''' Misc ''' from __future__ import absolute_import, division import collections import sys import fractions import numpy as np from numpy import ( meshgrid, abs, amax, any, logical_and, arange, linspace, atleast_1d, asarray, ceil, floor, frexp, hypot, sqrt, arctan2, sin, cos, exp, log, log1p, mod, diff, finfo, inf, pi, interp, isnan, isscalar, zeros, ones, linalg, r_, sign, unique, hstack, vstack, nonzero, where, extract) from scipy.special import gammaln, gamma, psi from scipy.integrate import trapz, simps import warnings from time import strftime, gmtime from .plotbackend import plotbackend from collections import OrderedDict try: from . import c_library as clib # @UnresolvedImport except ImportError: warnings.warn('c_library not found. Check its compilation.') clib = None floatinfo = finfo(float) _TINY = np.finfo(float).tiny _EPS = np.finfo(float).eps __all__ = ['now', 'spaceline', 'narg_smallest', 'args_flat', 'is_numlike', 'JITImport', 'DotDict', 'Bunch', 'printf', 'sub_dict_select', 'parse_kwargs', 'detrendma', 'ecross', 'findcross', 'findextrema', 'findpeaks', 'findrfc', 'rfcfilter', 'findtp', 'findtc', 'findoutliers', 'common_shape', 'argsreduce', 'stirlerr', 'getshipchar', 'betaloge', 'gravity', 'nextpow2', 'discretize', 'polar2cart', 'cart2polar', 'meshgrid', 'ndgrid', 'trangood', 'tranproc', 'plot_histgrm', 'num2pistr', 'test_docstrings', 'lazywhere', 'piecewise', 'valarray'] def valarray(shape, value=np.NaN, typecode=None): """Return an array of all value. """ if typecode is None: typecode = bool out = ones(shape, dtype=typecode) * value if not isinstance(out, np.ndarray): out = asarray(out) return out def piecewise(xi, condlist, funclist, fill_value=0.0, 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 ---------- xi : tuple input arguments to the functions in funclist, i.e., (x0, x1,...., xn) condlist : list of bool arrays Each boolean array corresponds to a function in `funclist`. Wherever `condlist[i]` is True, `funclist[i](x0,x1,...,xn)` is used as the output value. Each boolean array in `condlist` selects a piece of `xi`, and should therefore be of the same shape as `xi`. The length of `condlist` must correspond to that of `funclist`. If one extra function is given, i.e. if ``len(funclist) - len(condlist) == 1``, then that extra function is the default value, used wherever all conditions are false. funclist : list of callables, f(*(xi + args), **kw), or scalars Each function is evaluated over `x` wherever its corresponding condition is True. It should take an array as input and give an array or a scalar value as output. If, instead of a callable, a scalar is provided then a constant function (``lambda x: scalar``) is assumed. fill_value : scalar fill value for out of range values. Default 0. args : tuple, optional Any further arguments given here passed to the functions upon execution, i.e., if called ``piecewise(..., ..., args=(1, 'a'))``, then each function is called as ``f(x0, x1,..., xn, 1, 'a')``. kw : dict, optional Keyword arguments used in calling `piecewise` are passed to the functions upon execution, i.e., if called ``piecewise(..., ..., lambda=1)``, then each function is called as ``f(x0, x1,..., xn, lambda=1)``. Returns ------- out : ndarray The output is the same shape and type as x and is found by calling the functions in `funclist` on the appropriate portions of `x`, as defined by the boolean arrays in `condlist`. Portions not covered by any condition have undefined values. See Also -------- choose, select, where Notes ----- This is similar to choose or select, except that functions are evaluated on elements of `xi` that satisfy the corresponding condition from `condlist`. The result is:: |-- |funclist[0](x0[condlist[0]],x1[condlist[0]],...,xn[condlist[0]]) out = |funclist[1](x0[condlist[1]],x1[condlist[1]],...,xn[condlist[1]]) |... |funclist[n2](x0[condlist[n2]],x1[condlist[n2]],...,xn[condlist[n2]]) |-- Examples -------- Define the sigma function, which is -1 for ``x < 0`` and +1 for ``x >= 0``. >>> x = np.linspace(-2.5, 2.5, 6) >>> piecewise(x, [x < 0, x >= 0], [-1, 1]) array([-1., -1., -1., 1., 1., 1.]) Define the absolute value, which is ``-x`` for ``x <0`` and ``x`` for ``x >= 0``. >>> 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]) Define