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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)

    * conv     - convolution
    * corrcoef - The matrix of correlation coefficients

    * 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
    * linspace -- Linear spaced array from min to max

    * hist -- Histogram
    * polyfit - least squares best polynomial fit of x to y

    * polyval - evaluate a vector for a vector of polynomial coeffs

    * prctile - find the percentiles of a sequence
    * prepca - Principal Component's Analysis
    * psd - Power spectral density uing Welch's average periodogram

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

    * trapz - trapeziodal integration
  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 alot of pairs.  This function is optimized to do
      this efficiently by caching the direct FFTs.


  Unless otherwise noted, these functions were written by
  Author: John D. Hunter <jdhunter@ace.bsd.uchicago.edu>

  Some others are from the Numeric documentation, or imported from
  MLab or other Numeric packages


from __future__ import division
import sys, random
from matplotlib import verbose
import numerix
import numerix.mlab 
from numerix import linear_algebra

from numerix import array, asarray, arange, divide, exp, arctan2, \
     multiply, transpose, ravel, repeat, resize, reshape, floor, ceil,\
     absolute, matrixmultiply, power, take, where, Float, Int, asum,\
     dot, convolve, pi, Complex, ones, zeros, diagonal, Matrix, nonzero, \
     log, searchsorted, concatenate, sort, ArrayType, clip, size, indices,\

from numerix.mlab import hanning, cov, diff, svd, rand, std
from numerix.fft import fft

from cbook import iterable

def mean(x, dim=None):
   if len(x)==0: return None
   elif dim is None:
      return numerix.mlab.mean(x)
   else: return numerix.mlab.mean(x, dim)

def linspace(xmin, xmax, N):
   if N==1: return xmax
   dx = (xmax-xmin)/(N-1)
   return xmin + dx*arange(N)

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

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

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

def conv(x, y, mode=2):
    'convolve x with y'
    return 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 detrend_mean(x):
    "Return x minus the mean(x)"
    return x - mean(x)

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

def detrend_linear(x):
    "Return x minus best fit line; 'linear' detrending "

    # I'm going to regress x on xx=range(len(x)) and return x -
    # (b*xx+a).  Now that I have polyfit working, I could convert the
    # code here, but if it ain't broke, don't fix it!
    xx = arange(float(len(x)))
    X = transpose(array([xx]+[x]))
    C = cov(X)
    b = C[0,1]/C[0,0]
    a = mean(x) - b*mean(xx)
    return x-(b*xx+a)

00133 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.

    -- NFFT must be a power of 2
    -- detrend and window are functions, unlike in matlab where they are
    -- 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)


    if NFFT % 2:
        raise ValueError, 'NFFT must be a power of 2'

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

    # for real x, ignore the negative frequencies
    if x.typecode()==Complex: numFreqs = NFFT
    else: numFreqs = NFFT//2+1
    windowVals = window(ones((NFFT,),x.typecode()))
    step = NFFT-noverlap
    ind = range(0,len(x)-NFFT+1,step)
    n = len(ind)
    Pxx = zeros((numFreqs,n), Float)
    # do the ffts of the slices
    for i in range(n):
        thisX = x[ind[i]:ind[i]+NFFT]
        thisX = windowVals*detrend(thisX)
        fx = absolute(fft(thisX))**2
        Pxx[:,i] = divide(fx[:numFreqs], norm(windowVals)**2)

    # Scale the spectrum by the norm of the window to compensate for
    # windowing loss; see Bendat & Piersol Sec 11.5.2
    if n>1:
       Pxx = mean(Pxx,1)

    freqs = Fs/NFFT*arange(numFreqs)
    Pxx.shape = len(freqs),

    return Pxx, freqs

00194 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 a power of 2

    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 a power of 2'

