[SciPy-dev] Is this a bugfix for scipy.hilbert?
Ariel Rokem
arokem at berkeley.edu
Mon Jan 18 18:59:03 EST 2010
Hi Josef - thanks!
I have to say that I don't quite understand what is happening in either
hilbert2 or in fftpack.hilbert, but just from looking at it, at least
hilbert2 doesn't have the same problem of inconsistency among the 'axis'
arguments that hilbert had. Maybe someone else understands these two things
better than me?
Cheers,
Ariel
On Sun, Jan 17, 2010 at 9:54 PM, <josef.pktd at gmail.com> wrote:
> On Sun, Jan 17, 2010 at 10:49 PM, <josef.pktd at gmail.com> wrote:
> > On Sun, Jan 17, 2010 at 12:31 AM, <josef.pktd at gmail.com> wrote:
> >> On Fri, Jan 15, 2010 at 4:20 PM, Ariel Rokem <arokem at berkeley.edu>
> wrote:
> >>> Hi - I've never done this before, so it would be great if I could 'look
> over
> >>> your shoulder' (in the sense that I know how this ticket came about
> :D), as
> >>> you submit a ticket on this.
> >>
> >> Done in http://projects.scipy.org/scipy/ticket/1093
> >> tests pass at 14 decimals
> >
> >
> > While adding some tests, I got one more question
> >
> > According to Wikipedia
> http://en.wikipedia.org/wiki/Analytic_signal#Definition
> > the imaginary part of the analytical signal is equal to the Hilbert
> transform
> >
> > however, fftpack.hilbert has the opposite sign.
> >
> > From the examples signal.hilbert looks correct, so does
> > fftpack.hilbert have a sign mistake or is it based on a different
> > definition?
> >
> >
> >>>> r = np.random.randn(20)
> >>>> fftpack.hilbert(r)
> > array([-0.27285468, -1.39747965, 1.7991044 , -0.16609304, -1.84459577,
> > 0.48696479, -0.33190553, 0.59383033, 2.15361055, -0.89341275,
> > -0.13730369, 0.84046658, 1.38110384, -1.7595949 , -0.04869402,
> > 0.59871558, -1.09627219, 0.59375139, -1.6021929 , 1.10285168])
> >>>> hilbert(r).imag
> > array([ 0.27285468, 1.39747965, -1.7991044 , 0.16609304, 1.84459577,
> > -0.48696479, 0.33190553, -0.59383033, -2.15361055, 0.89341275,
> > 0.13730369, -0.84046658, -1.38110384, 1.7595949 , 0.04869402,
> > -0.59871558, 1.09627219, -0.59375139, 1.6021929 , -1.10285168])
> >
> >
> > I'm just checking definitions for the tests,
>
> committed in http://projects.scipy.org/scipy/changeset/6205
>
> There is also a hilbert2 for 2d convolution. It is not in the
> documentation (at least in not my old ones), and I didn't try whether
> it is correct.
>
> Josef
>
>
> >
> > Josef
> >>
> >> Josef
> >>
> >>>
> >>> Thanks --
> >>>
> >>> Ariel
> >>>
> >>> On Fri, Jan 15, 2010 at 11:48 AM, <josef.pktd at gmail.com> wrote:
> >>>>
> >>>> On Fri, Jan 15, 2010 at 2:34 PM, Ariel Rokem <arokem at berkeley.edu>
> wrote:
> >>>> > Hi -
> >>>> >
> >>>> > attached is a file with a couple of tests. I am not sure this tests
> the
> >>>> > issues we were dealing with previously (the axis issues, etc.), but
> it
> >>>> > has
> >>>> > some sensible test-cases, which compare to what Matlab would give
> you
> >>>> > (not
> >>>> > quite 10by3 or 10by6, but as you can see, they make sense). Also -
> all
> >>>> > the
> >>>> > assertions are assert_almost_equal. Do you think that's OK? I think
> >>>> > there
> >>>> > are float-precision issues here, which would make assert_equal fail,
> but
> >>>> > I
> >>>> > am not sure - I would be happy to get any general comments on these
> >>>> > tests,
> >>>> > in case I am doing this all wrong.
