mlab.py

"""

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

Matlab(TM) compatible functions
-------------------------------

:func:`cohere`
  Coherence (normalized cross spectral density)

:func:`csd`
  Cross spectral density uing Welch's average periodogram

:func:`detrend`
  Remove the mean or best fit line from an array

:func:`find`
  Return the indices where some condition is true;
         numpy.nonzero is similar but more general.

:func:`griddata`
  interpolate irregularly distributed data to a
             regular grid.

:func:`prctile`
  find the percentiles of a sequence

:func:`prepca`
  Principal Component Analysis

:func:`psd`
  Power spectral density uing Welch's average periodogram

:func:`rk4`
  A 4th order runge kutta integrator for 1D or ND systems

:func:`specgram`
  Spectrogram (power spectral density over segments of time)

Miscellaneous functions
-------------------------

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

:meth:`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.

:meth:`rk4`
    A 4th order Runge-Kutta ODE integrator in case you ever find
    yourself stranded without scipy (and the far superior
    scipy.integrate tools)

record array helper functions
-------------------------------

A collection of helper methods for numpyrecord arrays

.. _htmlonly::

    See :ref:`misc-examples-index`

:meth:`rec2txt`
    pretty print a record array

:meth:`rec2csv`
    store record array in CSV file

:meth:`csv2rec`
    import record array from CSV file with type inspection

:meth:`rec_append_fields`
    adds  field(s)/array(s) to record array

:meth:`rec_drop_fields`
    drop fields from record array

:meth:`rec_join`
    join two record arrays on sequence of fields

:meth:`rec_groupby`
    summarize data by groups (similar to SQL GROUP BY)

:meth:`rec_summarize`
    helper code to filter rec array fields into new fields

For the rec viewer functions(e rec2csv), 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()
    win.set_size_request(600,800)
    win.add(scroll)
    win.show_all()
    gtk.main()


Deprecated functions
---------------------

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


:meth:`conv`
    convolution  (numpy.convolve)

:meth:`corrcoef`
    The matrix of correlation coefficients

:meth:`hist`
    Histogram (numpy.histogram)

:meth:`linspace`
    Linear spaced array from min to max

:meth:`load`
    load ASCII file - use numpy.loadtxt

:meth:`meshgrid`
    Make a 2D grid from 2 1 arrays (numpy.meshgrid)

:meth:`polyfit`
    least squares best polynomial fit of x to y (numpy.polyfit)

:meth:`polyval`
    evaluate a vector for a vector of polynomial coeffs (numpy.polyval)

:meth:`save`
    save ASCII file - use numpy.savetxt

:meth:`trapz`
    trapeziodal integration (trapz(x,y) -> numpy.trapz(y,x))

:meth:`vander`
    the Vandermonde matrix (numpy.vander)

"""

from __future__ import division
import csv, warnings, copy, os

import numpy as np
ma = np.ma
from matplotlib import verbose

import matplotlib.nxutils as nxutils
import matplotlib.cbook as 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 np.linspace(*args, **kw)

def meshgrid(x,y):
    warnings.warn("use numpy.meshgrid", DeprecationWarning)
    return np.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 np.mean(x, axis=dim)


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

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

def window_hanning(x):
    "return x times the hanning window of len(x)"
    return np.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 np.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 = np.asarray(x)
    if axis:
        ind = [slice(None)] * axis
        ind.append(np.newaxis)
        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 = np.arange(len(y), dtype=np.float_)
    C = np.cov(x, y, bias=1)
    b = C[0,1]/C[0,0]
    a = y.mean() - b*x.mean()
    return y - (b*x + a)

#This is a helper function that implements the commonality between the
#psd, csd, and spectrogram.  It is *NOT* meant to be used outside of mlab
def _spectral_helper(x, y, NFFT=256, Fs=2, detrend=detrend_none,
        window=window_hanning, noverlap=0, pad_to=None, sides='default',
        scale_by_freq=None):
    #The checks for if y is x are so that we can use the same function to
    #implement the core of psd(), csd(), and spectrogram() without doing
    #extra calculations.  We return the unaveraged Pxy, freqs, and t.
    same_data = y is x

    #Make sure we're dealing with a numpy array. If y and x were the same
    #object to start with, keep them that way

    x = np.asarray(x)
    if not same_data:
        y = np.asarray(y)

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

    if not same_data and len(y)<NFFT:
        n = len(y)
        y = np.resize(y, (NFFT,))
        y[n:] = 0

    if pad_to is None:
        pad_to = NFFT

    if scale_by_freq is None:
        warnings.warn("psd, csd, and specgram have changed to scale their "
            "densities by the sampling frequency for better MatLab "
            "compatibility. You can pass scale_by_freq=False to disable "
            "this behavior.  Also, one-sided densities are scaled by a "
            "factor of 2.")
        scale_by_freq = True

    # For real x, ignore the negative frequencies unless told otherwise
    if (sides == 'default' and np.iscomplexobj(x)) or sides == 'twosided':
        numFreqs = pad_to
        scaling_factor = 1.
    elif sides in ('default', 'onesided'):
        numFreqs = pad_to//2 + 1
        scaling_factor = 2.
    else:
        raise ValueError("sides must be one of: 'default', 'onesided', or "
            "'twosided'")

    # Matlab divides by the sampling frequency so that density function
    # has units of dB/Hz and can be integrated by the plotted frequency
    # values. Perform the same scaling here.
    if scale_by_freq:
        scaling_factor /= Fs

    if cbook.iterable(window):
        assert(len(window) == NFFT)
        windowVals = window
    else:
        windowVals = window(np.ones((NFFT,), x.dtype))

    step = NFFT - noverlap
    ind = np.arange(0, len(x) - NFFT + 1, step)
    n = len(ind)
    Pxy = np.zeros((numFreqs,n), np.complex_)

    # do the ffts of the slices
    for i in range(n):
        thisX = x[ind[i]:ind[i]+NFFT]
        thisX = windowVals * detrend(thisX)
        fx = np.fft.fft(thisX, n=pad_to)

        if same_data:
            fy = fx
        else:
            thisY = y[ind[i]:ind[i]+NFFT]
            thisY = windowVals * detrend(thisY)
            fy = np.fft.fft(thisY, n=pad_to)
        Pxy[:,i] = np.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.  Also include
    # scaling factors for one-sided densities and dividing by the sampling
    # frequency, if desired.
    Pxy *= scaling_factor / (np.abs(windowVals)**2).sum()
    t = 1./Fs * (ind + NFFT / 2.)
    freqs = float(Fs) / pad_to * np.arange(numFreqs)

    return Pxy, freqs, t

#Split out these keyword docs so that they can be used elsewhere
kwdocd = dict()
kwdocd['PSD'] ="""
    Keyword arguments:

      *NFFT*: integer
          The number of data points used in each block for the FFT.
          Must be even; a power 2 is most efficient.  The default value is 256.

