[Numpy-discussion] Cross-covariance function
josef.pktd@gmai...
josef.pktd@gmai...
Thu Jan 26 18:43:14 CST 2012
On Thu, Jan 26, 2012 at 3:58 PM, Bruce Southey <bsouthey@gmail.com> wrote:
> On Thu, Jan 26, 2012 at 12:45 PM, <josef.pktd@gmail.com> wrote:
>> On Thu, Jan 26, 2012 at 1:25 PM, Bruce Southey <bsouthey@gmail.com> wrote:
>>> On Thu, Jan 26, 2012 at 10:07 AM, Pierre Haessig
>>> <pierre.haessig@crans.org> wrote:
>>>> Le 26/01/2012 15:57, Bruce Southey a écrit :
>>>>> Can you please provide a
>>>>> couple of real examples with expected output that clearly show what
>>>>> you want?
>>>>>
>>>> Hi Bruce,
>>>>
>>>> Thanks for your ticket feedback ! It's precisely because I see a big
>>>> potential impact of the proposed change that I send first a ML message,
>>>> second a ticket before jumping to a pull-request like a Sergio Leone's
>>>> cowboy (sorry, I watched "for a few dollars more" last weekend...)
>>>>
>>>> Now, I realize that in the ticket writing I made the wrong trade-off
>>>> between conciseness and accuracy which led to some of the errors you
>>>> raised. Let me try to use your example to try to share what I have in mind.
>>>>
>>>>> >> X = array([-2.1, -1. , 4.3])
>>>>> >> Y = array([ 3. , 1.1 , 0.12])
>>>>
>>>> Indeed, with today's cov behavior we have a 2x2 array:
>>>>> >> cov(X,Y)
>>>> array([[ 11.71 , -4.286 ],
>>>> [ -4.286 , 2.14413333]])
>>>>
>>>> Now, when I used the word 'concatenation', I wasn't precise enough
>>>> because I meant assembling X and Y in the sense of 2 vectors of
>>>> observations from 2 random variables X and Y.
>>>> This is achieved by concatenate(X,Y) *when properly playing with
>>>> dimensions* (which I didn't mentioned) :
>>>>> >> XY = np.concatenate((X[None, :], Y[None, :]))
>>>> array([[-2.1 , -1. , 4.3 ],
>>>> [ 3. , 1.1 , 0.12]])
>>>
>>> In this context, I find stacking, np.vstack((X,Y)), more appropriate
>>> than concatenate.
>>>
>>>>
>>>> In this case, I can indeed say that "cov(X,Y) is equivalent to cov(XY)".
>>>>> >> np.cov(XY)
>>>> array([[ 11.71 , -4.286 ],
>>>> [ -4.286 , 2.14413333]])
>>>>
>>> Sure the resulting array is the same but whole process is totally different.
>>>
>>>
>>>> (And indeed, the actual cov Python code does use concatenate() )
>>> Yes, but the user does not see that. Whereas you are forcing the user
>>> to do the stacking in the correct dimensions.
>>>
>>>
>>>>
>>>>
>>>> Now let me come back to my assertion about this behavior *usefulness*.
>>>> You'll acknowledge that np.cov(XY) is made of four blocks (here just 4
>>>> simple scalars blocks).
>>> No there are not '4' blocks just rows and columns.
>>
>> Sturla showed the 4 blocks in his first message.
>>
> Well, I could not follow that because the code is wrong.
> X = np.array([-2.1, -1. , 4.3])
>>>> cX = X - X.mean(axis=0)[np.newaxis,:]
>
> Traceback (most recent call last):
> File "<pyshell#6>", line 1, in <module>
> cX = X - X.mean(axis=0)[np.newaxis,:]
> IndexError: 0-d arrays can only use a single () or a list of newaxes
> (and a single ...) as an index
> whoops!
>
> Anyhow, variance-covariance matrix is symmetric but numpy or scipy
> lacks lapac's symmetrix matrix
> (http://www.netlib.org/lapack/explore-html/de/d9e/group___s_y.html)
>
>>>
>>>> * diagonal blocks are just cov(X) and cov(Y) (which in this case comes
>>>> to var(X) and var(Y) when setting ddof to 1)
>>> Sure but variances are still covariances.
