__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. Return num evenly spaced samples from start to stop. If endpoint is True, the last sample is stop. If retstep is True then return (seq, step_value), where step_value used. Parameters ---------- start : {float} The value the sequence starts at. stop : {float} The value the sequence stops at. If ``endpoint`` is false, then this is not included in the sequence. Otherwise it is guaranteed to be the last value. num : {integer} Number of samples to generate. Default is 50. endpoint : {boolean} If true, ``stop`` is the last sample. Otherwise, it is not included. Default is true. retstep : {boolean} If true, return ``(samples, step)``, where ``step`` is the spacing used in generating the samples. Returns ------- samples : {array} ``num`` equally spaced samples from the range [start, stop] or [start, stop). step : {float} (Only if ``retstep`` is true) Size of spacing between samples. See Also -------- arange : Similiar to linspace, however, when used with a float endpoint, that endpoint may or may not be included. logspace """ 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): """Evenly spaced numbers on a logarithmic scale. Computes int(num) evenly spaced exponents from base**start to base**stop. If endpoint=True, then last number is base**stop """ 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=False): """Compute the histogram from a set of data. Parameters ---------- a : array The data to histogram. bins : int or sequence If an int, then the number of equal-width bins in the given range. If new=True, bins can also be the bin edges, allowing for non-constant bin widths. range : (float, float) The lower and upper range of the bins. If not provided, range is simply (a.min(), a.max()). Using new=False, lower than range are ignored, and values higher than range are tallied in the rightmost bin. Using new=True, both lower and upper outliers are ignored. normed : bool If False, the result array will contain the number of samples in each bin. If True, the result array 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 all of the histogram values will not usually be 1; it is not a probability *mass* function. weights : array An array of weights, the same shape as a. If normed is False, the histogram is computed by summing the weights of the values falling into each bin. If normed is True, the weights are normalized, so that the integral of the density over the range is 1. This option is only available with new=True. new : bool Compatibility argument to transition from the old version (v1.1) to the new version (v1.2). Returns ------- hist : array The values of the histogram. See `normed` and `weights` for a description of the possible semantics. bin_edges : float array With new=False, return the left bin edges (length(hist)). With new=True, return the bin edges (length(hist)+1). See Also -------- histogramdd """ # Old behavior if new is False: warnings.warn(""" The semantics of histogram will be modified in release 1.2 to improve outlier handling. The new behavior can be obtained using new=True. Note that the new version accepts/returns the bin edges instead of the left bin edges. Please read the docstring for more information.""", FutureWarning) 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()) else: warnings.warn(""" Outliers handling will change in version 1.2. Please read the docstring for details.""", FutureWarning) 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.' warnings.warn(""" The semantic for bins will change in version 1.2. The bins will become the bin edges, instead of the left bin edges. """, FutureWarning) 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 is True: 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): """histogramdd(sample, bins=10, range=None, normed=False, weights=None) Return the N-dimensional histogram of the sample. Parameters ---------- sample : sequence or array A sequence containing N arrays or an NxM array. Input data. bins : sequence or scalar A sequence of edge arrays, a sequence of bin counts, or a scalar which is the bin count for all dimensions. Default is 10. range : sequence A sequence of lower and upper bin edges. Default is [min, max]. normed : boolean If False, return the number of samples in each bin, if True, returns the density. weights : array Array of weights. The weights are normed only if normed is True. Should the sum of the weights not equal N, the total bin count will not be equal to the number of samples. Returns ------- hist : array Histogram array. edges : list List of arrays defining the lower bin edges. See Also -------- histogram Examples -------- >>> x = np.random.randn(100,3) >>> hist3d, edges = np.lib.histogramdd(x, bins = (5, 6, 7)) """ 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 a over the given axis. Parameters ---------- a : array_like Data to be averaged. axis : {None, integer}, optional Axis along which to average a. If None, averaging is done over the entire array irrespective of its shape. weights : {None, array_like}, optional The importance each datum has in the computation of the average. The weights array can either be 1D, 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, all data are assumed to have weight equal to one. returned :{False, boolean}, optional If True, the tuple (average, sum_of_weights) is returned, otherwise only the average is returmed. Note that if weights=None, then the sum of the weights is also the number of elements averaged over. 