      *Fs*: scalar
          The sampling frequency (samples per time unit).  It is used
          to calculate the Fourier frequencies, freqs, in cycles per time
          unit. The default value is 2.

      *detrend*: callable
          The function applied to each segment before fft-ing,
          designed to remove the mean or linear trend.  Unlike in
          matlab, where the *detrend* parameter is a vector, in
          matplotlib is it a function.  The :mod:`~matplotlib.pylab`
          module defines :func:`~matplotlib.pylab.detrend_none`,
          :func:`~matplotlib.pylab.detrend_mean`, and
          :func:`~matplotlib.pylab.detrend_linear`, but you can use
          a custom function as well.

      *window*: callable or ndarray
          A function or a vector of length *NFFT*. To create window
          vectors see :func:`window_hanning`, :func:`window_none`,
          :func:`numpy.blackman`, :func:`numpy.hamming`,
          :func:`numpy.bartlett`, :func:`scipy.signal`,
          :func:`scipy.signal.get_window`, etc. The default is
          :func:`window_hanning`.  If a function is passed as the
          argument, it must take a data segment as an argument and
          return the windowed version of the segment.

      *noverlap*: integer
          The number of points of overlap between blocks.  The default value
          is 0 (no overlap).

      *pad_to*: integer
          The number of points to which the data segment is padded when
          performing the FFT.  This can be different from *NFFT*, which
          specifies the number of data points used.  While not increasing
          the actual resolution of the psd (the minimum distance between
          resolvable peaks), this can give more points in the plot,
          allowing for more detail. This corresponds to the *n* parameter
          in the call to fft(). The default is None, which sets *pad_to*
          equal to *NFFT*

      *sides*: [ 'default' | 'onesided' | 'twosided' ]
          Specifies which sides of the PSD to return.  Default gives the
          default behavior, which returns one-sided for real data and both
          for complex data.  'onesided' forces the return of a one-sided PSD,
          while 'twosided' forces two-sided.

      *scale_by_freq*: boolean
          Specifies whether the resulting density values should be scaled
          by the scaling frequency, which gives density in units of Hz^-1.
          This allows for integration over the returned frequency values.
          The default is True for MatLab compatibility.
"""

def psd(x, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning,
        noverlap=0, pad_to=None, sides='default', scale_by_freq=None):
    """
    The power spectral density by Welch's average periodogram method.
    The vector *x* is divided into *NFFT* length blocks.  Each block
    is detrended by the function *detrend* and windowed by the function
    *window*.  *noverlap* gives the length of the overlap between blocks.
    The absolute(fft(block))**2 of each segment are averaged to compute
    *Pxx*, with a scaling to correct for power loss due to windowing.

    If len(*x*) < *NFFT*, it will be zero padded to *NFFT*.

    *x*
        Array or sequence containing the data
    %(PSD)s
    Returns the tuple (*Pxx*, *freqs*).

    Refs:
        Bendat & Piersol -- Random Data: Analysis and Measurement
        Procedures, John Wiley & Sons (1986)
    """
    Pxx,freqs = csd(x, x, NFFT, Fs, detrend, window, noverlap, pad_to, sides,
        scale_by_freq)
    return Pxx.real,freqs

psd.__doc__ = psd.__doc__ % kwdocd

def csd(x, y, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning,
        noverlap=0, pad_to=None, sides='default', scale_by_freq=None):
    """
    The cross power spectral density by Welch's average periodogram
    method.  The vectors *x* and *y* are divided into *NFFT* length
    blocks.  Each block is detrended by the function *detrend* and
    windowed by the function *window*.  *noverlap* gives the length
    of the overlap between blocks.  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.

    If len(*x*) < *NFFT* or len(*y*) < *NFFT*, they will be zero
    padded to *NFFT*.

    *x*, *y*
        Array or sequence containing the data
    %(PSD)s
    Returns the tuple (*Pxy*, *freqs*).

    Refs:
        Bendat & Piersol -- Random Data: Analysis and Measurement
        Procedures, John Wiley & Sons (1986)
    """
    Pxy, freqs, t = _spectral_helper(x, y, NFFT, Fs, detrend, window,
        noverlap, pad_to, sides, scale_by_freq)

    if len(Pxy.shape) == 2 and Pxy.shape[1]>1:
        Pxy = Pxy.mean(axis=1)
    return Pxy, freqs

csd.__doc__ = csd.__doc__ % kwdocd

def specgram(x, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning,
        noverlap=128, pad_to=None, sides='default', scale_by_freq=None):
    """
    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*.

    If *x* is real (i.e. non-complex) only the spectrum of the positive
    frequencie is returned.  If *x* is complex then the complete
    spectrum is returned.
    %(PSD)s
    Returns a tuple (*Pxx*, *freqs*, *t*):

         - *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
           segments.

    .. seealso::
        :func:`psd`:
            :func:`psd` differs in the default overlap; in returning
            the mean of the segment periodograms; and in not returning
            times.
    """
    assert(NFFT > noverlap)

    Pxx, freqs, t = _spectral_helper(x, x, NFFT, Fs, detrend, window,
        noverlap, pad_to, sides, scale_by_freq)
    Pxx = Pxx.real #Needed since helper implements generically

    if (np.iscomplexobj(x) and sides == 'default') or sides == 'twosided':
        # center the frequency range at zero
        freqs = np.concatenate((freqs[NFFT/2:]-Fs,freqs[:NFFT/2]))
        Pxx   = np.concatenate((Pxx[NFFT/2:,:],Pxx[:NFFT/2,:]),0)

    return Pxx, freqs, t

specgram.__doc__ = specgram.__doc__ % kwdocd

_coh_error = """Coherence is calculated by averaging over *NFFT*
length segments.  Your signal is too short for your choice of *NFFT*.
"""
def cohere(x, y, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning,
        noverlap=0, pad_to=None, sides='default', scale_by_freq=None):
    """
    The coherence between *x* and *y*.  Coherence is the normalized
    cross spectral density:

    .. math::

        C_{xy} = \\frac{|P_{xy}|^2}{P_{xx}P_{yy}}

    *x*, *y*
        Array or sequence containing the data
    %(PSD)s
    The return value is the tuple (*Cxy*, *f*), where *f* are the
    frequencies of the coherence vector. For cohere, scaling the
    individual densities by the sampling frequency has no effect, since
    the factors cancel out.

    .. seealso::
        :func:`psd` and :func:`csd`:
            For information about the methods used to compute
            :math:`P_{xy}`, :math:`P_{xx}` and :math:`P_{yy}`.
    """

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

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

cohere.__doc__ = cohere.__doc__ % kwdocd

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

    .. math::

      r_{ij} = \\frac{C_{ij}}{\\sqrt{C_{ii}C_{jj}}}
    """
    warnings.warn("Use numpy.corrcoef", DeprecationWarning)
    kw = dict(rowvar=False)
    return np.corrcoef(*args, **kw)


def polyfit(*args, **kwargs):
    u"""
    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
    `vandermonds
    <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*) \N{MULTIPLICATION SIGN} *N* + 1) matrix,
    *p* is a *N*+1 length vector, and *y* is a (len(*x*)
    \N{MULTIPLICATION SIGN} 1) vector.

