -- uses data on both sides. Also, whether there are sudden jumps depends on weighting. In many fields (e.g again climatology) if you use weights that diminish towards the edges of a window, then even a very different value has little impact when it enters or leaves a window It's a matter of history that Tukey borrowed the name Hanning from moving averages with weights 1:2:1 as used by the climatologist J. von Hann (punning on the similar name of his colleague Hamming). $\endgroup$
    – Nick Cox
    Feb 19, 2018 at 19:21

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