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00001 """

Numerical python functions written for compatability with matlab(TM)
commands with the same names.

  Matlab(TM) compatible functions:

    * cohere - Coherence (normalized cross spectral density)

    * csd - Cross spectral density uing Welch's average periodogram

    * detrend -- Remove the mean or best fit line from an array

    * find - Return the indices where some condition is true;
             numpy.nonzero is similar but more general.

    * prctile - find the percentiles of a sequence

    * prepca - Principal Component Analysis

    * psd - Power spectral density uing Welch's average periodogram

    * rk4 - A 4th order runge kutta integrator for 1D or ND systems

  The following are deprecated; please import directly from numpy
  (with care--function signatures may differ):

    * conv     - convolution  (numpy.convolve)
    * corrcoef - The matrix of correlation coefficients
    * hist -- Histogram (numpy.histogram)
    * linspace -- Linear spaced array from min to max
    * meshgrid
    * polyfit - least squares best polynomial fit of x to y
    * polyval - evaluate a vector for a vector of polynomial coeffs
    * trapz - trapeziodal integration (trapz(x,y) -> numpy.trapz(y,x))
    * vander - the Vandermonde matrix

  Functions that don't exist in matlab(TM), but are useful anyway:

    * cohere_pairs - Coherence over all pairs.  This is not a matlab
      function, but we compute coherence a lot in my lab, and we
      compute it for a lot of pairs.  This function is optimized to do
      this efficiently by caching the direct FFTs.

= record array helper functions =

   * rec2csv          : store record array in CSV file
   * rec2excel        : store record array in excel worksheet - required pyExcelerator

   * csv2rec          : import record array from CSV file with type inspection
   * rec_append_field : add a field/array to record array
   * rec_drop_fields  : drop fields from record array
   * rec_join         : join two record arrays on sequence of fields

For the rec viewer clases (rec2csv, rec2excel), there are
a bunch of Format objects you can pass into the functions that will do
things like color negative values red, set percent formatting and
scaling, etc.

Example usage:

    r = csv2rec('somefile.csv', checkrows=0)

    formatd = dict(
        weight = FormatFloat(2),
        change = FormatPercent(2),
        cost   = FormatThousands(2),

    rec2excel(r, 'test.xls', formatd=formatd)
    rec2csv(r, 'test.csv', formatd=formatd)
    scroll = rec2gtk(r, formatd=formatd)

    win = gtk.Window()


from __future__ import division
import sys, datetime, csv, warnings, copy

import numpy as npy

from matplotlib import nxutils
from matplotlib import cbook

# set is a new builtin function in 2.4; delete the following when
# support for 2.3 is dropped.
try: set
except NameError:
    from sets import Set as set

def linspace(*args, **kw):
    warnings.warn("use numpy.linspace", DeprecationWarning)
    return npy.linspace(*args, **kw)

def meshgrid(x,y):
    warnings.warn("use numpy.meshgrid", DeprecationWarning)
    return npy.meshgrid(x,y)

def mean(x, dim=None):
    warnings.warn("Use numpy.mean(x) or x.mean()", DeprecationWarning)
    if len(x)==0: return None
    return npy.mean(x, axis=dim)

def logspace(xmin,xmax,N):
    return npy.exp(npy.linspace(npy.log(xmin), npy.log(xmax), N))

def _norm(x):
    "return sqrt(x dot x)"
    return npy.sqrt(npy.dot(x,x))

def window_hanning(x):
    "return x times the hanning window of len(x)"
    return npy.hanning(len(x))*x

def window_none(x):
    "No window function; simply return x"
    return x

#from numpy import convolve as conv
def conv(x, y, mode=2):
    'convolve x with y'
    warnings.warn("Use numpy.convolve(x, y, mode='full')", DeprecationWarning)
    return npy.convolve(x,y,mode)

def detrend(x, key=None):
    if key is None or key=='constant':
        return detrend_mean(x)
    elif key=='linear':
        return detrend_linear(x)

def demean(x, axis=0):
    "Return x minus its mean along the specified axis"
    x = npy.asarray(x)
    if axis:
        ind = [slice(None)] * axis
        return x - x.mean(axis)[ind]
    return x - x.mean(axis)

def detrend_mean(x):
    "Return x minus the mean(x)"
    return x - x.mean()

def detrend_none(x):
    "Return x: no detrending"
    return x

def detrend_linear(y):
    "Return y minus best fit line; 'linear' detrending "
    # This is faster than an algorithm based on linalg.lstsq.
    x = npy.arange(len(y), dtype=npy.float_)
    C = npy.cov(x, y, bias=1)
    b = C[0,1]/C[0,0]
    a = y.mean() - b*x.mean()
    return y - (b*x + a)

00170 def psd(x, NFFT=256, Fs=2, detrend=detrend_none,
        window=window_hanning, noverlap=0):
    The power spectral density by Welches average periodogram method.
    The vector x is divided into NFFT length segments.  Each segment
    is detrended by function detrend and windowed by function window.
    noperlap gives the length of the overlap between segments.  The
    absolute(fft(segment))**2 of each segment are averaged to compute Pxx,
    with a scaling to correct for power loss due to windowing.

    Fs is the sampling frequency (samples per time unit).  It is used
    to calculate the Fourier frequencies, freqs, in cycles per time

    -- NFFT must be even; a power 2 is most efficient.
    -- detrend is a functions, unlike in matlab where it is a vector.
    -- window can be a function or a vector of length NFFT. To create window
       vectors see numpy.blackman, numpy.hamming, numpy.bartlett,
       scipy.signal, scipy.signal.get_window etc.
    -- if length x < NFFT, it will be zero padded to NFFT

    Returns the tuple Pxx, freqs

      Bendat & Piersol -- Random Data: Analysis and Measurement
        Procedures, John Wiley & Sons (1986)

    # I think we could remove this condition without hurting anything.
    if NFFT % 2:
        raise ValueError('NFFT must be even')

    x = npy.asarray(x) # make sure we're dealing with a numpy array

    # zero pad x up to NFFT if it is shorter than NFFT
    if len(x)<NFFT:
        n = len(x)
        x = npy.resize(x, (NFFT,))    # Can't use resize method.
        x[n:] = 0

    # for real x, ignore the negative frequencies
    if npy.iscomplexobj(x): numFreqs = NFFT
    else: numFreqs = NFFT//2+1

    if cbook.iterable(window):
        assert(len(window) == NFFT)
        windowVals = window
        windowVals = window(npy.ones((NFFT,),x.dtype))
    step = NFFT-noverlap
    ind = range(0,len(x)-NFFT+1,step)
    n = len(ind)
    Pxx = npy.zeros((numFreqs,n), npy.float_)
    # do the ffts of the slices
    for i in range(n):
        thisX = x[ind[i]:ind[i]+NFFT]
        thisX = windowVals * detrend(thisX)
        fx = npy.absolute(npy.fft.fft(thisX))**2
        Pxx[:,i] = fx[:numFreqs]

    if n>1:
        Pxx = Pxx.mean(axis=1)
    # Scale the spectrum by the norm of the window to compensate for
    # windowing loss; see Bendat & Piersol Sec 11.5.2
    Pxx /= (npy.abs(windowVals)**2).sum()

    freqs = Fs/NFFT * npy.arange(numFreqs)

    return Pxx, freqs

00241 def csd(x, y, NFFT=256, Fs=2, detrend=detrend_none,
        window=window_hanning, noverlap=0):
    The cross spectral density Pxy by Welches average periodogram
    method.  The vectors x and y are divided into NFFT length
    segments.  Each segment is detrended by function detrend and
    windowed by function window.  noverlap gives the length of the
    overlap between segments.  The product of the direct FFTs of x and
    y are averaged over each segment to compute Pxy, with a scaling to
    correct for power loss due to windowing.  Fs is the sampling

    NFFT must be even; a power of 2 is most efficient

    window can be a function or a vector of length NFFT. To create
    window vectors see numpy.blackman, numpy.hamming, numpy.bartlett,
    scipy.signal, scipy.signal.get_window etc.