> >>>>
> >>>> nice test cases, I like theoretical tests even better than verified
> >>>> numbers from other packages.
> >>>>
> >>>> Besides some cosmetic changes to get them into a test function, the
> >>>> only part to add is the precision of the tests.
> >>>> The default precision of assert_almost_equal is only 6 decimals.
> >>>>
> >>>> For these kind of cases, I usually go to 12 to 15 depending on the
> >>>> numerical precision of the algorithm. Usually, I go by trial and error
> >>>> until the test breaks, or calculate max abs of the error.
> >>>>
> >>>> I can add some tests for the axis argument.
> >>>>
> >>>> Can you open a ticket for the record or shall I ?
> >>>>
> >>>> Josef
> >>>>
> >>>>
> >>>> >
> >>>> > Cheers,
> >>>> >
> >>>> > Ariel
> >>>> >
> >>>> > On Thu, Jan 14, 2010 at 11:10 PM, <josef.pktd at gmail.com> wrote:
> >>>> >>
> >>>> >> On Fri, Jan 15, 2010 at 1:44 AM, Ariel Rokem <arokem at berkeley.edu>
> >>>> >> wrote:
> >>>> >> > Yes - looks good. Except I would prefer to eventually set the
> axis to
> >>>> >> > default to -1, to be consistent with signal.fft (and also
> np.fft.fft)
> >>>> >> > which
> >>>> >> > has axis=-1.
> >>>> >>
> >>>> >> I'm indifferent to the default axis, from a quick look and my
> >>>> >> experience there are not many functions with axis arguments in
> signal.
> >>>> >> So I'm fine with switching to axis=-1. We should do it with this
> >>>> >> bugfix, since until now the function wasn't correct anyway for 2d.
> >>>> >>
> >>>> >> >
> >>>> >> > As for whether it's doing what it's supposed to do, for what it's
> >>>> >> > worth
> >>>> >> > - it
> >>>> >> > seems to do similar things to what Matlab's 'hilbert' function
> does
> >>>> >> > on a
> >>>> >> > few
> >>>> >> > simple examples I tried out.
> >>>> >>
> >>>> >> I was reading briefly on wikipedia, and checked with
> fftpack.hilbert,
> >>>> >> which returns the same array as signal.hilbert(a).imag, but I
> didn't
> >>>> >> manage to figure out why fftpack.hilbert only allows 1d (i got lost
> >>>> >> starting at convolve.pyf)
> >>>> >>
> >>>> >> Could you write a simple test case compared to matlab, e.g. 10by3
> as
> >>>> >> in my example, for both axis, or 10by6 if 10by3 doesn't make sense?
> >>>> >>
> >>>> >> If nobody objects, I can commit the change with axis=-1.
> >>>> >>
> >>>> >> Josef
> >>>> >>
> >>>> >> >
> >>>> >> > Cheers,
> >>>> >> >
> >>>> >> > Ariel
> >>>> >> >
> >>>> >> >
> >>>> >> >
> >>>> >> > On Thu, Jan 14, 2010 at 8:53 PM, <josef.pktd at gmail.com> wrote:
> >>>> >> >>
> >>>> >> >> On Thu, Jan 14, 2010 at 11:27 PM, <josef.pktd at gmail.com>
> wrote:
> >>>> >> >> > On Thu, Jan 14, 2010 at 11:02 PM, <josef.pktd at gmail.com>
> wrote:
> >>>> >> >> >> On Thu, Jan 14, 2010 at 10:54 PM, <josef.pktd at gmail.com>
> wrote:
> >>>> >> >> >>> On Thu, Jan 14, 2010 at 10:24 PM, Ariel Rokem
> >>>> >> >> >>> <arokem at berkeley.edu>