the absolute value, which is ``-x*y`` for ``x*y <0`` and ``x*y`` for ``x*y >= 0`` >>> X, Y = np.meshgrid(x, x) >>> piecewise((X, Y), [X * Y < 0, ], ... [lambda x, y: -x * y, lambda x, y: x * y]) array([[ 6.25, 3.75, 1.25, 1.25, 3.75, 6.25], [ 3.75, 2.25, 0.75, 0.75, 2.25, 3.75], [ 1.25, 0.75, 0.25, 0.25, 0.75, 1.25], [ 1.25, 0.75, 0.25, 0.25, 0.75, 1.25], [ 3.75, 2.25, 0.75, 0.75, 2.25, 3.75], [ 6.25, 3.75, 1.25, 1.25, 3.75, 6.25]]) """ def otherwise_condition(condlist): return ~np.logical_or.reduce(condlist, axis=0) def check_shapes(condlist, funclist): nc, nf = len(condlist), len(funclist) if nc not in [nf-1, nf]: raise ValueError("function list and condition list" + " must be the same length") check_shapes(condlist, funclist) if not isinstance(xi, tuple): xi = (xi,) condlist = np.broadcast_arrays(*condlist) if len(condlist) == len(funclist)-1: condlist.append(otherwise_condition(condlist)) arrays = np.broadcast_arrays(*xi) dtype = np.result_type(*arrays) out = valarray(arrays[0].shape, fill_value, dtype) for cond, func in zip(condlist, funclist): if isinstance(func, collections.Callable): temp = tuple(np.extract(cond, arr) for arr in arrays) + args np.place(out, cond, func(*temp, **kw)) else: # func is a scalar value np.place(out, cond, func) return out def lazywhere(cond, arrays, f, fillvalue=None, f2=None): """ np.where(cond, x, fillvalue) always evaluates x even where cond is False. This one only evaluates f(arr1[cond], arr2[cond], ...). For example, >>> a, b = np.array([1, 2, 3, 4]), np.array([5, 6, 7, 8]) >>> def f(a, b): ... return a*b >>> lazywhere(a > 2, (a, b), f, np.nan) array([ nan, nan, 21., 32.]) Notice it assumes that all `arrays` are of the same shape, or can be broadcasted together. """ if fillvalue is None: if f2 is None: raise ValueError("One of (fillvalue, f2) must be given.") else: fillvalue = np.nan else: if f2 is not None: raise ValueError("Only one of (fillvalue, f2) can be given.") arrays = np.broadcast_arrays(*arrays) temp = tuple(np.extract(cond, arr) for arr in arrays) out = valarray(np.shape(arrays[0]), value=fillvalue) np.place(out, cond, f(*temp)) if f2 is not None: temp = tuple(np.extract(~cond, arr) for arr in arrays) np.place(out, ~cond, f2(*temp)) return out def rotation_matrix(heading, pitch, roll): ''' Examples >>> import numpy as np >>> rotation_matrix(heading=0, pitch=0, roll=0) array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]]) >>> np.all(np.abs(rotation_matrix(heading=180, pitch=0, roll=0)- ... np.array([[ -1.000000e+00, -1.224647e-16, 0.000000e+00], ... [ 1.224647e-16, -1.000000e+00, 0.000000e+00], ... [ -0.000000e+00, 0.000000e+00, 1.000000e+00]]))<1e-7) True >>> np.all(np.abs(rotation_matrix(heading=0, pitch=180, roll=0)- ... np.array([[ -1.000000e+00, 0.000000e+00, 1.224647e-16], ... [ -0.000000e+00, 1.000000e+00, 0.000000e+00], ... [ -1.224647e-16, -0.000000e+00, -1.000000e+00]]))<1e-7) True >>> np.all(np.abs(rotation_matrix(heading=0, pitch=0, roll=180)- ... np.array([[ 1.000000e+00, 0.000000e+00, 0.000000e+00], ... [ 0.000000e+00, -1.000000e+00, -1.224647e-16], ... [ -0.000000e+00, 1.224647e-16, -1.000000e+00]]))<1e-7) True ''' data = np.diag(np.ones(3)) # No transform if H=P=R=0 if heading != 0 or pitch != 0 or roll != 0: deg2rad = np.pi / 180 H = heading * deg2rad P = pitch * deg2rad R = roll * deg2rad # Convert to radians data.put(0, cos(H) * cos(P)) data.put(1, cos(H) * sin(P) * sin(R) - sin(H) * cos(R)) data.put(2, cos(H) * sin(P) * cos(R) + sin(H) * sin(R)) data.put(3, sin(H) * cos(P)) data.put(4, sin(H) * sin(P) * sin(R) + cos(H) * cos(R)) data.put(5, sin(H) * sin(P) * cos(R) - cos(H) * sin(R)) data.put(6, -sin(P)) data.put(7, cos(P) * sin(R)) data.put(8, cos(P) * cos(R)) return data def rotate(x, y, z, heading=0, pitch=0, roll=0): rot_param = rotation_matrix(heading, pitch, roll).ravel() X = x * rot_param[0] + y * rot_param[1] + z * rot_param[2] Y = x * rot_param[3] + y * rot_param[4] + z * rot_param[5] Z = x * rot_param[6] + y * rot_param[7] + z * rot_param[8] return X, Y, Z def rotate_2d(x, y, angle_deg): ''' Rotate points in the xy cartesian plane counter clockwise Examples -------- >>> rotate_2d(x=1, y=0, angle_deg=0) (1.0, 0.0) >>> rotate_2d(x=1, y=0, angle_deg=90) (6.123233995736766e-17, 