How to calculate moving average without keeping the count and data-total

Question

I am trying to find a way to calculate a moving cumulative average without storing the count and total data that is received so far   I came up with two algorithms but both need to store the count    new average     old count   old data    next data    next count new average   old average    next data - old average    next count   The problem with these methods is that the count gets bigger and bigger resulting in losing precision in the resulting average   The first method uses the old count and next count which are obviously 1 apart  This got me thinking that perhaps there is a way to remove the count but unfortunately I haven t found it yet  It did get me a bit further though  resulting in the second method but still count is present   Is it possible  or am I just searching for the impossible

User · Answer

The answer of Flip is computationally more consistent than the Muis one.

Using double number format, you could see the roundoff problem in the Muis approach:

When you divide and subtract, a roundoff appears in the previous stored value, changing it.

However, the Flip approach preserves the stored value and reduces the number of divisions, hence, reducing the roundoff, and minimizing the error propagated to the stored value. Adding only will bring up roundoffs if there is something to add (when N is big, there is nothing to add)

Those changes are remarkable when you make a mean of big values tend their mean to zero.

I show you the results using a spreadsheet program:

Firstly, the results obtained:

The A and B columns are the n and X_n values, respectively.

The C column is the Flip approach, and the D one is the Muis approach, the result stored in the mean. The E column corresponds with the medium value used in the computation.

A graph showing the mean of even values is the next one:

As you can see, there is big differences between both approachs.

User · Answer

You can simply do   double approxRollingAverage  double avg  double new sample         avg -  avg   N      avg    new sample   N       return avg      Where N is the number of samples where you want to average over  Note that this approximation is equivalent to an exponential moving average  See  Calculate rolling   moving average in C

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In Java8   LongSummaryStatistics movingAverage   new LongSummaryStatistics    movingAverage accept new data       average   movingAverage getAverage      you have also IntSummaryStatistics  DoubleSummaryStatistics

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New average   old average    n-1  n   new value  n   This is assuming the count only changed by one value  In case it is changed by M values then   new average   old average    n-len M   n    sum of values in M  n     This is the mathematical formula  I believe the most efficient one   believe you can do further code by yourselves

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An example using javascript  for comparison   https   jsfiddle net drzaus Lxsa4rpz   function calcNormalAvg list           sum list    len list      return list reduce function a  b    return a   b       list length    function calcRunningAvg previousAverage  currentNumber  index             avg     n-1    x     n     return   previousAverage    index - 1    currentNumber     index       x000D   x000D   function    x000D       populate base list x000D  var list       x000D  function getSeedNumber     return Math random   100    x000D  for var i   0  i  lt  50  i    list push  getSeedNumber      x000D   x000D       our calculation functions  for comparison x000D  function calcNormalAvg list    x000D        sum list    len list  x000D   return list reduce function a  b    return a   b       list length  x000D    x000D  function calcRunningAvg previousAverage  currentNumber  index    x000D          avg     n-1    x     n x000D   return   previousAverage    index - 1    currentNumber     index  x000D    x000D    function calcMovingAvg accumulator  new value  alpha    x000D     return  alpha   new value     1 0 - alpha    accumulator  x000D    x000D   x000D       start our baseline x000D  var baseAvg   calcNormalAvg list   x000D  var runningAvg   baseAvg  movingAvg   baseAvg  x000D  console log  base avg   d   baseAvg   x000D     x000D    var okay   true  x000D     x000D       table of output  cleaner console view x000D    var results       x000D   x000D       add 10 more numbers to the list and compare calculations x000D  for var n   list length  i   0  i  lt  10  i    n      x000D   var newNumber   getSeedNumber    x000D   x000D   runningAvg   calcRunningAvg runningAvg  newNumber  n 1   x000D   movingAvg   calcMovingAvg movingAvg  newNumber  1  n 1    x000D   x000D   list push newNumber   x000D   baseAvg   calcNormalAvg list   x000D   x000D      assert and inspect x000D   console log  added   d  to list at pos  d  running avg    d vs  regular avg    d   s   vs  moving avg    d   s   x000D      newNumber  list length  runningAvg  baseAvg  runningAvg    baseAvg  movingAvg  movingAvg    baseAvg x000D     x000D  results push   x  newNumber  n list length  regular  baseAvg  running  runningAvg  moving  movingAvg  eqRun  baseAvg    runningAvg  eqMov  baseAvg    movingAvg      x000D   x000D  if runningAvg    baseAvg  console warn  Fail     x000D  okay   okay  amp  amp   runningAvg    baseAvg       x000D    x000D     x000D    console log  Everything matched for running avg   s   okay   x000D    if console table  console table results   x000D        x000D   x000D   x000D

User · Answer

Here s yet another answer offering commentary on how Muis  Abdullah Al-Ageel and Flip s answer are all mathematically the same thing except written differently   Sure  we have Jos   Manuel Ramos s analysis explaining how rounding errors affect each slightly differently  but that s implementation dependent and would change based on how each answer were applied to code   There is however a rather big difference  It s in Muis s N  Flip s k  and Abdullah Al-Ageel s n   Abdullah Al-Ageel doesn t quite explain what n should be  but N and k differ in that N is  the number of samples where you want to average over  while k is the count of values sampled    Although I have doubts to whether calling N the number of samples is accurate    And here we come to the answer below   It s essentially the same old exponential weighted moving average as the others  so if you were looking for an alternative  stop right here   Exponential weighted moving average  Initially   average   0 counter   0   For each value   counter    1 average   average    value - average    min counter  FACTOR    The difference is the min counter  FACTOR  part   This is the same as saying min Flip s k  Muis s N    FACTOR is a constant that affects how quickly the average  catches up  to the latest trend   Smaller the number the faster    At 1 it s no longer an average and just becomes the latest value    This answer requires the running counter counter   If problematic  the min counter  FACTOR  can be replaced with just FACTOR  turning it into Muis s answer   The problem with doing this is the moving average is affected by whatever average is initiallized to   If it was initialized to 0  that zero can take a long time to work its way out of the average   How it ends up looking

User · Answer

A neat Python solution based on the above answers  class RunningAverage        def   init   self           self average   0         self n   0              def   call   self  new value           self n    1         self average    self average    self n-1    new value    self n               def   float   self           return self average          def   repr   self           return  quot average   quot    str self average   usage  x   RunningAverage   x 0  x 2  x 4  print x

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From a blog on running sample variance calculations  where the mean is also calculated using Welford s method     Too bad we can t upload SVG images

[moving-average] How to calculate moving average without keeping the count and data-total?

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