    This equation can be solved as

    .. math::

      p = (X_t X)^-1 X_t y

    where :math:`X_t` is the transpose of *X* and -1 denotes the
    inverse.  Numerically, however, this is not a good method, so we
    use :func:`numpy.linalg.lstsq`.

    For more info, see `least squares fitting
    <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.

    .. seealso::
        :func:`polyval`
    """
    warnings.warn("use numpy.poyfit", DeprecationWarning)
    return np.polyfit(*args, **kwargs)




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

    .. seealso::
        :func:`polyfit`
    """
    warnings.warn("use numpy.polyval", DeprecationWarning)
    return np.polyval(*args, **kwargs)

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 np.vander(*args, **kwargs)


def donothing_callback(*args):
    pass

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

    u"""
    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 :func:`psd`.

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

      - *Cxy*: a dictionary of (*i*, *j*) tuples -> coherence vector for that
        pair.  I.e., ``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.  The 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, :func:`cohere_pairs` can be much more
    efficient than just calling :func:`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\N{SUPERSCRIPT TWO}).  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*, :func:`cohere_pairs` will
    still give significant performace gains over calling
    :func:`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 non-redundant pairs,
    *preferSpeedOverMemory* = *True* delivered a 33% performace boost
    on a 1.7GHZ Athlon with 512MB RAM compared with
    *preferSpeedOverMemory* = *False*.  But both solutions were more
    than 10x faster than naievly crunching all possible pairs through
    cohere.

    .. seealso::
        :file:`test/cohere_pairs_test.py` in the src tree:
            For an example script that shows that this
            :func:`cohere_pairs` and :func:`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 = np.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 np.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
    else:
        windowVals = window(np.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 = np.zeros( (numSlices,numFreqs), dtype=np.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] = np.divide(np.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] * np.conjugate(FFTSlices[j])
        if numSlices>1: Pxy = np.mean(Pxy)
        Pxy = np.divide(Pxy, normVal)
        Cxy[(i,j)] = np.divide(np.absolute(Pxy)**2, Pxx[i]*Pxx[j])
        Phase[(i,j)] =  np.arctan2(Pxy.imag, Pxy.real)

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



def entropy(y, bins):
    r"""
    Return the entropy of the data in *y*.

    .. math::

      \sum p_i \log_2(p_i)

    where :math:`p_i` is the probability of observing *y* in the
    :math:`i^{th}` bin of *bins*.  *bins* can be a number of bins or a
    range of bins; see :func:`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 = np.histogram(y, bins)
    n = n.astype(np.float_)

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

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

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

def hist(y, bins=10, normed=0):
    """
    Return the histogram of *y* with *bins* equally sized bins.  If
    bins is an array, use those 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 returned tuple.  If *normed* is *True*, return the probability
    density :math:`\\frac{n}{(len(y)\mathrm{dbin}}`.

    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 np.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./(np.sqrt(2*np.pi)*sigma)*np.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 = np.linspace(minx, maxx, N), where N is even'


    dx = x[1]-x[0]


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

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

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


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

def trapz(x, y):
    """
    Trapezoidal integral of *y*(*x*).
    """
    warnings.warn("Use numpy.trapz(y,x) instead of trapz(x,y)", DeprecationWarning)
    return np.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 np.sum(0.5*np.diff(x)*(y[1:]+y[:-1]))



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

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

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

    return ind

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

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

    Return value is a tuple of the form (*Pcomponents*, *Trans*,
    *fracVar*) where:

      - *Pcomponents* : a (numVars, 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 = np.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 = np.dot(Trans,P)
    return Pcomponents, Trans, fracVar[ind]

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 ith
    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 = np.array(x).ravel()  # we need a copy
    x.sort()
    Nx = len(x)

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

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

def prctile_rank(x, p):
    """
    Return the rank 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 = np.arange(100.0/p, 100.0, 100.0/p)
    else:
        p = np.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 np.searchsorted(ptiles, x)

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 = np.asarray(M, np.float_)
    if dim:
        M = (M - M.mean(axis=0)) / M.std(axis=0)
    else:
        M = (M - M.mean(axis=1)[:,np.newaxis])
        M = M / M.std(axis=1)[:,np.newaxis]
    return M



def rk4(derivs, y0, t):
    """
    Integrate 1D or ND system of ODEs using 4-th order Runge-Kutta.
    This is a toy implementation which may be useful if you find
    yourself stranded on a system w/o scipy.  Otherwise use
    :func:`scipy.integrate`.

    *y0*
        initial state vector

    *t*
        sample times

    *derivs*
        returns the derivative of the system and has the
        signature ``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 = np.zeros( (len(t),), np.float_)
    else:
        yout = np.zeros( (len(t), Ny), np.float_)


    yout[0] = y0
    i = 0

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

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

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


def bivariate_normal(X, Y, sigmax=1.0, sigmay=1.0,
                     mux=0.0, muy=0.0, sigmaxy=0.0):
    """
    Bivariate Gaussian distribution for equal shape *X*, *Y*.

    See `bivariate normal
    <http://mathworld.wolfram.com/BivariateNormalDistribution.html>`_
    at mathworld.
    """
    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*np.pi*sigmax*sigmay*np.sqrt(1-rho**2)
    return np.exp( -z/(2*(1-rho**2))) / denom

def get_xyz_where(Z, Cond):
    """
    *Z* and *Cond* are *M* x *N* 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*, and *z* are
    1D arrays.
    """
    X,Y = np.indices(Z.shape)
    return X[Cond], Y[Cond], Z[Cond]

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

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

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
    http://softsurfer.com/Archive/algorithm_0102/algorithm_0102.htm#Distance%20to%20Ray%20or%20Segment
    """
    p = np.asarray(p, np.float_)
    s0 = np.asarray(s0, np.float_)
    s1 = np.asarray(s1, np.float_)
    v = s1 - s0
    w = p - s0

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

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

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

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


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)
    else:
        x = window(detrend(x))
    z = np.fft.fft(x)
    a = 2.*np.pi*1j
    phase = a * np.random.rand(len(x))
    z = z*np.exp(phase)
    return np.fft.ifft(z).real


def liaupunov(x, fprime):
    """
    *x* is a very long trajectory from a map, and *fprime* returns the
    derivative of *x*.

    Returns :
    .. math::

        \lambda = \\frac{1}{n}\\sum \\ln|f^'(x_i)|

    .. seealso::
        Sec 10.5 Strogatz (1994) "Nonlinear Dynamics and Chaos".

        `Wikipedia article on Lyapunov Exponent
        <http://en.wikipedia.org/wiki/Lyapunov_exponent>`_.

    .. note::
        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 np.mean(np.log(np.absolute(fprime(x))))

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 to grab data from the buffer
    and plot it to screen less freqeuently than the incoming.