1.0) >>> rotate_2d(x=1, y=0, angle_deg=180) (-1.0, 1.2246467991473532e-16) >>> rotate_2d(x=1, y=0, angle_deg=360) (1.0, -2.4492935982947064e-16) ''' angle_rad = angle_deg * pi / 180 ch = cos(angle_rad) sh = sin(angle_rad) return ch * x - sh * y, sh * x + ch * y def now(show_seconds=True): ''' Return current date and time as a string ''' if show_seconds: return strftime("%a, %d %b %Y %H:%M:%S", gmtime()) else: return strftime("%a, %d %b %Y %H:%M", gmtime()) def _assert(cond, txt=''): if not cond: raise ValueError(txt) def spaceline(start_point, stop_point, num=10): '''Return `num` evenly spaced points between the start and stop points. Parameters ---------- start_point : vector, size=3 The starting point of the sequence. stop_point : vector, size=3 The end point of the sequence. num : int, optional Number of samples to generate. Default is 10. Returns ------- space_points : ndarray of shape n x 3 There are `num` equally spaced points in the closed interval ``[start, stop]``. See Also -------- linspace : similar to spaceline, but in 1D. arange : Similiar to `linspace`, but uses a step size (instead of the number of samples). logspace : Samples uniformly distributed in log space. Example ------- >>> import wafo.misc as pm >>> pm.spaceline((2,0,0), (3,0,0), num=5) array([[ 2. , 0. , 0. ], [ 2.25, 0. , 0. ], [ 2.5 , 0. , 0. ], [ 2.75, 0. , 0. ], [ 3. , 0. , 0. ]]) ''' num = int(num) e1, e2 = np.atleast_1d(start_point, stop_point) e2m1 = e2 - e1 length = np.sqrt((e2m1 ** 2).sum()) # length = sqrt((E2[0]-E1(1))^2 + (E2(2)-E1(2))^2 + (E2(3)-E1(3))^2) C = e2m1 / length delta = length / float(num - 1) return np.array([e1 + n * delta * C for n in range(num)]) def narg_smallest(n, arr): ''' Return the n smallest indicis to the arr ''' return np.array(arr).argsort()[:n] def args_flat(*args): ''' Return x,y,z positions as a N x 3 ndarray Parameters ---------- pos : array-like, shape N x 3 [x,y,z] positions or x,y,z : array-like [x,y,z] positions Returns ------ pos : ndarray, shape N x 3 [x,y,z] positions common_shape : None or tuple common shape of x, y and z variables if given as triple input. Examples -------- >>> x = [1,2,3] >>> pos, c_shape =args_flat(x,2,3) >>> pos array([[1, 2, 3], [2, 2, 3], [3, 2, 3]]) >>> c_shape (3,) >>> pos1, c_shape1 = args_flat([1,2,3]) >>> pos1 array([[1, 2, 3]]) >>> c_shape1 is None True >>> pos1, c_shape1 = args_flat(1,2,3) >>> pos1 array([[1, 2, 3]]) >>> c_shape1 () >>> pos1, c_shape1 = args_flat([1],2,3) >>> pos1 array([[1, 2, 3]]) >>> c_shape1 (1,) ''' nargin = len(args) if (nargin == 1): # pos pos = np.atleast_2d(args[0]) _assert((pos.shape[1] == 3) and (pos.ndim == 2), 'POS array must be of shape N x 3!') return pos, None elif nargin == 3: x, y, z = np.broadcast_arrays(*args[:3]) c_shape = x.shape return np.vstack((x.ravel(), y.ravel(), z.ravel())).T, c_shape else: raise ValueError('Number of arguments must be 1 or 3!') def index2sub(shape, index, order='C'): ''' Returns Multiple subscripts from linear index. Parameters ---------- shape : array-like shape of array index : linear index into array order : {'C','F'}, optional The order of the linear index. 'C' means C (row-major) order. 'F' means Fortran (column-major) order. By default, 'C' order is used. This function is used to determine the equivalent subscript values corresponding to a given single index into an array. Example ------- >>> shape = (3,3,4) >>> a = np.arange(np.prod(shape)).reshape(shape) >>> order = 'C' >>> a[1, 2, 3] 23 >>> i = sub2index(shape, 1, 2, 3, order=order) >>> a.ravel(order)[i] 23 >>> index2sub(shape, i, order=order) (1, 2, 3) See also -------- sub2index ''' return np.unravel_index(index, shape, order=order) def sub2index(shape, *subscripts, **kwds): ''' Returns linear index from multiple subscripts. Parameters ---------- shape : array-like shape of array *subscripts : subscripts into array order : {'C','F'}, optional The order of the linear index. 'C' means C (row-major) order. 