[Numpy-discussion] def of var of complex

[Charles R Harris]

On Jan 8, 2008 6:54 PM, Robert Kern <robert.kern at gmail.com> wrote:

> Neal Becker wrote:
> > I noticed that if I generate complex rv i.i.d. with var=1, that numpy
> says:
> >
> > var (<real part>) -> (close to 1.0)
> > var (<imag part>) -> (close to 1.0)
> >
> > but
> >
> > var (complex array) -> (close to complex 0)
> >
> > Is that not a strange definition?
>
> There is some discussion on this in the tracker.
>
>   http://projects.scipy.org/scipy/numpy/ticket/638
>
> The current state of affairs is that the implementation of var() just
> naively
> applies the standard formula for real numbers.
>
>   mean((x - mean(x)) ** 2)
>
> I think this is pretty obviously wrong prima facie. AFAIK, no one
> considers this
> a valid definition of variance for complex RVs or in fact a useful value.
> I
> think we should change this. Unfortunately, there is no single alternative
> but
> several.
>
> 1. Punt. Complex numbers are inherently multidimensional, and a single
> scale
> parameter doesn't really describe most distributions of complex numbers.
> Instead, you need a real covariance matrix which you can get with cov([
> z.real,
> z.imag]). This estimates the covariance matrix of a 2-D Gaussian
> distribution
> over RR^2 (interpreted as CC).
>
> 2. Take a slightly less naive formula for the variance which seems to show
> up in
> some texts:
>
>   mean(absolute(z - mean(z)) ** 2)
>
> This estimates the single parameter of a circular Gaussian over RR^2
> (interpreted as CC). It is also the trace of the covariance matrix above.
>
> 3. Take the variances of the real and imaginary components independently.
> This
> is equivalent to taking the diagonal of the covariance matrix above. This
> wouldn't be the definition of "*the* complex variance" that anyone else
> uses,
> but rather another form of punting. "There isn't a single complex variance
> to
> give you, but in the spirit of broadcasting, we'll compute the marginal
> variances of each dimension independently."
>
> Personally, I like 1 a lot. I'm hesitant to support 2 until I've seen an
> actual
> application of that definition. The references I have been given in the
> ticket
> comments are all early parts of books where the authors are laying out
> definitions without applications. Personally, it feels to me like the
> authors
> are just sticking in the absolute()'s ex post facto just so they can
> extend the
> definition they already have to complex numbers. I'm also not a fan of the
> expectation-centric treatments of random variables. IMO, the variance of
> an
> arbitrary RV isn't an especially important quantity. It's a parameter of a
> Gaussian distribution, and in this case, I see no reason to favor circular
> Gaussians in CC over general ones.
>

> But if someone shows me an actual application of the definition, I can
> amend my
> view.
>

Suppose you have a set of z_i and want to choose z to minimize the average
square error $ \sum_i |z_i - z|^2 $. The solution is that $z=\mean{z_i}$ and
the resulting average error is given by 2). Note that I didn't mention
Gaussians anywhere. No distribution is needed to justify the argument, just
the idea of minimizing the squared distance. Leaving out the ^2 would yield
another metric, or one could ask for a minmax solution. It is a question of
the distance function, not probability. Anyway, that is one justification
for the approach in 2) and it is one that makes a lot of applied math
simple. Whether of not a least squares fit is useful is different question.

Chuck
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