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

    # for real x, ignore the negative frequencies
    if x.typecode()==Complex: numFreqs = NFFT
    else: numFreqs = NFFT//2+1
    windowVals = window(ones((NFFT,),x.typecode()))
    step = NFFT-noverlap
    ind = range(0,len(x)-NFFT+1,step)
    n = len(ind)
    Pxy = zeros((numFreqs,n), 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 = fft(thisX)
        fy = fft(thisY)
        Pxy[:,i] = 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 = mean(Pxy,1)
    Pxy = divide(Pxy, norm(windowVals)**2)
    freqs = Fs/NFFT*arange(numFreqs)
    Pxy.shape = len(freqs),
    return Pxy, freqs

00261 def cohere(x, y, NFFT=256, Fs=2, detrend=detrend_none,
           window=window_hanning, noverlap=0):
    cohere 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, windowm noverlap, as
    well as the methods used to compute Pxy, Pxx and Pyy.

    Returns the tuple Cxy, freqs

    if len(x)<2*NFFT:
       raise RuntimeError('Coherence is calculated by averaging over NFFT length segments.  Your signal is too short for your choice of NFFT')
    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 = divide(absolute(Pxy)**2, Pxx*Pyy)
    Cxy.shape = len(f),
    return Cxy, f

00288 def corrcoef(*args):
    corrcoef(X) where X is a matrix returns a matrix of correlation
    coefficients for each numrows observations and numcols variables.
    corrcoef(x,y) where x and y are vectors returns the matrix or
    correlation coefficients for x and y.

    Numeric 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])

    if len(args)==2:
        X = transpose(array([args[0]]+[args[1]]))
    elif len(args)==1:
        X = args[0]
        raise RuntimeError, 'Only expecting 1 or 2 arguments'

    C = cov(X)

    if len(args)==2:
       d = resize(diagonal(C), (2,1))
       denom = numerix.mlab.sqrt(matrixmultiply(d,transpose(d)))
       dc = diagonal(C)
       N = len(dc)       
       shape = N,N
       vi = resize(dc, shape)
       denom = numerix.mlab.sqrt(vi*transpose(vi)) # element wise multiplication

    r = divide(C,denom)
    try: return r.real
    except AttributeError: return r

00332 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.

    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


    x = asarray(x)+0.
    y = asarray(y)+0.
    y = reshape(y, (len(y),1))
    X = Matrix(vander(x, N+1))
    Xt = Matrix(transpose(X))
    c = array(linear_algebra.inverse(Xt*X)*Xt*y)  # convert back to array
    c.shape = (N+1,)
    return c


00381 def polyval(p,x):
    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
    x = asarray(x)+0.
    p = reshape(p, (len(p),1))
    X = vander(x,len(p))
    y =  matrixmultiply(X,p)
    return reshape(y, x.shape)

00404 def vander(x,N=None):
    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).

    if N is None: N=len(x)
    X = ones( (len(x),N), x.typecode())
    for i in range(N-1):
        X[:,i] = x**(N-i-1)
    return X

def donothing_callback(*args):

00424 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 Numeric 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 = zeros( (NFFT, numCols), X.typecode())
        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 X.typecode()==Complex: 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
    windowVals = window(ones((NFFT,), X.typecode()))
    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 = zeros( (numSlices,numFreqs), 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] = divide(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] * conjugate(FFTSlices[j])
        if numSlices>1: Pxy = mean(Pxy)
        Pxy = divide(Pxy, normVal)
        Cxy[(i,j)] = divide(absolute(Pxy)**2, Pxx[i]*Pxx[j])
        Phase[(i,j)] =  arctan2(Pxy.imag, Pxy.real)

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

00568 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 hist

   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 = hist(y, bins)
   n = n.astype(Float)

   n = take(n, nonzero(n))         # get the positive

   p = divide(n, len(y))

