## Thursday, November 25, 2010

### Running Variance

Variance - kinda the bread and butter for analysis work on a time series. Doesn't get much respect though. But, take the square root of the variance and you get the almighty standard deviation. Today, though, let's give variance its due...
For an intro into variance...check out these posts:
Problem with variance is calculating it in the traditional sense. Its costly to compute across a time series. It can be quite a drag on your simulation engine's performance. The way to reduce the cost is to calculate the running variance. And that's when you get into quite a briar patch - loss of precision and overflow issues. See John D. Cook's post covering the variance briar patch:
And a few more posts by John covering different variance formulas and their outcomes:
John does great work and I learn a lot from his posts. But, I was still having problems finding a variance formula that fit my needs:
• Reduced the precision loss issue as much as possible;
• Allowed an easy way to window the running variance;
• Allowed an easy way to memoize the call.
Thankfully, I found a post by Subluminal Messages covering his very cool Running Standard Deviations formula. The code doesn't work as is - needs correcting on a few items - but you can get the gist of the formula just fine. The formula uses the power sum of the squared differences of the values versus Welford's approach of using the sum of the squared differences of the mean. Which makes it a bit easier to memoize. Not sure if its as good in solving the precision loss and overflow issues as Welford's does....but so far I haven't found any issues with it.

So, let's start with the formula for the Power Sum Average ($$PSA$$):

$$PSA = PSA_{yesterday} + ( ( (x_{today} * x_{today}) - x_{yesterday} ) ) / n)$$

Where:
• $$x$$ = value in your time series
• $$n$$ = number of values you've analyzed so far
You also need the Simple Moving Average, which you can find in one of my previous posts here.
Once you have the $$PSA$$ and $$SMA$$; you can tackle the Running Population Variance ($$Var$$ ):

$$Population Var = (PSA_{today} * n - n * SMA_{today} * SMA_{today}) / n$$

Now, one problem with all these formulas - they don't cover how to window the running variance. Windowing the variance gives you the ability to view the 20 period running variance at bar 150. All the formulas I've mentioned above only give you the running cumulative variance. Deriving the running windowed variance is just a matter of using the same SMA I've posted about before and adjusting the Power Sum Average to the following:

$$PSA = PSA_{yesterday} + (((x_{today} * x_{today}) - (x_{yesterday} * x_{yesterday}) / n)$$

Where:
• $$x$$ = value in your time series
• $$n$$ = the period
[Update] If you want the sample Variance you just need to adjust the Var formula to the following:

$$Sample Var = (PSA_{today} * n - n * SMA_{today} * SMA_{today}) / (n - 1)$$

Okay, on to the code.

Code for the Power Sum Average:
def powersumavg(bar, series, period, pval=None):
"""
Returns the power sum average based on the blog post from
Subliminal Messages.  Use the power sum average to help derive the running
variance.
sources: http://subluminal.wordpress.com/2008/07/31/running-standard-deviations/

Keyword arguments:
bar     --  current index or location of the value in the series
series  --  list or tuple of data to average
period  -- number of values to include in average
pval    --  previous powersumavg (n - 1) of the series.
"""

if period < 1:
raise ValueError("period must be 1 or greater")

if bar < 0:
bar = 0

if pval == None:
if bar > 0:
raise ValueError("pval of None invalid when bar > 0")

pval = 0.0

newamt = float(series[bar])

if bar < period:
result = pval + (newamt * newamt - pval) / (bar + 1.0)

else:
oldamt = float(series[bar - period])
result = pval + (((newamt * newamt) - (oldamt * oldamt)) / period)

return result
Code for the Running Windowed Variance:
def running_var(bar, series, period, asma, apowsumavg):
"""
Returns the running variance based on a given time period.
sources: http://subluminal.wordpress.com/2008/07/31/running-standard-deviations/

