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__docformat__ = "restructuredtext en" __all__ = ['select', 'piecewise', 'trim_zeros', 'copy', 'iterable', 'diff', 'gradient', 'angle', 'unwrap', 'sort_complex', 'disp', 'extract', 'place', 'nansum', 'nanmax', 'nanargmax', 'nanargmin', 'nanmin', 'vectorize', 'asarray_chkfinite', 'average', 'histogram', 'histogramdd', 'bincount', 'digitize', 'cov', 'corrcoef', 'msort', 'median', 'sinc', 'hamming', 'hanning', 'bartlett', 'blackman', 'kaiser', 'trapz', 'i0', 'add_newdoc', 'add_docstring', 'meshgrid', 'delete', 'insert', 'append', 'interp' ] import warnings
import types import numpy.core.numeric as _nx from numpy.core import linspace from numpy.core.numeric import ones, zeros, arange, concatenate, array, \ asarray, asanyarray, empty, empty_like, ndarray, around from numpy.core.numeric import ScalarType, dot, where, newaxis, intp, \ integer, isscalar from numpy.core.umath import pi, multiply, add, arctan2, \ frompyfunc, isnan, cos, less_equal, sqrt, sin, mod, exp, log10 from numpy.core.fromnumeric import ravel, nonzero, choose, sort, mean from numpy.core.numerictypes import typecodes, number from numpy.core import atleast_1d, atleast_2d from numpy.lib.twodim_base import diag from _compiled_base import _insert, add_docstring from _compiled_base import digitize, bincount, interp as compiled_interp from arraysetops import setdiff1d from utils import deprecate import numpy as np
#end Fernando's utilities
def iterable(y): try: iter(y) except: return 0 return 1
def histogram(a, bins=10, range=None, normed=False, weights=None, new=None): """ Compute the histogram of a set of data.
Parameters ---------- a : array_like Input data. The histogram is computed over the flattened array. bins : int or sequence of scalars, optional If `bins` is an int, it defines the number of equal-width bins in the given range (10, by default). If `bins` is a sequence, it defines the bin edges, including the rightmost edge, allowing for non-uniform bin widths. range : (float, float), optional The lower and upper range of the bins. If not provided, range is simply ``(a.min(), a.max())``. Values outside the range are ignored. Note that with `new` set to False, values below the range are ignored, while those above the range are tallied in the rightmost bin. normed : bool, optional If False, the result will contain the number of samples in each bin. If True, the result is the value of the probability *density* function at the bin, normalized such that the *integral* over the range is 1. Note that the sum of the histogram values will not be equal to 1 unless bins of unity width are chosen; it is not a probability *mass* function. weights : array_like, optional An array of weights, of the same shape as `a`. Each value in `a` only contributes its associated weight towards the bin count (instead of 1). If `normed` is True, the weights are normalized, so that the integral of the density over the range remains 1. The `weights` keyword is only available with `new` set to True. new : {None, True, False}, optional Whether to use the new semantics for histogram: * None : the new behaviour is used, no warning is printed. * True : the new behaviour is used and a warning is raised about the future removal of the `new` keyword. * False : the old behaviour is used and a DeprecationWarning is raised. As of NumPy 1.3, this keyword should not be used explicitly since it will disappear in NumPy 2.0.
Returns ------- hist : array The values of the histogram. See `normed` and `weights` for a description of the possible semantics. bin_edges : array of dtype float Return the bin edges ``(length(hist)+1)``. With ``new=False``, return the left bin edges (``length(hist)``).
See Also -------- histogramdd, bincount, searchsorted
Notes ----- All but the last (righthand-most) bin is half-open. In other words, if `bins` is::
[1, 2, 3, 4]
then the first bin is ``[1, 2)`` (including 1, but excluding 2) and the second ``[2, 3)``. The last bin, however, is ``[3, 4]``, which *includes* 4.
Examples -------- >>> np.histogram([1, 2, 1], bins=[0, 1, 2, 3]) (array([0, 2, 1]), array([0, 1, 2, 3])) >>> np.histogram(np.arange(4), bins=np.arange(5), normed=True) (array([ 0.25, 0.25, 0.25, 0.25]), array([0, 1, 2, 3, 4])) >>> np.histogram([[1, 2, 1], [1, 0, 1]], bins=[0,1,2,3]) (array([1, 4, 1]), array([0, 1, 2, 3])) ]), array([0, 1, 2, 3]))
>>> a = np.arange(5) >>> hist, bin_edges = np.histogram(a, normed=True) >>> hist array([ 0.5, 0. , 0.5, 0. , 0. , 0.5, 0. , 0.5, 0. , 0.5]) >>> hist.sum() 2.4999999999999996 >>> np.sum(hist*np.diff(bin_edges)) 1.0
""" # Old behavior if new == False: warnings.warn(""" The histogram semantics being used is now deprecated and will disappear in NumPy 2.0. Please update your code to use the default semantics. """, DeprecationWarning)
a = asarray(a).ravel()
if (range is not None): mn, mx = range if (mn > mx): raise AttributeError, \ 'max must be larger than min in range parameter.'
if not iterable(bins): if range is None: range = (a.min(), a.max()) mn, mx = [mi+0.0 for mi in range] if mn == mx: mn -= 0.5 mx += 0.5 bins = linspace(mn, mx, bins, endpoint=False) else: if normed: raise ValueError, 'Use new=True to pass bin edges explicitly.' raise ValueError, 'Use new=True to pass bin edges explicitly.' bins = asarray(bins) if (np.diff(bins) < 0).any(): raise AttributeError, 'bins must increase monotonically.'
if weights is not None: raise ValueError, 'weights are only available with new=True.'
# best block size probably depends on processor cache size block = 65536 n = sort(a[:block]).searchsorted(bins) for i in xrange(block, a.size, block): n += sort(a[i:i+block]).searchsorted(bins) n = concatenate([n, [len(a)]]) n = n[1:]-n[:-1]
if normed: db = bins[1] - bins[0] return 1.0/(a.size*db) * n, bins else: return n, bins
# New behavior elif new in [True, None]: if new is True: warnings.warn(""" The new semantics of histogram is now the default and the `new` keyword will be removed in NumPy 2.0. """, Warning) a = asarray(a) if weights is not None: weights = asarray(weights) if np.any(weights.shape != a.shape): raise ValueError, 'weights should have the same shape as a.' weights = weights.ravel() a = a.ravel()
if (range is not None): mn, mx = range if (mn > mx): raise AttributeError, \ 'max must be larger than min in range parameter.'
if not iterable(bins): if range is None: range = (a.min(), a.max()) mn, mx = [mi+0.0 for mi in range] if mn == mx: mn -= 0.5 mx += 0.5 bins = linspace(mn, mx, bins+1, endpoint=True) else: bins = asarray(bins) if (np.diff(bins) < 0).any(): raise AttributeError, 'bins must increase monotonically.'
# Histogram is an integer or a float array depending on the weights. if weights is None: ntype = int else: ntype = weights.dtype n = np.zeros(bins.shape, ntype)
block = 65536 if weights is None: for i in arange(0, len(a), block): sa = sort(a[i:i+block]) n += np.r_[sa.searchsorted(bins[:-1], 'left'), \ sa.searchsorted(bins[-1], 'right')] else: zero = array(0, dtype=ntype) for i in arange(0, len(a), block): tmp_a = a[i:i+block] tmp_w = weights[i:i+block] sorting_index = np.argsort(tmp_a) sa = tmp_a[sorting_index] sw = tmp_w[sorting_index] cw = np.concatenate(([zero,], sw.cumsum())) bin_index = np.r_[sa.searchsorted(bins[:-1], 'left'), \ sa.searchsorted(bins[-1], 'right')] n += cw[bin_index]
n = np.diff(n)
if normed: db = array(np.diff(bins), float) return n/(n*db).sum(), bins else: return n, bins
def histogramdd(sample, bins=10, range=None, normed=False, weights=None): """ Compute the multidimensional histogram of some data.
Parameters ---------- sample : array_like The data to be histogrammed. It must be an (N,D) array or data that can be converted to such. The rows of the resulting array are the coordinates of points in a D dimensional polytope. bins : sequence or int, optional The bin specification:
* A sequence of arrays describing the bin edges along each dimension. * The number of bins for each dimension (nx, ny, ... =bins) * The number of bins for all dimensions (nx=ny=...=bins).
range : sequence, optional A sequence of lower and upper bin edges to be used if the edges are not given explicitely in `bins`. Defaults to the minimum and maximum values along each dimension. normed : boolean, optional If False, returns the number of samples in each bin. If True, returns the bin density, ie, the bin count divided by the bin hypervolume. weights : array_like (N,), optional An array of values `w_i` weighing each sample `(x_i, y_i, z_i, ...)`. Weights are normalized to 1 if normed is True. If normed is False, the values of the returned histogram are equal to the sum of the weights belonging to the samples falling into each bin.
Returns ------- H : ndarray The multidimensional histogram of sample x. See normed and weights for the different possible semantics. edges : list A list of D arrays describing the bin edges for each dimension.