    Returns the tuple Pxy, freqs

      Bendat & Piersol -- Random Data: Analysis and Measurement
        Procedures, John Wiley & Sons (1986)


    if NFFT % 2:
        raise ValueError, 'NFFT must be even'

    x = npy.asarray(x) # make sure we're dealing with a numpy array
    y = npy.asarray(y) # make sure we're dealing with a numpy array

    # zero pad x and y up to NFFT if they are shorter than NFFT
    if len(x)<NFFT:
        n = len(x)
        x = npy.resize(x, (NFFT,))
        x[n:] = 0
    if len(y)<NFFT:
        n = len(y)
        y = npy.resize(y, (NFFT,))
        y[n:] = 0

    # for real x, ignore the negative frequencies
    if npy.iscomplexobj(x): numFreqs = NFFT
    else: numFreqs = NFFT//2+1

    if cbook.iterable(window):
        assert(len(window) == NFFT)
        windowVals = window
        windowVals = window(npy.ones((NFFT,), x.dtype))
    step = NFFT-noverlap
    ind = range(0,len(x)-NFFT+1,step)
    n = len(ind)
    Pxy = npy.zeros((numFreqs,n), npy.complex_)

    # do the ffts of the slices
    for i in range(n):
        thisX = x[ind[i]:ind[i]+NFFT]
        thisX = windowVals*detrend(thisX)
        thisY = y[ind[i]:ind[i]+NFFT]
        thisY = windowVals*detrend(thisY)
        fx = npy.fft.fft(thisX)
        fy = npy.fft.fft(thisY)
        Pxy[:,i] = npy.conjugate(fx[:numFreqs])*fy[:numFreqs]

    # Scale the spectrum by the norm of the window to compensate for
    # windowing loss; see Bendat & Piersol Sec 11.5.2
    if n>1:
        Pxy = Pxy.mean(axis=1)
    Pxy /= (npy.abs(windowVals)**2).sum()
    freqs = Fs/NFFT*npy.arange(numFreqs)
    return Pxy, freqs

00317 def specgram(x, NFFT=256, Fs=2, detrend=detrend_none,
             window=window_hanning, noverlap=128):
    Compute a spectrogram of data in x.  Data are split into NFFT
    length segements and the PSD of each section is computed.  The
    windowing function window is applied to each segment, and the
    amount of overlap of each segment is specified with noverlap.

    window can be a function or a vector of length NFFT. To create
    window vectors see numpy.blackman, numpy.hamming, numpy.bartlett,
    scipy.signal, scipy.signal.get_window etc.

    See psd for more info. (psd differs in the default overlap;
    in returning the mean of the segment periodograms; and in not
    returning times.)

    If x is real (i.e. non-Complex) only the positive spectrum is
    given.  If x is Complex then the complete spectrum is given.

         Pxx -  2-D array, columns are the periodograms of
              successive segments
         freqs - 1-D array of frequencies corresponding to
              the rows in Pxx
         t - 1-D array of times corresponding to midpoints of

    x = npy.asarray(x)
    #if npy.log(NFFT)/npy.log(2) != int(npy.log(NFFT)/npy.log(2)):
    #   raise ValueError, 'NFFT must be a power of 2'
    if NFFT % 2:
        raise ValueError('NFFT must be even')

    # zero pad x up to NFFT if it is shorter than NFFT
    if len(x)<NFFT:
        n = len(x)
        x = npy.resize(x, (NFFT,))
        x[n:] = 0

    # for real x, ignore the negative frequencies
    if npy.iscomplexobj(x):
        numFreqs = NFFT//2+1

    if cbook.iterable(window):
        assert(len(window) == NFFT)
        windowVals = npy.asarray(window)
        windowVals = window(npy.ones((NFFT,),x.dtype))
    step = NFFT-noverlap
    ind = npy.arange(0,len(x)-NFFT+1,step)
    n = len(ind)
    Pxx = npy.zeros((numFreqs,n), npy.float_)
    # do the ffts of the slices

    for i in range(n):
        thisX = x[ind[i]:ind[i]+NFFT]
        thisX = windowVals*detrend(thisX)
        fx = npy.absolute(npy.fft.fft(thisX))**2
        Pxx[:,i] = fx[:numFreqs]
    # Scale the spectrum by the norm of the window to compensate for
    # windowing loss; see Bendat & Piersol Sec 11.5.2
    Pxx /= (npy.abs(windowVals)**2).sum()
    t = 1/Fs*(ind+NFFT/2)
    freqs = Fs/NFFT*npy.arange(numFreqs)

    if npy.iscomplexobj(x):
        # center the frequency range at zero
        freqs = npy.concatenate((freqs[NFFT/2:]-Fs,freqs[:NFFT/2]))
        Pxx   = npy.concatenate((Pxx[NFFT/2:,:],Pxx[:NFFT/2,:]),0)

    return Pxx, freqs, t

_coh_error = """Coherence is calculated by averaging over NFFT
length segments.  Your signal is too short for your choice of NFFT.
00400 def cohere(x, y, NFFT=256, Fs=2, detrend=detrend_none,
           window=window_hanning, noverlap=0):
    The coherence between x and y.  Coherence is the normalized
    cross spectral density

    Cxy = |Pxy|^2/(Pxx*Pyy)

    The return value is (Cxy, f), where f are the frequencies of the
    coherence vector.  See the docs for psd and csd for information
    about the function arguments NFFT, detrend, window, noverlap, as
    well as the methods used to compute Pxy, Pxx and Pyy.

    Returns the tuple Cxy, freqs


    if len(x)<2*NFFT:
        raise ValueError(_coh_error)
    Pxx, f = psd(x, NFFT, Fs, detrend, window, noverlap)
    Pyy, f = psd(y, NFFT, Fs, detrend, window, noverlap)
    Pxy, f = csd(x, y, NFFT, Fs, detrend, window, noverlap)

    Cxy = npy.divide(npy.absolute(Pxy)**2, Pxx*Pyy)
    Cxy.shape = (len(f),)
    return Cxy, f

00427 def corrcoef(*args):
    corrcoef(X) where X is a matrix returns a matrix of correlation
    coefficients for the columns of X.

    corrcoef(x,y) where x and y are vectors returns the matrix of
    correlation coefficients for x and y.

    Numpy arrays can be real or complex

    The correlation matrix is defined from the covariance matrix C as

    r(i,j) = C[i,j] / sqrt(C[i,i]*C[j,j])
    warnings.warn("Use numpy.corrcoef", DeprecationWarning)
    kw = dict(rowvar=False)
    return npy.corrcoef(*args, **kw)

00446 def polyfit(*args, **kwargs):
    def polyfit(x,y,N)

    Do a best fit polynomial of order N of y to x.  Return value is a
    vector of polynomial coefficients [pk ... p1 p0].  Eg, for N=2

      p2*x0^2 +  p1*x0 + p0 = y1
      p2*x1^2 +  p1*x1 + p0 = y1
      p2*x2^2 +  p1*x2 + p0 = y2
      p2*xk^2 +  p1*xk + p0 = yk

    Method: if X is a the Vandermonde Matrix computed from x (see
    http://mathworld.wolfram.com/VandermondeMatrix.html), then the
    polynomial least squares solution is given by the 'p' in

      X*p = y

    where X is a len(x) x N+1 matrix, p is a N+1 length vector, and y
    is a len(x) x 1 vector

    This equation can be solved as

      p = (XT*X)^-1 * XT * y

    where XT is the transpose of X and -1 denotes the inverse.
    Numerically, however, this is not a good method, so we use

    For more info, see
    but note that the k's and n's in the superscripts and subscripts
    on that page.  The linear algebra is correct, however.

    See also polyval

    warnings.warn("use numpy.poyfit", DeprecationWarning)
    return npy.polyfit(*args, **kwargs)

00491 def polyval(*args, **kwargs):
    y = polyval(p,x)

    p is a vector of polynomial coeffients and y is the polynomial
    evaluated at x.

    Example code to remove a polynomial (quadratic) trend from y:

      p = polyfit(x, y, 2)
      trend = polyval(p, x)
      resid = y - trend

    See also polyfit

    warnings.warn("use numpy.polyval", DeprecationWarning)
    return npy.polyval(*args, **kwargs)

00510 def vander(*args, **kwargs):
    X = vander(x,N=None)

    The Vandermonde matrix of vector x.  The i-th column of X is the
    the i-th power of x.  N is the maximum power to compute; if N is
    None it defaults to len(x).

    warnings.warn("Use numpy.vander()", DeprecationWarning)
    return npy.vander(*args, **kwargs)

def donothing_callback(*args):

00526 def cohere_pairs( X, ij, NFFT=256, Fs=2, detrend=detrend_none,
                  window=window_hanning, noverlap=0,

    Cxy, Phase, freqs = cohere_pairs( X, ij, ...)

    Compute the coherence for all pairs in ij.  X is a
    numSamples,numCols numpy array.  ij is a list of tuples (i,j).
    Each tuple is a pair of indexes into the columns of X for which
    you want to compute coherence.  For example, if X has 64 columns,
    and you want to compute all nonredundant pairs, define ij as

      ij = []
      for i in range(64):
          for j in range(i+1,64):
              ij.append( (i,j) )

    The other function arguments, except for 'preferSpeedOverMemory'
    (see below), are explained in the help string of 'psd'.

    Return value is a tuple (Cxy, Phase, freqs).

      Cxy -- a dictionary of (i,j) tuples -> coherence vector for that
        pair.  Ie, Cxy[(i,j) = cohere(X[:,i], X[:,j]).  Number of
        dictionary keys is len(ij)

      Phase -- a dictionary of phases of the cross spectral density at
        each frequency for each pair.  keys are (i,j).

      freqs -- a vector of frequencies, equal in length to either the
        coherence or phase vectors for any i,j key.  Eg, to make a coherence
        Bode plot:

          plot( freqs, Cxy[(12,19)])
          plot( freqs, Phase[(12,19)])

    For a large number of pairs, cohere_pairs can be much more
    efficient than just calling cohere for each pair, because it
    caches most of the intensive computations.  If N is the number of
    pairs, this function is O(N) for most of the heavy lifting,
    whereas calling cohere for each pair is O(N^2).  However, because
    of the caching, it is also more memory intensive, making 2
    additional complex arrays with approximately the same number of
    elements as X.