> >>>> >> >> >>> wrote:
> >>>> >> >> >>>> Hi everyone,
> >>>> >> >> >>>>
> >>>> >> >> >>>> I have been trying to use scipy.signal.hilbert and I got
> the
> >>>> >> >> >>>> following
> >>>> >> >> >>>> puzzling result:
> >>>> >> >> >>>>
> >>>> >> >> >>>> In [22]: import scipy
> >>>> >> >> >>>>
> >>>> >> >> >>>> In [23]: scipy.__version__ #I have r6182
> >>>> >> >> >>>> Out[23]: '0.8.0.dev'
> >>>> >> >> >>>>
> >>>> >> >> >>>> In [24]: import scipy.signal as signal
> >>>> >> >> >>>>
> >>>> >> >> >>>> In [25]: a = np.random.rand(100,100)
> >>>> >> >> >>>>
> >>>> >> >> >>>> In [26]: np.abs(signal.hilbert(a[-1]))
> >>>> >> >> >>>> Out[26]:
> >>>> >> >> >>>> array([ 0.57567681, 0.25918624, 0.50207097, 0.51834052,
> >>>> >> >> >>>> 0.24293389,
> >>>> >> >> >>>> 0.5779464 , 0.6515758 , 0.89973173, 1.00275444,
> >>>> >> >> >>>> 0.37352935,
> >>>> >> >> >>>> 0.62332717, 0.93599749, 0.40651376, 0.65088756,
> >>>> >> >> >>>> 0.8332281
> >>>> >> >> >>>> ,
> >>>> >> >> >>>> 0.5770101 , 0.9288512 , 0.46671906, 0.41536055,
> >>>> >> >> >>>> 0.71418068,
> >>>> >> >> >>>> 0.81250913, 0.07652627, 0.72939072, 0.26755626,
> >>>> >> >> >>>> 0.36396146,
> >>>> >> >> >>>> 0.59725999, 1.02264694, 0.41227986, 0.98122853,
> >>>> >> >> >>>> 0.71906675,
> >>>> >> >> >>>> 0.58582611, 0.77288117, 0.3217015 , 0.65261394,
> >>>> >> >> >>>> 0.11947618,
> >>>> >> >> >>>> 0.75632703, 0.43432935, 0.52182485, 1.0277177 ,
> >>>> >> >> >>>> 1.01104986,
> >>>> >> >> >>>> 0.3023265 , 0.6024772 , 0.69257548, 0.55418735,
> >>>> >> >> >>>> 0.46259052,
> >>>> >> >> >>>> 0.25832231, 0.38278355, 0.45508532, 0.26215872,
> >>>> >> >> >>>> 0.34207947,
> >>>> >> >> >>>> 0.80704729, 0.80755477, 0.95317178, 0.97458885,
> >>>> >> >> >>>> 0.58762294,
> >>>> >> >> >>>> 0.82540618, 0.62005585, 0.82494646, 1.04221293,
> >>>> >> >> >>>> 0.14983027,
> >>>> >> >> >>>> 1.01571579, 0.99381328, 0.24158714, 0.84256569,
> >>>> >> >> >>>> 0.53418924,
> >>>> >> >> >>>> 0.24067628, 0.90489883, 1.02217747, 0.34988034,
> >>>> >> >> >>>> 0.5310065
> >>>> >> >> >>>> ,
> >>>> >> >> >>>> 0.48135002, 1.03020269, 0.6013679 , 0.46062485,
> >>>> >> >> >>>> 0.3918485
> >>>> >> >> >>>> ,
> >>>> >> >> >>>> 0.21554545, 0.31704519, 0.04868385, 0.1787766 ,
> >>>> >> >> >>>> 0.37361852,
> >>>> >> >> >>>> 0.21977912, 0.7649772 , 0.77867281, 0.37684278,
> >>>> >> >> >>>> 0.64432638,
> >>>> >> >> >>>> 0.77494951, 0.87106309, 0.77611484, 0.52666801,
> >>>> >> >> >>>> 0.88683667,
> >>>> >> >> >>>> 0.69164967, 0.98618191, 0.84811375, 0.35934198,
> >>>> >> >> >>>> 0.32650478,
> >>>> >> >> >>>> 0.1752677 , 0.60574454, 0.5109132 , 0.52332287,
> >>>> >> >> >>>> 0.99777805])
> >>>> >> >> >>>>
> >>>> >> >> >>>> In [27]: np.abs(signal.hilbert(a))[-1]
> >>>> >> >> >>>> Out[27]:
> >>>> >> >> >>>> array([ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0.,
> >>>> >> >> >>>> 0., 0., 0., 0., 0., 0., 0., 0., 0.])