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

00595 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
    Credits: the Numeric 22 documentation


    y = asarray(y)
    if len(y.shape)>1: y = ravel(y)

    if not iterable(bins):       
        ymin, ymax = min(y), max(y)
        if ymin==ymax:
            ymin -= 0.5
            ymax += 0.5

        if bins==1: bins=ymax
        dy = (ymax-ymin)/bins 
        bins = ymin + dy*arange(bins)

    n = searchsorted(sort(y), bins)
    n = diff(concatenate([n, [len(y)]]))
    if normed:
       db = bins[1]-bins[0]
       return 1/(len(y)*db)*n, bins
       return n, bins

def normpdf(x, *args):
   "Return the normal pdf evaluated at x; args provides mu, sigma"
   mu, sigma = args
   return 1/(numerix.mlab.sqrt(2*pi)*sigma)*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 = linspace(minx, maxx, N), where N is even'

   dx = x[1]-x[0]

   f = 1/(N*dx)*arange(-N/2, N/2, Float)

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

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


def find(condition):
   "Return the indices where condition is true"
   return nonzero(condition)

def trapz(x, y):
   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 asum(0.5*diff(x)*(y[1:]+y[:-1]))

00683 def longest_contiguous_ones(x):
    return the indicies of the longest stretch of contiguous ones in x,
    assuming x is a vector of zeros and ones.
    if len(x)==0: return array([])

    ind = find(x==0)
    if len(ind)==0:  return arange(len(x))
    if len(ind)==len(x): return array([])

    y = zeros( (len(x)+2,),  x.typecode())
    y[1:-1] = x
    dif = diff(y)
    up = find(dif ==  1);
    dn = find(dif == -1);
    ind = find( dn-up == max(dn - up))
    ind = arange(take(up, ind), take(dn, ind))

    return ind

00705 def longest_ones(x):
    return the indicies 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 = asarray(x)
    if len(x)==0: return array([])

    #print 'x', x
    ind = find(x==0)
    if len(ind)==0:  return arange(len(x))
    if len(ind)==len(x): return array([])

    y = zeros( (len(x)+2,), Int)
    y[1:-1] = x
    d = diff(y)
    #print 'd', d
    up = find(d ==  1);
    dn = find(d == -1);

    #print 'dn', dn, 'up', up, 
    ind = find( dn-up == max(dn - up))
    # pick the first
    if iterable(ind): ind = ind[0]
    ind = arange(up[ind], dn[ind])

    return ind

00735 def prepca(P, frac=0):
    Compute the principal components of P.  P is a numVars x
    numObservations numeric array.  frac is the minimum fraction of
    variance that a component must contain to be included

    Return value are
    Pcomponents : a num components x num observations numeric array
    Trans       : the weights matrix, ie, Pcomponents = Trans*P
    fracVar     : the fraction of the variance accounted for by each
                  component returned
    U,s,v = svd(P)
    varEach = s**2/P.shape[1]
    totVar = asum(varEach)
    fracVar = divide(varEach,totVar)
    ind = int(asum(fracVar>=frac))

    # select the components that are greater
    Trans = transpose(U[:,:ind])
    # The transformed data
    Pcomponents = matrixmultiply(Trans,P)
    return Pcomponents, Trans, fracVar[:ind]

# 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.

_eps_approx = 1e-13

00777 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.
    dim = numerix.shape(x)
    if numerix.mlab.rank(x)==2:
        y = reshape(x,(1,dim[0]*dim[1]))[0]
        y = y.tolist()
    elif numerix.mlab.rank(x)==1:
        y = x
        y = [x]
    for i in range(len(y)):
      if y[i]>0:
            y[i] = floor(y[i])
            y[i] = ceil(y[i])
    if numerix.mlab.rank(x)==2:
        x = reshape(y,dim)
    elif numerix.mlab.rank(x)==0:
        x = y[0]
    return x

00805 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, rem(x,0) returns None.
    We keep the convention by Matlab:
    "The input x and y must be real arrays of the same size, or real scalars."
    x,y = asarray(x),asarray(y)
    if numerix.shape(x) == numerix.shape(y) or numerix.shape(y) == ():
            return x - y * fix(x/y)
        except OverflowError:
            return None
    raise RuntimeError('Dimension error')