    The parameter 'preferSpeedOverMemory', if false, limits the
    caching by only making one, rather than two, complex cache arrays.
    This is useful if memory becomes critical.  Even when
    preferSpeedOverMemory is false, cohere_pairs will still give
    significant performace gains over calling cohere for each pair,
    and will use subtantially less memory than if
    preferSpeedOverMemory is true.  In my tests with a 43000,64 array
    over all nonredundant pairs, preferSpeedOverMemory=1 delivered a
    33% performace boost on a 1.7GHZ Athlon with 512MB RAM compared
    with preferSpeedOverMemory=0.  But both solutions were more than
    10x faster than naievly crunching all possible pairs through

    See test/cohere_pairs_test.py in the src tree for an example
    script that shows that this cohere_pairs and cohere give the same
    results for a given pair.

    numRows, numCols = X.shape

    # zero pad if X is too short
    if numRows < NFFT:
        tmp = X
        X = npy.zeros( (NFFT, numCols), X.dtype)
        X[:numRows,:] = tmp
        del tmp

    numRows, numCols = X.shape
    # get all the columns of X that we are interested in by checking
    # the ij tuples
    seen = {}
    for i,j in ij:
        seen[i]=1; seen[j] = 1
    allColumns = seen.keys()
    Ncols = len(allColumns)
    del seen

    # for real X, ignore the negative frequencies
    if npy.iscomplexobj(X): numFreqs = NFFT
    else: numFreqs = NFFT//2+1

    # cache the FFT of every windowed, detrended NFFT length segement
    # of every channel.  If preferSpeedOverMemory, cache the conjugate
    # as well
    if cbook.iterable(window):
        assert(len(window) == NFFT)
        windowVals = window
        windowVals = window(npy.ones((NFFT,), typecode(X)))
    ind = range(0, numRows-NFFT+1, NFFT-noverlap)
    numSlices = len(ind)
    FFTSlices = {}
    FFTConjSlices = {}
    Pxx = {}
    slices = range(numSlices)
    normVal = norm(windowVals)**2
    for iCol in allColumns:
        progressCallback(i/Ncols, 'Cacheing FFTs')
        Slices = npy.zeros( (numSlices,numFreqs), dtype=npy.complex_)
        for iSlice in slices:
            thisSlice = X[ind[iSlice]:ind[iSlice]+NFFT, iCol]
            thisSlice = windowVals*detrend(thisSlice)
            Slices[iSlice,:] = fft(thisSlice)[:numFreqs]

        FFTSlices[iCol] = Slices
        if preferSpeedOverMemory:
            FFTConjSlices[iCol] = conjugate(Slices)
        Pxx[iCol] = npy.divide(npy.mean(absolute(Slices)**2), normVal)
    del Slices, ind, windowVals

    # compute the coherences and phases for all pairs using the
    # cached FFTs
    Cxy = {}
    Phase = {}
    count = 0
    N = len(ij)
    for i,j in ij:
        count +=1
        if count%10==0:
            progressCallback(count/N, 'Computing coherences')

        if preferSpeedOverMemory:
            Pxy = FFTSlices[i] * FFTConjSlices[j]
            Pxy = FFTSlices[i] * npy.conjugate(FFTSlices[j])
        if numSlices>1: Pxy = npy.mean(Pxy)
        Pxy = npy.divide(Pxy, normVal)
        Cxy[(i,j)] = npy.divide(npy.absolute(Pxy)**2, Pxx[i]*Pxx[j])
        Phase[(i,j)] =  npy.arctan2(Pxy.imag, Pxy.real)

    freqs = Fs/NFFT*npy.arange(numFreqs)
    if returnPxx:
        return Cxy, Phase, freqs, Pxx
        return Cxy, Phase, freqs

00674 def entropy(y, bins):
    Return the entropy of the data in y

    \sum p_i log2(p_i) where p_i is the probability of observing y in
    the ith bin of bins.  bins can be a number of bins or a range of
    bins; see numpy.histogram

    Compare S with analytic calculation for a Gaussian
    x = mu + sigma*randn(200000)
    Sanalytic = 0.5  * ( 1.0 + log(2*pi*sigma**2.0) )

    n,bins = npy.histogram(y, bins)
    n = n.astype(npy.float_)

    n = npy.take(n, npy.nonzero(n)[0])         # get the positive

    p = npy.divide(n, len(y))

    delta = bins[1]-bins[0]
    S = -1.0*npy.sum(p*log(p)) + log(delta)
    #S = -1.0*npy.sum(p*log(p))
    return S

00699 def hist(y, bins=10, normed=0):
    Return the histogram of y with bins equally sized bins.  If bins
    is an array, use the bins.  Return value is
    (n,x) where n is the count for each bin in x

    If normed is False, return the counts in the first element of the
    return tuple.  If normed is True, return the probability density

    If y has rank>1, it will be raveled.  If y is masked, only
    the unmasked values will be used.
    Credits: the Numeric 22 documentation
    warnings.warn("Use numpy.histogram()", DeprecationWarning)
    return npy.histogram(y, bins=bins, range=None, normed=normed)

def normpdf(x, *args):
    "Return the normal pdf evaluated at x; args provides mu, sigma"
    mu, sigma = args
    return 1/(npy.sqrt(2*npy.pi)*sigma)*npy.exp(-0.5 * (1/sigma*(x - mu))**2)

def levypdf(x, gamma, alpha):
    "Returm the levy pdf evaluated at x for params gamma, alpha"

    N = len(x)

    if N%2 != 0:
        raise ValueError, 'x must be an event length array; try\n' + \
              'x = npy.linspace(minx, maxx, N), where N is even'

    dx = x[1]-x[0]

    f = 1/(N*dx)*npy.arange(-N/2, N/2, npy.float_)

    ind = npy.concatenate([npy.arange(N/2, N, int),
                           npy.arange(0, N/2, int)])
    df = f[1]-f[0]
    cfl = exp(-gamma*npy.absolute(2*pi*f)**alpha)

    px = npy.fft.fft(npy.take(cfl,ind)*df).astype(npy.float_)
    return npy.take(px, ind)

def find(condition):
    "Return the indices where ravel(condition) is true"
    res, = npy.nonzero(npy.ravel(condition))
    return res

00751 def trapz(x, y):
    Trapezoidal integral of y(x).
    warnings.warn("Use numpy.trapz(y,x) instead of trapz(x,y)", DeprecationWarning)
    return npy.trapz(y, x)
    #if len(x)!=len(y):
    #    raise ValueError, 'x and y must have the same length'
    #if len(x)<2:
    #    raise ValueError, 'x and y must have > 1 element'
    #return npy.sum(0.5*npy.diff(x)*(y[1:]+y[:-1]))

00765 def longest_contiguous_ones(x):
    return the indices of the longest stretch of contiguous ones in x,
    assuming x is a vector of zeros and ones.
    If there are two equally long stretches, pick the first

    x = npy.ravel(x)
    if len(x)==0:
        return npy.array([])

    ind = (x==0).nonzero()[0]
    if len(ind)==0:
        return npy.arange(len(x))
    if len(ind)==len(x):
        return npy.array([])

    y = npy.zeros( (len(x)+2,), x.dtype)
    y[1:-1] = x
    dif = npy.diff(y)
    up = (dif ==  1).nonzero()[0];
    dn = (dif == -1).nonzero()[0];
    i = (dn-up == max(dn - up)).nonzero()[0][0]
    ind = npy.arange(up[i], dn[i])

    return ind

00792 def longest_ones(x):
    '''alias for longest_contiguous_ones'''
    return longest_contiguous_ones(x)

00796 def prepca(P, frac=0):
    Compute the principal components of P.  P is a numVars x
    numObs array.  frac is the minimum fraction of
    variance that a component must contain to be included.