> >>>> >> >> >>>>
> >>>> >> >> >>>>
> >>>> >> >> >>>>
> >>>> >> >> >>>>
> >>>> >> >> >>>>
> >>>> >> >> >>>>
> ----------------------------------------------------------------------
> >>>> >> >> >>>>
> >>>> >> >> >>>> I was expecting both of these to have the same values - am
> I
> >>>> >> >> >>>> missing
> >>>> >> >> >>>> something?
> >>>> >> >> >>>>
> >>>> >> >> >>>> I think that the following solves this issue, but now I am
> not
> >>>> >> >> >>>> that
> >>>> >> >> >>>> sure
> >>>> >> >> >>>> whether it does what it is supposed to do and I couldn't
> find a
> >>>> >> >> >>>> test
> >>>> >> >> >>>> for
> >>>> >> >> >>>> this in test_signaltools.py. Does anyone know of a good
> >>>> >> >> >>>> test-case
> >>>> >> >> >>>> for
> >>>> >> >> >>>> the
> >>>> >> >> >>>> analytic signal, that I could create for this?
> >>>> >> >> >>>>
> >>>> >> >> >>>> Index: scipy/signal/signaltools.py
> >>>> >> >> >>>>
> >>>> >> >> >>>>
> >>>> >> >> >>>>
> ===================================================================
> >>>> >> >> >>>> --- scipy/signal/signaltools.py (revision 6182)
> >>>> >> >> >>>> +++ scipy/signal/signaltools.py (working copy)
> >>>> >> >> >>>> @@ -1062,13 +1062,13 @@
> >>>> >> >> >>>> """
> >>>> >> >> >>>> x = asarray(x)
> >>>> >> >> >>>> if N is None:
> >>>> >> >> >>>> - N = len(x)
> >>>> >> >> >>>> + N = x.shape[-1]
> >>>> >> >> >>>> if N <=0:
> >>>> >> >> >>>> raise ValueError, "N must be positive."
> >>>> >> >> >>>> if iscomplexobj(x):
> >>>> >> >> >>>> print "Warning: imaginary part of x ignored."
> >>>> >> >> >>>> x = real(x)
> >>>> >> >> >>>> - Xf = fft(x,N,axis=0)
> >>>> >> >> >>>> + Xf = fft(x,N,axis=-1)
> >>>> >> >> >>>> h = zeros(N)
> >>>> >> >> >>>> if N % 2 == 0:
> >>>> >> >> >>>> h[0] = h[N/2] = 1
> >>>> >> >> >>>> @@ -1078,7 +1078,7 @@
> >>>> >> >> >>>> h[1:(N+1)/2] = 2
> >>>> >> >> >>>>
> >>>> >> >> >>>> if len(x.shape) > 1:
> >>>> >> >> >>>> - h = h[:, newaxis]
> >>>> >> >> >>>> + h = h[newaxis,:]
> >>>> >> >> >>>> x = ifft(Xf*h)
> >>>> >> >> >>>> return x
> >>>> >> >> >>>
> >>>> >> >> >>> I think your change would break the currently advertised
> >>>> >> >> >>> behavior,
> >>>> >> >> >>> axis=0 (The transformation is done along the first axis)
> >>>> >> >> >>>
> >>>> >> >> >>> but fft and ifft have default axis=-1
> >>>> >> >> >>>
> >>>> >> >> >>> fft in hilbert uses axis=0 as in docstring
> >>>> >> >> >>> but ifft uses default axis=-1
> >>>> >> >> >>>
> >>>> >> >> >>> so, I would think the fix should be x = ifft(Xf*h, axis=0)
> >>>> >> >> >>>
> >>>> >> >> >>> But as it currently looks like the axis argument doesn't
> work
> >>>> >> >> >>> anyway,
> >>>> >> >> >>> there wouldn't be much breakage if the axis would be
> included as
> >>>> >> >> >>> an
> >>>> >> >> >>> argument and default to -1.
> >>>> >> >> >>> However, I don't know what the "standard" for scipy.signal
> is
> >>>> >> >> >>> for
> >>>> >> >> >>> default axis.
