common.py

"""
Functions which are common and require SciPy Base and Level 1 SciPy
(special, linalg)
"""

from __future__ import division, print_function, absolute_import

from numpy import exp, log, asarray, arange, newaxis, hstack, product, array, \
                  zeros, eye, poly1d, r_, rollaxis, sum, fromstring

__all__ = ['logsumexp', 'central_diff_weights', 'derivative', 'pade', 'lena',
           'ascent', 'face']

# XXX: the factorial functions could move to scipy.special, and the others
# to numpy perhaps?


def logsumexp(a, axis=None, b=None):
    """Compute the log of the sum of exponentials of input elements.

    Parameters
    ----------
    a : array_like
        Input array.
    axis : int, optional
        Axis over which the sum is taken. By default `axis` is None,
        and all elements are summed.

        .. versionadded:: 0.11.0
    b : array-like, optional
        Scaling factor for exp(`a`) must be of the same shape as `a` or
        broadcastable to `a`.

        .. versionadded:: 0.12.0

    Returns
    -------
    res : ndarray
        The result, ``np.log(np.sum(np.exp(a)))`` calculated in a numerically
        more stable way. If `b` is given then ``np.log(np.sum(b*np.exp(a)))``
        is returned.

    See Also
    --------
    numpy.logaddexp, numpy.logaddexp2

    Notes
    -----
    Numpy has a logaddexp function which is very similar to `logsumexp`, but
    only handles two arguments. `logaddexp.reduce` is similar to this
    function, but may be less stable.

    Examples
    --------
    >>> from scipy.misc import logsumexp
    >>> a = np.arange(10)
    >>> np.log(np.sum(np.exp(a)))
    9.4586297444267107
    >>> logsumexp(a)
    9.4586297444267107

    With weights

    >>> a = np.arange(10)
    >>> b = np.arange(10, 0, -1)
    >>> logsumexp(a, b=b)
    9.9170178533034665
    >>> np.log(np.sum(b*np.exp(a)))
    9.9170178533034647
    """
    a = asarray(a)
    if axis is None:
        a = a.ravel()
    else:
        a = rollaxis(a, axis)
    a_max = a.max(axis=0)
    if b is not None:
        b = asarray(b)
        if axis is None:
            b = b.ravel()
        else:
            b = rollaxis(b, axis)
        out = log(sum(b * exp(a - a_max), axis=0))
    else:
        out = log(sum(exp(a - a_max), axis=0))
    out += a_max
    return out


def central_diff_weights(Np, ndiv=1):
    """
    Return weights for an Np-point central derivative.

    Assumes equally-spaced function points.

    If weights are in the vector w, then
    derivative is w[0] * f(x-ho*dx) + ... + w[-1] * f(x+h0*dx)

    Parameters
    ----------
    Np : int
        Number of points for the central derivative.
    ndiv : int, optional
        Number of divisions.  Default is 1.

    Notes
    -----
    Can be inaccurate for large number of points.

    """
    if Np < ndiv + 1:
        raise ValueError("Number of points must be at least the derivative order + 1.")
    if Np % 2 == 0:
        raise ValueError("The number of points must be odd.")
    from scipy import linalg
    ho = Np >> 1
    x = arange(-ho,ho+1.0)
    x = x[:,newaxis]
    X = x**0.0
    for k in range(1,Np):
        X = hstack([X,x**k])
    w = product(arange(1,ndiv+1),axis=0)*linalg.inv(X)[ndiv]
    return w


def derivative(func, x0, dx=1.0, n=1, args=(), order=3):
    """
    Find the n-th derivative of a function at a point.

    Given a function, use a central difference formula with spacing `dx` to
    compute the `n`-th derivative at `x0`.

    Parameters
    ----------
    func : function
        Input function.
    x0 : float
        The point at which `n`-th derivative is found.
    dx : int, optional
        Spacing.
    n : int, optional
        Order of the derivative. Default is 1.
    args : tuple, optional
        Arguments
    order : int, optional
        Number of points to use, must be odd.

    Notes
    -----
    Decreasing the step size too small can result in round-off error.

    Examples
    --------
    >>> def f(x):
    ...     return x**3 + x**2
    ...
    >>> derivative(f, 1.0, dx=1e-6)
    4.9999999999217337

    """
    if order < n + 1:
        raise ValueError("'order' (the number of points used to compute the derivative), "
                         "must be at least the derivative order 'n' + 1.")
    if order % 2 == 0:
        raise ValueError("'order' (the number of points used to compute the derivative) "
                         "must be odd.")
    # pre-computed for n=1 and 2 and low-order for speed.
    if n == 1:
        if order == 3:
            weights = array([-1,0,1])/2.0
        elif order == 5:
            weights = array([1,-8,0,8,-1])/12.0
        elif order == 7:
            weights = array([-1,9,-45,0,45,-9,1])/60.0
        elif order == 9:
            weights = array([3,-32,168,-672,0,672,-168,32,-3])/840.0
        else:
            weights = central_diff_weights(order,1)
    elif n == 2:
        if order == 3:
            weights = array([1,-2.0,1])
        elif order == 5:
            weights = array([-1,16,-30,16,-1])/12.0
        elif order == 7:
            weights = array([2,-27,270,-490,270,-27,2])/180.0
        elif order == 9:
            weights = array([-9,128,-1008,8064,-14350,8064,-1008,128,-9])/5040.0
        else:
            weights = central_diff_weights(order,2)
    else:
        weights = central_diff_weights(order, n)
    val = 0.0
    ho = order >> 1
    for k in range(order):
        val += weights[k]*func(x0+(k-ho)*dx,*args)
    return val / product((dx,)*n,axis=0)


def pade(an, m):
    """
    Return Pade approximation to a polynomial as the ratio of two polynomials.

