# Can white noise be removed using moving averages?

In the book Time Series Analysis by R, the author mentions the use of moving average to smooth out the white noise.

Can moving averages be used to remove white noise or are there better methods?

• Moving averages can reduce noise but not necessarily completely remove it. Mar 28, 2017 at 12:43
• You have marked this with the Time-Series tag and with the Moving-Average tag. The Moving-Average tag specifically addresses a running average approach. Within Time Series, there is a Box-Cox approach to time series modeling (aka ARIMA) where "MA" has a specific and specialized meaning. Can answers assume you are referring to the windowed, running-mean-kind of moving average and not the ARIMA-style MA? Mar 28, 2017 at 19:19
• I was referring to Time-series analysis(Springer) Mar 29, 2017 at 5:28
• @ Wayne I am a cs student , I just started learning statistics few days ago with the book, Time Series Analysis by r ,and i have no proper grip or idea on this subject . . I think i was referring to moving average ,Sorry . Mar 29, 2017 at 5:51

The role of the ARIMA structure is to convert colored noise to white noise. Smoothing out white noise would seem to be potentially creating colored noise. The concept of an oxymoron pops up in my head and I am not sure why.

• I am not following what you are saying this time Dave. Random noise is white noise. Mar 28, 2017 at 12:46
• I agree random noise is white noise . To me smoothing white noise runs up against the Slutsky Effect thus possibly injecting structure Mar 28, 2017 at 13:18
• Does a moving average introduce correlations as an improperly-used differencing can? Mar 28, 2017 at 14:27
• @Wayne Yes. For intuition, consider a moving average over a long window. Two adjacent windows will involve exactly the same data, except that one value at the beginning of one window is dropped and replaced by a value at the end of the next window. The window averages must therefore be very positively correlated indeed.
– whuber
Mar 28, 2017 at 15:02
• @whuber: Thanks! I'm not sure that an MA model is actually a moving average in any meaningful way, but given that an actual moving average introduces correlation, does an MA model avoid this or also do this? (I.e. is there any advantage to an MA model rather than a literal moving average, which is what I think the OP is asking? Oops, just looked again and perhaps the question was edited to say "moving averages" rather than "MA Model", which is what I thought I read.) Mar 28, 2017 at 16:09

Some thoughts on moving averages. [I think you edited your question to move from "MA model", part of the ARIMA (Box Jenkins) time series modeling family, to talking about "moving" or "running" averages, which is something different. Is this correct?]

First, per whuber's comments in IrishStat's answer, a literal moving average introduces correlation into your data. If you intend to do any analysis of your data after smoothing, you will be analyzing data with erroneous correlations added to it.

Second, a literal moving average is a binary operation: your samples pop into the window and out of the window 100%, which creates artifacts. These artifacts are a bit technical and have to do with the phase of your data -- engineers looking at signal waveforms care a lot about them. To avoid such artifacts, you should prefer a weighted moving average. (A moving average is a rectangular convolution, and you want something with smooth sides that go more gradually to zero effect.)

Third, a naive, centered moving average will use data from the future, which might be okay if you're just graphing something -- or it might not be -- but would be a big problem in any real analysis.

Fourth, you need to ask yourself: why you want to remove or "smooth out" the noise? If all you want to do is make a less-busy plot, this may be okay, but if you intend to smooth your data and then analyze it, you are modifying your data in multiple ways that will distort any analysis: introducing spurious correlations, changing phases, fooling analysis techniques into being over-confident of their results, etc.

If you state your analytical goal -- rather than just wanting to eliminate noise -- you'll get a better answer.

• When talking about the ARIMA framework, you meant Box Jenkins and not Box Cox. Mar 28, 2017 at 20:53