'F' means Fortran (column-major) order. By default, 'C' order is used. This function is used to determine the equivalent single index corresponding to a given set of subscript values into an array. Example ------- >>> shape = (3,3,4) >>> a = np.arange(np.prod(shape)).reshape(shape) >>> order = 'C' >>> i = sub2index(shape, 1, 2, 3, order=order) >>> a[1, 2, 3] 23 >>> a.ravel(order)[i] 23 >>> index2sub(shape, i, order=order) (1, 2, 3) See also -------- index2sub ''' return np.ravel_multi_index(subscripts, shape, **kwds) def is_numlike(obj): 'return true if *obj* looks like a number' try: obj + 1 except TypeError: return False else: return True class JITImport(object): ''' Just In Time Import of module Example ------- >>> np = JITImport('numpy') >>> np.exp(0)==1.0 True ''' def __init__(self, module_name): self._module_name = module_name self._module = None def __getattr__(self, attr): try: return getattr(self._module, attr) except: if self._module is None: self._module = __import__(self._module_name, None, None, ['*']) # assert(isinstance(self._module, types.ModuleType), 'module') return getattr(self._module, attr) else: raise class DotDict(dict): ''' Implement dot access to dict values Example ------- >>> d = DotDict(test1=1,test2=3) >>> d.test1 1 ''' __getattr__ = dict.__getitem__ class Bunch(object): ''' Implement keyword argument initialization of class Example ------- >>> d = Bunch(test1=1,test2=3) >>> d.test1 1 ''' def __init__(self, **kwargs): self.__dict__.update(kwargs) def keys(self): return self.__dict__.keys() def update(self, ** kwargs): self.__dict__.update(kwargs) def printf(format, *args): # @ReservedAssignment sys.stdout.write(format % args) def sub_dict_select(somedict, somekeys): ''' Extracting a Subset from Dictionary Example -------- # Update options dict from keyword arguments if # the keyword exists in options >>> opt = dict(arg1=2, arg2=3) >>> kwds = dict(arg2=100,arg3=1000) >>> sub_dict = sub_dict_select(kwds,opt.keys()) >>> opt.update(sub_dict) >>> opt {'arg1': 2, 'arg2': 100} See also -------- dict_intersection ''' # slower: validKeys = set(somedict).intersection(somekeys) return dict((k, somedict[k]) for k in somekeys if k in somedict) def parse_kwargs(options, **kwargs): ''' Update options dict from keyword arguments if the keyword exists in options Example >>> opt = dict(arg1=2, arg2=3) >>> opt = parse_kwargs(opt,arg2=100) >>> print(opt) {'arg1': 2, 'arg2': 100} >>> opt2 = dict(arg2=101) >>> opt = parse_kwargs(opt,**opt2) See also sub_dict_select ''' newopts = sub_dict_select(kwargs, options.keys()) if len(newopts) > 0: options.update(newopts) return options def detrendma(x, L): """ Removes a trend from data using a moving average of size 2*L+1. If 2*L+1 > len(x) then the mean is removed Parameters ---------- x : vector or matrix of column vectors of data L : scalar, integer defines the size of the moving average window Returns ------- y : ndarray detrended data Examples -------- >>> import wafo.misc as wm >>> import pylab as plt >>> exp = plt.exp; cos = plt.cos; randn = plt.randn >>> x = plt.linspace(0,1,200) >>> y = exp(x)+cos(5*2*pi*x)+1e-1*randn(x.size) >>> y0 = wm.detrendma(y,20); tr = y-y0 >>> h = plt.plot(x, y, x, y0, 'r', x, exp(x), 'k', x, tr, 'm') >>> plt.close('all') See also -------- Reconstruct """ if L <= 0: raise ValueError('L must be positive') if L != round(L): raise ValueError('L must be an integer') x1 = np.atleast_1d(x) if x1.shape[0] == 1: x1 = x1.ravel() n = x1.shape[0] if n < 2 * L + 1: # only able to remove the mean return x1 - x1.mean(axis=0) mn = x1[0:2 * L + 1].mean(axis=0) y = np.empty_like(x1) y[0:L] = x1[0:L] - mn ix = np.r_[L:(n - L)] trend = ((x1[ix + L] - x1[ix - L]) / (2 * L + 1)).cumsum(axis=0) + mn y[ix] = x1[ix] - trend y[n - L::] = x1[n - L::] - trend[-1] return y def ecross(t, f, ind, v=0): ''' Extracts exact level v crossings ECROSS interpolates t and f linearly to find the exact level v crossings, i.e., the points where f(t0) = v Parameters ---------- t,f : vectors of arguments and functions values, respectively. ind : ndarray of integers indices to level v crossings as found by findcross. v : scalar or vector (of size(ind)) defining the level(s) to cross. Returns ------- t0 : vector of exact level v crossings. Example ------- >>> from matplotlib import pylab as plt >>> import wafo.misc as wm >>> ones = np.ones >>> t = np.linspace(0,7*np.pi,250) >>> x = np.sin(t) >>> ind = wm.findcross(x,0.75) >>> ind array([ 9, 25, 80, 97, 151, 168, 223, 239]) >>> t0 = wm.ecross(t,x,ind,0.75) >>> np.abs(t0 - np.array([0.84910514, 2.2933879 , 7.13205663, 8.57630119, ... 