00823 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 = asarray(x)
    if numerix.mlab.rank(x)==2:
        if y==2:
            return numerix.mlab.max(numerix.mlab.svd(x)[1])
        elif y==1:
            return numerix.mlab.max(asum(absolute((x))))
        elif y=='inf':
            return numerix.mlab.max(asum(absolute((transpose(x)))))
        elif y=='fro':
            return numerix.mlab.sqrt(asum(numerix.mlab.diag(matrixmultiply(transpose(x),x))))
            raise RuntimeError('Second argument not permitted for matrices')
        if y == 'inf':
            return numerix.mlab.max(absolute(x))
        elif y == '-inf':
            return numerix.mlab.min(absolute(x))
            return power(asum(power(absolute(x),y)),1/float(y))

00867 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     = array(A)
    U,S,V = numerix.mlab.svd(A)

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

    tol = numerix.mlab.max((m,n)) * numerix.mlab.max(s) * _eps_approx
    r = asum(s > tol)
    Q = take(U,range(r),1)

    return Q

00895 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      = asarray(x)
      u,s,v  = numerix.mlab.svd(x)
      # maxabs = numerix.mlab.max(numerix.absolute(s)) is also possible.
      maxabs = norm(x)  
      maxdim = numerix.mlab.max(numerix.shape(x))
      tol    = maxabs*maxdim*_eps_approx
      r      = s>tol
      return asum(r)

00914 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(numerix.mlab.sqrt, x)

00922 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      = array(x) 
      (v, u) = numerix.mlab.eig(x)
      uT     = transpose(u)
      V      = numerix.mlab.diag(f(v+0j))
      y      = matrixmultiply(
           uT, matrixmultiply(
           V, linear_algebra.inverse(uT)))
      return approx_real(y)

00938 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.

      if numerix.mlab.max(numerix.mlab.max(absolute(x.imag))) <= numerix.mlab.max(numerix.mlab.max(absolute(x.real))) * _eps_approx:
            return x.real
            return x

00952 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
    percentil 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
    x = sort(ravel(x))
    Nx = len(x)

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

    p = multiply(array(p), Nx/100.0)
    ind = p.astype(Int)
    ind = where(ind>=Nx, Nx-1, ind)        
    return take(x, ind)

00970 def center_matrix(M, dim=0):
    Return the matrix M with each row having zero mean and unit std

    if dim=1, center columns rather than rows
    # todo: implement this w/o loop.  Allow optional arg to specify
    # dimension to remove the mean from
    if dim==1: M = transpose(M)
    M = array(M, Float)
    if len(M.shape)==1 or M.shape[0]==1 or M.shape[1]==1:
       M = M-mean(M)
       sigma = std(M)
       if sigma>0:
          M = divide(M, sigma)
       if dim==1: M=transpose(M)
       return M
    for i in range(M.shape[0]):
        M[i] -= mean(M[i])
        sigma = std(M[i])
        if sigma>0:
           M[i] = divide(M[i], sigma)
    if dim==1: M=transpose(M)
    return M

00996 def meshgrid(x,y):
    For vectors x, y with lengths Nx=len(x) and Ny=len(y), return X, Y
    where X and Y are (Ny, Nx) shaped arrays with the elements of x
    and y repeated to fill the matrix


      [X, Y] = meshgrid([1,2,3], [4,5,6,7])

       X =
         1   2   3
         1   2   3
         1   2   3
         1   2   3

       Y =
         4   4   4
         5   5   5
         6   6   6
         7   7   7
    x = array(x)
    y = array(y)
    numRows, numCols = len(y), len(x)  # yes, reversed
    x.shape = 1, numCols
    X = repeat(x, numRows)

    y.shape = numRows,1
    Y = repeat(y, numCols, 1)
    return X, Y

01032 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
        # Numeric solution
        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)

    try: Ny = len(y0)
    except TypeError:
        yout = zeros( (len(t),), Float)
        yout = zeros( (len(t), Ny), Float)
    yout[0] = y0
    i = 0
    for i in arange(len(t)-1):

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

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

01093 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

    See pdf for more info.