Keyword arguments:
bar     --  current index or location of the value in the series
series  --  list or tuple of data to average
asma    --  current average of the given period
apowsumavg -- current powersumavg of the given period
"""
if period < 1:
raise ValueError("period must be 1 or greater")

if bar <= 0:
return 0.0

if asma == None:
raise ValueError("asma of None invalid when bar > 0")

if apowsumavg == None:
raise ValueError("powsumavg of None invalid when bar > 0")

windowsize = bar + 1.0
if windowsize >= period:
windowsize = period

return (apowsumavg * windowsize - windowsize * asma * asma) / windowsize


Example call and results:
list_of_values = [3, 5, 8, 10, 4, 8, 12, 15, 11, 9]
prev_powersumavg = None
prev_sma = None
prev_sma = None
period = 3
for bar, price in enumerate(list_of_values):
new_sma = running_sma(bar, list_of_values, period, prev_sma)
new_powersumavg = powersumavg(bar, list_of_values, period, prev_powersumavg)
new_var = running_var(bar, list_of_values, period, new_sma, new_powersumavg)

msg = "SMA=%.4f, PSA=%.4f, Var=%.4f" % (new_sma, new_powersumavg, new_var)
print "bar %i: %s" % (bar, msg)

prev_sma = new_sma
prev_powersumavg = new_powersumavg

----------------------------------------------------------
Results of call:
bar 0: SMA=3.0000, PSA=9.0000, Var=0.0000
bar 1: SMA=4.0000, PSA=17.0000, Var=1.0000
bar 2: SMA=5.3333, PSA=32.6667, Var=4.2222
bar 3: SMA=7.6667, PSA=63.0000, Var=4.2222
bar 4: SMA=7.3333, PSA=60.0000, Var=6.2222
bar 5: SMA=7.3333, PSA=60.0000, Var=6.2222
bar 6: SMA=8.0000, PSA=74.6667, Var=10.6667
bar 7: SMA=11.6667, PSA=144.3333, Var=8.2222
bar 8: SMA=12.6667, PSA=163.3333, Var=2.8889
bar 9: SMA=11.6667, PSA=142.3333, Var=6.2222


Of course, as I said in the beginning of this post, just take the square root of this Running Windowed Variance to obtain the Standard Deviation.

MT

Asf said...

Great article,thanks.  I think there's an error in line 28 of running_var.  The line:return (apowsumavg * windowsize - windowsize * asma * asma) / windowsizeshould be:return (apowsumavg * windowsize - windowsize * asma * asma) / (windowsize-1)

Mike Taylor said...

You are correct if you want to obtain the sample variance instead of the population variance - which most people including myself do.

Mike Taylor said...

You are correct if you want to obtain the sample variance instead of the population variance – which most people including myself do.

Yogi said...

Thanks for the algo.
Is there a way to calculate the value without requiring the
series[bar - period] data point ? This requires me to store previous data points in a sliding window calculation.

Mike Taylor said...

No way I know of to avoid storing the previous data point in order to obtain the running variance. Just have to decide where you'd like to store it. You could store it and then use it in the call to your function. Or you could create a class for your variance logic and memoize the previous data point (cache the value) in the class. Then the object will keep track of it for you through the window.

Peter Hillyard said...
This comment has been removed by the author.
Peter Hillyard said...

Thanks for posting this. Any chance you have the code for the SMA? The two links provided no longer work.

Mike Taylor said...

Peter, sorry about the bad links. I have updated the links for the SMA post. Also, you might be interested in the code I have over on github that contains both the Running SMA and EMA: statio. Take care. MT

Mike Rider said...

Mike, Great work thank you! One question... Should line 25 of the Power Sum Average be indented so it is part of the if statement. As is it now, it looks like pval is always set to 0.0. Thanks, David

Mike Taylor said...

Thanks for catching the error in the code Mike. I have corrected the Power Sum Average accordingly. Thanks again! MT

Anonymous said...

return (apowsumavg * windowsize - windowsize * asma * asma) / windowsize

could be simplified to:

return apowsumavg - asma * asma