    If you set the *dataLim* attr to
    :class:`~matplotlib.transforms.BBox` (eg
    :attr:`matplotlib.Axes.dataLim`), the *dataLim* will be updated as
    new data come in.

    TODO: add a grow method that will extend nmax

    .. note::

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

    def register(self, func, N):
        """
        Call *func* every time *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:
                    func(self)

        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]

    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 = np.empty((n,), dtype=np.float_)
    w[:] = 1.0/n
    return np.convolve(x, w, mode='valid')

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 :func:`load` function 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')
        else:
            fh = file(fname,'w')
    elif hasattr(fname, 'seek'):
        fh = fname
    else:
        raise ValueError('fname must be a string or file handle')


    X = np.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




def load(fname,comments='#',delimiter=None, converters=None,skiprows=0,
         usecols=None, unpack=False, dtype=np.float_):
    """
    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; for that, use :mod:`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 (or the optional
      *dtype* if specified).  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)

    - *dtype*: the array will have this dtype.  default: ``numpy.float_``

    .. seealso::
        See :file:`examples/pylab_examples/load_converter.py` in the source tree:
           Exercises 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()
    else:
        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 = splitfunc(line)
            row = [converterseq[j](vals[j]) for j in usecols]
        else:
            row = [converterseq[j](val)
                      for j,val in enumerate(splitfunc(line))]
        thisLen = len(row)
        X.append(row)

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


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
    of A CONSISTENTLY WELL BEHAVED METHOD OF INTERPOLATION by Russel
    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
    arbitrary.

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

    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=np.asarray(x, np.float_)
    y=np.asarray(y, np.float_)

    yp=np.zeros(y.shape, np.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


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);
      plot(x,y,'o',xi,yi)

    The interpolation method is described in the article A
    CONSISTENTLY WELL BEHAVED METHOD OF INTERPOLATION by Russell
    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=np.asarray(x, np.float_)
    y=np.asarray(y, np.float_)
    assert x.shape == y.shape
    N=len(y)

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

    xi=np.asarray(xi, np.float_)
    yi=np.zeros(xi.shape, np.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 = np.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 * np.choose(np.array(np.sign(dy1dy2), np.int32)+1,
                                 ((2*xi-xidx-xidxp1)/((dy1-dy2)*(xidxp1-xidx)),
                                  0.0,
                                  1/(dy1+dy2),))
    return yi

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, =  np.nonzero(nxutils.points_inside_poly(points, verts))
    return res

def poly_below(ymin, xs, ys):
    """
    given a arrays *xs* and *ys*, return the vertices of a polygon
    that has a scalar lower bound *ymin* and an upper bound at the *ys*.

    intended for use with Axes.fill, eg::

      xv, yv = poly_below(0, x, y)
      ax.fill(xv, yv)
    """
    return poly_between(xs, ys, xmin)


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*np.ones(Nx)

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

    x = np.concatenate( (x, x[::-1]) )
    y = np.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:
http://www.opensource.org/licenses/bsd-license.php

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.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

"""

import operator
import math


#*****************************************************************************
# Globals

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

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
    hardware.
    """

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

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

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


#from numpy import zeros_like
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 np.zeros_like(a)

#from numpy import sum as sum_flat
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 np.sum(a)

#from numpy import mean as mean_flat
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 np.mean(a)

def rms_flat(a):
    """
    Return the root mean square of all the elements of *a*, flattened out.
    """
    return np.sqrt(np.mean(np.absolute(a)**2))

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

    Implemented as a separate function (not a call to :func:`norm` for speed).
    """
    return np.sum(np.absolute(a))

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

    Implemented as a separate function (not a call to :func:`norm` for speed).
    """
    return np.sqrt(np.sum(np.absolute(a)**2))

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 np.amax(np.absolute(a))
    else:
        return (np.sum(np.absolute(a)**p))**(1.0/p)

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
    :func:`numpy.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 :func:`range` and
    :func:`numpy.arange`. This is deliberate, since :func:`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 :func:`range` can be obtained by
    setting the keyword *closed* = 0, in this case, :func:`frange`
    basically becomes :func:numpy.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*.

    :func:`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). :func:`numpy.arange`
    doesn't offer this option.

    Examples::

      >>> 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  ])
    """

    #defaults
    kw.setdefault('closed',1)
    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
    try:
        npts=kw['npts']
        delta=(xfin-xini)/float(npts-endpoint)
    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 np.arange(npts)*delta+xini
# end frange()

#import numpy.diag as diagonal_matrix
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 np.diag(diag)

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

    For ranks higher than 2, this object is simply a multi-index Kronecker
    delta::

                            /  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",
                       DeprecationWarning)
        dtype = typecode
    iden = np.zeros((n,)*rank, dtype)
    for i in range(n):
        idx = (i,)*rank
        iden[idx] = 1
    return iden

def base_repr (number, base = 2, padding = 0):
    """
    Return the representation of a *number* in any given *base*.
    """
    chars = '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ'
    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))

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

    This is more efficient than using :func:`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','')

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

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
    else:
        return len(bin_n)

def isvector(X):
    """
    Like the Matlab (TM) function with the same name, returns *True*
    if the supplied numpy array or matrix *X* looks like a vector,
    meaning it has a one non-singleton axis (i.e., it can have
    multiple axes, but all must have length 1, except for one of
    them).

    If you just want to see if the array has 1 axis, use X.ndim == 1.
    """
    return np.prod(X.shape)==np.max(X.shape)

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

    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 np.fromfunction(function, dimensions, **kwargs)

### end fperez numutils code


def rem(x,y):
    """
    Deprecated - see :func:`numpy.remainder`
    """
    raise NotImplementedError('Deprecated - see numpy.remainder')

def norm(x,y=2):
    """
    Deprecated - see :func:`numpy.linalg.norm`
    """
    raise NotImplementedError('Deprecated - see numpy.linalg.norm')


def orth(A):
    """
    Deprecated - needs clean room implementation
    """
    raise NotImplementedError('Deprecated - needs clean room implementation')

def rank(x):
    """
    Deprecated - see :func:`numpy.rank`
    """
    raise NotImplementedError('Deprecated - see numpy.rank')

def sqrtm(x):
    """
    Deprecated - needs clean room implementation
    """
    raise NotImplementedError('Deprecated - see scipy.linalg.sqrtm')


def mfuncC(f, x):
    """
    Deprecated
    """
    raise NotImplementedError('Deprecated - needs clean room implementation')

def approx_real(x):
    """
    Deprecated - needs clean room implementation
    """
    raise NotImplementedError('Deprecated - needs clean room implementation')