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 has the same type as the average. Examples -------- >>> np.average(range(1,11), weights=range(10,0,-1)) 4.0 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. """ 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): """Like asarray, but check that no NaNs or Infs are present. """ 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): """Return a piecewise-defined function. x is the domain condlist is a list of boolean arrays or a single boolean array The length of the condition list must be n2 or n2-1 where n2 is the length of the function list. If len(condlist)==n2-1, then an 'otherwise' condition is formed by |'ing all the conditions and inverting. funclist is a list of functions to call of length (n2). Each function should return an array output for an array input Each function can take (the same set) of extra arguments and keyword arguments which are passed in after the function list. A constant may be used in funclist for a function that returns a constant (e.g. val and lambda x: val are equivalent in a funclist). The output is the same shape and type as x and is found by calling the functions on the appropriate portions of x. Note: This is similar to choose or select, except the the functions are only evaluated on elements of x that satisfy the corresponding condition. The result is |-- | f1(x) for condition1 y = --| f2(x) for condition2 | ... | fn(x) for conditionn |-- """ 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 composed of different elements in choicelist, depending on the list of 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. Equivalent to: output = [] for m in range(M): output += [V[m] for V,C in zip(values,cond) if C[m]] or [default] """ 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. """ return array(a, copy=True) # Basic operations def gradient(f, *varargs): """Calculate the gradient of an N-dimensional scalar function. Uses central differences on the interior and first differences on boundaries to give the same shape. Inputs: f -- An N-dimensional array giving samples of a scalar function varargs -- 0, 1, or N scalars giving the sample distances in each direction Outputs: N arrays of the same shape as f giving the derivative of f with respect to each dimension. """ 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. """ 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): """Return the value of a piecewise-linear function at each value in x. The piecewise-linear function, f, is defined by the known data-points fp=f(xp). The xp points must be sorted in increasing order but this is not checked. For values of x < xp[0] return the value given by left. If left is None, then return fp[0]. For values of x > xp[-1] return the value given by right. If right is None, then return fp[-1]. """ 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 z. Examples -------- >>> np.angle(1+1j) # in radians 0.78539816339744828 >>> 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 radian phase p by changing absolute jumps greater than 'discont' to their 2*pi complement along the given axis. """ 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)>> 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 sorted unique items from an array or sequence. Examples -------- >>> np.unique([5,2,4,0,4,4,2,2,1]) array([0, 1, 2, 4, 5]) """ 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 ravel(arr) where ravel(condition) is True (in 1D). Equivalent to compress(ravel(condition), ravel(arr)). """ return _nx.take(ravel(arr), nonzero(ravel(condition))[0]) def place(arr, mask, vals): """Similar to putmask arr[mask] = vals but the 1D array vals has the same number of elements as the non-zero values of mask. Inverse of extract. """ return _insert(arr, mask, vals) def nansum(a, axis=None): """Sum the array over the given axis, treating NaNs as 0. """ y = array(a,subok=True) if not issubclass(y.dtype.type, _nx.integer): y[isnan(a)] = 0 return y.sum(axis) def nanmin(a, axis=None): """Find the minimium over the given axis, ignoring NaNs. """ y = array(a,subok=True) if not issubclass(y.dtype.type, _nx.integer): y[isnan(a)] = _nx.inf return y.min(axis) def nanargmin(a, axis=None): """Find the indices of the minimium over the given axis ignoring NaNs. """ y = array(a, subok=True) if not issubclass(y.dtype.type, _nx.integer): y[isnan(a)] = _nx.inf return y.argmin(axis) def nanmax(a, axis=None): """Find the maximum over the given axis ignoring NaNs. """ y = array(a, subok=True) if not issubclass(y.dtype.type, _nx.integer): y[isnan(a)] = -_nx.inf return