See Also -------- histogram: 1D histogram histogram2d: 2D histogram
Examples -------- >>> r = np.random.randn(100,3) >>> H, edges = np.histogramdd(r, bins = (5, 8, 4)) >>> H.shape, edges[0].size, edges[1].size, edges[2].size ((5,8,4), 6, 9, 5)
"""
try: # Sample is an ND-array. N, D = sample.shape except (AttributeError, ValueError): # Sample is a sequence of 1D arrays. sample = atleast_2d(sample).T N, D = sample.shape
nbin = empty(D, int) edges = D*[None] dedges = D*[None] if weights is not None: weights = asarray(weights)
try: M = len(bins) if M != D: raise AttributeError, 'The dimension of bins must be equal ' \ 'to the dimension of the sample x.' except TypeError: bins = D*[bins]
# Select range for each dimension # Used only if number of bins is given. if range is None: smin = atleast_1d(array(sample.min(0), float)) smax = atleast_1d(array(sample.max(0), float)) else: smin = zeros(D) smax = zeros(D) for i in arange(D): smin[i], smax[i] = range[i]
# Make sure the bins have a finite width. for i in arange(len(smin)): if smin[i] == smax[i]: smin[i] = smin[i] - .5 smax[i] = smax[i] + .5
# Create edge arrays for i in arange(D): if isscalar(bins[i]): nbin[i] = bins[i] + 2 # +2 for outlier bins edges[i] = linspace(smin[i], smax[i], nbin[i]-1) else: edges[i] = asarray(bins[i], float) nbin[i] = len(edges[i])+1 # +1 for outlier bins dedges[i] = diff(edges[i])
nbin = asarray(nbin)
# Compute the bin number each sample falls into. Ncount = {} for i in arange(D): Ncount[i] = digitize(sample[:,i], edges[i])
# Using digitize, values that fall on an edge are put in the right bin. # For the rightmost bin, we want values equal to the right # edge to be counted in the last bin, and not as an outlier. outliers = zeros(N, int) for i in arange(D): # Rounding precision decimal = int(-log10(dedges[i].min())) +6 # Find which points are on the rightmost edge. on_edge = where(around(sample[:,i], decimal) == around(edges[i][-1], decimal))[0] # Shift these points one bin to the left. Ncount[i][on_edge] -= 1
# Flattened histogram matrix (1D) # Reshape is used so that overlarge arrays # will raise an error. hist = zeros(nbin, float).reshape(-1)
# Compute the sample indices in the flattened histogram matrix. ni = nbin.argsort() shape = [] xy = zeros(N, int) for i in arange(0, D-1): xy += Ncount[ni[i]] * nbin[ni[i+1:]].prod() xy += Ncount[ni[-1]]
# Compute the number of repetitions in xy and assign it to the # flattened histmat. if len(xy) == 0: return zeros(nbin-2, int), edges
flatcount = bincount(xy, weights) a = arange(len(flatcount)) hist[a] = flatcount
# Shape into a proper matrix hist = hist.reshape(sort(nbin)) for i in arange(nbin.size): j = ni.argsort()[i] hist = hist.swapaxes(i,j) ni[i],ni[j] = ni[j],ni[i]
# Remove outliers (indices 0 and -1 for each dimension). core = D*[slice(1,-1)] hist = hist[core]
# Normalize if normed is True if normed: s = hist.sum() for i in arange(D): shape = ones(D, int) shape[i] = nbin[i]-2 hist = hist / dedges[i].reshape(shape) hist /= s
if (hist.shape != nbin-2).any(): raise RuntimeError('Internal Shape Error') return hist, edges
def average(a, axis=None, weights=None, returned=False): """ Compute the weighted average along the specified axis.
Parameters ---------- a : array_like Array containing data to be averaged. If `a` is not an array, a conversion is attempted. axis : int, optional Axis along which to average `a`. If `None`, averaging is done over the flattened array. weights : array_like, optional An array of weights associated with the values in `a`. Each value in `a` contributes to the average according to its associated weight. The weights array can either be 1-D (in which case its length must be the size of `a` along the given axis) or of the same shape as `a`. If `weights=None`, then all data in `a` are assumed to have a weight equal to one. returned : bool, optional Default is `False`. If `True`, the tuple (`average`, `sum_of_weights`) is returned, otherwise only the average is returned. If `weights=None`, `sum_of_weights` is equivalent to the number of elements over which the average is taken.
Returns ------- average, [sum_of_weights] : {array_type, double} Return the average along the specified axis. When returned is `True`, return a tuple with the average as the first element and the sum of the weights as the second element. The return type is `Float` if `a` is of integer type, otherwise it is of the same type as `a`. `sum_of_weights` is of the same type as `average`.
Raises ------ ZeroDivisionError When all weights along axis are zero. See `numpy.ma.average` for a version robust to this type of error. TypeError When the length of 1D `weights` is not the same as the shape of `a` along axis.
See Also -------- mean
ma.average : average for masked arrays
Examples -------- >>> data = range(1,5) >>> data [1, 2, 3, 4] >>> np.average(data) 2.5 >>> np.average(range(1,11), weights=range(10,0,-1)) 4.0
>>> data = np.arange(6).reshape((3,2)) >>> data array([[0, 1], [2, 3], [4, 5]]) >>> np.average(data, axis=1, weights=[1./4, 3./4]) array([ 0.75, 2.75, 4.75]) >>> np.average(data, weights=[1./4, 3./4]) Traceback (most recent call last): ... TypeError: Axis must be specified when shapes of a and weights differ.
""" if not isinstance(a, np.matrix) : a = np.asarray(a)
if weights is None : avg = a.mean(axis) scl = avg.dtype.type(a.size/avg.size) else : a = a + 0.0 wgt = np.array(weights, dtype=a.dtype, copy=0)
# Sanity checks if a.shape != wgt.shape : if axis is None : raise TypeError, "Axis must be specified when shapes of a and weights differ." if wgt.ndim != 1 : raise TypeError, "1D weights expected when shapes of a and weights differ." if wgt.shape[0] != a.shape[axis] : raise ValueError, "Length of weights not compatible with specified axis."
# setup wgt to broadcast along axis wgt = np.array(wgt, copy=0, ndmin=a.ndim).swapaxes(-1,axis)
scl = wgt.sum(axis=axis) if (scl == 0.0).any(): raise ZeroDivisionError, "Weights sum to zero, can't be normalized"
avg = np.multiply(a,wgt).sum(axis)/scl
if returned: scl = np.multiply(avg,0) + scl return avg, scl else: return avg
def asarray_chkfinite(a): """ Convert the input to an array, checking for NaNs or Infs.
Parameters ---------- a : array_like Input data, in any form that can be converted to an array. This includes lists, lists of tuples, tuples, tuples of tuples, tuples of lists and ndarrays. Success requires no NaNs or Infs. dtype : data-type, optional By default, the data-type is inferred from the input data. order : {'C', 'F'}, optional Whether to use row-major ('C') or column-major ('FORTRAN') memory representation. Defaults to 'C'.
Returns ------- out : ndarray Array interpretation of `a`. No copy is performed if the input is already an ndarray. If `a` is a subclass of ndarray, a base class ndarray is returned.
Raises ------ ValueError Raises ValueError if `a` contains NaN (Not a Number) or Inf (Infinity).
See Also -------- asarray : Create and array. asanyarray : Similar function which passes through subclasses. ascontiguousarray : Convert input to a contiguous array. asfarray : Convert input to a floating point ndarray. asfortranarray : Convert input to an ndarray with column-major memory order. fromiter : Create an array from an iterator. fromfunction : Construct an array by executing a function on grid positions.
Examples -------- Convert a list into an array. If all elements are finite ``asarray_chkfinite`` is identical to ``asarray``.
>>> a = [1, 2] >>> np.asarray_chkfinite(a) array([1, 2])
Raises ValueError if array_like contains Nans or Infs.
>>> a = [1, 2, np.inf] >>> try: ... np.asarray_chkfinite(a) ... except ValueError: ... print 'ValueError' ... ValueError
""" a = asarray(a) if (a.dtype.char in typecodes['AllFloat']) \ and (_nx.isnan(a).any() or _nx.isinf(a).any()): raise ValueError, "array must not contain infs or NaNs" return a
def piecewise(x, condlist, funclist, *args, **kw): """ Evaluate a piecewise-defined function.
Given a set of conditions and corresponding functions, evaluate each function on the input data wherever its condition is true.
Parameters ---------- x : ndarray The input domain. condlist : list of bool arrays Each boolean array corresponds to a function in `funclist`. Wherever `condlist[i]` is True, `funclist[i](x)` is used as the output value.
Each boolean array in `condlist` selects a piece of `x`, and should therefore be of the same shape as `x`.
The length of `condlist` must correspond to that of `funclist`. If one extra function is given, i.e. if ``len(funclist) - len(condlist) == 1``, then that extra function is the default value, used wherever all conditions are false. funclist : list of callables, f(x,*args,**kw), or scalars Each function is evaluated over `x` wherever its corresponding condition is True. It should take an array as input and give an array or a scalar value as output. If, instead of a callable, a scalar is provided then a constant function (``lambda x: scalar``) is assumed. args : tuple, optional Any further arguments given to `piecewise` are passed to the functions upon execution, i.e., if called ``piecewise(..., ..., 1, 'a')``, then each function is called as ``f(x, 1, 'a')``. kw : dict, optional Keyword arguments used in calling `piecewise` are passed to the functions upon execution, i.e., if called ``piecewise(..., ..., lambda=1)``, then each function is called as ``f(x, lambda=1)``.
Returns ------- out : ndarray The output is the same shape and type as x and is found by calling the functions in `funclist` on the appropriate portions of `x`, as defined by the boolean arrays in `condlist`. Portions not covered by any condition have undefined values.
See Also -------- choose, select, where
Notes ----- This is similar to choose or select, except that functions are evaluated on elements of `x` that satisfy the corresponding condition from `condlist`.
The result is::
|-- |funclist[0](x[condlist[0]]) out = |funclist[1](x[condlist[1]]) |... |funclist[n2](x[condlist[n2]]) |--
Examples -------- Define the sigma function, which is -1 for ``x < 0`` and +1 for ``x >= 0``.
>>> x = np.arange(6) - 2.5 >>> np.piecewise(x, [x < 0, x >= 0], [-1, 1]) array([-1., -1., -1., 1., 1., 1.])
Define the absolute value, which is ``-x`` for ``x <0`` and ``x`` for ``x >= 0``.
>>> np.piecewise(x, [x < 0, x >= 0], [lambda x: -x, lambda x: x]) array([ 2.5, 1.5, 0.5, 0.5, 1.5, 2.5])
""" x = asanyarray(x) n2 = len(funclist) if isscalar(condlist) or \ not (isinstance(condlist[0], list) or isinstance(condlist[0], ndarray)): condlist = [condlist] condlist = [asarray(c, dtype=bool) for c in condlist] n = len(condlist) if n == n2-1: # compute the "otherwise" condition. totlist = condlist[0] for k in range(1, n): totlist |= condlist[k] condlist.append(~totlist) n += 1 if (n != n2): raise ValueError, "function list and condition list " \ "must be the same" zerod = False # This is a hack to work around problems with NumPy's # handling of 0-d arrays and boolean indexing with # numpy.bool_ scalars if x.ndim == 0: x = x[None] zerod = True newcondlist = [] for k in range(n): if condlist[k].ndim == 0: condition = condlist[k][None] else: condition = condlist[k] newcondlist.append(condition) condlist = newcondlist
y = zeros(x.shape, x.dtype) for k in range(n): item = funclist[k] if not callable(item): y[condlist[k]] = item else: vals = x[condlist[k]] if vals.size > 0: y[condlist[k]] = item(vals, *args, **kw) if zerod: y = y.squeeze() return y
def select(condlist, choicelist, default=0): """ Return an array drawn from elements in choicelist, depending on conditions.
Parameters ---------- condlist : list of bool ndarrays The list of conditions which determine from which array in `choicelist` the output elements are taken. When multiple conditions are satisfied, the first one encountered in `condlist` is used. choicelist : list of ndarrays The list of arrays from which the output elements are taken. It has to be of the same length as `condlist`. default : scalar, optional The element inserted in `output` when all conditions evaluate to False.