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


def safe_isnan(x):
    ':func:`numpy.isnan` for arbitrary types'
    if cbook.is_string_like(x):
        return False
    try: b = np.isnan(x)
    except NotImplementedError: return False
    except TypeError: return False
    else: return b

def safe_isinf(x):
    ':func:`numpy.isinf` for arbitrary types'
    if cbook.is_string_like(x):
        return False
    try: b = np.isinf(x)
    except NotImplementedError: return False
    except TypeError: return False
    else: return b

def rec_view(rec):
    """
    Return a view of an ndarray as a recarray

    .. seealso::

       http://projects.scipy.org/pipermail/numpy-discussion/2008-August/036429.html
    """
    return rec.view(np.recarray)
    #return rec.view(dtype=(np.record, rec.dtype), type=np.recarray)

def rec_append_field(rec, name, arr, dtype=None):
    """
    Return a new record array with field name populated with data from
    array *arr*.  This function is Deprecated. Please use
    :func:`rec_append_fields`.
    """
    warnings.warn("use rec_append_fields", DeprecationWarning)
    return rec_append_fields(rec, name, arr, dtype)

def rec_append_fields(rec, names, arrs, dtypes=None):
    """
    Return a new record array with field names populated with data
    from arrays in *arrs*.  If appending a single field, then *names*,
    *arrs* and *dtypes* do not have to be lists. They can just be the
    values themselves.
    """
    if (not cbook.is_string_like(names) and cbook.iterable(names) \
            and len(names) and cbook.is_string_like(names[0])):
        if len(names) != len(arrs):
            raise ValueError, "number of arrays do not match number of names"
    else: # we have only 1 name and 1 array
        names = [names]
        arrs = [arrs]
    arrs = map(np.asarray, arrs)
    if dtypes is None:
        dtypes = [a.dtype for a in arrs]
    elif not cbook.iterable(dtypes):
        dtypes = [dtypes]
    if len(arrs) != len(dtypes):
        if len(dtypes) == 1:
            dtypes = dtypes * len(arrs)
        else:
            raise ValueError, "dtypes must be None, a single dtype or a list"

    newdtype = np.dtype(rec.dtype.descr + zip(names, dtypes))
    newrec = np.empty(rec.shape, dtype=newdtype)
    for field in rec.dtype.fields:
        newrec[field] = rec[field]
    for name, arr in zip(names, arrs):
        newrec[name] = arr
    return rec_view(newrec)


def rec_drop_fields(rec, names):
    """
    Return a new numpy record array with fields in *names* dropped.
    """

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

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

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

    return rec_view(newrec)



def rec_groupby(r, groupby, stats):
    """
    *r* is a numpy record array

    *groupby* is a sequence of record array attribute names that
    together form the grouping key.  eg ('date', 'productcode')

    *stats* is a sequence of (*attr*, *func*, *outname*) tuples which
    will call ``x = func(attr)`` and assign *x* to the record array
    output with attribute *outname*.  For example::

      stats = ( ('sales', len, 'numsales'), ('sales', np.mean, 'avgsale') )

    Return record array has *dtype* names for each attribute name in
    the the *groupby* argument, with the associated group values, and
    for each outname name in the *stats* argument, with the associated
    stat summary output.
    """
    # build a dictionary from groupby keys-> list of indices into r with
    # those keys
    rowd = dict()
    for i, row in enumerate(r):
        key = tuple([row[attr] for attr in groupby])
        rowd.setdefault(key, []).append(i)

    # sort the output by groupby keys
    keys = rowd.keys()
    keys.sort()

    rows = []
    for key in keys:
        row = list(key)
        # get the indices for this groupby key
        ind = rowd[key]
        thisr = r[ind]
        # call each stat function for this groupby slice
        row.extend([func(thisr[attr]) for attr, func, outname in stats])
        rows.append(row)

    # build the output record array with groupby and outname attributes
    attrs, funcs, outnames = zip(*stats)
    names = list(groupby)
    names.extend(outnames)
    return np.rec.fromrecords(rows, names=names)



def rec_summarize(r, summaryfuncs):
    """
    *r* is a numpy record array

    *summaryfuncs* is a list of (*attr*, *func*, *outname*) tuples
    which will apply *func* to the the array *r*[attr] and assign the
    output to a new attribute name *outname*.  The returned record
    array is identical to *r*, with extra arrays for each element in
    *summaryfuncs*.

    """

    names = list(r.dtype.names)
    arrays = [r[name] for name in names]

    for attr, func, outname in summaryfuncs:
        names.append(outname)
        arrays.append(np.asarray(func(r[attr])))

    return np.rec.fromarrays(arrays, names=names)


def rec_join(key, r1, r2, jointype='inner', defaults=None, r1postfix='1', r2postfix='2'):
    """
    Join record arrays *r1* and *r2* on *key*; *key* is a tuple of
    field names -- if *key* is a string it is assumed to be a single
    attribute name. 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*.

    *r1* (also *r2*) must not have any duplicate keys.

    The *jointype* keyword can be 'inner', 'outer', 'leftouter'.  To
    do a rightouter join just reverse *r1* and *r2*.

    The *defaults* keyword is a dictionary filled with
    ``{column_name:default_value}`` pairs.

    The keywords *r1postfix* and *r2postfix* are postfixed to column names
    (other than keys) that are both in *r1* and *r2*.
    """

    if cbook.is_string_like(key):
        key = (key, )

    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])

    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())

    common_keys = r1keys & r2keys

    r1ind = np.array([r1d[k] for k in common_keys])
    r2ind = np.array([r2d[k] for k in common_keys])

    common_len = len(common_keys)
    left_len = right_len = 0
    if jointype == "outer" or jointype == "leftouter":
        left_keys = r1keys.difference(r2keys)
        left_ind = np.array([r1d[k] for k in left_keys])
        left_len = len(left_ind)
    if jointype == "outer":
        right_keys = r2keys.difference(r1keys)
        right_ind = np.array([r2d[k] for k in right_keys])
        right_len = len(right_ind)

    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 != np.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])
        else:
            return (name, dt2.descr[0][1])


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

    def mapped_r1field(name):
        """
        The column name in *newrec* that corresponds to the column in *r1*.
        """
        if name in key or name not in r2.dtype.names: return name
        else: return name + r1postfix

    def mapped_r2field(name):
        """
        The column name in *newrec* that corresponds to the column in *r2*.
        """
        if name in key or name not in r1.dtype.names: return name
        else: return name + r2postfix

    r1desc = [(mapped_r1field(desc[0]), desc[1]) for desc in r1.dtype.descr if desc[0] not in key]
    r2desc = [(mapped_r2field(desc[0]), desc[1]) for desc in r2.dtype.descr if desc[0] not in key]
    newdtype = np.dtype(keydesc + r1desc + r2desc)

    newrec = np.empty(common_len + left_len + right_len, dtype=newdtype)

    if jointype != 'inner' and defaults is not None: # fill in the defaults enmasse
        newrec_fields = newrec.dtype.fields.keys()
        for k, v in defaults.items():
            if k in newrec_fields:
                newrec[k] = v

    for field in r1.dtype.names:
        newfield = mapped_r1field(field)
        if common_len:
            newrec[newfield][:common_len] = r1[field][r1ind]
        if (jointype == "outer" or jointype == "leftouter") and left_len:
            newrec[newfield][common_len:(common_len+left_len)] = r1[field][left_ind]

    for field in r2.dtype.names:
        newfield = mapped_r2field(field)
        if field not in key and common_len:
            newrec[newfield][:common_len] = r2[field][r2ind]
        if jointype == "outer" and right_len:
            newrec[newfield][-right_len:] = r2[field][right_ind]

    newrec.sort(order=key)

    return rec_view(newrec)


def csv2rec(fname, comments='#', skiprows=0, checkrows=0, delimiter=',',
            converterd=None, names=None, missing='', missingd=None,
            use_mrecords=True):
    """
    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.

