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

MATLAB 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, 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)

:meth:`contiguous_regions`
    return the indices of the regions spanned by some logical mask

:meth:`cross_from_below`
    return the indices where a 1D array crosses a threshold from below

:meth:`cross_from_above`
    return the indices where a 1D array crosses a threshold from above


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:`recs_join`
    a simple join of multiple recarrays using a single column as a key

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

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

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    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.
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    The power spectral density by Welch's average periodogram method.
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    *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)

    (tcsdtreal(R
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    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

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    Returns the tuple (*Pxy*, *freqs*).

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

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    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.
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        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.

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    Call signature::

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

    Compute the coherence and phase for all pairs *ij*, in *X*.

    *X* is a *numSamples* * *numCols* array

    *ij* is a list of tuples.  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) )

    *preferSpeedOverMemory* is an optional bool. Defaults to true. 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 nonredundant pairs,
    *preferSpeedOverMemory* = True delivered a 33% performance boost
    on a 1.7GHZ Athlon with 512MB RAM compared with
    *preferSpeedOverMemory* = False.  But both solutions were more
    than 10x faster than naively crunching all possible pairs through
    :func:`cohere`.

    Returns::

       (Cxy, Phase, freqs)

    where:

      - *Cxy*: 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*: dictionary of phases of the cross spectral density at
        each frequency for each pair.  Keys are (*i*, *j*).

      - *freqs*: 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 :math:`N` is the
    number of pairs, this function is :math:`O(N)` for most of the
    heavy lifting, whereas calling cohere for each pair is
    :math:`O(N^2)`.  However, because of the caching, it is also more
    memory intensive, making 2 additional complex arrays with
    approximately the same number of elements as *X*.

    See :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.

    .. seealso::

        :func:`psd`
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            :math:`P_{xy}`, :math:`P_{xx}` and :math:`P_{yy}`.
    Nii
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    This function provides simple (but somewhat less so than
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    :func:`simple_linear_interpolation` will give a list of point
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    interpolation at an arbitrary set of points.

    This is very inefficient linear interpolation meant to be used
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    :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,
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    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
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    *points* is a sequence of *x*, *y* points.
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    return the indices into *x* where *x* crosses some threshold from
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        import matplotlib.pyplot as plt

        t = np.arange(0.0, 2.0, 0.1)
        s = np.sin(2*np.pi*t)

        fig = plt.figure()
        ax = fig.add_subplot(111)
        ax.plot(t, s, '-o')
        ax.axhline(0.5)
        ax.axhline(-0.5)

        ind = cross_from_below(s, 0.5)
        ax.vlines(t[ind], -1, 1)

        ind = cross_from_above(s, -0.5)
        ax.vlines(t[ind], -1, 1)

        plt.show()

    .. seealso::

        :func:`cross_from_above` and :func:`contiguous_regions`

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    Finds the length of a set of vectors in *n* dimensions.  This is
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    Converts a quadratic Bezier curve to a cubic approximation.

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