    Return value are
    Pcomponents : a numVars x numObs array
    Trans       : the weights matrix, ie, Pcomponents = Trans*P
    fracVar     : the fraction of the variance accounted for by each
                  component returned

    A similar function of the same name was in the Matlab (TM)
    R13 Neural Network Toolbox but is not found in later versions;
    its successor seems to be called "processpcs".
    U,s,v = npy.linalg.svd(P)
    varEach = s**2/P.shape[1]
    totVar = varEach.sum()
    fracVar = varEach/totVar
    ind = slice((fracVar>=frac).sum())
    # select the components that are greater
    Trans = U[:,ind].transpose()
    # The transformed data
    Pcomponents = npy.dot(Trans,P)
    return Pcomponents, Trans, fracVar[ind]

00823 def prctile(x, p = (0.0, 25.0, 50.0, 75.0, 100.0)):
    Return the percentiles of x.  p can either be a sequence of
    percentile values or a scalar.  If p is a sequence the i-th element
    of the return sequence is the p(i)-th percentile of x.
    If p is a scalar, the largest value of x less than or equal
    to the p percentage point in the sequence is returned.

    x = npy.array(x).ravel()  # we need a copy
    Nx = len(x)

    if not cbook.iterable(p):
        return x[int(p*Nx/100.0)]

    p = npy.asarray(p)* Nx/100.0
    ind = p.astype(int)
    ind = npy.where(ind>=Nx, Nx-1, ind)
    return x.take(ind)

00845 def prctile_rank(x, p):
    return the for each element in x, return the rank 0..len(p) .  Eg
    if p=(25, 50, 75), the return value will be a len(x) array with
    values in [0,1,2,3] where 0 indicates the value is less than the
    25th percentile, 1 indicates the value is >= the 25th and < 50th
    percentile, ... and 3 indicates the value is above the 75th
    percentile cutoff

    p is either an array of percentiles in [0..100] or a scalar which
    indicates how many quantiles of data you want ranked

    if not cbook.iterable(p):
        p = npy.arange(100.0/p, 100.0, 100.0/p)
        p = npy.asarray(p)

    if p.max()<=1 or p.min()<0 or p.max()>100:
        raise ValueError('percentiles should be in range 0..100, not 0..1')

    ptiles = prctile(x, p)
    return npy.searchsorted(ptiles, x)

00869 def center_matrix(M, dim=0):
    Return the matrix M with each row having zero mean and unit std

    if dim=1 operate on columns instead of rows.  (dim is opposite
    to the numpy axis kwarg.)
    M = npy.asarray(M, npy.float_)
    if dim:
        M = (M - M.mean(axis=0)) / M.std(axis=0)
        M = (M - M.mean(axis=1)[:,npy.newaxis])
        M = M / M.std(axis=1)[:,npy.newaxis]
    return M

00886 def rk4(derivs, y0, t):
    Integrate 1D or ND system of ODEs from initial state y0 at sample
    times t.  derivs returns the derivative of the system and has the

     dy = derivs(yi, ti)

    Example 1 :

        ## 2D system

        def derivs6(x,t):
            d1 =  x[0] + 2*x[1]
            d2 =  -3*x[0] + 4*x[1]
            return (d1, d2)
        dt = 0.0005
        t = arange(0.0, 2.0, dt)
        y0 = (1,2)
        yout = rk4(derivs6, y0, t)

    Example 2:

        ## 1D system
        alpha = 2
        def derivs(x,t):
            return -alpha*x + exp(-t)

        y0 = 1
        yout = rk4(derivs, y0, t)

    If you have access to scipy, you should probably be using the
    scipy.integrate tools rather than this function.


    try: Ny = len(y0)
    except TypeError:
        yout = npy.zeros( (len(t),), npy.float_)
        yout = npy.zeros( (len(t), Ny), npy.float_)

    yout[0] = y0
    i = 0

    for i in npy.arange(len(t)-1):

        thist = t[i]
        dt = t[i+1] - thist
        dt2 = dt/2.0
        y0 = yout[i]

        k1 = npy.asarray(derivs(y0, thist))
        k2 = npy.asarray(derivs(y0 + dt2*k1, thist+dt2))
        k3 = npy.asarray(derivs(y0 + dt2*k2, thist+dt2))
        k4 = npy.asarray(derivs(y0 + dt*k3, thist+dt))
        yout[i+1] = y0 + dt/6.0*(k1 + 2*k2 + 2*k3 + k4)
    return yout

00948 def bivariate_normal(X, Y, sigmax=1.0, sigmay=1.0,
                     mux=0.0, muy=0.0, sigmaxy=0.0):
    Bivariate gaussan distribution for equal shape X, Y

    Xmu = X-mux
    Ymu = Y-muy

    rho = sigmaxy/(sigmax*sigmay)
    z = Xmu**2/sigmax**2 + Ymu**2/sigmay**2 - 2*rho*Xmu*Ymu/(sigmax*sigmay)
    denom = 2*npy.pi*sigmax*sigmay*npy.sqrt(1-rho**2)
    return npy.exp( -z/(2*(1-rho**2))) / denom

00966 def get_xyz_where(Z, Cond):
    Z and Cond are MxN matrices.  Z are data and Cond is a boolean
    matrix where some condition is satisfied.  Return value is x,y,z
    where x and y are the indices into Z and z are the values of Z at
    those indices.  x,y,z are 1D arrays
    X,Y = npy.indices(Z.shape)
    return X[Cond], Y[Cond], Z[Cond]

def get_sparse_matrix(M,N,frac=0.1):
    'return a MxN sparse matrix with frac elements randomly filled'
    data = npy.zeros((M,N))*0.
    for i in range(int(M*N*frac)):
        x = npy.random.randint(0,M-1)
        y = npy.random.randint(0,N-1)
        data[x,y] = npy.random.rand()
    return data

def dist(x,y):
    'return the distance between two points'
    d = x-y
    return npy.sqrt(npy.dot(d,d))

00990 def dist_point_to_segment(p, s0, s1):
    get the distance of a point to a segment.

    p, s0, s1 are xy sequences

    This algorithm from
    p = npy.asarray(p, npy.float_)
    s0 = npy.asarray(s0, npy.float_)
    s1 = npy.asarray(s1, npy.float_)
    v = s1 - s0
    w = p - s0

    c1 = npy.dot(w,v);
    if ( c1 <= 0 ):
        return dist(p, s0);

    c2 = npy.dot(v,v)
    if ( c2 <= c1 ):
        return dist(p, s1);

    b = c1 / c2
    pb = s0 + b * v;
    return dist(p, pb)

01017 def segments_intersect(s1, s2):
    Return True if s1 and s2 intersect.
    s1 and s2 are defined as

    s1: (x1, y1), (x2, y2)
    s2: (x3, y3), (x4, y4)

    (x1, y1), (x2, y2) = s1
    (x3, y3), (x4, y4) = s2

    den = ((y4-y3) * (x2-x1)) - ((x4-x3)*(y2-y1))

    n1 = ((x4-x3) * (y1-y3)) - ((y4-y3)*(x1-x3))
    n2 = ((x2-x1) * (y1-y3)) - ((y2-y1)*(x1-x3))

    if den == 0:
        # lines parallel
        return False

    u1 = n1/den
    u2 = n2/den

    return 0.0 <= u1 <= 1.0 and 0.0 <= u2 <= 1.0

01044 def fftsurr(x, detrend=detrend_none, window=window_none):
    Compute an FFT phase randomized surrogate of x
    if cbook.iterable(window):
        x = window(detrend(x))
    z = npy.fft.fft(x)
    a = 2.*npy.pi*1j
    phase = a * npy.random.rand(len(x))
    z = z*npy.exp(phase)
    return npy.fft.ifft(z).real

01059 def liaupunov(x, fprime):
    x is a very long trajectory from a map, and fprime returns the
    derivative of x.  Return lambda = 1/n\sum ln|fprime(x_i)|.  See Sec
    10.5 Strogatz (1994)"Nonlinear Dynamics and Chaos".
    See also http://en.wikipedia.org/wiki/Lyapunov_exponent.
    What the function here calculates may not be what you really want;
    caveat emptor.
    It also seems that this function's name is badly misspelled.
    return npy.mean(npy.log(npy.absolute(fprime(x))))

01071 class FIFOBuffer:
    A FIFO queue to hold incoming x, y data in a rotating buffer using
    numpy arrays under the hood.  It is assumed that you will call
    asarrays much less frequently than you add data to the queue --
    otherwise another data structure will be faster

    This can be used to support plots where data is added from a real
    time feed and the plot object wants grab data from the buffer and
    plot it to screen less freqeuently than the incoming

    If you set the dataLim attr to a matplotlib BBox (eg ax.dataLim),
    the dataLim will be updated as new data come in

    TODI: add a grow method that will extend nmax

    mlab seems like the wrong place for this class.
    def __init__(self, nmax):
        'buffer up to nmax points'
        self._xa = npy.zeros((nmax,), npy.float_)
        self._ya = npy.zeros((nmax,), npy.float_)
        self._xs = npy.zeros((nmax,), npy.float_)
        self._ys = npy.zeros((nmax,), npy.float_)
        self._ind = 0
        self._nmax = nmax
        self.dataLim = None
        self.callbackd = {}

    def register(self, func, N):
        'call func everytime N events are passed; func signature is func(fifo)'
        self.callbackd.setdefault(N, []).append(func)

    def add(self, x, y):
        'add scalar x and y to the queue'
        if self.dataLim is not None:
            xys = ((x,y),)
            self.dataLim.update(xys, -1) #-1 means use the default ignore setting
        ind = self._ind % self._nmax
        #print 'adding to fifo:', ind, x, y
        self._xs[ind] = x
        self._ys[ind] = y

        for N,funcs in self.callbackd.items():
            if (self._ind%N)==0:
                for func in funcs:

        self._ind += 1

    def last(self):
        'get the last x, y or None, None if no data set'
        if self._ind==0: return None, None
        ind = (self._ind-1) % self._nmax
        return self._xs[ind], self._ys[ind]

01127     def asarrays(self):
        return x and y as arrays; their length will be the len of data
        added or nmax
        if self._ind<self._nmax:
            return self._xs[:self._ind], self._ys[:self._ind]
        ind = self._ind % self._nmax

        self._xa[:self._nmax-ind] = self._xs[ind:]
        self._xa[self._nmax-ind:] = self._xs[:ind]
        self._ya[:self._nmax-ind] = self._ys[ind:]
        self._ya[self._nmax-ind:] = self._ys[:ind]

        return self._xa, self._ya

    def update_datalim_to_current(self):
        'update the datalim in the current data in the fifo'
        if self.dataLim is None:
            raise ValueError('You must first set the dataLim attr')
        x, y = self.asarrays()
        self.dataLim.update_numerix(x, y, True)

def movavg(x,n):
    'compute the len(n) moving average of x'
    w = npy.empty((n,), dtype=npy.float_)
    w[:] = 1.0/n
    return npy.convolve(x, w, mode='valid')

01156 def save(fname, X, fmt='%.18e',delimiter=' '):
    Save the data in X to file fname using fmt string to convert the
    data to strings

    fname can be a filename or a file handle.  If the filename ends in .gz,
    the file is automatically saved in compressed gzip format.  The load()
    command understands gzipped files transparently.