    - *converted*: 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

    - *missingd* is a dictionary mapping munged column names to field values
      which signify that the field does not contain actual data and should
      be masked, e.g. '0000-00-00' or 'unused'

    - *missing*: a string whose value signals a missing field regardless of
      the column it appears in

    - *use_mrecords*: if True, return an mrecords.fromrecords record array if any of the data are missing

      If no rows are found, *None* is returned -- see :file:`examples/loadrec.py`
    """

    if converterd is None:
        converterd = dict()

    if missingd is None:
        missingd = {}

    import dateutil.parser
    import datetime
    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):
            self.fh.close()

        def seek(self, arg):
            self.fh.seek(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

    process_skiprows(reader)

    def ismissing(name, val):
        "Should the value val in column name be masked?"

        if val == missing or val == missingd.get(name) or val == '':
            return True
        else:
            return False

    def with_default_value(func, default):
        def newfunc(name, val):
            if ismissing(name, val):
                return default
            else:
                return func(val)
        return newfunc


    def mybool(x):
        if x=='True': return True
        elif x=='False': return False
        else: raise ValueError('invalid bool')

    dateparser = dateutil.parser.parse
    mydateparser = with_default_value(dateparser, datetime.date(1,1,1))
    myfloat = with_default_value(float, np.nan)
    myint = with_default_value(int, -1)
    mystr = with_default_value(str, '')
    mybool = with_default_value(mybool, None)

    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()
    mydate = with_default_value(mydate, datetime.date(1,1,1))

    def get_func(name, item, func):
        # promote functions in this order
        funcmap = {mybool:myint,myint:myfloat, myfloat:mydate, mydate:mydateparser, mydateparser:mystr}
        try: func(name, item)
        except:
            if func==mystr:
                raise ValueError('Could not find a working conversion function')
            else: return get_func(name, 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 = [mybool]*len(row)
            if checkrows and i>checkrows:
                break
            #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(name, item, func)
                else:
                    # how should we handle custom converters and defaults?
                    func = with_default_value(func, None)
                converters[j] = func
        return converters

    # Get header and remove invalid characters
    needheader = names is None

    if needheader:
        for row in reader:
            #print 'csv2rec', row
            if len(row) and row[0].startswith(comments):
                continue
            headers = row
            break

        # remove these chars
        delete = set("""~!@#$%^&*()-=+~\|]}[{';: /?.>,<""")
        delete.add('"')

        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)
            else:
                names.append(item)
            seen[item] = cnt+1

    else:
        if cbook.is_string_like(names):
            names = [n.strip() for n in names.split(',')]

    # 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
    fh.seek(0)
    reader = csv.reader(fh, delimiter=delimiter)
    process_skiprows(reader)
    if needheader:
        skipheader = reader.next()

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

    if not len(rows):
        return None

    if use_mrecords and np.any(rowmasks):
        try: from numpy.ma import mrecords
        except ImportError:
            raise RuntimeError('numpy 1.05 or later is required for masked array support')
        else:
            r = mrecords.fromrecords(rows, names=names, mask=rowmasks)
    else:
        r = np.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)

    def fromstr(self, s):
        return s

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

    def fromstr(self, s):
        return float(s)/self.scale


class FormatInt(FormatObj):

    def tostr(self, x):
        return '%d'%int(x)

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

    def fromstr(self, s):
        return int(s)

class FormatBool(FormatObj):


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

    def fromstr(self, s):
        return bool(s)

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)

    def fromstr(self, x):
        import dateutil.parser
        return dateutil.parser.parse(x).date()

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

    def fromstr(self, x):
        import dateutil.parser
        return dateutil.parser.parse(x)




defaultformatd = {
    np.bool_ : FormatBool(),
    np.int16 : FormatInt(),
    np.int32 : FormatInt(),
    np.int64 : FormatInt(),
    np.float32 : FormatFloat(),
    np.float64 : FormatFloat(),
    np.object_ : FormatObj(),
    np.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

def rec2txt(r, header=None, padding=3, precision=3):
    """
    Returns a textual representation of a record array.

    *r*: numpy recarray

    *header*: list of column headers

    *padding*: space between each column

    *precision*: number of decimal places to use for floats.
        Set to an integer to apply to all floats.  Set to a
        list of integers to apply precision individually.
        Precision for non-floats is simply ignored.

    Example::

      precision=[0,2,3]

    Output::

      ID    Price   Return
      ABC   12.54    0.234
      XYZ    6.32   -0.076
    """

    if cbook.is_numlike(precision):
        precision = [precision]*len(r.dtype)

    def get_type(item,atype=int):
        tdict = {None:int, int:float, float:str}
        try: atype(str(item))
        except: return get_type(item,tdict[atype])
        return atype

    def get_justify(colname, column, precision):
        ntype = type(column[0])

        if ntype==np.str or ntype==np.str_ or ntype==np.string0 or ntype==np.string_:
            length = max(len(colname),column.itemsize)
            return 0, length+padding, "%s" # left justify

        if ntype==np.int or ntype==np.int16 or ntype==np.int32 or ntype==np.int64 or ntype==np.int8 or ntype==np.int_:
            length = max(len(colname),np.max(map(len,map(str,column))))
            return 1, length+padding, "%d" # right justify

        # JDH: my powerbook does not have np.float96 using np 1.3.0
        """
        In [2]: np.__version__
        Out[2]: '1.3.0.dev5948'

        In [3]: !uname -a
        Darwin Macintosh-5.local 9.4.0 Darwin Kernel Version 9.4.0: Mon Jun  9 19:30:53 PDT 2008; root:xnu-1228.5.20~1/RELEASE_I386 i386 i386

