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mlab.py

00001 """

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

  Matlab compatible functions:

    * cohere - Coherence (normalized cross spectral density)

    * 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, 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.

Credits:

  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 MLab
import sys
from MLab import *
from Numeric import *
from LinearAlgebra import inverse, eigenvectors
from FFT import fft
from Matrix import Matrix
from cbook import iterable


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


00118 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
       vectors.
    -- if length x < NFFT, it will be zero padded to NFFT
    

    Returns the tuple Pxx, freqs
    
    Refs:
      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] = fx[: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: Pxx = mean(Pxx,1)
    Pxx = divide(Pxx, norm(windowVals)**2)

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

00178 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
    frequency.

    NFFT must be a power of 2

    Returns the tuple Pxy, freqs

    Refs:
      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

00243 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:
       print >>sys.stderr, '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

00270 def corrcoef(*args):
    """
    
    corrcoef(X) where X is a matrix returns a matrix of correlation
    coefficients for each row of X.
    
    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] / (C[i,i]*C[j,j])
    """

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

    
    C = cov(X)
    d = resize(diagonal(C), (2,1))
    r = divide(C,sqrt(matrixmultiply(d,transpose(d))))
    try: return r.real
    except AttributeError: return r




00303 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
    http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html,
    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

    """

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

    

00350 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
    
    """

    p = reshape(p, (len(p),1))
    X = vander(x,len(p))
    y =  matrixmultiply(X,p)
    return reshape(y, x.shape)


00373 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):
    pass

00393 def cohere_pairs( X, ij, NFFT=256, Fs=2, detrend=detrend_none,
                  window=window_hanning, noverlap=0,
                  preferSpeedOverMemory=1,
                  progressCallback=donothing_callback):

    """
    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:

          subplot(211)
          plot( freqs, Cxy[(12,19)])
          subplot(212)
          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
    cohere.

    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.

    """

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

    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]
        else:
            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)
    return Cxy, Phase, freqs



00538 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
    n/(len(y)*dbin)
    
    Credits: the Numeric 22 documentation


    """
    if not iterable(bins):
        ymin, ymax = min(y), max(y)
        if ymin==ymax:
            ymin -= 0.5
            ymax += 0.5
        bins = linspace(ymin, ymax, 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
    else:
       return n, bins


def normpdf(x, *args):
   "Return the normal pdf evaluated at x; args provides mu, sigma"
   mu, sigma = args
   return 1/(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]

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

   ind = concatenate([arange(N/2, N, typecode=Int),
                      arange(N/2,typecode=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 sum(0.5*diff(x)*(y[1:]+y[:-1]))
   
   

00617 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(d ==  1);
    dn = find(d == -1);
    ind = find( dn-up == max(dn - up))
    ind = arange(take(up, ind), take(dn, ind))

    return ind


00639 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

00669 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 = sum(varEach)
    fracVar = divide(varEach,totVar)
    ind = int(sum(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
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.
"""

import Numeric, LinearAlgebra, MLab
_eps_approx = 1e-13

00712 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 = Numeric.shape(x)
    if MLab.rank(x)==2:
        y = Numeric.reshape(x,(1,dim[0]*dim[1]))[0]
        y = y.tolist()
    elif MLab.rank(x)==1:
        y = x
    else:
        y = [x]
    for i in range(len(y)):
      if y[i]>0:
            y[i] = Numeric.floor(y[i])
      else:
            y[i] = Numeric.ceil(y[i])
    if MLab.rank(x)==2:
        x = Numeric.reshape(y,dim)
    elif MLab.rank(x)==0:
        x = y[0]
    return x

00740 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 = Numeric.asarray(x),Numeric.asarray(y)
    if Numeric.shape(x) == Numeric.shape(y) or Numeric.shape(y) == ():
        try:
            return x - y * fix(x/y)
        except OverflowError:
            return None
    print >>sys.stderr, 'Dimension error'
    return None

00758 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 = Numeric.asarray(x)
    if MLab.rank(x)==2:
        if y==2:
            return MLab.max(MLab.svd(x)[1])
        elif y==1:
            return MLab.max(MLab.sum(Numeric.absolute((x))))
        elif y=='inf':
            return MLab.max(MLab.sum(Numeric.absolute((MLab.transpose(x)))))
        elif y=='fro':
            return MLab.sqrt(MLab.sum(MLab.diag(Numeric.matrixmultiply(MLab.transpose(x),x))))
        else:
            print >>sys.stderr, 'Second argument not permitted for matrices'
            return None
        
    else:
        if y == 'inf':
            return MLab.max(Numeric.absolute(x))
        elif y == '-inf':
            return MLab.min(Numeric.absolute(x))
        else:
            return Numeric.power(MLab.sum(Numeric.power(Numeric.absolute(x),y)),1/float(y))


00803 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     = Numeric.array(A)
    U,S,V = MLab.svd(A)

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

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

    return Q

00831 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
        matrix).
        Note that MLab.rank() is not equivalent to Matlab's rank.
        This function is!
        """
        
      x      = Numeric.asarray(x)
      u,s,v  = MLab.svd(x)
      # maxabs = MLab.max(Numeric.absolute(s)) is also possible.
      maxabs = norm(x)  
      maxdim = MLab.max(Numeric.shape(x))
      tol    = maxabs*maxdim*_eps_approx
      r      = s>tol
      return MLab.sum(r)

00850 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(MLab.sqrt, x)

00858 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      = Numeric.array(x) 
      (v, u) = MLab.eig(x)
      uT     = MLab.transpose(u)
      V      = MLab.diag(f(v+0j))
      y      = Numeric.matrixmultiply(
           uT, Numeric.matrixmultiply(
           V, LinearAlgebra.inverse(uT)))
      return approx_real(y)

00874 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 MLab.max(MLab.max(Numeric.absolute(x.imag))) <= MLab.max(MLab.max(Numeric.absolute(x.real))) * _eps_approx:
            return x.real
      else:
            return x



00888 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(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)


00906 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

00932 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

    EG,

      [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 = asarray(x)
    y = asarray(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
 

00967 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
    signature

     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)
    else:
        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



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