    Example usage:

    save('test.out', X)         # X is an array
    save('test1.out', (x,y,z))  # x,y,z equal sized 1D arrays
    save('test2.out', x)        # x is 1D
    save('test3.out', x, fmt='%1.4e')  # use exponential notation

    delimiter is used to separate the fields, eg delimiter ',' for
    comma-separated values

    if cbook.is_string_like(fname):
        if fname.endswith('.gz'):
            import gzip
            fh = gzip.open(fname,'wb')
            fh = file(fname,'w')
    elif hasattr(fname, 'seek'):
        fh = fname
        raise ValueError('fname must be a string or file handle')

    X = npy.asarray(X)
    origShape = None
    if X.ndim == 1:
        origShape = X.shape
        X.shape = len(X), 1
    for row in X:
        fh.write(delimiter.join([fmt%val for val in row]) + '\n')

    if origShape is not None:
        X.shape = origShape

01202 def load(fname,comments='#',delimiter=None, converters=None,skiprows=0,
         usecols=None, unpack=False):
    Load ASCII data from fname into an array and return the array.

    The data must be regular, same number of values in every row

    fname can be a filename or a file handle.  Support for gzipped files is
    automatic, if the filename ends in .gz

    matfile data is not supported; use scipy.io.mio module

    Example usage:

      X = load('test.dat')  # data in two columns
      t = X[:,0]
      y = X[:,1]

    Alternatively, you can do the same with "unpack"; see below

      X = load('test.dat')    # a matrix of data
      x = load('test.dat')    # a single column of data

    comments - the character used to indicate the start of a comment
    in the file

    delimiter is a string-like character used to seperate values in the
    file. If delimiter is unspecified or none, any whitespace string is
    a separator.

    converters, if not None, is a dictionary mapping column number to
    a function that will convert that column to a float.  Eg, if
    column 0 is a date string: converters={0:datestr2num}

    skiprows is the number of rows from the top to skip

    usecols, if not None, is a sequence of integer column indexes to
    extract where 0 is the first column, eg usecols=(1,4,5) to extract
    just the 2nd, 5th and 6th columns

    unpack, if True, will transpose the matrix allowing you to unpack
    into named arguments on the left hand side

        t,y = load('test.dat', unpack=True) # for  two column data
        x,y,z = load('somefile.dat', usecols=(3,5,7), unpack=True)

    See examples/load_demo.py which exeercises many of these options.

    if converters is None: converters = {}
    fh = cbook.to_filehandle(fname)
    X = []

    if delimiter==' ':
        # space splitting is a special case since x.split() is what
        # you want, not x.split(' ')
        def splitfunc(x):
            return x.split()
        def splitfunc(x):
            return x.split(delimiter)

    converterseq = None
    for i,line in enumerate(fh):
        if i<skiprows: continue
        line = line.split(comments, 1)[0].strip()
        if not len(line): continue
        if converterseq is None:
            converterseq = [converters.get(j,float)
                               for j,val in enumerate(splitfunc(line))]
        if usecols is not None:
            vals = line.split(delimiter)
            row = [converterseq[j](vals[j]) for j in usecols]
            row = [converterseq[j](val)
                      for j,val in enumerate(splitfunc(line))]
        thisLen = len(row)

    X = npy.array(X, npy.float_)
    r,c = X.shape
    if r==1 or c==1:
        X.shape = max(r,c),
    if unpack: return X.transpose()
    else: return X

01289 def slopes(x,y):
    SLOPES calculate the slope y'(x) Given data vectors X and Y SLOPES
    calculates Y'(X), i.e the slope of a curve Y(X). The slope is
    estimated using the slope obtained from that of a parabola through
    any three consecutive points.

    This method should be superior to that described in the appendix
    W. Stineman (Creative Computing July 1980) in at least one aspect:

    Circles for interpolation demand a known aspect ratio between x-
    and y-values.  For many functions, however, the abscissa are given
    in different dimensions, so an aspect ratio is completely

    The parabola method gives very similar results to the circle
    method for most regular cases but behaves much better in special

    Norbert Nemec, Institute of Theoretical Physics, University or
    Regensburg, April 2006 Norbert.Nemec at physik.uni-regensburg.de

    (inspired by a original implementation by Halldor Bjornsson,
    Icelandic Meteorological Office, March 2006 halldor at vedur.is)
    # Cast key variables as float.
    x=npy.asarray(x, npy.float_)
    y=npy.asarray(y, npy.float_)

    yp=npy.zeros(y.shape, npy.float_)

    dx=x[1:] - x[:-1]
    dy=y[1:] - y[:-1]
    dydx = dy/dx
    yp[1:-1] = (dydx[:-1] * dx[1:] + dydx[1:] * dx[:-1])/(dx[1:] + dx[:-1])
    yp[0] = 2.0 * dy[0]/dx[0] - yp[1]
    yp[-1] = 2.0 * dy[-1]/dx[-1] - yp[-2]
    return yp

01330 def stineman_interp(xi,x,y,yp=None):
    STINEMAN_INTERP Well behaved data interpolation.  Given data
    vectors X and Y, the slope vector YP and a new abscissa vector XI
    the function stineman_interp(xi,x,y,yp) uses Stineman
    interpolation to calculate a vector YI corresponding to XI.

    Here's an example that generates a coarse sine curve, then
    interpolates over a finer abscissa:

      x = linspace(0,2*pi,20);  y = sin(x); yp = cos(x)
      xi = linspace(0,2*pi,40);
      yi = stineman_interp(xi,x,y,yp);

    The interpolation method is described in the article A
    W. Stineman. The article appeared in the July 1980 issue of
    Creative Computing with a note from the editor stating that while
    they were

      not an academic journal but once in a while something serious
      and original comes in adding that this was
      "apparently a real solution" to a well known problem.

    For yp=None, the routine automatically determines the slopes using
    the "slopes" routine.

    X is assumed to be sorted in increasing order

    For values xi[j] < x[0] or xi[j] > x[-1], the routine tries a
    extrapolation.  The relevance of the data obtained from this, of
    course, questionable...

    original implementation by Halldor Bjornsson, Icelandic
    Meteorolocial Office, March 2006 halldor at vedur.is

    completely reworked and optimized for Python by Norbert Nemec,
    Institute of Theoretical Physics, University or Regensburg, April
    2006 Norbert.Nemec at physik.uni-regensburg.de


    # Cast key variables as float.
    x=npy.asarray(x, npy.float_)
    y=npy.asarray(y, npy.float_)
    assert x.shape == y.shape

    if yp is None:
        yp = slopes(x,y)
        yp=npy.asarray(yp, npy.float_)

    xi=npy.asarray(xi, npy.float_)
    yi=npy.zeros(xi.shape, npy.float_)

    # calculate linear slopes
    dx = x[1:] - x[:-1]
    dy = y[1:] - y[:-1]
    s = dy/dx  #note length of s is N-1 so last element is #N-2

    # find the segment each xi is in
    # this line actually is the key to the efficiency of this implementation
    idx = npy.searchsorted(x[1:-1], xi)

    # now we have generally: x[idx[j]] <= xi[j] <= x[idx[j]+1]
    # except at the boundaries, where it may be that xi[j] < x[0] or xi[j] > x[-1]

    # the y-values that would come out from a linear interpolation:
    sidx = s.take(idx)
    xidx = x.take(idx)
    yidx = y.take(idx)
    xidxp1 = x.take(idx+1)
    yo = yidx + sidx * (xi - xidx)