        In [4]: np.float96
        ---------------------------------------------------------------------------
        AttributeError                            Traceback (most recent call la
        """
        if ntype==np.float or ntype==np.float32 or ntype==np.float64 or (hasattr(np, 'float96') and (ntype==np.float96)) or ntype==np.float_:
            fmt = "%." + str(precision) + "f"
            length = max(len(colname),np.max(map(len,map(lambda x:fmt%x,column))))
            return 1, length+padding, fmt   # right justify

        return 0, max(len(colname),np.max(map(len,map(str,column))))+padding, "%s"

    if header is None:
        header = r.dtype.names

    justify_pad_prec = [get_justify(header[i],r.__getitem__(colname),precision[i]) for i, colname in enumerate(r.dtype.names)]

    justify_pad_prec_spacer = []
    for i in range(len(justify_pad_prec)):
        just,pad,prec = justify_pad_prec[i]
        if i == 0:
            justify_pad_prec_spacer.append((just,pad,prec,0))
        else:
            pjust,ppad,pprec = justify_pad_prec[i-1]
            if pjust == 0 and just == 1:
                justify_pad_prec_spacer.append((just,pad-padding,prec,0))
            elif pjust == 1 and just == 0:
                justify_pad_prec_spacer.append((just,pad,prec,padding))
            else:
                justify_pad_prec_spacer.append((just,pad,prec,0))

    def format(item, just_pad_prec_spacer):
        just, pad, prec, spacer = just_pad_prec_spacer
        if just == 0:
            return spacer*' ' + str(item).ljust(pad)
        else:
            if get_type(item) == float:
                item = (prec%float(item))
            elif get_type(item) == int:
                item = (prec%int(item))

            return item.rjust(pad)

    textl = []
    textl.append(''.join([format(colitem,justify_pad_prec_spacer[j]) for j, colitem in enumerate(header)]))
    for i, row in enumerate(r):
        textl.append(''.join([format(colitem,justify_pad_prec_spacer[j]) for j, colitem in enumerate(row)]))
        if i==0:
            textl[0] = textl[0].rstrip()

    text = os.linesep.join(textl)
    return text



def rec2csv(r, fname, delimiter=',', formatd=None, missing='',
            missingd=None):
    """
    Save the data from numpy recarray *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'

    .. seealso::
        :func:`csv2rec`:
            For information about *missing* and *missingd*, which can
            be used to fill in masked values into your CSV file.
    """

    if missingd is None:
        missingd = dict()

    def with_mask(func):
        def newfunc(val, mask, mval):
            if mask:
                return mval
            else:
                return func(val)
        return newfunc

    formatd = get_formatd(r, formatd)
    funcs = []
    for i, name in enumerate(r.dtype.names):
        funcs.append(with_mask(csvformat_factory(formatd[name]).tostr))

    fh, opened = cbook.to_filehandle(fname, 'w', return_opened=True)
    writer = csv.writer(fh, delimiter=delimiter)
    header = r.dtype.names
    writer.writerow(header)

    # Our list of specials for missing values
    mvals = []
    for name in header:
        mvals.append(missingd.get(name, missing))

    ismasked = False
    if len(r):
        row = r[0]
        ismasked = hasattr(row, '_fieldmask')

    for row in r:
        if ismasked:
            row, rowmask = row.item(), row._fieldmask.item()
        else:
            rowmask = [False] * len(row)
        writer.writerow([func(val, mask, mval) for func, val, mask, mval
                         in zip(funcs, row, rowmask, mvals)])
    if opened:
        fh.close()

def griddata(x,y,z,xi,yi):
    """
    ``zi = griddata(x,y,z,xi,yi)`` fits a surface of the form *z* =
    *f*(*x*, *y*) to the data in the (usually) nonuniformly spaced
    vectors (*x*, *y*, *z*).  :func:`griddata` interpolates this
    surface at the points specified by (*xi*, *yi*) to produce
    *zi*. *xi* and *yi* must describe a regular grid, can be either 1D
    or 2D, but must be monotonically increasing.

    A masked array is returned if any grid points are outside convex
    hull defined by input data (no extrapolation is done).

    Uses natural neighbor interpolation based on Delaunay
    triangulation.  By default, this algorithm is provided by the
    :mod:`matplotlib.delaunay` package, written by Robert Kern.  The
    triangulation algorithm in this package is known to fail on some
    nearly pathological cases. For this reason, a separate toolkit
    (:mod:`mpl_tookits.natgrid`) has been created that provides a more
    robust algorithm fof triangulation and interpolation.  This
    toolkit is based on the NCAR natgrid library, which contains code
    that is not redistributable under a BSD-compatible license.  When
    installed, this function will use the :mod:`mpl_toolkits.natgrid`
    algorithm, otherwise it will use the built-in
    :mod:`matplotlib.delaunay` package.

    The natgrid matplotlib toolkit can be downloaded from
    http://sourceforge.net/project/showfiles.php?group_id=80706&package_id=142792
    """
    try:
        from mpl_toolkits.natgrid import _natgrid, __version__
        _use_natgrid = True
    except ImportError:
        import matplotlib.delaunay as delaunay
        from matplotlib.delaunay import  __version__
        _use_natgrid = False
    if not griddata._reported:
        if _use_natgrid:
            verbose.report('using natgrid version %s' % __version__)
        else:
            verbose.report('using delaunay version %s' % __version__)
        griddata._reported = True
    if xi.ndim != yi.ndim:
        raise TypeError("inputs xi and yi must have same number of dimensions (1 or 2)")
    if xi.ndim != 1 and xi.ndim != 2:
        raise TypeError("inputs xi and yi must be 1D or 2D.")
    if not len(x)==len(y)==len(z):
        raise TypeError("inputs x,y,z must all be 1D arrays of the same length")
    # remove masked points.
    if hasattr(z,'mask'):
        x = x.compress(z.mask == False)
        y = y.compress(z.mask == False)
        z = z.compressed()
    if _use_natgrid: # use natgrid toolkit if available.
        if xi.ndim == 2:
            xi = xi[0,:]
            yi = yi[:,0]
        # override default natgrid internal parameters.
        _natgrid.seti('ext',0)
        _natgrid.setr('nul',np.nan)
        # cast input arrays to doubles (this makes a copy)
        x = x.astype(np.float)
        y = y.astype(np.float)
        z = z.astype(np.float)
        xo = xi.astype(np.float)
        yo = yi.astype(np.float)
        if min(xo[1:]-xo[0:-1]) < 0 or min(yo[1:]-yo[0:-1]) < 0:
            raise ValueError, 'output grid defined by xi,yi must be monotone increasing'
        # allocate array for output (buffer will be overwritten by nagridd)
        zo = np.empty((yo.shape[0],xo.shape[0]), np.float)
        _natgrid.natgridd(x,y,z,xo,yo,zo)
    else: # use Robert Kern's delaunay package from scikits (default)
        if xi.ndim != yi.ndim:
            raise TypeError("inputs xi and yi must have same number of dimensions (1 or 2)")
        if xi.ndim != 1 and xi.ndim != 2:
            raise TypeError("inputs xi and yi must be 1D or 2D.")
        if xi.ndim == 1:
            xi,yi = np.meshgrid(xi,yi)
        # triangulate data
        tri = delaunay.Triangulation(x,y)
        # interpolate data
        interp = tri.nn_interpolator(z)
        zo = interp(xi,yi)
    # mask points on grid outside convex hull of input data.
    if np.any(np.isnan(zo)):
        zo = np.ma.masked_where(np.isnan(zo),zo)
    return zo
griddata._reported = False

##################################################
# Linear interpolation algorithms
##################################################
def less_simple_linear_interpolation( x, y, xi, extrap=False ):
    """
    This function provides simple (but somewhat less so than
    :func:`cbook.simple_linear_interpolation`) linear interpolation.
    :func:`simple_linear_interpolation` will give a list of point
    between a start and an end, while this does true linear
    interpolation at an arbitrary set of points.