Returns ------- output : ndarray The output at position m is the m-th element of the array in `choicelist` where the m-th element of the corresponding array in `condlist` is True.
See Also -------- where : Return elements from one of two arrays depending on condition. take, choose, compress, diag, diagonal
Examples -------- >>> x = np.arange(10) >>> condlist = [x<3, x>5] >>> choicelist = [x, x**2] >>> np.select(condlist, choicelist) array([ 0, 1, 2, 0, 0, 0, 36, 49, 64, 81])
""" n = len(condlist) n2 = len(choicelist) if n2 != n: raise ValueError, "list of cases must be same length as list of conditions" choicelist = [default] + choicelist S = 0 pfac = 1 for k in range(1, n+1): S += k * pfac * asarray(condlist[k-1]) if k < n: pfac *= (1-asarray(condlist[k-1])) # handle special case of a 1-element condition but # a multi-element choice if type(S) in ScalarType or max(asarray(S).shape)==1: pfac = asarray(1) for k in range(n2+1): pfac = pfac + asarray(choicelist[k]) if type(S) in ScalarType: S = S*ones(asarray(pfac).shape, type(S)) else: S = S*ones(asarray(pfac).shape, S.dtype) return choose(S, tuple(choicelist))
def copy(a): """ Return an array copy of the given object.
Parameters ---------- a : array_like Input data.
Returns ------- arr : ndarray Array interpretation of `a`.
Notes ----- This is equivalent to
>>> np.array(a, copy=True)
Examples -------- Create an array x, with a reference y and a copy z:
>>> x = np.array([1, 2, 3]) >>> y = x >>> z = np.copy(x)
Note that, when we modify x, y changes, but not z:
>>> x[0] = 10 >>> x[0] == y[0] True >>> x[0] == z[0] False
""" return array(a, copy=True)
# Basic operations
def gradient(f, *varargs): """ Return the gradient of an N-dimensional array.
The gradient is computed using central differences in the interior and first differences at the boundaries. The returned gradient hence has the same shape as the input array.
Parameters ---------- f : array_like An N-dimensional array containing samples of a scalar function. `*varargs` : scalars 0, 1, or N scalars specifying the sample distances in each direction, that is: `dx`, `dy`, `dz`, ... The default distance is 1.
Returns ------- g : ndarray N arrays of the same shape as `f` giving the derivative of `f` with respect to each dimension.
Examples -------- >>> x = np.array([1, 2, 4, 7, 11, 16], dtype=np.float) >>> np.gradient(x) array([ 1. , 1.5, 2.5, 3.5, 4.5, 5. ]) >>> np.gradient(x, 2) array([ 0.5 , 0.75, 1.25, 1.75, 2.25, 2.5 ])
>>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=np.float)) [array([[ 2., 2., -1.], [ 2., 2., -1.]]), array([[ 1. , 2.5, 4. ], [ 1. , 1. , 1. ]])]
""" N = len(f.shape) # number of dimensions n = len(varargs) if n == 0: dx = [1.0]*N elif n == 1: dx = [varargs[0]]*N elif n == N: dx = list(varargs) else: raise SyntaxError, "invalid number of arguments"
# use central differences on interior and first differences on endpoints
outvals = []
# create slice objects --- initially all are [:, :, ..., :] slice1 = [slice(None)]*N slice2 = [slice(None)]*N slice3 = [slice(None)]*N
otype = f.dtype.char if otype not in ['f', 'd', 'F', 'D']: otype = 'd'
for axis in range(N): # select out appropriate parts for this dimension out = zeros(f.shape, f.dtype.char) slice1[axis] = slice(1, -1) slice2[axis] = slice(2, None) slice3[axis] = slice(None, -2) # 1D equivalent -- out[1:-1] = (f[2:] - f[:-2])/2.0 out[slice1] = (f[slice2] - f[slice3])/2.0 slice1[axis] = 0 slice2[axis] = 1 slice3[axis] = 0 # 1D equivalent -- out[0] = (f[1] - f[0]) out[slice1] = (f[slice2] - f[slice3]) slice1[axis] = -1 slice2[axis] = -1 slice3[axis] = -2 # 1D equivalent -- out[-1] = (f[-1] - f[-2]) out[slice1] = (f[slice2] - f[slice3])
# divide by step size outvals.append(out / dx[axis])
# reset the slice object in this dimension to ":" slice1[axis] = slice(None) slice2[axis] = slice(None) slice3[axis] = slice(None)
if N == 1: return outvals[0] else: return outvals
def diff(a, n=1, axis=-1): """ Calculate the n-th order discrete difference along given axis.
The first order difference is given by ``out[n] = a[n+1] - a[n]`` along the given axis, higher order differences are calculated by using `diff` recursively.
Parameters ---------- a : array_like Input array n : int, optional The number of times values are differenced. axis : int, optional The axis along which the difference is taken, default is the last axis.
Returns ------- out : ndarray The `n` order differences. The shape of the output is the same as `a` except along `axis` where the dimension is smaller by `n`.
See Also -------- gradient, ediff1d
Examples -------- >>> x = np.array([1, 2, 4, 7, 0]) >>> np.diff(x) array([ 1, 2, 3, -7]) >>> np.diff(x, n=2) array([ 1, 1, -10])
>>> x = np.array([[1, 3, 6, 10], [0, 5, 6, 8]]) >>> np.diff(x) array([[2, 3, 4], [5, 1, 2]]) >>> np.diff(x, axis=0) array([[-1, 2, 0, -2]])
""" if n == 0: return a if n < 0: raise ValueError, 'order must be non-negative but got ' + repr(n) a = asanyarray(a) nd = len(a.shape) slice1 = [slice(None)]*nd slice2 = [slice(None)]*nd slice1[axis] = slice(1, None) slice2[axis] = slice(None, -1) slice1 = tuple(slice1) slice2 = tuple(slice2) if n > 1: return diff(a[slice1]-a[slice2], n-1, axis=axis) else: return a[slice1]-a[slice2]
def interp(x, xp, fp, left=None, right=None): """ One-dimensional linear interpolation.
Returns the one-dimensional piecewise linear interpolant to a function with given values at discrete data-points.
Parameters ---------- x : array_like The x-coordinates of the interpolated values.
xp : 1-D sequence of floats The x-coordinates of the data points, must be increasing.
fp : 1-D sequence of floats The y-coordinates of the data points, same length as `xp`.
left : float, optional Value to return for `x < xp[0]`, default is `fp[0]`.
right : float, optional Value to return for `x > xp[-1]`, defaults is `fp[-1]`.
Returns ------- y : {float, ndarray} The interpolated values, same shape as `x`.
Raises ------ ValueError If `xp` and `fp` have different length
Notes ----- Does not check that the x-coordinate sequence `xp` is increasing. If `xp` is not increasing, the results are nonsense. A simple check for increasingness is::
np.all(np.diff(xp) > 0)
Examples -------- >>> xp = [1, 2, 3] >>> fp = [3, 2, 0] >>> np.interp(2.5, xp, fp) 1.0 >>> np.interp([0, 1, 1.5, 2.72, 3.14], xp, fp) array([ 3. , 3. , 2.5, 0.56, 0. ]) >>> UNDEF = -99.0 >>> np.interp(3.14, xp, fp, right=UNDEF) -99.0
Plot an interpolant to the sine function:
>>> x = np.linspace(0, 2*np.pi, 10) >>> y = np.sin(x) >>> xvals = np.linspace(0, 2*np.pi, 50) >>> yinterp = np.interp(xvals, x, y) >>> import matplotlib.pyplot as plt >>> plt.plot(x, y, 'o') >>> plt.plot(xvals, yinterp, '-x') >>> plt.show()
""" if isinstance(x, (float, int, number)): return compiled_interp([x], xp, fp, left, right).item() else: return compiled_interp(x, xp, fp, left, right)
def angle(z, deg=0): """ Return the angle of the complex argument.
Parameters ---------- z : array_like A complex number or sequence of complex numbers. deg : bool, optional Return angle in degrees if True, radians if False (default).
Returns ------- angle : {ndarray, scalar} The counterclockwise angle from the positive real axis on the complex plane, with dtype as numpy.float64.
See Also -------- arctan2 absolute
Examples -------- >>> np.angle([1.0, 1.0j, 1+1j]) # in radians array([ 0. , 1.57079633, 0.78539816]) >>> np.angle(1+1j, deg=True) # in degrees 45.0
""" if deg: fact = 180/pi else: fact = 1.0 z = asarray(z) if (issubclass(z.dtype.type, _nx.complexfloating)): zimag = z.imag zreal = z.real else: zimag = 0 zreal = z return arctan2(zimag, zreal) * fact
def unwrap(p, discont=pi, axis=-1): """ Unwrap by changing deltas between values to 2*pi complement.
Unwrap radian phase `p` by changing absolute jumps greater than `discont` to their 2*pi complement along the given axis.
Parameters ---------- p : array_like Input array. discont : float, optional Maximum discontinuity between values, default is ``pi``. axis : int, optional Axis along which unwrap will operate, default is the last axis.
Returns ------- out : ndarray Output array.
See Also -------- rad2deg, deg2rad
Notes ----- If the discontinuity in `p` is smaller than ``pi``, but larger than `discont`, no unwrapping is done because taking the 2*pi complement would only make the discontinuity larger.
Examples -------- >>> phase = np.linspace(0, np.pi, num=5) >>> phase[3:] += np.pi >>> phase array([ 0. , 0.78539816, 1.57079633, 5.49778714, 6.28318531]) >>> np.unwrap(phase) array([ 0. , 0.78539816, 1.57079633, -0.78539816, 0. ])
""" p = asarray(p) nd = len(p.shape) dd = diff(p, axis=axis) slice1 = [slice(None, None)]*nd # full slices slice1[axis] = slice(1, None) ddmod = mod(dd+pi, 2*pi)-pi _nx.putmask(ddmod, (ddmod==-pi) & (dd > 0), pi) ph_correct = ddmod - dd; _nx.putmask(ph_correct, abs(dd)<discont, 0) up = array(p, copy=True, dtype='d') up[slice1] = p[slice1] + ph_correct.cumsum(axis) return up
def sort_complex(a): """ Sort a complex array using the real part first, then the imaginary part.
Parameters ---------- a : array_like Input array
Returns ------- out : complex ndarray Always returns a sorted complex array.
