__docformat__ = "restructuredtext en" __all__ = ['logspace', 'linspace', 'select', 'piecewise', 'trim_zeros', 'copy', 'iterable', 'diff', 'gradient', 'angle', 'unwrap', 'sort_complex', 'disp', 'unique', '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.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.lib.shape_base 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 import numpy as np #end Fernando's utilities def linspace(start, stop, num=50, endpoint=True, retstep=False): """ Return evenly spaced numbers over a specified interval. Returns `num` evenly spaced samples, calculated over the interval [`start`, `stop` ]. The endpoint of the interval can optionally be excluded. Parameters ---------- start : {float, int} The starting value of the sequence. stop : {float, int} The end value of the sequence, unless `endpoint` is set to False. In that case, the sequence consists of all but the last of ``num + 1`` evenly spaced samples, so that `stop` is excluded. Note that the step size changes when `endpoint` is False. num : int, optional Number of samples to generate. Default is 50. endpoint : bool, optional If True, `stop` is the last sample. Otherwise, it is not included. Default is True. retstep : bool, optional If True, return (`samples`, `step`), where `step` is the spacing between samples. Returns ------- samples : ndarray There are `num` equally spaced samples in the closed interval ``[start, stop]`` or the half-open interval ``[start, stop)`` (depending on whether `endpoint` is True or False). step : float (only if `retstep` is True) Size of spacing between samples. See Also -------- arange : Similiar to `linspace`, but uses a step size (instead of the number of samples). logspace : Samples uniformly distributed in log space. Examples -------- >>> np.linspace(2.0, 3.0, num=5) array([ 2. , 2.25, 2.5 , 2.75, 3. ]) >>> np.linspace(2.0, 3.0, num=5, endpoint=False) array([ 2. , 2.2, 2.4, 2.6, 2.8]) >>> np.linspace(2.0, 3.0, num=5, retstep=True) (array([ 2. , 2.25, 2.5 , 2.75, 3. ]), 0.25) Graphical illustration: >>> import matplotlib.pyplot as plt >>> N = 8 >>> y = np.zeros(N) >>> x1 = np.linspace(0, 10, N, endpoint=True) >>> x2 = np.linspace(0, 10, N, endpoint=False) >>> plt.plot(x1, y, 'o') >>> plt.plot(x2, y + 0.5, 'o') >>> plt.ylim([-0.5, 1]) >>> plt.show() """ num = int(num) if num <= 0: return array([], float) if endpoint: if num == 1: return array([float(start)]) step = (stop-start)/float((num-1)) y = _nx.arange(0, num) * step + start y[-1] = stop else: step = (stop-start)/float(num) y = _nx.arange(0, num) * step + start if retstep: return y, step else: return y def logspace(start,stop,num=50,endpoint=True,base=10.0): """ Return numbers spaced evenly on a log scale. In linear space, the sequence starts at ``base ** start`` (`base` to the power of `start`) and ends with ``base ** stop`` (see `endpoint` below). Parameters ---------- start : float ``base ** start`` is the starting value of the sequence. stop : float ``base ** stop`` is the final value of the sequence, unless `endpoint` is False. In that case, ``num + 1`` values are spaced over the interval in log-space, of which all but the last (a sequence of length ``num``) are returned. num : integer, optional Number of samples to generate. Default is 50. endpoint : boolean, optional If true, `stop` is the last sample. Otherwise, it is not included. Default is True. base : float, optional The base of the log space. The step size between the elements in ``ln(samples) / ln(base)`` (or ``log_base(samples)``) is uniform. Default is 10.0. Returns ------- samples : ndarray `num` samples, equally spaced on a log scale. See Also -------- arange : Similiar to linspace, with the step size specified instead of the number of samples. Note that, when used with a float endpoint, the endpoint may or may not be included. linspace : Similar to logspace, but with the samples uniformly distributed in linear space, instead of log space. Notes ----- Logspace is equivalent to the code >>> y = linspace(start, stop, num=num, endpoint=endpoint) >>> power(base, y) Examples -------- >>> np.logspace(2.0, 3.0, num=4) array([ 100. , 215.443469 , 464.15888336, 1000. ]) >>> np.logspace(2.0, 3.0, num=4, endpoint=False) array([ 100. , 177.827941 , 316.22776602, 562.34132519]) >>> np.logspace(2.0, 3.0, num=4, base=2.0) array([ 4. , 5.0396842 , 6.34960421, 8. ]) Graphical illustration: >>> import matplotlib.pyplot as plt >>> N = 10 >>> x1 = np.logspace(0.1, 1, N, endpoint=True) >>> x2 = np.logspace(0.1, 1, N, endpoint=False) >>> y = np.zeros(N) >>> plt.plot(x1, y, 'o') >>> plt.plot(x2, y + 0.5, 'o') >>> plt.ylim([-0.5, 1]) >>> plt.show() """ y = linspace(start,stop,num=num,endpoint=endpoint) return _nx.power(base,y) 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. 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 often not be equal to 1; 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 1.4. 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 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])) """ # Old behavior if new == False: warnings.warn(""" The histogram semantics being used is now deprecated and will disappear in NumPy 1.4. 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 1.4. """, 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 is False: return n, bins elif normed is True: db = array(np.diff(bins), float) return n/(n*db).sum(), bins def histogramdd(sample, bins=10, range=None, normed=False, weights=None): """ Compute the multidimensional histogram of some data. Parameters ---------- sample : array_like Data to histogram passed as a sequence of D arrays of length N, or as an (N,D) array. 