    The returned times are the midpoints of the intervals over which
    the ffts are calculated

    if log(NFFT)/log(2) != int(log(NFFT)/log(2)):
       raise ValueError, 'NFFT must be a power of 2'

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

    # for real x, ignore the negative frequencies
    if x.typecode()==Complex: numFreqs = NFFT
    else: numFreqs = NFFT//2+1
    windowVals = window(ones((NFFT,),x.typecode()))
    step = NFFT-noverlap
    ind = arange(0,len(x)-NFFT+1,step)
    n = len(ind)
    Pxx = zeros((numFreqs,n), Float)
    # do the ffts of the slices

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

    return Pxx, freqs, t

01141 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*rho*Xmu*Ymu/(sigmax*sigmay)
    return 1.0/(2*pi*sigmax*sigmay*(1-rho**2)) * exp( -z/(2*(1-rho**2)))

01158 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
    M,N = Z.shape
    z = ravel(Z)
    ind = nonzero( ravel(Cond) )

    x = arange(M); x.shape = M,1
    X = repeat(x, N, 1)
    x = ravel(X)

    y = arange(N); y.shape = 1,N
    Y = repeat(y, M)
    y = ravel(Y)

    x = take(x, ind)
    y = take(y, ind)
    z = take(z, ind)
    return x,y,z

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

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

01197 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 = asarray(p, Float)
    s0 = asarray(s0, Float)
    s1 = asarray(s1, Float)    
    v = s1 - s0
    w = p - s0

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

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

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

01224 def segments_intersect(s1, s2):
    Return True if s1 and s2 intersect.
    s1 and s2 are defines 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

### the following code was written and submitted by Fernando Perez
### from the ipython numutils package under a BSD license
A set of convenient utilities for numerical work.

Most of this module requires Numerical Python or is meant to be used with it.
See http://www.pfdubois.com/numpy for details.

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

01303 def exp_safe(x):
    """Compute exponentials which safely underflow to zero.

    Slow but convenient to use. Note that NumArray will introduce proper
    floating point exception handling with access to the underlying

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

01315 def amap(fn,*args):
    """amap(function, sequence[, sequence, ...]) -> array.

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

01323 def zeros_like(a):
    """Return an array of zeros of the shape and typecode of a."""

    return zeros(a.shape,a.typecode())

01328 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."""

    if a.iscontiguous():
        return asum(a.flat)
        return asum(ravel(a))

01338 def mean_flat(a):
    """Return the mean of all the elements of a, flattened out."""

    return sum_flat(a)/float(size(a))

01343 def rms_flat(a):
    """Return the root mean square of all the elements of a, flattened out."""

    return numerix.mlab.sqrt(sum_flat(absolute(a)**2)/float(size(a)))

01348 def l1norm(a):
    """Return the l1 norm of a, flattened out.

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

    return sum_flat(absolute(a))

01355 def l2norm(a):
    """Return the l2 norm of a, flattened out.

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

    return numerix.mlab.sqrt(sum_flat(absolute(a)**2))

def norm(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."""
    if p=='Infinity':
        return max(absolute(a).flat)
        return (sum_flat(absolute(a)**p))**(1.0/p)    
01375 def frange(xini,xfin=None,delta=None,**kw):
    """frange([start,] stop[, step, keywords]) -> array of floats

    Return a Numeric array() 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])
    >>> 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:
        # round() gets npts right even with the vagaries of floating point.

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

01434 def diagonal_matrix(diag):
    """Return square diagonal matrix whose non-zero elements are given by the
    input array."""

    return diag*identity(len(diag))

01440 def identity(n,rank=2,typecode='l'):
    """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 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 the Numeric identity function."""
    iden = zeros((n,)*rank,typecode=typecode)
    for i in range(n):
        idx = (i,)*rank
        iden[idx] = 1
    return iden

01460 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))

01472 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','')

01490 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 ineger 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)

01506 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)

01517 def fromfunction_kw(function, dimensions, **kwargs):
    """Drop-in replacement for fromfunction() from Numerical Python.
    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 recieve."""

    return function(tuple(indices(dimensions)),**kwargs)

### end Perez numutils code

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