Examples -------- >>> np.sort_complex([5, 3, 6, 2, 1]) array([ 1.+0.j, 2.+0.j, 3.+0.j, 5.+0.j, 6.+0.j])
>>> np.sort_complex([1 + 2j, 2 - 1j, 3 - 2j, 3 - 3j, 3 + 5j]) array([ 1.+2.j, 2.-1.j, 3.-5.j, 3.-3.j, 3.+2.j])
""" b = array(a,copy=True) b.sort() if not issubclass(b.dtype.type, _nx.complexfloating): if b.dtype.char in 'bhBH': return b.astype('F') elif b.dtype.char == 'g': return b.astype('G') else: return b.astype('D') else: return b
def trim_zeros(filt, trim='fb'): """ Trim the leading and/or trailing zeros from a 1-D array or sequence.
Parameters ---------- filt : 1-D array or sequence Input array. trim : str, optional A string with 'f' representing trim from front and 'b' to trim from back. Default is 'fb', trim zeros from both front and back of the array.
Returns ------- trimmed : 1-D array or sequence The result of trimming the input. The input data type is preserved.
Examples -------- >>> a = np.array((0, 0, 0, 1, 2, 3, 0, 2, 1, 0)) >>> np.trim_zeros(a) array([1, 2, 3, 0, 2, 1])
>>> np.trim_zeros(a, 'b') array([0, 0, 0, 1, 2, 3, 0, 2, 1])
The input data type is preserved, list/tuple in means list/tuple out.
>>> np.trim_zeros([0, 1, 2, 0]) [1, 2]
""" first = 0 trim = trim.upper() if 'F' in trim: for i in filt: if i != 0.: break else: first = first + 1 last = len(filt) if 'B' in trim: for i in filt[::-1]: if i != 0.: break else: last = last - 1 return filt[first:last]
import sys if sys.hexversion < 0x2040000: from sets import Set as set
@deprecate def unique(x): """ This function is deprecated. Use numpy.lib.arraysetops.unique() instead. """ try: tmp = x.flatten() if tmp.size == 0: return tmp tmp.sort() idx = concatenate(([True],tmp[1:]!=tmp[:-1])) return tmp[idx] except AttributeError: items = list(set(x)) items.sort() return asarray(items)
def extract(condition, arr): """ Return the elements of an array that satisfy some condition.
This is equivalent to ``np.compress(ravel(condition), ravel(arr))``. If `condition` is boolean ``np.extract`` is equivalent to ``arr[condition]``.
Parameters ---------- condition : array_like An array whose nonzero or True entries indicate the elements of `arr` to extract. arr : array_like Input array of the same size as `condition`.
See Also -------- take, put, putmask, compress
Examples -------- >>> arr = np.arange(12).reshape((3, 4)) >>> arr array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) >>> condition = np.mod(arr, 3)==0 >>> condition array([[ True, False, False, True], [False, False, True, False], [False, True, False, False]], dtype=bool) >>> np.extract(condition, arr) array([0, 3, 6, 9])
If `condition` is boolean:
>>> arr[condition] array([0, 3, 6, 9])
""" return _nx.take(ravel(arr), nonzero(ravel(condition))[0])
def place(arr, mask, vals): """ Change elements of an array based on conditional and input values.
Similar to ``np.putmask(a, mask, vals)``, the difference is that `place` uses the first N elements of `vals`, where N is the number of True values in `mask`, while `putmask` uses the elements where `mask` is True.
Note that `extract` does the exact opposite of `place`.
Parameters ---------- a : array_like Array to put data into. mask : array_like Boolean mask array. Must have the same size as `a`. vals : 1-D sequence Values to put into `a`. Only the first N elements are used, where N is the number of True values in `mask`. If `vals` is smaller than N it will be repeated.
See Also -------- putmask, put, take, extract
Examples -------- >>> x = np.arange(6).reshape(2, 3) >>> np.place(x, x>2, [44, 55]) >>> x array([[ 0, 1, 2], [44, 55, 44]])
""" return _insert(arr, mask, vals)
def _nanop(op, fill, a, axis=None): """ General operation on arrays with not-a-number values.
Parameters ---------- op : callable Operation to perform. fill : float NaN values are set to fill before doing the operation. a : array-like Input array. axis : {int, None}, optional Axis along which the operation is computed. By default the input is flattened.
Returns ------- y : {ndarray, scalar} Processed data.
""" y = array(a, subok=True) mask = isnan(a)
# We only need to take care of NaN's in floating point arrays if not np.issubdtype(y.dtype, int): y[mask] = fill
res = op(y, axis=axis) mask_all_along_axis = mask.all(axis=axis)
# Along some axes, only nan's were encountered. As such, any values # calculated along that axis should be set to nan. if mask_all_along_axis.any(): if np.isscalar(res): res = np.nan else: res[mask_all_along_axis] = np.nan
return res
def nansum(a, axis=None): """ Return the sum of array elements over a given axis treating Not a Numbers (NaNs) as zero.
Parameters ---------- a : array_like Array containing numbers whose sum is desired. If `a` is not an array, a conversion is attempted. axis : int, optional Axis along which the sum is computed. The default is to compute the sum of the flattened array.
Returns ------- y : ndarray An array with the same shape as a, with the specified axis removed. If a is a 0-d array, or if axis is None, a scalar is returned with the same dtype as `a`.
See Also -------- numpy.sum : Sum across array including Not a Numbers. isnan : Shows which elements are Not a Number (NaN). isfinite: Shows which elements are not: Not a Number, positive and negative infinity
Notes ----- Numpy uses the IEEE Standard for Binary Floating-Point for Arithmetic (IEEE 754). This means that Not a Number is not equivalent to infinity. If positive or negative infinity are present the result is positive or negative infinity. But if both positive and negative infinity are present, the result is Not A Number (NaN).
Arithmetic is modular when using integer types (all elements of `a` must be finite i.e. no elements that are NaNs, positive infinity and negative infinity because NaNs are floating point types), and no error is raised on overflow.
Examples -------- >>> np.nansum(1) 1 >>> np.nansum([1]) 1 >>> np.nansum([1, np.nan]) 1.0 >>> a = np.array([[1, 1], [1, np.nan]]) >>> np.nansum(a) 3.0 >>> np.nansum(a, axis=0) array([ 2., 1.])
When positive infinity and negative infinity are present
>>> np.nansum([1, np.nan, np.inf]) inf >>> np.nansum([1, np.nan, np.NINF]) -inf >>> np.nansum([1, np.nan, np.inf, np.NINF]) nan
""" return _nanop(np.sum, 0, a, axis)
def nanmin(a, axis=None): """ Return the minimum of array elements over the given axis ignoring any NaNs.
Parameters ---------- a : array_like Array containing numbers whose sum is desired. If `a` is not an array, a conversion is attempted. axis : int, optional Axis along which the minimum is computed.The default is to compute the minimum of the flattened array.
Returns ------- y : {ndarray, scalar} An array with the same shape as `a`, with the specified axis removed. If `a` is a 0-d array, or if axis is None, a scalar is returned. The the same dtype as `a` is returned.
See Also -------- numpy.amin : Minimum across array including any Not a Numbers. numpy.nanmax : Maximum across array ignoring any Not a Numbers. isnan : Shows which elements are Not a Number (NaN). isfinite: Shows which elements are not: Not a Number, positive and negative infinity
Notes ----- Numpy uses the IEEE Standard for Binary Floating-Point for Arithmetic (IEEE 754). This means that Not a Number is not equivalent to infinity. Positive infinity is treated as a very large number and negative infinity is treated as a very small (i.e. negative) number.
If the input has a integer type, an integer type is returned unless the input contains NaNs and infinity.
Examples -------- >>> a = np.array([[1, 2], [3, np.nan]]) >>> np.nanmin(a) 1.0 >>> np.nanmin(a, axis=0) array([ 1., 2.]) >>> np.nanmin(a, axis=1) array([ 1., 3.])
When positive infinity and negative infinity are present:
>>> np.nanmin([1, 2, np.nan, np.inf]) 1.0 >>> np.nanmin([1, 2, np.nan, np.NINF]) -inf
""" return _nanop(np.min, np.inf, a, axis)
def nanargmin(a, axis=None): """ Return indices of the minimum values over an axis, ignoring NaNs.
Parameters ---------- a : array_like Input data. axis : int, optional Axis along which to operate. By default flattened input is used.
Returns ------- index_array : ndarray An array of indices or a single index value.
See Also -------- argmin, nanargmax
Examples -------- >>> a = np.array([[np.nan, 4], [2, 3]]) >>> np.argmin(a) 0 >>> np.nanargmin(a) 2 >>> np.nanargmin(a, axis=0) array([1, 1]) >>> np.nanargmin(a, axis=1) array([1, 0])
""" return _nanop(np.argmin, np.inf, a, axis)
def nanmax(a, axis=None): """ Return the maximum of array elements over the given axis ignoring any NaNs.
Parameters ---------- a : array_like Array containing numbers whose maximum is desired. If `a` is not an array, a conversion is attempted. axis : int, optional Axis along which the maximum is computed.The default is to compute the maximum of the flattened array.
Returns ------- y : ndarray An array with the same shape as `a`, with the specified axis removed. If `a` is a 0-d array, or if axis is None, a scalar is returned. The the same dtype as `a` is returned.
See Also -------- numpy.amax : Maximum across array including any Not a Numbers. numpy.nanmin : Minimum across array ignoring any Not a Numbers. isnan : Shows which elements are Not a Number (NaN). isfinite: Shows which elements are not: Not a Number, positive and negative infinity
Notes ----- Numpy uses the IEEE Standard for Binary Floating-Point for Arithmetic (IEEE 754). This means that Not a Number is not equivalent to infinity. Positive infinity is treated as a very large number and negative infinity is treated as a very small (i.e. negative) number.
If the input has a integer type, an integer type is returned unless the input contains NaNs and infinity.
Examples -------- >>> a = np.array([[1, 2], [3, np.nan]]) >>> np.nanmax(a) 3.0 >>> np.nanmax(a, axis=0) array([ 3., 2.]) >>> np.nanmax(a, axis=1) array([ 2., 3.])
When positive infinity and negative infinity are present:
>>> np.nanmax([1, 2, np.nan, np.NINF]) 2.0 >>> np.nanmax([1, 2, np.nan, np.inf]) inf
""" return _nanop(np.max, -np.inf, a, axis)
def nanargmax(a, axis=None): """ Return indices of the maximum values over an axis, ignoring NaNs.
Parameters ---------- a : array_like Input data. axis : int, optional Axis along which to operate. By default flattened input is used.
Returns ------- index_array : ndarray An array of indices or a single index value.
See Also -------- argmax, nanargmin
Examples -------- >>> a = np.array([[np.nan, 4], [2, 3]]) >>> np.argmax(a) 0 >>> np.nanargmax(a) 1 >>> np.nanargmax(a, axis=0) array([1, 0]) >>> np.nanargmax(a, axis=1) array([1, 1])
""" return _nanop(np.argmax, -np.inf, a, axis)
def disp(mesg, device=None, linefeed=True): """ Display a message on a device.