> >>>> >> >> >>>
> >>>> >> >> >>> Josef
> >>>> >> >> >>
> >>>> >> >> >> after adding axis to ifft:
> >>>> >> >> >>>>> print hilbert(aa).real
> >>>> >> >> >> [[ 0.82584851 0.15215031 0.14767381]
> >>>> >> >> >> [ 0.95021675 0.16803995 0.43562964]
> >>>> >> >> >> [ 0.13033881 0.06198952 0.70729614]
> >>>> >> >> >> [ 0.69409563 0.06962778 0.72552601]
> >>>> >> >> >> [ 0.34297612 0.50579001 0.86463304]
> >>>> >> >> >> [ 0.28355261 0.21626889 0.85165102]
> >>>> >> >> >> [ 0.49481491 0.21290645 0.71416814]
> >>>> >> >> >> [ 0.2645843 0.95783096 0.77514016]
> >>>> >> >> >> [ 0.38735994 0.14274852 0.56344808]
> >>>> >> >> >> [ 0.88084015 0.39879649 0.64949951]]
> >>>> >> >> >>>>> print hilbert(aa[:,:1]).real
> >>>> >> >> >> [[ 0.82584851]
> >>>> >> >> >> [ 0.95021675]
> >>>> >> >> >> [ 0.13033881]
> >>>> >> >> >> [ 0.69409563]
> >>>> >> >> >> [ 0.34297612]
> >>>> >> >> >> [ 0.28355261]
> >>>> >> >> >> [ 0.49481491]
> >>>> >> >> >> [ 0.2645843 ]
> >>>> >> >> >> [ 0.38735994]
> >>>> >> >> >> [ 0.88084015]]
> >>>> >> >> >>
> >>>> >> >> >> but it treats a 1d array as row vector and transforms along
> zero
> >>>> >> >> >> axis
> >>>> >> >> >> of length 1, and not along the length of the array.
> >>>> >> >> >> so another fix to handle 1d arrays correctly should be done
> >>>> >> >> >>
> >>>> >> >> >>>>> print hilbert(aa[:,1]).real
> >>>> >> >> >> [ 0.15215031 0.16803995 0.06198952 0.06962778 0.50579001
> >>>> >> >> >> 0.21626889
> >>>> >> >> >> 0.21290645 0.95783096 0.14274852 0.39879649]
> >>>> >> >> >>>>> aa[:,1]
> >>>> >> >> >> array([ 0.15215031, 0.16803995, 0.06198952, 0.06962778,
> >>>> >> >> >> 0.50579001,
> >>>> >> >> >> 0.21626889, 0.21290645, 0.95783096, 0.14274852,
> >>>> >> >> >> 0.39879649])
> >>>> >> >> >>>>>
> >>>> >> >> >
> >>>> >> >> > there's something wrong with my example, the real part is the
> same
> >>>> >> >> > which confused me
> >>>> >> >> >
> >>>> >> >> > it works correctly with 1d
> >>>> >> >> >
> >>>> >> >> >>>> np.abs(hilbert(aa[:,0]))
> >>>> >> >> > array([ 0.83251128, 1.04487091, 0.27702083, 0.69901499,
> >>>> >> >> > 0.49170197,
> >>>> >> >> > 0.31227114, 0.49505637, 0.26461488, 0.61385196,
> >>>> >> >> > 0.90716272])
> >>>> >> >> >
> >>>> >> >> >>>> np.abs(hilbert(aa[:,:1])).T
> >>>> >> >> > array([[ 0.83251128, 1.04487091, 0.27702083, 0.69901499,
> >>>> >> >> > 0.49170197,
> >>>> >> >> > 0.31227114, 0.49505637, 0.26461488, 0.61385196,
> >>>> >> >> > 0.90716272]])
> >>>> >> >> >
> >>>> >> >> >>>> np.abs(hilbert(aa))[:,0]
> >>>> >> >> > array([ 0.83251128, 1.04487091, 0.27702083, 0.69901499,
> >>>> >> >> > 0.49170197,
> >>>> >> >> > 0.31227114, 0.49505637, 0.26461488, 0.61385196,
> >>>> >> >> > 0.90716272])
> >>>> >> >> >
> >>>> >> >> > besides reading the docstring, I don't know what hilbert is
> >>>> >> >> > supposed
> >>>> >> >> > to be good for.