13.41484739, 14.85909194, 19.69776067, 21.14204343]))<1e-7 array([ True, True, True, True, True, True, True, True], dtype=bool) >>> a = plt.plot(t, x, '.', t[ind], x[ind], 'r.', t, ones(t.shape)*0.75, ... t0, ones(t0.shape)*0.75, 'g.') >>> plt.close('all') See also -------- findcross ''' # Tested on: Python 2.5 # revised pab Feb2004 # By pab 18.06.2001 return (t[ind] + (v - f[ind]) * (t[ind + 1] - t[ind]) / (f[ind + 1] - f[ind])) def _findcross(xn): '''Return indices to zero up and downcrossings of a vector ''' if clib is not None: ind, m = clib.findcross(xn, 0.0) return ind[:m] n = len(xn) iz, = (xn == 0).nonzero() if len(iz) > 0: # Trick to avoid turning points on the crossinglevel. if iz[0] == 0: if len(iz) == n: warnings.warn('All values are equal to crossing level!') return zeros(0, dtype=np.int) diz = diff(iz) if len(diz) > 0 and (diz > 1).any(): ix = iz[(diz > 1).argmax()] else: ix = iz[-1] # x(ix) is a up crossing if x(1:ix) = v and x(ix+1) > v. # x(ix) is a downcrossing if x(1:ix) = v and x(ix+1) < v. xn[0:ix + 1] = -xn[ix + 1] iz = iz[ix + 1::] for ix in iz.tolist(): xn[ix] = xn[ix - 1] # indices to local level crossings ( without turningpoints) ind, = (xn[:n - 1] * xn[1:] < 0).nonzero() return ind def xor(a, b): """ Return True only when inputs differ. """ return a ^ b def findcross(x, v=0.0, kind=None): ''' Return indices to level v up and/or downcrossings of a vector Parameters ---------- x : array_like vector with sampled values. v : scalar, real level v. kind : string defines type of wave or crossing returned. Possible options are 'dw' : downcrossing wave 'uw' : upcrossing wave 'cw' : crest wave 'tw' : trough wave 'd' : downcrossings only 'u' : upcrossings only None : All crossings will be returned Returns ------- ind : array-like indices to the crossings in the original sequence x. Example ------- >>> from matplotlib import pylab as plt >>> import wafo.misc as wm >>> ones = np.ones >>> findcross([0, 1, -1, 1],0) array([0, 1, 2]) >>> v = 0.75 >>> t = np.linspace(0,7*np.pi,250) >>> x = np.sin(t) >>> ind = wm.findcross(x,v) # all crossings >>> ind array([ 9, 25, 80, 97, 151, 168, 223, 239]) >>> t0 = plt.plot(t,x,'.',t[ind],x[ind],'r.', t, ones(t.shape)*v) >>> ind2 = wm.findcross(x,v,'u') >>> ind2 array([ 9, 80, 151, 223]) >>> t0 = plt.plot(t[ind2],x[ind2],'o') >>> plt.close('all') See also -------- crossdef wavedef ''' xn = np.int8(sign(atleast_1d(x).ravel() - v)) # @UndefinedVariable ind = _findcross(xn) if ind.size == 0: warnings.warn('No level v = %0.5g crossings found in x' % v) return ind if kind not in ('du', 'all', None): if kind == 'd': # downcrossings only t_0 = int(xn[ind[0] + 1] > 0) ind = ind[t_0::2] elif kind == 'u': # upcrossings only t_0 = int(xn[ind[0] + 1] < 0) ind = ind[t_0::2] elif kind in ('dw', 'uw', 'tw', 'cw'): # make sure the first is a level v down-crossing # if wdef=='dw' or wdef=='tw' # or make sure the first is a level v up-crossing # if wdef=='uw' or wdef=='cw' first_is_down_crossing = int(xn[ind[0]] > xn[ind[0] + 1]) if xor(first_is_down_crossing, kind in ('dw', 'tw')): ind = ind[1::] # make sure the number of troughs and crests are according to the # wavedef, i.e., make sure length(ind) is odd if dw or uw # and even if tw or cw is_odd = mod(ind.size, 2) if xor(is_odd, kind in ('dw', 'uw')): ind = ind[:-1] else: raise ValueError('Unknown wave/crossing definition!') return ind def findextrema(x): ''' Return indices to minima and maxima of a vector Parameters ---------- x : vector with sampled values. Returns ------- ind : indices to minima and maxima in the original sequence x. Examples -------- >>> import numpy as np >>> import pylab as plt >>> import wafo.misc as wm >>> t = np.linspace(0,7*np.pi,250) >>> x = np.sin(t) >>> ind = wm.findextrema(x) >>> a = plt.plot(t,x,'.',t[ind],x[ind],'r.') >>> plt.close('all') See also -------- findcross crossdef ''' xn = atleast_1d(x).ravel() return findcross(diff(xn), 0.0) + 1 def findpeaks(data, n=2, min_h=None, min_p=0.0): ''' Find peaks of vector or matrix possibly rainflow filtered Parameters ---------- data = matrix or vector n = The n highest peaks are found (if exist). (default 2) min_h = The threshold in the rainflowfilter (default 0.05*range(S(:))). A zero value will