    Parameters
    ----------
    an : (N,) array_like
        Taylor series coefficients.
    m : int
        The order of the returned approximating polynomials.

    Returns
    -------
    p, q : Polynomial class
        The pade approximation of the polynomial defined by `an` is
        `p(x)/q(x)`.

    Examples
    --------
    >>> from scipy import misc
    >>> e_exp = [1.0, 1.0, 1.0/2.0, 1.0/6.0, 1.0/24.0, 1.0/120.0]
    >>> p, q = misc.pade(e_exp, 2)

    >>> e_exp.reverse()
    >>> e_poly = np.poly1d(e_exp)

    Compare ``e_poly(x)`` and the pade approximation ``p(x)/q(x)``

    >>> e_poly(1)
    2.7166666666666668

    >>> p(1)/q(1)
    2.7179487179487181

    """
    from scipy import linalg
    an = asarray(an)
    N = len(an) - 1
    n = N - m
    if n < 0:
        raise ValueError("Order of q <m> must be smaller than len(an)-1.")
    Akj = eye(N+1, n+1)
    Bkj = zeros((N+1, m), 'd')
    for row in range(1, m+1):
        Bkj[row,:row] = -(an[:row])[::-1]
    for row in range(m+1, N+1):
        Bkj[row,:] = -(an[row-m:row])[::-1]
    C = hstack((Akj, Bkj))
    pq = linalg.solve(C, an)
    p = pq[:n+1]
    q = r_[1.0, pq[n+1:]]
    return poly1d(p[::-1]), poly1d(q[::-1])


def lena():
    """
    Get classic image processing example image, Lena, at 8-bit grayscale
    bit-depth, 512 x 512 size.

    Parameters
    ----------
    None

    Returns
    -------
    lena : ndarray
        Lena image

    Examples
    --------
    >>> import scipy.misc
    >>> lena = scipy.misc.lena()
    >>> lena.shape
    (512, 512)
    >>> lena.max()
    245
    >>> lena.dtype
    dtype('int32')

    >>> import matplotlib.pyplot as plt
    >>> plt.gray()
    >>> plt.imshow(lena)
    >>> plt.show()

    """
    import pickle
    import os
    fname = os.path.join(os.path.dirname(__file__),'lena.dat')
    f = open(fname,'rb')
    lena = array(pickle.load(f))
    f.close()
    return lena


def ascent():
    """
    Get an 8-bit grayscale bit-depth, 512 x 512 derived image for easy use in demos

    The image is derived from accent-to-the-top.jpg at
    http://www.public-domain-image.com/people-public-domain-images-pictures/

    Parameters
    ----------
    None

    Returns
    -------
    ascent : ndarray
       convenient image to use for testing and demonstration

    Examples
    --------
    >>> import scipy.misc
    >>> ascent = scipy.misc.ascent()
    >>> ascent.shape
    (512, 512)
    >>> ascent.max()
    255

    >>> import matplotlib.pyplot as plt
    >>> plt.gray()
    >>> plt.imshow(ascent)
    >>> plt.show()

    """
    import pickle
    import os
    fname = os.path.join(os.path.dirname(__file__),'ascent.dat')
    with open(fname, 'rb') as f:
        ascent = array(pickle.load(f))
    return ascent


def face(gray=False):
    """
    Get a 1024 x 768, color image of a raccoon face.

    raccoon-procyon-lotor.jpg at http://www.public-domain-image.com

    Parameters
    ----------
    gray : bool, optional
        If True then return color image, otherwise return an 8-bit gray-scale

    Returns
    -------
    face : ndarray
        image of a racoon face

    Examples
    --------
    >>> import scipy.misc
    >>> face = scipy.misc.face()
    >>> face.shape
    (768, 1024, 3)
    >>> face.max()
    230
    >>> face.dtype
    dtype('uint8')

    >>> import matplotlib.pyplot as plt
    >>> plt.gray()
    >>> plt.imshow(face)
    >>> plt.show()

    """
    import bz2
    import os
    with open(os.path.join(os.path.dirname(__file__), 'face.dat'), 'rb') as f:
        rawdata = f.read()
    data = bz2.decompress(rawdata)
    face = fromstring(data, dtype='uint8')
    face.shape = (768, 1024, 3)
    if gray is True:
        face = (0.21 * face[:,:,0] + 0.71 * face[:,:,1] + 0.07 * face[:,:,2]).astype('uint8')
    return face