    This is very inefficient linear interpolation meant to be used
    only for a small number of points in relatively non-intensive use
    cases.  For real linear interpolation, use scipy.
    """
    if cbook.is_scalar(xi): xi = [xi]

    x = np.asarray(x)
    y = np.asarray(y)
    xi = np.asarray(xi)

    s = list(y.shape)
    s[0] = len(xi)
    yi = np.tile( np.nan, s )

    for ii,xx in enumerate(xi):
        bb = x == xx
        if np.any(bb):
            jj, = np.nonzero(bb)
            yi[ii] = y[jj[0]]
        elif xx<x[0]:
            if extrap:
                yi[ii] = y[0]
        elif xx>x[-1]:
            if extrap:
                yi[ii] = y[-1]
        else:
            jj, = np.nonzero(x<xx)
            jj = max(jj)

            yi[ii] = y[jj] + (xx-x[jj])/(x[jj+1]-x[jj]) * (y[jj+1]-y[jj])

    return yi

def slopes(x,y):
    """
    :func:`slopes` calculates the slope *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
    of A CONSISTENTLY WELL BEHAVED METHOD OF INTERPOLATION by Russel
    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 arbitrary.

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

    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=np.asarray(x, np.float_)
    y=np.asarray(y, np.float_)

    yp=np.zeros(y.shape, np.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


def stineman_interp(xi,x,y,yp=None):
    """
    Given data vectors *x* and *y*, the slope vector *yp* and a new
    abscissa vector *xi*, the function :func:`stineman_interp` 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);
      plot(x,y,'o',xi,yi)

    The interpolation method is described in the article A
    CONSISTENTLY WELL BEHAVED METHOD OF INTERPOLATION by Russell
    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 :func:`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 an extrapolation.  The relevance of the data obtained from
    this, of course, is 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=np.asarray(x, np.float_)
    y=np.asarray(y, np.float_)
    assert x.shape == y.shape
    N=len(y)

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

    xi=np.asarray(xi, np.float_)
    yi=np.zeros(xi.shape, np.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 = np.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 * np.choose(np.array(np.sign(dy1dy2), np.int32)+1,
                                 ((2*xi-xidx-xidxp1)/((dy1-dy2)*(xidxp1-xidx)),
                                  0.0,
                                  1/(dy1+dy2),))
    return yi

##################################################
# Code related to things in and around polygons
##################################################
def inside_poly(points, verts):
    """
    *points* is a sequence of *x*, *y* points.
    *verts* is a sequence of *x*, *y* vertices of a polygon.

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

def poly_below(xmin, xs, ys):
    """
    Given a sequence of *xs* and *ys*, return the vertices of a
    polygon that has a horizontal base at *xmin* and an upper bound at
    the *ys*.  *xmin* is a scalar.

    Intended for use with :meth:`matplotlib.axes.Axes.fill`, eg::

      xv, yv = poly_below(0, x, y)
      ax.fill(xv, yv)
    """
    if ma.isMaskedArray(xs) or ma.isMaskedArray(ys):
        nx = ma
    else:
        nx = np

    xs = nx.asarray(xs)
    ys = nx.asarray(ys)
    Nx = len(xs)
    Ny = len(ys)
    assert(Nx==Ny)
    x = xmin*nx.ones(2*Nx)
    y = nx.ones(2*Nx)
    x[:Nx] = xs
    y[:Nx] = ys
    y[Nx:] = ys[::-1]
    return x, y



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
    :meth:`matplotlib.axes.Axes.fill`.
    """
    if ma.isMaskedArray(ylower) or ma.isMaskedArray(yupper) or ma.isMaskedArray(x):
        nx = ma
    else:
        nx = np

    Nx = len(x)
    if not cbook.iterable(ylower):
        ylower = ylower*nx.ones(Nx)

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

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


def is_closed_polygon(X):
    """
    Tests whether first and last object in a sequence are the same.  These are
    presumably coordinates on a polygonal curve, in which case this function
    tests if that curve is closed.
    """
    return np.all(X[0] == X[-1])


def contiguous_regions(mask):
    """
    return a list of (ind0, ind1) such that mask[ind0:ind1].all() is
    True and we cover all such regions

    TODO: this is a pure python implementation which probably has a much faster numpy impl
    """

    in_region = None
    boundaries = []
    for i, val in enumerate(mask):
        if in_region is None and val:
            in_region = i
        elif in_region is not None and not val:
            boundaries.append((in_region, i))
            in_region = None

    if in_region is not None:
        boundaries.append((in_region, i+1))
    return boundaries

##################################################
# Vector and path length geometry calculations
##################################################
def vector_lengths( X, P=2., axis=None ):
    """
    Finds the length of a set of vectors in *n* dimensions.  This is
    like the :func:`numpy.norm` function for vectors, but has the ability to
    work over a particular axis of the supplied array or matrix.

    Computes ``(sum((x_i)^P))^(1/P)`` for each ``{x_i}`` being the
    elements of *X* along the given axis.  If *axis* is *None*,
    compute over all elements of *X*.
    """
    X = np.asarray(X)
    return (np.sum(X**(P),axis=axis))**(1./P)

def distances_along_curve( X ):
    """
    Computes the distance between a set of successive points in *N* dimensions.

    Where *X* is an *M* x *N* array or matrix.  The distances between
    successive rows is computed.  Distance is the standard Euclidean
    distance.
    """
    X = np.diff( X, axis=0 )
    return vector_lengths(X,axis=1)

def path_length(X):
    """
    Computes the distance travelled along a polygonal curve in *N* dimensions.

    Where *X* is an *M* x *N* array or matrix.  Returns an array of
    length *M* consisting of the distance along the curve at each point
    (i.e., the rows of *X*).
    """
    X = distances_along_curve(X)
    return np.concatenate( (np.zeros(1), np.cumsum(X)) )

def quad2cubic(q0x, q0y, q1x, q1y, q2x, q2y):
    """
    Converts a quadratic Bezier curve to a cubic approximation.

    The inputs are the *x* and *y* coordinates of the three control
    points of a quadratic curve, and the output is a tuple of *x* and
    *y* coordinates of the four control points of the cubic curve.
    """
    # c0x, c0y = q0x, q0y
    c1x, c1y = q0x + 2./3. * (q1x - q0x), q0y + 2./3. * (q1y - q0y)
    c2x, c2y = c1x + 1./3. * (q2x - q0x), c1y + 1./3. * (q2y - q0y)
    # c3x, c3y = q2x, q2y
    return q0x, q0y, c1x, c1y, c2x, c2y, q2x, q2y