    # the difference that comes when using the slopes given in yp
    dy1 = (yp.take(idx)- sidx) * (xi - xidx)       # using the yp slope of the left point
    dy2 = (yp.take(idx+1)-sidx) * (xi - xidxp1) # using the yp slope of the right point

    dy1dy2 = dy1*dy2
    # The following is optimized for Python. The solution actually
    # does more calculations than necessary but exploiting the power
    # of numpy, this is far more efficient than coding a loop by hand
    # in Python
    yi = yo + dy1dy2 * npy.choose(npy.array(npy.sign(dy1dy2), npy.int32)+1,
    return yi

01421 def inside_poly(points, verts):
    points is a sequence of x,y points
    verts is a sequence of x,y vertices of a poygon

    return value is a sequence of indices into points for the points
    that are inside the polygon
    res, =  npy.nonzero(nxutils.points_inside_poly(points, verts))
    return res

01432 def poly_below(xmin, xs, ys):
    given a sequence of xs and ys, return the vertices of a polygon
    that has a horzontal base at xmin and an upper bound at the ys.
    xmin is a scalar.

    intended for use with Axes.fill, eg
    xv, yv = poly_below(0, x, y)
    ax.fill(xv, yv)
    xs = npy.asarray(xs)
    ys = npy.asarray(ys)
    Nx = len(xs)
    Ny = len(ys)
    x = xmin*npy.ones(2*Nx)
    y = npy.ones(2*Nx)
    x[:Nx] = xs
    y[:Nx] = ys
    y[Nx:] = ys[::-1]
    return x, y

01455 def poly_between(x, ylower, yupper):
    given a sequence of x, ylower and yupper, return the polygon that
    fills the regions between them.  ylower or yupper can be scalar or
    iterable.  If they are iterable, they must be equal in length to x

    return value is x, y arrays for use with Axes.fill
    Nx = len(x)
    if not cbook.iterable(ylower):
        ylower = ylower*npy.ones(Nx)

    if not cbook.iterable(yupper):
        yupper = yupper*npy.ones(Nx)

    x = npy.concatenate( (x, x[::-1]) )
    y = npy.concatenate( (yupper, ylower[::-1]) )
    return x,y

### the following code was written and submitted by Fernando Perez
### from the ipython numutils package under a BSD license
# begin fperez functions

A set of convenient utilities for numerical work.

Most of this module requires numpy or is meant to be used with it.

Copyright (c) 2001-2004, Fernando Perez. <Fernando.Perez@colorado.edu>
All rights reserved.

This license was generated from the BSD license template as found in:

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

    * Redistributions of source code must retain the above copyright notice,
      this list of conditions and the following disclaimer.

    * Redistributions in binary form must reproduce the above copyright
      notice, this list of conditions and the following disclaimer in the
      documentation and/or other materials provided with the distribution.

    * Neither the name of the IPython project nor the names of its
      contributors may be used to endorse or promote products derived from
      this software without specific prior written permission.



import operator
import math

# Globals

# function definitions
exp_safe_MIN = math.log(2.2250738585072014e-308)
exp_safe_MAX = 1.7976931348623157e+308

01528 def exp_safe(x):
    """Compute exponentials which safely underflow to zero.

    Slow but convenient to use. Note that numpy provides proper
    floating point exception handling with access to the underlying

    if type(x) is npy.ndarray:
        return exp(npy.clip(x,exp_safe_MIN,exp_safe_MAX))
        return math.exp(x)

01540 def amap(fn,*args):
    """amap(function, sequence[, sequence, ...]) -> array.

    Works like map(), but it returns an array.  This is just a convenient
    shorthand for numpy.array(map(...))
    return npy.array(map(fn,*args))

#from numpy import zeros_like
01550 def zeros_like(a):
    """Return an array of zeros of the shape and typecode of a."""
    warnings.warn("Use numpy.zeros_like(a)", DeprecationWarning)
    return npy.zeros_like(a)

#from numpy import sum as sum_flat
01556 def sum_flat(a):
    """Return the sum of all the elements of a, flattened out.

    It uses a.flat, and if a is not contiguous, a call to ravel(a) is made."""
    warnings.warn("Use numpy.sum(a) or a.sum()", DeprecationWarning)
    return npy.sum(a)

#from numpy import mean as mean_flat
01564 def mean_flat(a):
    """Return the mean of all the elements of a, flattened out."""
    warnings.warn("Use numpy.mean(a) or a.mean()", DeprecationWarning)
    return npy.mean(a)

01569 def rms_flat(a):
    """Return the root mean square of all the elements of a, flattened out."""

    return npy.sqrt(npy.mean(npy.absolute(a)**2))

01574 def l1norm(a):
    """Return the l1 norm of a, flattened out.

    Implemented as a separate function (not a call to norm() for speed)."""

    return npy.sum(npy.absolute(a))

01581 def l2norm(a):
    """Return the l2 norm of a, flattened out.

    Implemented as a separate function (not a call to norm() for speed)."""

    return npy.sqrt(npy.sum(npy.absolute(a)**2))

01588 def norm_flat(a,p=2):
    """norm(a,p=2) -> l-p norm of a.flat

    Return the l-p norm of a, considered as a flat array.  This is NOT a true
    matrix norm, since arrays of arbitrary rank are always flattened.

    p can be a number or the string 'Infinity' to get the L-infinity norm."""
    # This function was being masked by a more general norm later in
    # the file.  We may want to simply delete it.
    if p=='Infinity':
        return npy.amax(npy.absolute(a))
        return (npy.sum(npy.absolute(a)**p))**(1.0/p)

01602 def frange(xini,xfin=None,delta=None,**kw):
    """frange([start,] stop[, step, keywords]) -> array of floats

    Return a numpy ndarray containing a progression of floats. Similar to
    arange(), but defaults to a closed interval.

    frange(x0, x1) returns [x0, x0+1, x0+2, ..., x1]; start defaults to 0, and
    the endpoint *is included*. This behavior is different from that of
    range() and arange(). This is deliberate, since frange will probably be
    more useful for generating lists of points for function evaluation, and
    endpoints are often desired in this use. The usual behavior of range() can
    be obtained by setting the keyword 'closed=0', in this case frange()
    basically becomes arange().

    When step is given, it specifies the increment (or decrement). All
    arguments can be floating point numbers.

    frange(x0,x1,d) returns [x0,x0+d,x0+2d,...,xfin] where xfin<=x1.

    frange can also be called with the keyword 'npts'. This sets the number of
    points the list should contain (and overrides the value 'step' might have
    been given). arange() doesn't offer this option.

    >>> frange(3)
    array([ 0.,  1.,  2.,  3.])
    >>> frange(3,closed=0)
    array([ 0.,  1.,  2.])
    >>> frange(1,6,2)
    array([1, 3, 5])   or 1,3,5,7, depending on floating point vagueries
    >>> frange(1,6.5,npts=5)
    array([ 1.   ,  2.375,  3.75 ,  5.125,  6.5  ])

    endpoint = kw['closed'] != 0

    # funny logic to allow the *first* argument to be optional (like range())
    # This was modified with a simpler version from a similar frange() found
    # at http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/66472
    if xfin == None:
        xfin = xini + 0.0
        xini = 0.0

    if delta == None:
        delta = 1.0

    # compute # of points, spacing and return final list
    except KeyError:
        npts = int(round((xfin-xini)/delta)) + endpoint
        #npts = int(floor((xfin-xini)/delta)*(1.0+1e-10)) + endpoint
        # round finds the nearest, so the endpoint can be up to
        # delta/2 larger than xfin.

    return npy.arange(npts)*delta+xini
# end frange()

#import numpy.diag as diagonal_matrix
01664 def diagonal_matrix(diag):
    """Return square diagonal matrix whose non-zero elements are given by the
    input array."""
    warnings.warn("Use numpy.diag(d)", DeprecationWarning)
    return npy.diag(diag)

01670 def identity(n, rank=2, dtype='l', typecode=None):
    """identity(n,r) returns the identity matrix of shape (n,n,...,n) (rank r).

    For ranks higher than 2, this object is simply a multi-index Kronecker
                        /  1  if i0=i1=...=iR,
    id[i0,i1,...,iR] = -|
                        \  0  otherwise.

    Optionally a dtype (or typecode) may be given (it defaults to 'l').

    Since rank defaults to 2, this function behaves in the default case (when
    only n is given) like numpy.identity(n)--but surprisingly, it is
    much faster.
    if typecode is not None:
        warnings.warn("Use dtype kwarg instead of typecode",
        dtype = typecode
    iden = npy.zeros((n,)*rank, dtype)
    for i in range(n):
        idx = (i,)*rank
        iden[idx] = 1
    return iden

01695 def base_repr (number, base = 2, padding = 0):
    """Return the representation of a number in any given base."""
    if number < base: \
       return (padding - 1) * chars [0] + chars [int (number)]
    max_exponent = int (math.log (number)/math.log (base))
    max_power = long (base) ** max_exponent
    lead_digit = int (number/max_power)
    return chars [lead_digit] + \
           base_repr (number - max_power * lead_digit, base, \
                      max (padding - 1, max_exponent))

01707 def binary_repr(number, max_length = 1025):
    """Return the binary representation of the input number as a string.