Parameters ---------- mesg : str Message to display. device : object Device to write message. If None, defaults to ``sys.stdout`` which is very similar to ``print``. `device` needs to have ``write()`` and ``flush()`` methods. linefeed : bool, optional Option whether to print a line feed or not. Defaults to True.
Raises ------ AttributeError If `device` does not have a ``write()`` or ``flush()`` method.
Examples -------- Besides ``sys.stdout``, a file-like object can also be used as it has both required methods:
>>> from StringIO import StringIO >>> buf = StringIO() >>> np.disp('"Display" in a file', device=buf) >>> buf.getvalue() '"Display" in a file\\n'
""" if device is None: import sys device = sys.stdout if linefeed: device.write('%s\n' % mesg) else: device.write('%s' % mesg) device.flush() return
# return number of input arguments and # number of default arguments import re def _get_nargs(obj): if not callable(obj): raise TypeError, "Object is not callable." if hasattr(obj,'func_code'): fcode = obj.func_code nargs = fcode.co_argcount if obj.func_defaults is not None: ndefaults = len(obj.func_defaults) else: ndefaults = 0 if isinstance(obj, types.MethodType): nargs -= 1 return nargs, ndefaults terr = re.compile(r'.*? takes exactly (?P<exargs>\d+) argument(s|) \((?P<gargs>\d+) given\)') try: obj() return 0, 0 except TypeError, msg: m = terr.match(str(msg)) if m: nargs = int(m.group('exargs')) ndefaults = int(m.group('gargs')) if isinstance(obj, types.MethodType): nargs -= 1 return nargs, ndefaults raise ValueError, 'failed to determine the number of arguments for %s' % (obj)
class vectorize(object): """ vectorize(pyfunc, otypes='', doc=None)
Generalized function class.
Define a vectorized function which takes a nested sequence of objects or numpy arrays as inputs and returns a numpy array as output. The vectorized function evaluates `pyfunc` over successive tuples of the input arrays like the python map function, except it uses the broadcasting rules of numpy.
The data type of the output of `vectorized` is determined by calling the function with the first element of the input. This can be avoided by specifying the `otypes` argument.
Parameters ---------- pyfunc : callable A python function or method. otypes : str or list of dtypes, optional The output data type. It must be specified as either a string of typecode characters or a list of data type specifiers. There should be one data type specifier for each output. doc : str, optional The docstring for the function. If None, the docstring will be the `pyfunc` one.
Examples -------- >>> def myfunc(a, b): ... \"\"\"Return a-b if a>b, otherwise return a+b\"\"\" ... if a > b: ... return a - b ... else: ... return a + b
>>> vfunc = np.vectorize(myfunc) >>> vfunc([1, 2, 3, 4], 2) array([3, 4, 1, 2])
The docstring is taken from the input function to `vectorize` unless it is specified
>>> vfunc.__doc__ 'Return a-b if a>b, otherwise return a+b' >>> vfunc = np.vectorize(myfunc, doc='Vectorized `myfunc`') >>> vfunc.__doc__ 'Vectorized `myfunc`'
The output type is determined by evaluating the first element of the input, unless it is specified
>>> out = vfunc([1, 2, 3, 4], 2) >>> type(out[0]) <type 'numpy.int32'> >>> vfunc = np.vectorize(myfunc, otypes=[np.float]) >>> out = vfunc([1, 2, 3, 4], 2) >>> type(out[0]) <type 'numpy.float64'>
""" def __init__(self, pyfunc, otypes='', doc=None): self.thefunc = pyfunc self.ufunc = None nin, ndefault = _get_nargs(pyfunc) if nin == 0 and ndefault == 0: self.nin = None self.nin_wo_defaults = None else: self.nin = nin self.nin_wo_defaults = nin - ndefault self.nout = None if doc is None: self.__doc__ = pyfunc.__doc__ else: self.__doc__ = doc if isinstance(otypes, str): self.otypes = otypes for char in self.otypes: if char not in typecodes['All']: raise ValueError, "invalid otype specified" elif iterable(otypes): self.otypes = ''.join([_nx.dtype(x).char for x in otypes]) else: raise ValueError, "output types must be a string of typecode characters or a list of data-types" self.lastcallargs = 0
def __call__(self, *args): # get number of outputs and output types by calling # the function on the first entries of args nargs = len(args) if self.nin: if (nargs > self.nin) or (nargs < self.nin_wo_defaults): raise ValueError, "mismatch between python function inputs"\ " and received arguments"
# we need a new ufunc if this is being called with more arguments. if (self.lastcallargs != nargs): self.lastcallargs = nargs self.ufunc = None self.nout = None
if self.nout is None or self.otypes == '': newargs = [] for arg in args: newargs.append(asarray(arg).flat[0]) theout = self.thefunc(*newargs) if isinstance(theout, tuple): self.nout = len(theout) else: self.nout = 1 theout = (theout,) if self.otypes == '': otypes = [] for k in range(self.nout): otypes.append(asarray(theout[k]).dtype.char) self.otypes = ''.join(otypes)
# Create ufunc if not already created if (self.ufunc is None): self.ufunc = frompyfunc(self.thefunc, nargs, self.nout)
# Convert to object arrays first newargs = [array(arg,copy=False,subok=True,dtype=object) for arg in args] if self.nout == 1: _res = array(self.ufunc(*newargs),copy=False, subok=True,dtype=self.otypes[0]) else: _res = tuple([array(x,copy=False,subok=True,dtype=c) \ for x, c in zip(self.ufunc(*newargs), self.otypes)]) return _res
def cov(m, y=None, rowvar=1, bias=0): """ Estimate a covariance matrix, given data.
Covariance indicates the level to which two variables vary together. If we examine N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]^T`, then the covariance matrix element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance of :math:`x_i`.
Parameters ---------- m : array_like A 1-D or 2-D array containing multiple variables and observations. Each row of `m` represents a variable, and each column a single observation of all those variables. Also see `rowvar` below. y : array_like, optional An additional set of variables and observations. `y` has the same form as that of `m`. rowvar : int, optional If `rowvar` is non-zero (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations. bias : int, optional Default normalization is by ``(N-1)``, where ``N`` is the number of observations given (unbiased estimate). If `bias` is 1, then normalization is by ``N``.
Returns ------- out : ndarray The covariance matrix of the variables.
See Also -------- corrcoef : Normalized covariance matrix
Examples -------- Consider two variables, :math:`x_0` and :math:`x_1`, which correlate perfectly, but in opposite directions:
>>> x = np.array([[0, 2], [1, 1], [2, 0]]).T >>> x array([[0, 1, 2], [2, 1, 0]])
Note how :math:`x_0` increases while :math:`x_1` decreases. The covariance matrix shows this clearly:
>>> np.cov(x) array([[ 1., -1.], [-1., 1.]])
Note that element :math:`C_{0,1}`, which shows the correlation between :math:`x_0` and :math:`x_1`, is negative.
Further, note how `x` and `y` are combined:
>>> x = [-2.1, -1, 4.3] >>> y = [3, 1.1, 0.12] >>> X = np.vstack((x,y)) >>> print np.cov(X) [[ 11.71 -4.286 ] [ -4.286 2.14413333]] >>> print np.cov(x, y) [[ 11.71 -4.286 ] [ -4.286 2.14413333]] >>> print np.cov(x) 11.71
"""
X = array(m, ndmin=2, dtype=float) if X.shape[0] == 1: rowvar = 1 if rowvar: axis = 0 tup = (slice(None),newaxis) else: axis = 1 tup = (newaxis, slice(None))
if y is not None: y = array(y, copy=False, ndmin=2, dtype=float) X = concatenate((X,y),axis)
X -= X.mean(axis=1-axis)[tup] if rowvar: N = X.shape[1] else: N = X.shape[0]
if bias: fact = N*1.0 else: fact = N-1.0
if not rowvar: return (dot(X.T, X.conj()) / fact).squeeze() else: return (dot(X, X.T.conj()) / fact).squeeze()
def corrcoef(x, y=None, rowvar=1, bias=0): """ Return correlation coefficients.
Please refer to the documentation for `cov` for more detail. The relationship between the correlation coefficient matrix, P, and the covariance matrix, C, is
.. math:: P_{ij} = \\frac{ C_{ij} } { \\sqrt{ C_{ii} * C_{jj} } }
The values of P are between -1 and 1.
Parameters ---------- m : array_like A 1-D or 2-D array containing multiple variables and observations. Each row of `m` represents a variable, and each column a single observation of all those variables. Also see `rowvar` below. y : array_like, optional An additional set of variables and observations. `y` has the same shape as `m`. rowvar : int, optional If `rowvar` is non-zero (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations. bias : int, optional Default normalization is by ``(N-1)``, where ``N`` is the number of observations given (unbiased estimate). If `bias` is 1, then normalization is by ``N``.
Returns ------- out : ndarray The correlation coefficient matrix of the variables.
See Also -------- cov : Covariance matrix
""" c = cov(x, y, rowvar, bias) try: d = diag(c) except ValueError: # scalar covariance return 1 return c/sqrt(multiply.outer(d,d))
def blackman(M): """ Return the Blackman window.
The Blackman window is a taper formed by using the the first three terms of a summation of cosines. It was designed to have close to the minimal leakage possible. It is close to optimal, only slightly worse than a Kaiser window.
Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned.
Returns ------- out : array The window, normalized to one (the value one appears only if the number of samples is odd).
See Also -------- bartlett, hamming, hanning, kaiser
Notes ----- The Blackman window is defined as
.. math:: w(n) = 0.42 - 0.5 \\cos(2\\pi n/M) + 0.08 \\cos(4\\pi n/M)
Most references to the Blackman window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. It is known as a "near optimal" tapering function, almost as good (by some measures) as the kaiser window.
References ---------- .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. [2] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [3] Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471.