>>>
>>>> * off diagonal blocks are symetric and are actually the covariance
>>>> estimate of X, Y observations (from
>>>> http://en.wikipedia.org/wiki/Covariance)
>>> Sure
>>>>
>>>> that is :
>>>>> >> ((X-X.mean()) * (Y-Y.mean())).sum()/ (3-1)
>>>> -4.2860000000000005
>>>>
>>>> The new proposed behaviour for cov is that cov(X,Y) would return :
>>>> array(-4.2860000000000005) instead of the 2*2 matrix.
>>>
>>> But how you interpret an 2D array where the rows are greater than 2?
>>>>>> Z=Y+X
>>>>>> np.cov(np.vstack((X,Y,Z)))
>>> array([[ 11.71 , -4.286 , 7.424 ],
>>> [ -4.286 , 2.14413333, -2.14186667],
>>> [ 7.424 , -2.14186667, 5.28213333]])
>>>
>>>
>>>>
>>>> * This would be in line with the cov(X,Y) mathematical definition, as
>>>> well as with R behavior.
>>> I don't care what R does because I am using Python and Python is
>>> infinitely better than R is!
>>>
>>> But I think that is only in the 1D case.
>>
>> I just checked R to make sure I remember correctly
>>
>>> xx = matrix((1:20)^2, nrow=4)
>>> xx
>> [,1] [,2] [,3] [,4] [,5]
>> [1,] 1 25 81 169 289
>> [2,] 4 36 100 196 324
>> [3,] 9 49 121 225 361
>> [4,] 16 64 144 256 400
>>> cov(xx, 2*xx[,1:2])
>> [,1] [,2]
>> [1,] 86.0000 219.3333
>> [2,] 219.3333 566.0000
>> [3,] 352.6667 912.6667
>> [4,] 486.0000 1259.3333
>> [5,] 619.3333 1606.0000
>>> cov(xx)
>> [,1] [,2] [,3] [,4] [,5]
>> [1,] 43.0000 109.6667 176.3333 243.0000 309.6667
>> [2,] 109.6667 283.0000 456.3333 629.6667 803.0000
>> [3,] 176.3333 456.3333 736.3333 1016.3333 1296.3333
>> [4,] 243.0000 629.6667 1016.3333 1403.0000 1789.6667
>> [5,] 309.6667 803.0000 1296.3333 1789.6667 2283.0000
>>
>>
>>>
>>>> * This would save memory and computing resources. (and therefore help
>>>> save the planet ;-) )
>>> Nothing that you have provided shows that it will.
>>
>> I don't know about saving the planet, but if X and Y have the same
>> number of columns, we save 3 quarters of the calculations, as Sturla
>> also explained in his first message.
>>
> Can not figure those savings:
> For a 2 by 2 output has 3 covariances (so 3/4 =0.75 is 'needed' not 25%)
> a 3 by 3 output has 6 covariances
> a 5 by 5 output 15 covariances
what numpy calculates are 4, 9 and 25 covariances, we might care only
about 1, 2 and 4 of them.
>
> If you want to save memory and calculation then use symmetric storage
> and associated methods.
actually for covariance matrix we stilll need to subtract means, so we
won't save 75%, but we save 75% in the cross-product.
suppose X and Y are (nobs, k_x) and (nobs, k_y) (means already subtracted)
(and ignoring that numpy "likes" rows instead of columns)
the partitioned dot product [X,Y]'[X,Y] is
[[ X'X, X'Y],
[Y'X, Y'Y]]
X'Y is (n_x, n_y)
total shape is (n_x + n_y, n_x + n_y)
If we are only interested in X'Y, we don't need the other three submatrices.
If n_x = 99 and n_y is 1, we save .... ?
(we get a (99,1) instead of a (100, 100) matrix)
and X'Y , np.dot(X, Y), doesn't have any duplicated symmetry, so
exploiting symmetry is a different issue.
Josef
>
> Bruce
>
>> Josef
>>
>>>
>>>>
>>>> However, I do understand that the impact for this change may be big.
>>>> This indeed requires careful reviewing.
>>>>
>>>> Pierre
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>>>
>>> Bruce
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