Al Chou wrote:
>>Date: Wed, 04 Jun 2003 21:05:14 0700
>>From: Phil Steitz <phil@steitz.com>
>>Subject: [math] more improvement to storage free mean, variance computation
>>
>>Check out procedure sum.2 and var.2 in
>>
>>http://www.stanford.edu/~glynn/PDF/0208.pdf
>>
>>The first looks like Brent's suggestion for a corrected mean
>>computation, with no memory required. The additional computational cost
>>that I complained about is docuemented to be 3x the flops cost of the
>>direct computation, but the computation is claimed to be more stable. So
>>the question is: do we pay the flops cost to get the numerical
>>stability? The example in the paper is compelling; but it uses small
>>words (err, numbers I mean  sorry, slipped in to my native Fortran for
>>a moment there ;)). So how do we go about deciding whether the
>>stability in the mean computation is worth the increased computational
>>effort? I would prefer not to answer "let the user decide". To make
>>the decision harder, we should note that it is actually worse than 3x,
>>since in the no storage version, the user may request the mean only
>>rarely (if at all) and the 3x comparison is against computiing the mean
>>for each value added.
>>
>>The variance formula looks better than what we have now, still requiring
>>no memory. Should we implement this for the no storage case?
>
>
> After implementing var.2 from the Stanford paper in UnivariateImpl and
> scratching my head for some time over why the variance calculation failed its
> JUnit test case, I realized there's a flaw in var.2 that I can't understand no
> one talks about. To update the variance (called S in the paper), the formula
> calculates
>
> z = y / i
> S = S + (i?1) * y * z
>
> where i is the number of data values (including the value just being added to
> the collection). It doesn't really matter how y is defined, because you will
> notice that
>
> S = S + (i?1) * y * y / i
> = S + (i?1) * y**2 / i
>
> which means that S can never decrease in magnitude (for real data, which is
> what we're talking about). But for the simple case of three data values {1, 2,
> 2} in the JUnit test case, the variance decreases between the addition of the
> second and third data values.
>
> Can anyone point out what I'm missing here?
>
>
I think that is OK, since if you look at the definition of S earlier in
the paper, S is not the variance, it is the sum of the squared
deviations from the mean. This should be always increasing.
>
> Al
>
> =====
> Albert Davidson Chou
>
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