Examples -------- >>> from numpy import blackman >>> blackman(12) array([ -1.38777878e-17, 3.26064346e-02, 1.59903635e-01, 4.14397981e-01, 7.36045180e-01, 9.67046769e-01, 9.67046769e-01, 7.36045180e-01, 4.14397981e-01, 1.59903635e-01, 3.26064346e-02, -1.38777878e-17])
Plot the window and the frequency response:
>>> from numpy import clip, log10, array, bartlett, linspace >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt
>>> window = blackman(51) >>> plt.plot(window) >>> plt.title("Blackman window") >>> plt.ylabel("Amplitude") >>> plt.xlabel("Sample") >>> plt.show()
>>> plt.figure() >>> A = fft(window, 2048) / 25.5 >>> mag = abs(fftshift(A)) >>> freq = linspace(-0.5,0.5,len(A)) >>> response = 20*log10(mag) >>> response = clip(response,-100,100) >>> plt.plot(freq, response) >>> plt.title("Frequency response of Blackman window") >>> plt.ylabel("Magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") >>> plt.axis('tight') >>> plt.show()
""" if M < 1: return array([]) if M == 1: return ones(1, float) n = arange(0,M) return 0.42-0.5*cos(2.0*pi*n/(M-1)) + 0.08*cos(4.0*pi*n/(M-1))
def bartlett(M): """ Return the Bartlett window.
The Bartlett window is very similar to a triangular window, except that the end points are at zero. It is often used in signal processing for tapering a signal, without generating too much ripple in the frequency domain.
Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned.
Returns ------- out : array The triangular window, normalized to one (the value one appears only if the number of samples is odd), with the first and last samples equal to zero.
See Also -------- blackman, hamming, hanning, kaiser
Notes ----- The Bartlett window is defined as
.. math:: w(n) = \\frac{2}{M-1} \\left( \\frac{M-1}{2} - \\left|n - \\frac{M-1}{2}\\right| \\right)
Most references to the Bartlett window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. Note that convolution with this window produces linear interpolation. It is also known as an apodization (which means"removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. The fourier transform of the Bartlett is the product of two sinc functions. Note the excellent discussion in Kanasewich.
References ---------- .. [1] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra", Biometrika 37, 1-16, 1950. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 109-110. .. [3] A.V. Oppenheim and R.W. Schafer, "Discrete-Time Signal Processing", Prentice-Hall, 1999, pp. 468-471. .. [4] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [5] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 429.
Examples -------- >>> np.bartlett(12) array([ 0. , 0.18181818, 0.36363636, 0.54545455, 0.72727273, 0.90909091, 0.90909091, 0.72727273, 0.54545455, 0.36363636, 0.18181818, 0. ])
Plot the window and its frequency response (requires SciPy and matplotlib):
>>> from numpy import clip, log10, array, bartlett, linspace >>> from numpy.fft import fft, fftshift >>> import matplotlib.pyplot as plt
>>> window = bartlett(51) >>> plt.plot(window) >>> plt.title("Bartlett window") >>> plt.ylabel("Amplitude") >>> plt.xlabel("Sample") >>> plt.show()
>>> plt.figure() >>> A = fft(window, 2048) / 25.5 >>> mag = abs(fftshift(A)) >>> freq = linspace(-0.5,0.5,len(A)) >>> response = 20*log10(mag) >>> response = clip(response,-100,100) >>> plt.plot(freq, response) >>> plt.title("Frequency response of Blackman window") >>> plt.ylabel("Magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") >>> plt.axis('tight') >>> plt.show()
""" if M < 1: return array([]) if M == 1: return ones(1, float) n = arange(0,M) return where(less_equal(n,(M-1)/2.0),2.0*n/(M-1),2.0-2.0*n/(M-1))
def hanning(M): """ Return the Hanning window.
The Hanning window is a taper formed by using a weighted cosine.
Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned.
Returns ------- out : ndarray, shape(M,) The window, normalized to one (the value one appears only if `M` is odd).
See Also -------- bartlett, blackman, hamming, kaiser
Notes ----- The Hanning window is defined as
.. math:: w(n) = 0.5 - 0.5cos\\left(\\frac{2\\pi{n}}{M-1}\\right) \\qquad 0 \\leq n \\leq M-1
The Hanning was named for Julius van Hann, an Austrian meterologist. It is also known as the Cosine Bell. Some authors prefer that it be called a Hann window, to help avoid confusion with the very similar Hamming window.
Most references to the Hanning window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function.
References ---------- .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 106-108. .. [3] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 425.
Examples -------- >>> from numpy import hanning >>> hanning(12) array([ 0. , 0.07937323, 0.29229249, 0.57115742, 0.82743037, 0.97974649, 0.97974649, 0.82743037, 0.57115742, 0.29229249, 0.07937323, 0. ])
Plot the window and its frequency response:
>>> from numpy.fft import fft, fftshift >>> import matplotlib.pyplot as plt
>>> window = np.hanning(51) >>> plt.plot(window) >>> plt.title("Hann window") >>> plt.ylabel("Amplitude") >>> plt.xlabel("Sample") >>> plt.show()
>>> plt.figure() >>> A = fft(window, 2048) / 25.5 >>> mag = abs(fftshift(A)) >>> freq = np.linspace(-0.5,0.5,len(A)) >>> response = 20*np.log10(mag) >>> response = np.clip(response,-100,100) >>> plt.plot(freq, response) >>> plt.title("Frequency response of the Hann window") >>> plt.ylabel("Magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") >>> plt.axis('tight') >>> plt.show()
""" if M < 1: return array([]) if M == 1: return ones(1, float) n = arange(0,M) return 0.5-0.5*cos(2.0*pi*n/(M-1))
def hamming(M): """ Return the Hamming window.
The Hamming window is a taper formed by using a weighted cosine.
Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned.
Returns ------- out : ndarray The window, normalized to one (the value one appears only if the number of samples is odd).
See Also -------- bartlett, blackman, hanning, kaiser
Notes ----- The Hamming window is defined as
.. math:: w(n) = 0.54 + 0.46cos\\left(\\frac{2\\pi{n}}{M-1}\\right) \\qquad 0 \\leq n \\leq M-1
The Hamming was named for R. W. Hamming, an associate of J. W. Tukey and is described in Blackman and Tukey. It was recommended for smoothing the truncated autocovariance function in the time domain. Most references to the Hamming window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function.
References ---------- .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 109-110. .. [3] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 425.
Examples -------- >>> np.hamming(12) array([ 0.08 , 0.15302337, 0.34890909, 0.60546483, 0.84123594, 0.98136677, 0.98136677, 0.84123594, 0.60546483, 0.34890909, 0.15302337, 0.08 ])
Plot the window and the frequency response:
>>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt
>>> window = np.hamming(51) >>> plt.plot(window) >>> plt.title("Hamming window") >>> plt.ylabel("Amplitude") >>> plt.xlabel("Sample") >>> plt.show()
>>> plt.figure() >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) >>> plt.title("Frequency response of Hamming window") >>> plt.ylabel("Magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") >>> plt.axis('tight') >>> plt.show()
""" if M < 1: return array([]) if M == 1: return ones(1,float) n = arange(0,M) return 0.54-0.46*cos(2.0*pi*n/(M-1))
## Code from cephes for i0
_i0A = [ -4.41534164647933937950E-18, 3.33079451882223809783E-17, -2.43127984654795469359E-16, 1.71539128555513303061E-15, -1.16853328779934516808E-14, 7.67618549860493561688E-14, -4.85644678311192946090E-13, 2.95505266312963983461E-12, -1.72682629144155570723E-11, 9.67580903537323691224E-11, -5.18979560163526290666E-10, 2.65982372468238665035E-9, -1.30002500998624804212E-8, 6.04699502254191894932E-8, -2.67079385394061173391E-7, 1.11738753912010371815E-6, -4.41673835845875056359E-6, 1.64484480707288970893E-5, -5.75419501008210370398E-5, 1.88502885095841655729E-4, -5.76375574538582365885E-4, 1.63947561694133579842E-3, -4.32430999505057594430E-3, 1.05464603945949983183E-2, -2.37374148058994688156E-2, 4.93052842396707084878E-2, -9.49010970480476444210E-2, 1.71620901522208775349E-1, -3.04682672343198398683E-1, 6.76795274409476084995E-1]
_i0B = [ -7.23318048787475395456E-18, -4.83050448594418207126E-18, 4.46562142029675999901E-17, 3.46122286769746109310E-17, -2.82762398051658348494E-16, -3.42548561967721913462E-16, 1.77256013305652638360E-15, 3.81168066935262242075E-15, -9.55484669882830764870E-15, -4.15056934728722208663E-14, 1.54008621752140982691E-14, 3.85277838274214270114E-13, 7.18012445138366623367E-13, -1.79417853150680611778E-12, -1.32158118404477131188E-11, -3.14991652796324136454E-11, 1.18891471078464383424E-11, 4.94060238822496958910E-10, 3.39623202570838634515E-9, 2.26666899049817806459E-8, 2.04891858946906374183E-7, 2.89137052083475648297E-6, 6.88975834691682398426E-5, 3.36911647825569408990E-3, 8.04490411014108831608E-1]
def _chbevl(x, vals): b0 = vals[0] b1 = 0.0
for i in xrange(1,len(vals)): b2 = b1 b1 = b0 b0 = x*b1 - b2 + vals[i]
return 0.5*(b0 - b2)
def _i0_1(x): return exp(x) * _chbevl(x/2.0-2, _i0A)
def _i0_2(x): return exp(x) * _chbevl(32.0/x - 2.0, _i0B) / sqrt(x)
def i0(x): """ Modified Bessel function of the first kind, order 0.
Usually denoted :math:`I_0`. This function does broadcast, but will *not* "up-cast" int dtype arguments unless accompanied by at least one float or complex dtype argument (see Raises below).
Parameters ---------- x : array_like, dtype float or complex Argument of the Bessel function.
Returns ------- out : ndarray, shape = x.shape, dtype = x.dtype The modified Bessel function evaluated at each of the elements of `x`.
Raises ------ TypeError: array cannot be safely cast to required type If argument consists exclusively of int dtypes.
See Also -------- scipy.special.iv, scipy.special.ive
Notes ----- We use the algorithm published by Clenshaw [1]_ and referenced by Abramowitz and Stegun [2]_, for which the function domain is partitioned into the two intervals [0,8] and (8,inf), and Chebyshev polynomial expansions are employed in each interval. Relative error on the domain [0,30] using IEEE arithmetic is documented [3]_ as having a peak of 5.8e-16 with an rms of 1.4e-16 (n = 30000).
References ---------- .. [1] C. W. Clenshaw, "Chebyshev series for mathematical functions," in *National Physical Laboratory Mathematical Tables*, vol. 5, London: Her Majesty's Stationery Office, 1962. .. [2] M. Abramowitz and I. A. Stegun, *Handbook of Mathematical Functions*, 10th printing, New York: Dover, 1964, pp. 379. http://www.math.sfu.ca/~cbm/aands/page_379.htm .. [3] http://kobesearch.cpan.org/htdocs/Math-Cephes/Math/Cephes.html
Examples -------- >>> np.i0([0.]) array(1.0) >>> np.i0([0., 1. + 2j]) array([ 1.00000000+0.j , 0.18785373+0.64616944j])
""" x = atleast_1d(x).copy() y = empty_like(x) ind = (x<0) x[ind] = -x[ind] ind = (x<=8.0) y[ind] = _i0_1(x[ind]) ind2 = ~ind y[ind2] = _i0_2(x[ind2]) return y.squeeze()
## End of cephes code for i0
def kaiser(M,beta): """ Return the Kaiser window.
