[Numpy-discussion] def of var of complex
Robert Kern
robert.kern@gmail....
Tue Jan 8 23:12:38 CST 2008
Travis E. Oliphant wrote:
> Robert Kern 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?
>>>
>>
>>
>> 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.
>>
>
> I tend to favor this interpretation because it is used quite heavily in
> signal processing applications where "circular" Gaussian random
> variables show up quite a bit --- so much so, in fact, that most EE
> folks would expect this as the output and you would have to explain to
> them why there may be other choices that make sense.
>
> So, #2 is kind of a nod to the signal-processing community (especially
> the communication section).
<sigh> Fair enough. I relent. You implement it; I'll document the correct^Wcov()
alternative. :-)
> But, there is also merit to me in #3 (although it may be harder to
> explain why the variance returns a complex number --- if that is what
> you meant).
Yes.
--
Robert Kern
"I have come to believe that the whole world is an enigma, a harmless enigma
that is made terrible by our own mad attempt to interpret it as though it had
an underlying truth."
-- Umberto Eco
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