Re: moving average function

• To: mathgroup at smc.vnet.net
• Subject: [mg126328] Re: moving average function
• From: "Oleksandr Rasputinov" <oleksandr_rasputinov at ymail.com>
• Date: Tue, 1 May 2012 05:23:49 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jnlj94\$n3k\$1@smc.vnet.net>

```On Mon, 30 Apr 2012 09:42:12 +0100, Robert McHugh <rtmphone09 at gmail.com>
wrote:

> Below is a moving average function that has the following features:
> 1. returns a list with the same length as the original length
> 2. provides a reasonable estimate for averages on the "sides" of the
> window.
>
> Have failed to figure out how to do this with ListConvolve and
> ListCorrelate, so I submit this with the hope that others can
> recommend how it might be improved. Also searched this website for
> alternatives but didn't find any that met the above criteria.
>
> I was motivated to do this in order to keep my code free of handling
> special cases related to the edges of the widow size.  Note that in
> one particular case, I have data measured every minute and would like
> to compare the results of using averaging the data over window sizes
> of 61, 121, and 181.
>
> Recommendations for how to improve the function or alternatives are
> appreciated.
> Bob.
>
>
> movingAverageBalanced[list_List, nAvg_Integer?OddQ ] :=
>  Module[{nHang, middle, left, right, all},
>   nHang = (nAvg - 1)/2;
>
>   middle = MovingAverage[list, nAvg];
>
>   left = Total[ Take[list, 2 # - 1]] /(2 # - 1) & /@ Range[nHang];
>   right =
>    Reverse[Total[ Take[list, -( 2 # - 1)]] /(2 # - 1) & /@
>      Range[nHang]];
>   all = Join[left, middle, right] ;
>   Return[all];
>   ]
>
> Example
> listTest = {a, b, c, d, e, f, g, h, i, j, k};
> r = movingAverageBalanced[listTest, 5];
> r // TableForm
>
> {
>  {a},
>  {1/3 (a + b + c)},
>  {1/5 (a + b + c + d + e)},
>  {1/5 (b + c + d + e + f)},
>  {1/5 (c + d + e + f + g)},
>  {1/5 (d + e + f + g + h)},
>  {1/5 (e + f + g + h + i)},
>  {1/5 (f + g + h + i + j)},
>  {1/5 (g + h + i + j + k)},
>  {1/3 (i + j + k)},
>  {k}
> }
>

This is almost a special case of the Savitzky-Golay filter, code for which
I posted here recently (see
http://forums.wolfram.com/mathgroup/archive/2012/Feb/msg00036.html). The
main difference is in the treatment of the endpoints; while you chose to
use a shortened window in dealing with these, I used a skewed window, i.e.
taking the endpoint values as given by the best-fitting n-th order
polynomial. The latter will obviously give worse results for the simple
moving average where n = 0 (i.e., the endpoints would be approximated by
their mean value) but can provide better results for moving polynomial
windows with larger n. For example, the effect of a 5-point quadratic
smoothing kernel is given by:

SavitzkyGolayFilter[
{a, b, c, d, e, f, g, h, i, j, k},
2, 2, 0
] // Factor

i.e.,

{1/35 (31 a + 9  b - 3  c - 5  d + 3  e),
1/35 (9  a + 13 b + 12 c + 6  d - 5  e),
1/35 (-3 a + 12 b + 17 c + 12 d - 3  e),
1/35 (-3 b + 12 c + 17 d + 12 e - 3  f),
1/35 (-3 c + 12 d + 17 e + 12 f - 3  g),
1/35 (-3 d + 12 e + 17 f + 12 g - 3  h),
1/35 (-3 e + 12 f + 17 g + 12 h - 3  i),
1/35 (-3 f + 12 g + 17 h + 12 i - 3  j),
1/35 (-3 g + 12 h + 17 i + 12 j - 3  k),
1/35 (-5 g + 6  h + 12 i + 13 j + 9  k),
1/35 (3  g - 5  h - 3  i + 9  j + 31 k)}

One may also take derivatives in this way; again using a 5-point kernel, a
piecewise linear approximation to the first derivative is:

SavitzkyGolayFilter[
{a, b, c, d, e, f, g, h, i, j, k},
2, 2, 1
] // Factor

or,

{1/70 (-34 a + 3  b + 20 c + 17 d - 6  e),
1/35 (-12 a -    b + 5  c + 6  d + 2  e),
1/10 (-2  a -    b +    d + 2  e),
1/10 (-2  b -    c +    e + 2  f),
1/10 (-2  c -    d +    f + 2  g),
1/10 (-2  d -    e +    g + 2  h),
1/10 (-2  e -    f +    h + 2  i),
1/10 (-2  f -    g +    i + 2  j),
1/10 (-2  g -    h +    j + 2  k),
1/35 (-2  g - 6  h - 5  i +    j + 12 k),
1/70 (6   g - 17 h - 20 i - 3  j + 34 k)}

and so on.

```

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