The Kaiser window is a taper formed by using a Bessel function.
Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. beta : float Shape parameter for window.
Returns ------- out : array The window, normalized to one (the value one appears only if the number of samples is odd).
See Also -------- bartlett, blackman, hamming, hanning
Notes ----- The Kaiser window is defined as
.. math:: w(n) = I_0\\left( \\beta \\sqrt{1-\\frac{4n^2}{(M-1)^2}} \\right)/I_0(\\beta)
with
.. math:: \\quad -\\frac{M-1}{2} \\leq n \\leq \\frac{M-1}{2},
where :math:`I_0` is the modified zeroth-order Bessel function.
The Kaiser was named for Jim Kaiser, who discovered a simple approximation to the DPSS window based on Bessel functions. The Kaiser window is a very good approximation to the Digital Prolate Spheroidal Sequence, or Slepian window, which is the transform which maximizes the energy in the main lobe of the window relative to total energy.
The Kaiser can approximate many other windows by varying the beta parameter.
==== ======================= beta Window shape ==== ======================= 0 Rectangular 5 Similar to a Hamming 6 Similar to a Hanning 8.6 Similar to a Blackman ==== =======================
A beta value of 14 is probably a good starting point. Note that as beta gets large, the window narrows, and so the number of samples needs to be large enough to sample the increasingly narrow spike, otherwise nans will get returned.
Most references to the Kaiser window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function.
References ---------- .. [1] J. F. Kaiser, "Digital Filters" - Ch 7 in "Systems analysis by digital computer", Editors: F.F. Kuo and J.F. Kaiser, p 218-285. John Wiley and Sons, New York, (1966). .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 177-178. .. [3] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function
Examples -------- >>> from numpy import kaiser >>> kaiser(12, 14) array([ 7.72686684e-06, 3.46009194e-03, 4.65200189e-02, 2.29737120e-01, 5.99885316e-01, 9.45674898e-01, 9.45674898e-01, 5.99885316e-01, 2.29737120e-01, 4.65200189e-02, 3.46009194e-03, 7.72686684e-06])
Plot the window and the frequency response:
>>> from numpy import clip, log10, array, kaiser, linspace >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt
>>> window = kaiser(51, 14) >>> plt.plot(window) >>> plt.title("Kaiser window") >>> plt.ylabel("Amplitude") >>> plt.xlabel("Sample") >>> plt.show()
>>> plt.figure() >>> A = fft(window, 2048) / 25.5 >>> mag = abs(fftshift(A)) >>> freq = linspace(-0.5,0.5,len(A)) >>> response = 20*log10(mag) >>> response = clip(response,-100,100) >>> plt.plot(freq, response) >>> plt.title("Frequency response of Kaiser window") >>> plt.ylabel("Magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") >>> plt.axis('tight') >>> plt.show()
""" from numpy.dual import i0 if M == 1: return np.array([1.]) n = arange(0,M) alpha = (M-1)/2.0 return i0(beta * sqrt(1-((n-alpha)/alpha)**2.0))/i0(float(beta))
def sinc(x): """ Return the sinc function.
The sinc function is :math:`\\sin(\\pi x)/(\\pi x)`.
Parameters ---------- x : ndarray Array (possibly multi-dimensional) of values for which to to calculate ``sinc(x)``.
Returns ------- out : ndarray ``sinc(x)``, which has the same shape as the input.
Notes ----- ``sinc(0)`` is the limit value 1.
The name sinc is short for "sine cardinal" or "sinus cardinalis".
The sinc function is used in various signal processing applications, including in anti-aliasing, in the construction of a Lanczos resampling filter, and in interpolation.
For bandlimited interpolation of discrete-time signals, the ideal interpolation kernel is proportional to the sinc function.
References ---------- .. [1] Weisstein, Eric W. "Sinc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SincFunction.html .. [2] Wikipedia, "Sinc function", http://en.wikipedia.org/wiki/Sinc_function
Examples -------- >>> x = np.arange(-20., 21.)/5. >>> np.sinc(x) array([ -3.89804309e-17, -4.92362781e-02, -8.40918587e-02, -8.90384387e-02, -5.84680802e-02, 3.89804309e-17, 6.68206631e-02, 1.16434881e-01, 1.26137788e-01, 8.50444803e-02, -3.89804309e-17, -1.03943254e-01, -1.89206682e-01, -2.16236208e-01, -1.55914881e-01, 3.89804309e-17, 2.33872321e-01, 5.04551152e-01, 7.56826729e-01, 9.35489284e-01, 1.00000000e+00, 9.35489284e-01, 7.56826729e-01, 5.04551152e-01, 2.33872321e-01, 3.89804309e-17, -1.55914881e-01, -2.16236208e-01, -1.89206682e-01, -1.03943254e-01, -3.89804309e-17, 8.50444803e-02, 1.26137788e-01, 1.16434881e-01, 6.68206631e-02, 3.89804309e-17, -5.84680802e-02, -8.90384387e-02, -8.40918587e-02, -4.92362781e-02, -3.89804309e-17])
>>> import matplotlib.pyplot as plt >>> plt.plot(x, np.sinc(x)) >>> plt.title("Sinc Function") >>> plt.ylabel("Amplitude") >>> plt.xlabel("X") >>> plt.show()
It works in 2-D as well:
>>> x = np.arange(-200., 201.)/50. >>> xx = np.outer(x, x) >>> plt.imshow(np.sinc(xx))
""" y = pi* where(x == 0, 1.0e-20, x) return sin(y)/y
def msort(a): """ Return a copy of an array sorted along the first axis.
Parameters ---------- a : array_like Array to be sorted.
Returns ------- sorted_array : ndarray Array of the same type and shape as `a`.
See Also -------- sort
Notes ----- ``np.msort(a)`` is equivalent to ``np.sort(a, axis=0)``.
""" b = array(a,subok=True,copy=True) b.sort(0) return b
def median(a, axis=None, out=None, overwrite_input=False): """ Compute the median along the specified axis.
Returns the median of the array elements.
Parameters ---------- a : array_like Input array or object that can be converted to an array. axis : {None, int}, optional Axis along which the medians are computed. The default (axis=None) is to compute the median along a flattened version of the array. out : ndarray, optional Alternative output array in which to place the result. It must have the same shape and buffer length as the expected output, but the type (of the output) will be cast if necessary. overwrite_input : {False, True}, optional If True, then allow use of memory of input array (a) for calculations. The input array will be modified by the call to median. This will save memory when you do not need to preserve the contents of the input array. Treat the input as undefined, but it will probably be fully or partially sorted. Default is False. Note that, if `overwrite_input` is True and the input is not already an ndarray, an error will be raised.
Returns ------- median : ndarray A new array holding the result (unless `out` is specified, in which case that array is returned instead). If the input contains integers, or floats of smaller precision than 64, then the output data-type is float64. Otherwise, the output data-type is the same as that of the input.
See Also -------- mean
Notes ----- Given a vector V of length N, the median of V is the middle value of a sorted copy of V, ``V_sorted`` - i.e., ``V_sorted[(N-1)/2]``, when N is odd. When N is even, it is the average of the two middle values of ``V_sorted``.
Examples -------- >>> a = np.array([[10, 7, 4], [3, 2, 1]]) >>> a array([[10, 7, 4], [ 3, 2, 1]]) >>> np.median(a) 3.5 >>> np.median(a, axis=0) array([ 6.5, 4.5, 2.5]) >>> np.median(a, axis=1) array([ 7., 2.]) >>> m = np.median(a, axis=0) >>> out = np.zeros_like(m) >>> np.median(a, axis=0, out=m) array([ 6.5, 4.5, 2.5]) >>> m array([ 6.5, 4.5, 2.5]) >>> b = a.copy() >>> np.median(b, axis=1, overwrite_input=True) array([ 7., 2.]) >>> assert not np.all(a==b) >>> b = a.copy() >>> np.median(b, axis=None, overwrite_input=True) 3.5 >>> assert not np.all(a==b)
""" if overwrite_input: if axis is None: sorted = a.ravel() sorted.sort() else: a.sort(axis=axis) sorted = a else: sorted = sort(a, axis=axis) if axis is None: axis = 0 indexer = [slice(None)] * sorted.ndim index = int(sorted.shape[axis]/2) if sorted.shape[axis] % 2 == 1: # index with slice to allow mean (below) to work indexer[axis] = slice(index, index+1) else: indexer[axis] = slice(index-1, index+1) # Use mean in odd and even case to coerce data type # and check, use out array. return mean(sorted[indexer], axis=axis, out=out)
def trapz(y, x=None, dx=1.0, axis=-1): """ Integrate along the given axis using the composite trapezoidal rule.
Integrate `y` (`x`) along given axis.
Parameters ---------- y : array_like Input array to integrate. x : array_like, optional If `x` is None, then spacing between all `y` elements is `dx`. dx : scalar, optional If `x` is None, spacing given by `dx` is assumed. Default is 1. axis : int, optional Specify the axis.
Returns ------- out : float Definite integral as approximated by trapezoidal rule.
See Also -------- sum, cumsum
Notes ----- Image [2]_ illustrates trapezoidal rule -- y-axis locations of points will be taken from `y` array, by default x-axis distances between points will be 1.0, alternatively they can be provided with `x` array or with `dx` scalar. Return value will be equal to combined area under the red lines.
References ---------- .. [1] Wikipedia page: http://en.wikipedia.org/wiki/Trapezoidal_rule
.. [2] Illustration image: http://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png
Examples -------- >>> np.trapz([1,2,3]) 4.0 >>> np.trapz([1,2,3], x=[4,6,8]) 8.0 >>> np.trapz([1,2,3], dx=2) 8.0 >>> a = np.arange(6).reshape(2, 3) >>> a array([[0, 1, 2], [3, 4, 5]]) >>> np.trapz(a, axis=0) array([ 1.5, 2.5, 3.5]) >>> np.trapz(a, axis=1) array([ 2., 8.])
""" y = asarray(y) if x is None: d = dx else: x = asarray(x) if x.ndim == 1: d = diff(x) # reshape to correct shape shape = [1]*y.ndim shape[axis] = d.shape[0] d = d.reshape(shape) else: d = diff(x, axis=axis) nd = len(y.shape) slice1 = [slice(None)]*nd slice2 = [slice(None)]*nd slice1[axis] = slice(1,None) slice2[axis] = slice(None,-1) return add.reduce(d * (y[slice1]+y[slice2])/2.0,axis)
#always succeed def add_newdoc(place, obj, doc): """Adds documentation to obj which is in module place.