    This is more efficient than using base_repr with base 2.

    Increase the value of max_length for very large numbers. Note that on
    32-bit machines, 2**1023 is the largest integer power of 2 which can be
    converted to a Python float."""

    #assert number < 2L << max_length
    shifts = map (operator.rshift, max_length * [number], \
                  range (max_length - 1, -1, -1))
    digits = map (operator.mod, shifts, max_length * [2])
    if not digits.count (1): return 0
    digits = digits [digits.index (1):]
    return ''.join (map (repr, digits)).replace('L','')

01725 def log2(x,ln2 = math.log(2.0)):
    """Return the log(x) in base 2.

    This is a _slow_ function but which is guaranteed to return the correct
    integer value if the input is an integer exact power of 2."""

        bin_n = binary_repr(x)[1:]
    except (AssertionError,TypeError):
        return math.log(x)/ln2
        if '1' in bin_n:
            return math.log(x)/ln2
            return len(bin_n)

01741 def ispower2(n):
    """Returns the log base 2 of n if n is a power of 2, zero otherwise.

    Note the potential ambiguity if n==1: 2**0==1, interpret accordingly."""

    bin_n = binary_repr(n)[1:]
    if '1' in bin_n:
        return 0
        return len(bin_n)

#from numpy import fromfunction as fromfunction_kw
01753 def fromfunction_kw(function, dimensions, **kwargs):
    """Drop-in replacement for fromfunction() from numpy

    Allows passing keyword arguments to the desired function.

    Call it as (keywords are optional):
    fromfunction_kw(MyFunction, dimensions, keywords)

    The function MyFunction() is responsible for handling the dictionary of
    keywords it will receive."""
    warnings.warn("Use numpy.fromfunction()", DeprecationWarning)
    return npy.fromfunction(function, dimensions, **kwargs)

### end fperez numutils code

### begin mlab2 functions
# From MLab2: http://pdilib.sourceforge.net/MLab2.py
readme = \
MLab2.py, release 1

Created on February 2003 by Thomas Wendler as part of the Emotionis Project.
This script is supposed to implement Matlab functions that were left out in
numerix.mlab.py (part of Numeric Python).
For further information on the Emotionis Project or on this script, please
contact their authors:
Rodrigo Benenson, rodrigob at elo dot utfsm dot cl
Thomas Wendler,   thomasw at elo dot utfsm dot cl
Look at: http://pdilib.sf.net for new releases.
## mlab2 functions numpified and checked 2007/08/04
01784 _eps_approx = 1e-13

#from numpy import fix
01787 def fix(x):
    Rounds towards zero.
    x_rounded = fix(x) rounds the elements of x to the nearest integers
    towards zero.
    For negative numbers is equivalent to ceil and for positive to floor.
    warnings.warn("Use numpy.fix()", DeprecationWarning)
    return npy.fix(x)

01797 def rem(x,y):
    Remainder after division.
    rem(x,y) is equivalent to x - y.*fix(x./y) in case y is not zero.
    By convention (but contrary to numpy), rem(x,0) returns None.
    This also differs from numpy.remainder, which uses floor instead of
    x,y = npy.asarray(x), npy.asarray(y)
    if npy.any(y == 0):
        return None
    return x - y * npy.fix(x/y)

01811 def norm(x,y=2):
    Norm of a matrix or a vector according to Matlab.
    The description is taken from Matlab:

        For matrices...
          NORM(X) is the largest singular value of X, max(svd(X)).
          NORM(X,2) is the same as NORM(X).
          NORM(X,1) is the 1-norm of X, the largest column sum,
                          = max(sum(abs((X)))).
          NORM(X,inf) is the infinity norm of X, the largest row sum,
                          = max(sum(abs((X')))).
          NORM(X,'fro') is the Frobenius norm, sqrt(sum(diag(X'*X))).
          NORM(X,P) is available for matrix X only if P is 1, 2, inf or 'fro'.

        For vectors...
          NORM(V,P) = sum(abs(V).^P)^(1/P).
          NORM(V) = norm(V,2).
          NORM(V,inf) = max(abs(V)).
          NORM(V,-inf) = min(abs(V)).

    x = npy.asarray(x)
    if x.ndim == 2:
        if y==2:
            return npy.max(npy.linalg.svd(x)[1])
        elif y==1:
            return npy.max(npy.sum(npy.absolute((x)), axis=0))
        elif y=='inf':
            return npy.max(npy.sum(npy.absolute((npy.transpose(x))), axis=0))
        elif y=='fro':
            xx = npy.dot(x.transpose(), x)
            return npy.sqrt(npy.sum(npy.diag(xx), axis=0))
            raise ValueError('Second argument not permitted for matrices')

        xa = npy.absolute(x)
        if y == 'inf':
            return npy.max(xa)
        elif y == '-inf':
            return npy.min(xa)
            return npy.power(npy.sum(npy.power(xa,y)),1/float(y))

01857 def orth(A):
    Orthogonalization procedure by Matlab.
    The description is taken from its help:

        Q = ORTH(A) is an orthonormal basis for the range of A.
        That is, Q'*Q = I, the columns of Q span the same space as
        the columns of A, and the number of columns of Q is the
        rank of A.

    A     = npy.asarray(A)
    U,S,V = npy.linalg.svd(A)

    m,n = A.shape
    if m > 1:
        s = S
    elif m == 1:
        s = S[0]
        s = 0

    tol = max(m,n) * npy.max(s) * _eps_approx
    r = npy.sum(s > tol)
    Q = npy.take(U,range(r),1)

    return Q

01885 def rank(x):
    Returns the rank of a matrix.
    The rank is understood here as the an estimation of the number of
    linearly independent rows or columns (depending on the size of the
    Note that numerix.mlab.rank() is not equivalent to Matlab's rank.
    This function is!
    x      = npy.asarray(x)
    s      = npy.linalg.svd(x, compute_uv=False)
    maxabs = npy.max(npy.absolute(s))
    maxdim = max(x.shape)
    tol    = maxabs * maxdim * _eps_approx
    return npy.sum(s > tol)

01901 def sqrtm(x):
    Returns the square root of a square matrix.
    This means that s=sqrtm(x) implies s*s = x.
    Note that s and x are matrices.
    return mfuncC(npy.sqrt, x)

01909 def mfuncC(f, x):
    mfuncC(f, x) : matrix function with possibly complex eigenvalues.
    Note: Numeric defines (v,u) = eig(x) => x*u.T = u.T * Diag(v)
    This function is needed by sqrtm and allows further functions.

    x      = npy.asarray(x)
    (v, u) = npy.linalg.eig(x)
    uT     = u.transpose()
    V      = npy.diag(f(v+0j))
    y      = npy.dot(uT, npy.dot(V, npy.linalg.inv(uT)))
    return approx_real(y)

01923 def approx_real(x):

    approx_real(x) : returns x.real if |x.imag| < |x.real| * _eps_approx.
    This function is needed by sqrtm and allows further functions.
    ai = npy.absolute(x.imag)
    ar = npy.absolute(x.real)
    if npy.max(ai) <= npy.max(ar) * _eps_approx:
        return x.real
        return x

### end mlab2 functions

#helpers for loading, saving, manipulating and viewing numpy record arrays

def safe_isnan(x):
    'isnan for arbitrary types'
    try: b = npy.isnan(x)
    except NotImplementedError: return False
    else: return b

def rec_append_field(rec, name, arr, dtype=None):
    'return a new record array with field name populated with data from array arr'
    arr = npy.asarray(arr)
    if dtype is None:
        dtype = arr.dtype
    newdtype = npy.dtype(rec.dtype.descr + [(name, dtype)])
    newrec = npy.empty(rec.shape, dtype=newdtype)
    for field in rec.dtype.fields:
        newrec[field] = rec[field]
    newrec[name] = arr
    return newrec.view(npy.recarray)

def rec_drop_fields(rec, names):
    'return a new numpy record array with fields in names dropped'

    names = set(names)
    Nr = len(rec)

    newdtype = npy.dtype([(name, rec.dtype[name]) for name in rec.dtype.names
                       if name not in names])

    newrec = npy.empty(Nr, dtype=newdtype)
    for field in newdtype.names:
        newrec[field] = rec[field]

    return newrec.view(npy.recarray)