> >>>> >> >>
> >>>> >> >> Would something like the function in the attachment do ?
> >>>> >> >>
> >>>> >> >>
> >>>> >> >>
> >>>> >> >> > Josef
> >>>> >> >> >
> >>>> >> >> >
> >>>> >> >> >> Josef
> >>>> >> >> >>
> >>>> >> >> >>
> >>>> >> >> >>>
> >>>> >> >> >>>>
> >>>> >> >> >>>>
> >>>> >> >> >>>> Cheers,
> >>>> >> >> >>>>
> >>>> >> >> >>>> Ariel
> >>>> >> >> >>>> --
> >>>> >> >> >>>> Ariel Rokem
> >>>> >> >> >>>> Helen Wills Neuroscience Institute
> >>>> >> >> >>>> University of California, Berkeley
> >>>> >> >> >>>> http://argentum.ucbso.berkeley.edu/ariel
> >>>> >> >> >>>>
> >>>> >> >> >>>> _______________________________________________
> >>>> >> >> >>>> SciPy-Dev mailing list
> >>>> >> >> >>>> SciPy-Dev at scipy.org
> >>>> >> >> >>>> http://mail.scipy.org/mailman/listinfo/scipy-dev
> >>>> >> >> >>>>
> >>>> >> >> >>>>
> >>>> >> >> >>>
> >>>> >> >> >>
> >>>> >> >> >
> >>>> >> >>
> >>>> >> >> _______________________________________________
> >>>> >> >> SciPy-Dev mailing list
> >>>> >> >> SciPy-Dev at scipy.org
> >>>> >> >> http://mail.scipy.org/mailman/listinfo/scipy-dev
> >>>> >> >>
> >>>> >> >
> >>>> >> >
> >>>> >> >
> >>>> >> > --
> >>>> >> > Ariel Rokem
> >>>> >> > Helen Wills Neuroscience Institute
> >>>> >> > University of California, Berkeley
> >>>> >> > http://argentum.ucbso.berkeley.edu/ariel
> >>>> >> >
> >>>> >> > _______________________________________________
> >>>> >> > SciPy-Dev mailing list
> >>>> >> > SciPy-Dev at scipy.org
> >>>> >> > http://mail.scipy.org/mailman/listinfo/scipy-dev
> >>>> >> >
> >>>> >> >
> >>>> >> _______________________________________________
> >>>> >> SciPy-Dev mailing list
> >>>> >> SciPy-Dev at scipy.org
> >>>> >> http://mail.scipy.org/mailman/listinfo/scipy-dev
> >>>> >
> >>>> >
> >>>> >
> >>>> > --
> >>>> > Ariel Rokem
> >>>> > Helen Wills Neuroscience Institute
> >>>> > University of California, Berkeley
> >>>> > http://argentum.ucbso.berkeley.edu/ariel
> >>>> >
> >>>> > _______________________________________________
> >>>> > SciPy-Dev mailing list
> >>>> > SciPy-Dev at scipy.org
> >>>> > http://mail.scipy.org/mailman/listinfo/scipy-dev
> >>>> >
> >>>> >
> >>>> _______________________________________________
> >>>> SciPy-Dev mailing list
> >>>> SciPy-Dev at scipy.org
> >>>> http://mail.scipy.org/mailman/listinfo/scipy-dev
> >>>
> >>>
> >>>
> >>> --
> >>> Ariel Rokem
> >>> Helen Wills Neuroscience Institute
> >>> University of California, Berkeley
> >>> http://argentum.ucbso.berkeley.edu/ariel
> >>>
> >>> _______________________________________________
> >>> SciPy-Dev mailing list
> >>> SciPy-Dev at scipy.org
> >>> http://mail.scipy.org/mailman/listinfo/scipy-dev
> >>>
> >>>
> >>
> >
> _______________________________________________
> SciPy-Dev mailing list
> SciPy-Dev at scipy.org
> http://mail.scipy.org/mailman/listinfo/scipy-dev
>
--
Ariel Rokem
Helen Wills Neuroscience Institute
University of California, Berkeley
http://argentum.ucbso.berkeley.edu/ariel
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