If doc is a string add it to obj as a docstring
If doc is a tuple, then the first element is interpreted as an attribute of obj and the second as the docstring (method, docstring)
If doc is a list, then each element of the list should be a sequence of length two --> [(method1, docstring1), (method2, docstring2), ...]
This routine never raises an error. """ try: new = {} exec 'from %s import %s' % (place, obj) in new if isinstance(doc, str): add_docstring(new[obj], doc.strip()) elif isinstance(doc, tuple): add_docstring(getattr(new[obj], doc[0]), doc[1].strip()) elif isinstance(doc, list): for val in doc: add_docstring(getattr(new[obj], val[0]), val[1].strip()) except: pass
# From matplotlib def meshgrid(x,y): """ Return coordinate matrices from two coordinate vectors.
Parameters ---------- x, y : ndarray Two 1-D arrays representing the x and y coordinates of a grid.
Returns ------- X, Y : ndarray For vectors `x`, `y` with lengths ``Nx=len(x)`` and ``Ny=len(y)``, return `X`, `Y` where `X` and `Y` are ``(Ny, Nx)`` shaped arrays with the elements of `x` and y repeated to fill the matrix along the first dimension for `x`, the second for `y`.
See Also -------- index_tricks.mgrid : Construct a multi-dimensional "meshgrid" using indexing notation. index_tricks.ogrid : Construct an open multi-dimensional "meshgrid" using indexing notation.
Examples -------- >>> X, Y = np.meshgrid([1,2,3], [4,5,6,7]) >>> X array([[1, 2, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3]]) >>> Y array([[4, 4, 4], [5, 5, 5], [6, 6, 6], [7, 7, 7]])
`meshgrid` is very useful to evaluate functions on a grid.
>>> x = np.arange(-5, 5, 0.1) >>> y = np.arange(-5, 5, 0.1) >>> xx, yy = np.meshgrid(x, y) >>> z = np.sin(xx**2+yy**2)/(xx**2+yy**2)
""" x = asarray(x) y = asarray(y) numRows, numCols = len(y), len(x) # yes, reversed x = x.reshape(1,numCols) X = x.repeat(numRows, axis=0)
y = y.reshape(numRows,1) Y = y.repeat(numCols, axis=1) return X, Y
def delete(arr, obj, axis=None): """ Return a new array with sub-arrays along an axis deleted.
Parameters ---------- arr : array_like Input array. obj : slice, int or array of ints Indicate which sub-arrays to remove. axis : int, optional The axis along which to delete the subarray defined by `obj`. If `axis` is None, `obj` is applied to the flattened array.
Returns ------- out : ndarray A copy of `arr` with the elements specified by `obj` removed. Note that `delete` does not occur in-place. If `axis` is None, `out` is a flattened array.
See Also -------- insert : Insert elements into an array. append : Append elements at the end of an array.
Examples -------- >>> arr = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]]) >>> arr array([[ 1, 2, 3, 4], [ 5, 6, 7, 8], [ 9, 10, 11, 12]]) >>> np.delete(arr, 1, 0) array([[ 1, 2, 3, 4], [ 9, 10, 11, 12]])
>>> np.delete(arr, np.s_[::2], 1) array([[ 2, 4], [ 6, 8], [10, 12]]) >>> np.delete(arr, [1,3,5], None) array([ 1, 3, 5, 7, 8, 9, 10, 11, 12])
""" wrap = None if type(arr) is not ndarray: try: wrap = arr.__array_wrap__ except AttributeError: pass
arr = asarray(arr) ndim = arr.ndim if axis is None: if ndim != 1: arr = arr.ravel() ndim = arr.ndim; axis = ndim-1; if ndim == 0: if wrap: return wrap(arr) else: return arr.copy() slobj = [slice(None)]*ndim N = arr.shape[axis] newshape = list(arr.shape) if isinstance(obj, (int, long, integer)): if (obj < 0): obj += N if (obj < 0 or obj >=N): raise ValueError, "invalid entry" newshape[axis]-=1; new = empty(newshape, arr.dtype, arr.flags.fnc) slobj[axis] = slice(None, obj) new[slobj] = arr[slobj] slobj[axis] = slice(obj,None) slobj2 = [slice(None)]*ndim slobj2[axis] = slice(obj+1,None) new[slobj] = arr[slobj2] elif isinstance(obj, slice): start, stop, step = obj.indices(N) numtodel = len(xrange(start, stop, step)) if numtodel <= 0: if wrap: return wrap(new) else: return arr.copy() newshape[axis] -= numtodel new = empty(newshape, arr.dtype, arr.flags.fnc) # copy initial chunk if start == 0: pass else: slobj[axis] = slice(None, start) new[slobj] = arr[slobj] # copy end chunck if stop == N: pass else: slobj[axis] = slice(stop-numtodel,None) slobj2 = [slice(None)]*ndim slobj2[axis] = slice(stop, None) new[slobj] = arr[slobj2] # copy middle pieces if step == 1: pass else: # use array indexing. obj = arange(start, stop, step, dtype=intp) all = arange(start, stop, dtype=intp) obj = setdiff1d(all, obj) slobj[axis] = slice(start, stop-numtodel) slobj2 = [slice(None)]*ndim slobj2[axis] = obj new[slobj] = arr[slobj2] else: # default behavior obj = array(obj, dtype=intp, copy=0, ndmin=1) all = arange(N, dtype=intp) obj = setdiff1d(all, obj) slobj[axis] = obj new = arr[slobj] if wrap: return wrap(new) else: return new
def insert(arr, obj, values, axis=None): """ Insert values along the given axis before the given indices.
Parameters ---------- arr : array_like Input array. obj : int, slice or sequence of ints Object that defines the index or indices before which `values` is inserted. values : array_like Values to insert into `arr`. If the type of `values` is different from that of `arr`, `values` is converted to the type of `arr`. axis : int, optional Axis along which to insert `values`. If `axis` is None then `arr` is flattened first.
Returns ------- out : ndarray A copy of `arr` with `values` inserted. Note that `insert` does not occur in-place: a new array is returned. If `axis` is None, `out` is a flattened array.
See Also -------- append : Append elements at the end of an array. delete : Delete elements from an array.
Examples -------- >>> a = np.array([[1, 1], [2, 2], [3, 3]]) >>> a array([[1, 1], [2, 2], [3, 3]]) >>> np.insert(a, 1, 5) array([1, 5, 1, 2, 2, 3, 3]) >>> np.insert(a, 1, 5, axis=1) array([[1, 5, 1], [2, 5, 2], [3, 5, 3]])
>>> b = a.flatten() >>> b array([1, 1, 2, 2, 3, 3]) >>> np.insert(b, [2, 2], [5, 6]) array([1, 1, 5, 6, 2, 2, 3, 3])
>>> np.insert(b, slice(2, 4), [5, 6]) array([1, 1, 5, 2, 6, 2, 3, 3])
>>> np.insert(b, [2, 2], [7.13, False]) # type casting array([1, 1, 7, 0, 2, 2, 3, 3])
>>> x = np.arange(8).reshape(2, 4) >>> idx = (1, 3) >>> np.insert(x, idx, 999, axis=1) array([[ 0, 999, 1, 2, 999, 3], [ 4, 999, 5, 6, 999, 7]])
""" wrap = None if type(arr) is not ndarray: try: wrap = arr.__array_wrap__ except AttributeError: pass
arr = asarray(arr) ndim = arr.ndim if axis is None: if ndim != 1: arr = arr.ravel() ndim = arr.ndim axis = ndim-1 if (ndim == 0): arr = arr.copy() arr[...] = values if wrap: return wrap(arr) else: return arr slobj = [slice(None)]*ndim N = arr.shape[axis] newshape = list(arr.shape) if isinstance(obj, (int, long, integer)): if (obj < 0): obj += N if obj < 0 or obj > N: raise ValueError, "index (%d) out of range (0<=index<=%d) "\ "in dimension %d" % (obj, N, axis) newshape[axis] += 1; new = empty(newshape, arr.dtype, arr.flags.fnc) slobj[axis] = slice(None, obj) new[slobj] = arr[slobj] slobj[axis] = obj new[slobj] = values slobj[axis] = slice(obj+1,None) slobj2 = [slice(None)]*ndim slobj2[axis] = slice(obj,None) new[slobj] = arr[slobj2] if wrap: return wrap(new) return new
elif isinstance(obj, slice): # turn it into a range object obj = arange(*obj.indices(N),**{'dtype':intp})
# get two sets of indices # one is the indices which will hold the new stuff # two is the indices where arr will be copied over
obj = asarray(obj, dtype=intp) numnew = len(obj) index1 = obj + arange(numnew) index2 = setdiff1d(arange(numnew+N),index1) newshape[axis] += numnew new = empty(newshape, arr.dtype, arr.flags.fnc) slobj2 = [slice(None)]*ndim slobj[axis] = index1 slobj2[axis] = index2 new[slobj] = values new[slobj2] = arr
if wrap: return wrap(new) return new
def append(arr, values, axis=None): """ Append values to the end of an array.
Parameters ---------- arr : array_like Values are appended to a copy of this array. values : array_like These values are appended to a copy of `arr`. It must be of the correct shape (the same shape as `arr`, excluding `axis`). If `axis` is not specified, `values` can be any shape and will be flattened before use. axis : int, optional The axis along which `values` are appended. If `axis` is not given, both `arr` and `values` are flattened before use.
Returns ------- out : ndarray A copy of `arr` with `values` appended to `axis`. Note that `append` does not occur in-place: a new array is allocated and filled. If `axis` is None, `out` is a flattened array.
See Also -------- insert : Insert elements into an array. delete : Delete elements from an array.
Examples -------- >>> np.append([1, 2, 3], [[4, 5, 6], [7, 8, 9]]) array([1, 2, 3, 4, 5, 6, 7, 8, 9])
When `axis` is specified, `values` must have the correct shape.
>>> np.append([[1, 2, 3], [4, 5, 6]], [[7, 8, 9]], axis=0) array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> np.append([[1, 2, 3], [4, 5, 6]], [7, 8, 9], axis=0) Traceback (most recent call last): ... ValueError: arrays must have same number of dimension
""" arr = asanyarray(arr) if axis is None: if arr.ndim != 1: arr = arr.ravel() values = ravel(values) axis = arr.ndim-1 return concatenate((arr, values), axis=axis)
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