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) hist = zeros(nbin.prod(), float) # 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 'Internal Shape Error' return hist, edges def average(a, axis=None, weights=None, returned=False): """ Return the weighted average of array over the specified axis. Parameters ---------- a : array_like Data to be averaged. axis : int, optional Axis along which to average `a`. If `None`, averaging is done over the entire array irrespective of its shape. weights : array_like, optional The importance that each datum has in the computation of the average. 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. Note that 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 -------- 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 """ 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 : (N,) ndarray The input domain. condlist : list of M (N,)-shaped boolean 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 the length of `funclist` is M+1, then that extra function is the default value, used wherever all conditions are false. funclist : list of M or M+1 callables, f(x,*args,**kw), or values 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 value is provided then a constant function (``lambda x: value``) 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 : dictionary, 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. 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 # x runs from -2.5 to 2.5 in steps of 1 >>> np.piecewise(x, [x < 0, x >= 0.5], [-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 N boolean arrays of length M The conditions C_0 through C_(N-1) which determine from which vector the output elements are taken. choicelist : list of N arrays of length M Th vectors V_0 through V_(N-1), from which the output elements are chosen. Returns ------- output : 1-dimensional array of length M The output at position m is the m-th element of the first vector V_n for which C_n[m] is non-zero. Note that the output depends on the order of conditions, since the first satisfied condition is used. Notes ----- Equivalent to: :: output = [] for m in range(M): output += [V[m] for V,C in zip(values,cond) if C[m]] or [default] Examples -------- >>> t = np.arange(10) >>> s = np.arange(10)*100 >>> condlist = [t == 4, t > 5] >>> choicelist = [s, t] >>> np.select(condlist, choicelist) array([ 0, 0, 0, 0, 400, 0, 6, 7, 8, 9]) """ 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 -------- >>> np.gradient(np.array([[1,1],[3,4]])) [array([[ 2., 3.], [ 2., 3.]]), array([[ 0., 0.], [ 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 nth order discrete difference along given axis. 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. Returns ------- out : ndarray The `n` order differences. The shape of the output is the same as `a` except along `axis` where the dimension is `n` less. Examples -------- >>> x = np.array([0,1,3,9,5,10]) >>> np.diff(x) array([ 1, 2, 6, -4, 5]) >>> np.diff(x,n=2) array([ 1, 4, -10, 9]) >>> 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] try: add_docstring(digitize, r"""digitize(x,bins) Return the index of the bin to which each value of x belongs. Each index i returned is such that bins[i-1] <= x < bins[i] if bins is monotonically increasing, or bins [i-1] > x >= bins[i] if bins is monotonically decreasing. Beyond the bounds of the bins 0 or len(bins) is returned as appropriate. """) except RuntimeError: pass try: add_docstring(bincount, r"""bincount(x,weights=None) Return the number of occurrences of each value in x. x must be a list of non-negative integers. The output, b[i], represents the number of times that i is found in x. If weights is specified, every occurrence of i at a position p contributes weights[p] instead of 1. See also: histogram, digitize, unique. """) except RuntimeError: pass try: add_docstring(add_docstring, r"""docstring(obj, docstring) Add a docstring to a built-in obj if possible. If the obj already has a docstring raise a RuntimeError If this routine does not know how to add a docstring to the object raise a TypeError """) except RuntimeError: pass 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 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 Maximum discontinuity between values. axis : integer Axis along which unwrap will operate. Returns ------- out : ndarray Output array """ 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)>> 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([5 + 2j, 3 - 1j, 6 - 2j, 2 - 3j, 1 - 5j]) array([ 1.-5.j, 2.-3.j, 3.-1.j, 5.+2.j, 6.-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 trailing zeros from a 1D array. Parameters ---------- filt : array_like Input array. trim : string, optional A string with 'f' representing trim from front and 'b' to trim from back. Examples -------- >>> a = np.array((0, 0, 0, 1, 2, 3, 2, 1, 0)) >>> np.trim_zeros(a) array([1, 2, 3, 2, 1]) """ 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 def unique(x): """ Return the sorted, unique elements of an array or sequence. Parameters ---------- x : array_like Input array. Returns ------- y : ndarray The sorted, unique elements are returned in a 1-D array. Examples -------- >>> np.unique([1, 1, 2, 2, 3, 3]) array([1, 2, 3]) >>> a = np.array([[1, 1], [2, 3]]) >>> np.unique(a) array([1, 2, 3]) """ 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 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]]) >>> condition = np.mod(arr, 3)==0 >>> condition array([[False, False, True, False], [False, True, False, False], [ True, False, False, True]], dtype=bool) >>> np.extract(condition, arr) array([ 3, 6, 9, 12]) If `condition` is boolean: >>> arr[condition] array([ 3, 6, 9, 12]) """ return _nx.take(ravel(arr), nonzero(ravel(condition))[0]) def place(arr, mask, vals): """ Changes elements of an array based on conditional and input values. Similar to ``putmask(a, mask, vals)`` but the 1D array `vals` has the same number of elements as the non-zero values of `mask`. Inverse of ``extract``. Sets `a`.flat[n] = `values`\\[n] for each n where `mask`.flat[n] is true. Parameters ---------- a : array_like Array to put data into. mask : array_like Boolean mask array. values : array_like, shape(number of non-zero `mask`, ) Values to put into `a`. See Also -------- putmask, put, take """ 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) if mask.all(): return np.nan if not issubclass(y.dtype.type, np.integer): y[mask] = fill return op(y, axis=axis) 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 along an axis, ignoring NaNs. See Also -------- nanargmax : corresponding function for maxima; see for details. """ 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, 1]) >>> np.nanargmax(a, axis=1) array([1, 0]) """ return _nanop(np.argmax, -np.inf, a, axis) def disp(mesg, device=None, linefeed=True): """ Display a message on a device Parameters ---------- mesg : string Message to display. device : device object with 'write' method Device to write message. If None, defaults to ``sys.stdout`` which is very similar to ``print``. linefeed : bool, optional Option whether to print a line feed or not. Defaults to True. """ 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\d+) argument(s|) \((?P\d+) given\)') 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(somefunction, otypes=None, doc=None) Generalized function class. Define a vectorized function which takes nested sequence of objects or numpy arrays as inputs and returns a numpy array as output, evaluating the function over successive tuples of the input arrays like the python map function except it uses the broadcasting rules of numpy. Data-type of 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 as either a string of typecode characters or a list of data-types specifiers. There should be one data-type specifier for each output. Parameters ---------- f : callable A Python function or method. Examples -------- >>> def myfunc(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]) """ 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. 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 >>> 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() >>> 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 Bartlett 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 >>> from numpy.fft import fft >>> import matplotlib.pyplot as plt >>> window = bartlett(51) >>> plt.plot(window) >>> plt.title("Bartlett window") >>> plt.ylabel("Amplitude") >>> plt.xlabel("Sample") >>> plt.show() >>> 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 Bartlett 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.subplot(121) >>> plt.plot(window) >>> plt.title("Hann window") >>> plt.ylabel("Amplitude") >>> plt.xlabel("Sample") >>> 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.subplot(122) >>> 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 -------- >>> from numpy import hamming >>> 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 numpy import clip, log10, array, hamming >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = hamming(51) >>> plt.plot(window) >>> plt.title("Hamming window") >>> plt.ylabel("Amplitude") >>> plt.xlabel("Sample") >>> plt.show() >>> 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 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`. Parameters ---------- x : array_like, dtype float or complex Argument of the Bessel function. Returns ------- out : ndarray, shape z.shape, dtype z.dtype The modified Bessel function evaluated at the elements of `x`. See Also -------- scipy.special.iv, scipy.special.ive References ---------- .. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions", 10th printing, 1964, pp. 374. http://www.math.sfu.ca/~cbm/aands/ .. [2] Wikipedia, "Bessel function", http://en.wikipedia.org/wiki/Bessel_function 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 >>> 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() >>> 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 n = arange(0,M) alpha = (M-1)/2.0 return i0(beta * sqrt(1-((n-alpha)/alpha)**2.0))/i0(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, 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(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 1. dx : scalar, optional If `x` is None, spacing given by `dx` is assumed. axis : int, optional Specify the axis. Examples -------- >>> np.trapz([1,2,3]) >>> 4.0 >>> np.trapz([1,2,3], [4,6,8]) >>> 8.0 """ y = asarray(y) if x is None: d = dx 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 1D arrays representing the x and y coordinates 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 -------- numpy.mgrid : Construct a multi-dimensional "meshgrid" using indexing notation. Examples -------- >>> X, Y = numpy.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]]) """ 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, integer or an array of integers Indicate which sub-arrays to remove. axis : integer, optional The axis along which to delete the subarray defined by `obj`. If `axis` is None, `obj` is applied to the flattened array. See Also -------- insert : Insert values into an array. append : Append values 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 : {integer, slice, integer array_like} Insert `values` before `obj` indices. values : Values to insert into `arr`. axis : int, optional Axis along which to insert `values`. If `axis` is None then ravel `arr` first. Examples -------- >>> a = np.array([[1,2,3], ... [4,5,6], ... [7,8,9]]) >>> np.insert(a, [1,2], [[4],[5]], axis=0) array([[1, 2, 3], [4, 4, 4], [4, 5, 6], [5, 5, 5], [7, 8, 9]]) """ 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. Examples -------- >>> np.append([1, 2, 3], [[4, 5, 6], [7, 8, 9]]) array([1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> np.append([[1, 2, 3], [4, 5, 6]], [[7, 8, 9]], axis=0) array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) """ 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)