y.max(axis) def nanargmax(a, axis=None): """Find the maximum over the given axis ignoring NaNs. """ y = array(a,subok=True) if not issubclass(y.dtype.type, _nx.integer): y[isnan(a)] = -_nx.inf return y.argmax(axis) def disp(mesg, device=None, linefeed=True): """Display a message to the given device (default is sys.stdout) with or without a linefeed. """ 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 the covariance matrix. If m is a vector, return the variance. For matrices return the covariance matrix. If y is given it is treated as an additional (set of) variable(s). Normalization is by (N-1) where N is the number of observations (unbiased estimate). If bias is 1 then normalization is by N. If rowvar is non-zero (default), then each row is a variable with observations in the columns, otherwise each column is a variable and the observations are in the rows. """ 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): """The correlation coefficients """ 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): """blackman(M) returns the M-point Blackman window. """ 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. References ---------- .. [1] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra", Biometrika 37, 1-16, 1950. .. [2] A.V. Oppenheim and R.W. Schafer, "Discrete-Time Signal Processing", Prentice-Hall, 1999, pp. 468-471. .. [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 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 scipy.fftpack import fft from matplotlib import pyplot as plt window = np.bartlett(51) plt.plot(window) #doctest: SKIP plt.title("Bartlett window") plt.ylabel("Amplitude") plt.xlabel("Sample") plt.show() A = fft(window, 2048) / 25.5 mag = abs(np.fft.fftshift(A)) freq = 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 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): """hanning(M) returns the M-point Hanning window. """ 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): """hamming(M) returns the M-point Hamming window. """ 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): 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): """kaiser(M, beta) returns a Kaiser window of length M with shape parameter beta. """ 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): """sinc(x) returns sin(pi*x)/(pi*x) at all points of array x. """ y = pi* where(x == 0, 1.0e-20, x) return sin(y)/y def msort(a): b = array(a,subok=True,copy=True) b.sort(0) return b def median(a, axis=0, out=None, overwrite_input=False): """Compute the median along the specified axis. Returns the median of the array elements. The median is taken over the first axis of the array by default, otherwise over the specified axis. Parameters ---------- a : array-like Input array or object that can be converted to an array axis : {int, None}, optional Axis along which the medians are computed. The default is to compute the median along the first dimension. axis=None returns the median of the flattened 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 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 is returned unless out is specified, in which case a reference to out is returned. Return datatype is float64 for ints and floats smaller than float64, or the input datatype otherwise. See Also ------- mean Notes ----- Given a vector V length N, the median of V is the middle value of a sorted copy of V (Vs) - i.e. Vs[(N-1)/2], when N is odd. It is the mean of the two middle values of Vs, when N is even. Examples -------- >>> a = np.array([[10, 7, 4], [3, 2, 1]]) >>> a array([[10, 7, 4], [ 3, 2, 1]]) >>> np.median(a) array([ 6.5, 4.5, 2.5]) >>> np.median(a, axis=None) 3.5 >>> np.median(a, axis=1) array([ 7., 2.]) >>> m = np.median(a) >>> out = np.zeros_like(m) >>> np.median(a, 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 y(x) using samples along the given axis and the composite trapezoidal rule. If x is None, spacing given by dx is assumed. """ 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): """ 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 EG, [X, Y] = meshgrid([1,2,3], [4,5,6,7]) X = 1 2 3 1 2 3 1 2 3 1 2 3 Y = 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. Return a new array with the sub-arrays (i.e. rows or columns) deleted along the given axis as specified by obj obj may be a slice_object (s_[3:5:2]) or an integer or an array of integers indicated which sub-arrays to remove. If axis is None, then ravel the array first. Examples -------- >>> arr = [[3,4,5], ... [1,2,3], ... [6,7,8]] >>> np.delete(arr, 1, 1) array([[3, 5], [1, 3], [6, 8]]) >>> np.delete(arr, 1, 0) array([[3, 4, 5], [6, 7, 8]]) """ 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): """Return a new array with values inserted along the given axis before the given indices If axis is None, then ravel the array first. The obj argument can be an integer, a slice, or a sequence of integers. 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 to the end of an array along axis (ravel first if None) """ 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)