01976 def rec_join(key, r1, r2):
    join record arrays r1 and r2 on key; key is a tuple of field
    names.  if r1 and r2 have equal values on all the keys in the key
    tuple, then their fields will be merged into a new record array
    containing the intersection of the fields of r1 and r2

    for name in key:
        if name not in r1.dtype.names:
            raise ValueError('r1 does not have key field %s'%name)
        if name not in r2.dtype.names:
            raise ValueError('r2 does not have key field %s'%name)

    def makekey(row):
        return tuple([row[name] for name in key])

    names = list(r1.dtype.names) + [name for name in r2.dtype.names if name not in set(r1.dtype.names)]

    r1d = dict([(makekey(row),i) for i,row in enumerate(r1)])
    r2d = dict([(makekey(row),i) for i,row in enumerate(r2)])

    r1keys = set(r1d.keys())
    r2keys = set(r2d.keys())

    keys = r1keys & r2keys

    r1ind = [r1d[k] for k in keys]
    r2ind = [r2d[k] for k in keys]

    r1 = r1[r1ind]
    r2 = r2[r2ind]

    r2 = rec_drop_fields(r2, r1.dtype.names)

    def key_desc(name):
        'if name is a string key, use the larger size of r1 or r2 before merging'
        dt1 = r1.dtype[name]
        if dt1.type != npy.string_:
            return (name, dt1.descr[0][1])

        dt2 = r1.dtype[name]
        assert dt2==dt1
        if dt1.num>dt2.num:
            return (name, dt1.descr[0][1])
            return (name, dt2.descr[0][1])

    keydesc = [key_desc(name) for name in key]

    newdtype = npy.dtype(keydesc +
                         [desc for desc in r1.dtype.descr if desc[0] not in key ] +
                         [desc for desc in r2.dtype.descr if desc[0] not in key ] )

    newrec = npy.empty(len(r1), dtype=newdtype)
    for field in r1.dtype.names:
        newrec[field] = r1[field]

    for field in r2.dtype.names:
        newrec[field] = r2[field]

    return newrec.view(npy.recarray)

02048 def csv2rec(fname, comments='#', skiprows=0, checkrows=0, delimiter=',',
            converterd=None, names=None, missing=None):
    Load data from comma/space/tab delimited file in fname into a
    numpy record array and return the record array.

    If names is None, a header row is required to automatically assign
    the recarray names.  The headers will be lower cased, spaces will
    be converted to underscores, and illegal attribute name characters
    removed.  If names is not None, it is a sequence of names to use
    for the column names.  In this case, it is assumed there is no header row.

    fname - can be a filename or a file handle.  Support for gzipped
    files is automatic, if the filename ends in .gz

    comments - the character used to indicate the start of a comment
    in the file

    skiprows  - is the number of rows from the top to skip

    checkrows - is the number of rows to check to validate the column
    data type.  When set to zero all rows are validated.

    converterd, if not None, is a dictionary mapping column number or
    munged column name to a converter function

    names, if not None, is a list of header names.  In this case, no
    header will be read from the file

    if no rows are found, None is returned -- see examples/loadrec.py

    if converterd is None:
        converterd = dict()

    import dateutil.parser
    parsedate = dateutil.parser.parse

    fh = cbook.to_filehandle(fname)

    class FH:
        for space delimited files, we want different behavior than
        comma or tab.  Generally, we want multiple spaces to be
        treated as a single separator, whereas with comma and tab we
        want multiple commas to return multiple (empty) fields.  The
        join/strip trick below effects this
        def __init__(self, fh):
            self.fh = fh

        def close(self):

        def seek(self, arg):

        def fix(self, s):
            return ' '.join(s.split())

        def next(self):
            return self.fix(self.fh.next())

        def __iter__(self):
            for line in self.fh:
                yield self.fix(line)

    if delimiter==' ':
        fh = FH(fh)

    reader = csv.reader(fh, delimiter=delimiter)
    def process_skiprows(reader):
        if skiprows:
            for i, row in enumerate(reader):
                if i>=(skiprows-1): break

        return fh, reader


    dateparser = dateutil.parser.parse

    def myfloat(x):
        if x==missing:
            return npy.nan
            return float(x)

    def mydate(x):
        # try and return a date object
        d = dateparser(x)

        if d.hour>0 or d.minute>0 or d.second>0:
            raise ValueError('not a date')
        return d.date()

    def get_func(item, func):
        # promote functions in this order
        funcmap = {int:myfloat, myfloat:mydate, mydate:dateparser, dateparser:str}
        try: func(item)
            if func==str:
                raise ValueError('Could not find a working conversion function')
            else: return get_func(item, funcmap[func])    # recurse
        else: return func

    # map column names that clash with builtins -- TODO - extend this list
    itemd = {
        'return' : 'return_',
        'file' : 'file_',
        'print' : 'print_',

    def get_converters(reader):

        converters = None
        for i, row in enumerate(reader):
            if i==0:
                converters = [int]*len(row)
            if checkrows and i>checkrows:
            #print i, len(names), len(row)
            #print 'converters', zip(converters, row)
            for j, (name, item) in enumerate(zip(names, row)):
                func = converterd.get(j)
                if func is None:
                    func = converterd.get(name)
                if func is None:
                    if not item.strip(): continue
                    func = converters[j]
                    if len(item.strip()):
                        func = get_func(item, func)
                converters[j] = func
        return converters

    # Get header and remove invalid characters
    needheader = names is None
    if needheader:
        headers = reader.next()
        # remove these chars
        delete = set("""~!@#$%^&*()-=+~\|]}[{';: /?.>,<""")

        names = []
        seen = dict()
        for i, item in enumerate(headers):
            item = item.strip().lower().replace(' ', '_')
            item = ''.join([c for c in item if c not in delete])
            if not len(item):
                item = 'column%d'%i

            item = itemd.get(item, item)
            cnt = seen.get(item, 0)
            if cnt>0:
                names.append(item + '%d'%cnt)
            seen[item] = cnt+1

    # get the converter functions by inspecting checkrows
    converters = get_converters(reader)
    if converters is None:
        raise ValueError('Could not find any valid data in CSV file')

    # reset the reader and start over
    reader = csv.reader(fh, delimiter=delimiter)
    if needheader:
        skipheader = reader.next()

    # iterate over the remaining rows and convert the data to date
    # objects, ints, or floats as approriate
    rows = []
    for i, row in enumerate(reader):
        if not len(row): continue
        if row[0].startswith(comments): continue
        rows.append([func(val) for func, val in zip(converters, row)])

    if not len(rows):
        return None
    r = npy.rec.fromrecords(rows, names=names)
    return r

# a series of classes for describing the format intentions of various rec views
class FormatObj:
    def tostr(self, x):
        return self.toval(x)

    def toval(self, x):
        return str(x)

class FormatString(FormatObj):
    def tostr(self, x):
        val = repr(x)
        return val[1:-1]

#class FormatString(FormatObj):
#    def tostr(self, x):
#        return '"%r"'%self.toval(x)

class FormatFormatStr(FormatObj):
    def __init__(self, fmt):
        self.fmt = fmt

    def tostr(self, x):
        if x is None: return 'None'
        return self.fmt%self.toval(x)

class FormatFloat(FormatFormatStr):
    def __init__(self, precision=4, scale=1.):
        FormatFormatStr.__init__(self, '%%1.%df'%precision)
        self.precision = precision
        self.scale = scale

    def toval(self, x):
        if x is not None:
            x = x * self.scale
        return x

class FormatInt(FormatObj):
    def toval(self, x):
        return x

class FormatPercent(FormatFloat):
    def __init__(self, precision=4):
        FormatFloat.__init__(self, precision, scale=100.)

class FormatThousands(FormatFloat):
    def __init__(self, precision=4):
        FormatFloat.__init__(self, precision, scale=1e-3)

class FormatMillions(FormatFloat):
    def __init__(self, precision=4):
        FormatFloat.__init__(self, precision, scale=1e-6)

class FormatDate(FormatObj):
    def __init__(self, fmt):
        self.fmt = fmt

    def toval(self, x):
        if x is None: return 'None'
        return x.strftime(self.fmt)

class FormatDatetime(FormatDate):
    def __init__(self, fmt='%Y-%m-%d %H:%M:%S'):
        FormatDate.__init__(self, fmt)

defaultformatd = {
    npy.int16 : FormatInt(),
    npy.int32 : FormatInt(),
    npy.int64 : FormatInt(),
    npy.float32 : FormatFloat(),
    npy.float64 : FormatFloat(),
    npy.object_ : FormatObj(),
    npy.string_ : FormatString(),

def get_formatd(r, formatd=None):
    'build a formatd guaranteed to have a key for every dtype name'
    if formatd is None:
        formatd = dict()

    for i, name in enumerate(r.dtype.names):
        dt = r.dtype[name]
        format = formatd.get(name)
        if format is None:
            format = defaultformatd.get(dt.type, FormatObj())
        formatd[name] = format
    return formatd

def csvformat_factory(format):
    format = copy.deepcopy(format)
    if isinstance(format, FormatFloat):
        format.scale = 1. # override scaling for storage
        format.fmt = '%r'
    return format

02337 def rec2csv(r, fname, delimiter=',', formatd=None):
    Save the data from numpy record array r into a comma/space/tab
    delimited file.  The record array dtype names will be used for
    column headers.

    fname - can be a filename or a file handle.  Support for gzipped
    files is automatic, if the filename ends in .gz
    formatd = get_formatd(r, formatd)
    funcs = []
    for i, name in enumerate(r.dtype.names):

    fh, opened = cbook.to_filehandle(fname, 'w', return_opened=True)
    writer = csv.writer(fh, delimiter=delimiter)
    header = r.dtype.names
    for row in r:
        writer.writerow([func(val) for func, val in zip(funcs, row)])
    if opened:

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