return all the peaks of S. min_p = 0..1, Only the peaks that are higher than min_p*max(max(S)) min_p*(the largest peak in S) are returned (default 0). Returns ix = linear index to peaks of S Example: Find highest 8 peaks that are not less that 0.3*"global max" and have rainflow amplitude larger than 5. >>> import numpy as np >>> import wafo.misc as wm >>> x = np.arange(0,10,0.01) >>> data = x**2+10*np.sin(3*x)+0.5*np.sin(50*x) >>> wm.findpeaks(data, n=8, min_h=5, min_p=0.3) array([908, 694, 481]) See also -------- findtp ''' S = np.atleast_1d(data) smax = S.max() if min_h is None: smin = S.min() min_h = 0.05 * (smax - smin) ndim = S.ndim S = np.atleast_2d(S) nrows, mcols = S.shape # Finding turningpoints of the spectrum # Returning only those with rainflowcycle heights greater than h_min indP = [] # indices to peaks ind = [] for iy in range(nrows): # % find all peaks TuP = findtp(S[iy], min_h) if len(TuP): ind = TuP[1::2] # ; % extract indices to maxima only else: # % did not find any , try maximum ind = np.atleast_1d(S[iy].argmax()) if ndim > 1: if iy == 0: ind2 = np.flatnonzero(S[iy, ind] > S[iy + 1, ind]) elif iy == nrows - 1: ind2 = np.flatnonzero(S[iy, ind] > S[iy - 1, ind]) else: ind2 = np.flatnonzero((S[iy, ind] > S[iy - 1, ind]) & (S[iy, ind] > S[iy + 1, ind])) if len(ind2): indP.append((ind[ind2] + iy * mcols)) if ndim > 1: ind = np.hstack(indP) if len(indP) else [] if len(ind) == 0: return [] peaks = S.take(ind) ind2 = peaks.argsort()[::-1] # keeping only the Np most significant peak frequencies. nmax = min(n, len(ind)) ind = ind[ind2[:nmax]] if (min_p > 0): # Keeping only peaks larger than min_p percent relative to the maximum # peak ind = ind[(S.take(ind) > min_p * smax)] return ind def findrfc_astm(tp): """ Return rainflow counted cycles Nieslony's Matlab implementation of the ASTM standard practice for rainflow counting ported to a Python C module. Parameters ---------- tp : array-like vector of turningpoints (NB! Only values, not sampled times) Returns ------- sig_rfc : array-like array of shape (n,3) with: sig_rfc[:,0] Cycles amplitude sig_rfc[:,1] Cycles mean value sig_rfc[:,2] Cycle type, half (=0.5) or full (=1.0) """ y1 = atleast_1d(tp).ravel() sig_rfc, cnr = clib.findrfc3_astm(y1) # the sig_rfc was constructed too big in rainflow.rf3, so # reduce the sig_rfc array as done originally by a matlab mex c function n = len(sig_rfc) sig_rfc = sig_rfc.__getslice__(0, n - cnr[0]) # sig_rfc holds the actual rainflow counted cycles, not the indices return sig_rfc def findrfc(tp, h=0.0, method='clib'): ''' Return indices to rainflow cycles of a sequence of TP. Parameters ----------- tp : array-like vector of turningpoints (NB! Only values, not sampled times) h : real scalar rainflow threshold. If h>0, then all rainflow cycles with height smaller than h are removed. method : string, optional 'clib' 'None' Specify 'clib' for calling the c_functions, otherwise fallback to the Python implementation. Returns ------- ind : ndarray of int indices to the rainflow cycles of the original sequence TP. Example: -------- >>> import matplotlib.pyplot as plt >>> import wafo.misc as wm >>> t = np.linspace(0,7*np.pi,250) >>> x = np.sin(t)+0.1*np.sin(50*t) >>> ind = wm.findextrema(x) >>> ti, tp = t[ind], x[ind] >>> a = plt.plot(t,x,'.',ti,tp,'r.') >>> ind1 = wm.findrfc(tp,0.3); ind1 array([ 0, 9, 32, 53, 74, 95, 116, 137]) >>> ind2 = wm.findrfc(tp,0.3, method=''); ind2 array([ 0, 9, 32, 53, 74, 95, 116, 137]) >>> a = plt.plot(ti[ind1],tp[ind1]) >>> plt.close('all') See also -------- rfcfilter, findtp. ''' # TODO: merge rfcfilter and findrfc y1 = atleast_1d(tp).ravel() n = len(y1) ind = zeros(0, dtype=np.int) ix = 0 if y1[0] > y1[1]: # first is a max, ignore it y = y1[1::] NC = floor((n - 1) / 2) - 1 Tstart = 1 else: y = y1 NC = floor(n / 2) - 1 Tstart = 0 if (NC < 1): return ind # No RFC cycles*/ if (y[0] > y[1]) and (y[1] > y[2]): warnings.warn('This is not a sequence of turningpoints, exit') return ind if (y[0] < y[1]) and (y[1] < y[2]): warnings.warn('This is not a sequence of turningpoints, exit') return ind if clib is None or method not in ('clib'): ind = zeros(n, dtype=np.int) NC = np.int(NC) for i in xrange(NC): Tmi = Tstart + 2 * i Tpl = Tstart + 2 * i + 2 xminus = y[2 * i] xplus = y[2 * i + 2] if(i != 0): j = i - 1 while ((j >= 0) and (y[2 * j + 1] <= y[2 * i + 1])): if (y[2 * j] < xminus): xminus = y[2 * j] Tmi = Tstart + 2 * j j -= 1 if (xminus >= xplus): if (y[2 * i + 1] - xminus >= h): ind[ix] = Tmi ix += 1 ind[ix] = (Tstart + 2 * i + 1) ix += 1 # goto L180 continue else: j = i + 1 while (j < NC): if (y[2 * j + 1] >= y[2 * i + 1]): break # goto L170 if((y[2 * j + 2] <= xplus)): xplus = y[2 * j + 2] Tpl = (Tstart + 2 * j + 2) j += 1 else: if ((y[2 * i + 1] - xminus) >= h): ind[ix] = Tmi ix += 1 ind[ix] = (Tstart + 2 * i + 1) ix += 1 # iy = i continue # goto L180 # L170: if (xplus <= xminus): if ((y[2 * i + 1] - xminus) >= h): ind[ix] = Tmi ix += 1 ind[ix] = (Tstart + 2 * i + 1) ix += 1 elif ((y[2 * i + 1] - xplus) >= h): ind[ix] = (Tstart + 2 * i + 1) ix += 1 ind[ix] = Tpl ix += 1 # L180: # iy=i # /* for i */ else: ind, ix = clib.findrfc(y, h) return np.sort(ind[:ix]) def mctp2rfc(fmM, fMm=None): ''' Return Rainflow matrix given a Markov chain of turning points computes f_rfc = f_mM + F_mct(f_mM). Parameters ---------- fmM = the min2max Markov matrix, fMm = the max2min Markov matrix, Returns ------- f_rfc = the rainflow matrix, Example: ------- >>> fmM = np.array([[ 0.0183, 0.0160, 0.0002, 0.0000, 0], ... [0.0178, 0.5405, 0.0952, 0, 0], ... [0.0002, 0.0813, 0, 0, 0], ... [0.0000, 0, 0, 0, 0], ... [ 0, 0, 0, 0, 0]]) >>> np.abs(mctp2rfc(fmM)-np.array([[2.669981e-02, 7.799700e-03, ... 4.906077e-07, 0.000000e+00, 0.000000e+00], ... [ 9.599629e-03, 5.485009e-01, 9.539951e-02, 0.000000e+00, ... 0.000000e+00], ... [ 5.622974e-07, 8.149944e-02, 0.000000e+00, 0.000000e+00, ... 0.000000e+00], ... [ 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, ... 0.000000e+00], ... [ 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, ... 0.000000e+00]]))<1.e-7 array([[ True, True, True, True, True], [ True, True, True, True, True], [ True, True, True, True, True], [ True, True, True, True, True], [ True, True, True, True, True]], dtype=bool) ''' if fMm is None: fmM = np.atleast_1d(fmM) fMm = fmM.copy() else: fmM, fMm = np.atleast_1d(fmM, fMm) f_mM, f_Mm = fmM.copy(), fMm.copy() N = max(f_mM.shape) f_max = np.sum(f_mM, axis=1) f_min = np.sum(f_mM, axis=0) f_rfc = zeros((N, N)) f_rfc[N - 2, 0] = f_max[N - 2] f_rfc[0, N - 2] = f_min[N - 2] for k in range(2, N - 1): for i in range(1, k): AA = f_mM[N - 1 - k:N - 1 - k + i, k - i:k] AA1 = f_Mm[N - 1 - k:N - 1 - k + i, k - i:k] RAA = f_rfc[N - 1 - k:N - 1 - k + i, k - i:k] nA = max(AA.shape) MA = f_max[N - 1 - k:N - 1 - k + i] mA = f_min[k - i:k] SA = AA.sum() SRA = RAA.sum() DRFC = SA - SRA NT = min(mA[0] - sum(RAA[:, 0]), MA[0] - sum(RAA[0, :])) # check! NT = max(NT, 0) # ??check if NT > 1e-6 * max(MA[0], mA[0]): NN = MA - np.sum(AA, axis=1) # T e = (mA - np.sum(AA, axis=0)) # T e = np.flipud(e) PmM = np.rot90(AA.copy()) for j in range(nA): norm = mA[nA - 1 - j] if norm != 0: PmM[j, :] = PmM[j, :] / norm e[j] = e[j] / norm # end # end fx = 0.0 if (max(abs(e)) > 1e-6 and max(abs(NN)) > 1e-6 * max(MA[0], mA[0])): PMm = AA1.copy() for j in range(nA): norm = MA[j] if norm != 0: PMm[j, :] = PMm[j, :] / norm # end # end PMm = np.fliplr(PMm) A = PMm B = PmM if nA == 1: fx = NN * (A / (1 - B * A) * e) else: rh = np.eye(A.shape[0]) - np.dot(B, A) # least squares fx = np.dot(NN, np.dot(A, linalg.solve(rh, e))) # end # end f_rfc[N - 1 - k, k - i] = fx + DRFC # check2=[ DRFC fx] # pause else: f_rfc[N - 1 - k, k - i] = 0.0 # end # end m0 = max(0, f_min[0] - np.sum(f_rfc[N - k + 1:N, 0])) M0 = max(0, f_max[N - 1 - k] - np.sum(f_rfc[N - 1 - k, 1:k])) f_rfc[N - 1 - k, 0] = min(m0, M0) # n_loops_left=N-k+1 # end for k in range(1, N): M0 = max(0, f_max[0] - np.sum(f_rfc[0, N - k:N])) m0 = max(0, f_min[N - 1 - k] - np.sum(f_rfc[1:k + 1, N - 1 - k])) f_rfc[0, N - 1 - k] = min(m0, M0) # end # %clf # %subplot(1,2,2) # %pcolor(levels(paramm),levels(paramM),flipud(f_mM)) # % title('Markov matrix') # % ylabel('max'), xlabel('min') # %axis([paramm(1) paramm(2) paramM(1) paramM(2)]) # %axis('square') # # %subplot(1,2,1) # %pcolor(levels(paramm),levels(paramM),flipud(f_rfc)) # % title('Rainflow matrix') # % ylabel('max'), xlabel('rfc-min') # %axis([paramm(1) paramm(2) paramM(1) paramM(2)]) # %axis('square') return f_rfc def rfcfilter(x, h, method=0): """ Rainflow filter a signal. Parameters ----------- x : vector Signal. [nx1] h : real, scalar Threshold for rainflow filter. method : scalar, integer 0 : removes cycles with range < h. (default) 1 : removes cycles with range <= h. Returns -------- y = Rainflow filtered signal. Examples: --------- # 1. Filtered signal y is the turning points of x. >>> import wafo.data as data >>> import wafo.misc as wm >>> x = data.sea() >>> y = wm.rfcfilter(x[:,1], h=0, method=1) >>> np.all(np.abs(y[0:5]-np.array([-1.2004945 , 0.83950546, -0.09049454, ... -0.02049454, -0.09049454]))<1e-7) True >>> y.shape (2172,) # 2. This removes all rainflow cycles with range less than 0.5. >>> y1 = wm.rfcfilter(x[:,1], h=0.5) >>> y1.shape (863,) >>> np.all(np.abs(y1[0:5]-np.array([-1.2004945 , 0.83950546, -0.43049454, ... 0.34950546, -0.51049454]))<1e-7) True >>> ind = wm.findtp(x[:,1], h=0.5) >>> y2 = x[ind,1] >>> y2[0:5] array([-1.2004945 , 0.83950546, -0.43049454, 0.34950546, -0.51049454]) >>> y2[-5::] array([ 0.83950546, -0.64049454, 0.65950546, -1.0004945 , 0.91950546]) See also -------- findrfc """ # TODO merge rfcfilter and findrfc y = atleast_1d(x).ravel() n = len(y) t = zeros(n, dtype=np.int) j = 0 t0 = 0 y0 = y[t0] z0 = 0 def aleb(a, b): return a <= b def altb(a, b): return a < b if method == 0: cmpfun1 = aleb cmpfun2 = altb else: cmpfun1 = altb cmpfun2 = aleb # The rainflow filter for tim1, yi in enumerate(y[1::]): fpi = y0 + h fmi = y0 - h ti = tim1 + 1 # yi = y[ti] if z0 == 0: if cmpfun1(yi, fmi): z1 = -1 elif cmpfun1(fpi, yi): z1 = +1 else: z1 = 0 t1, y1 = (t0, y0) if z1 == 0 else (ti, yi) else: if (((z0 == +1) & cmpfun1(yi, fmi)) | ((z0 == -1) & cmpfun2(yi, fpi))): z1 = -1 elif (((z0 == +1) & cmpfun2(fmi, yi)) | ((z0 == -1) & cmpfun1(fpi, yi))): z1 = +1 else: warnings.warn('Something wrong, i=%d' % tim1) # Update y1 if z1 != z0: t1, y1 = ti, yi elif z1 == -1: # y1 = min([y0 xi]) t1, y1 = (t0, y0) if y0 < yi else (ti, yi) elif z1 == +1: # y1 = max([y0 xi]) t1, y1 = (t0, y0) if y0 > yi else (ti, yi) # Update y if y0 is a turning point if abs(z0 - z1) == 2: j += 1 t[j] = t0 # Update t0, y0, z0 t0, y0, z0 = t1, y1, z1 # end # Update y if last y0 is greater than (or equal) threshold if cmpfun1(h, abs(y0 - y[t[j]])): j += 1 t[j] = t0 return y[t[:j + 1]] def findtp(x, h=0.0, kind=None): ''' Return indices to turning points (tp) of data, optionally rainflowfiltered. Parameters ---------- x : vector signal h : real, scalar rainflow threshold if h<0, then ind = range(len(x)) if h=0, then tp is a sequence of turning points (default) if h>0, then all rainflow cycles with height smaller than h are removed. kind : string defines the type of wave or indicate the ASTM rainflow counting method. Possible options are 'astm' 'mw' 'Mw' or 'none'. If None all rainflow filtered min and max will be returned, otherwise only the rainflow filtered min and max, which define a wave according to the wave definition, will be returned. Returns ------- ind : arraylike indices to the turning points in the original sequence. Example: -------- >>> import pylab as plt >>> import wafo.misc as wm >>> t = np.linspace(0,30,500).reshape((-1,1)) >>> x = np.hstack((t, np.cos(t) + 0.3 * np.sin(5*t))) >>> x1 = x[0:100,:] >>> itp = wm.findtp(x1[:,1],0,'Mw') >>> itph = wm.findtp(x1[:,1],0.3,'Mw') >>> tp = x1[itp,:] >>> tph = x1[itph,:] >>> a = plt.plot(x1[:,0],x1[:,1], ... tp[:,0],tp[:,1],'ro', ... tph[:,0],tph[:,1],'k.') >>> plt.close('all') >>> itp array([ 5, 18, 24, 38, 46, 57, 70, 76, 91, 98, 99]) >>> itph array([91]) See also --------- findtc findcross findextrema findrfc ''' n = len(x) if h < 0.0: return arange(n) ind = findextrema(x) if ind.size < 2: return None # In order to get the exact up-crossing intensity from rfc by # mm2lc(tp2mm(rfc)) we have to add the indices to the last value # (and also the first if the sequence of turning points does not start # with a minimum). if kind == 'astm': # the Nieslony approach always put the first loading point as the first # turning point. # add the first turning point is the first of the signal if x[ind[0]] != x[0]: ind = np.r_[0, ind, n - 1] else: # only add the last point of the signal ind = np.r_[ind, n - 1] else: if x[ind[0]] > x[ind[1]]: # adds indices to first and last value ind = r_[0, ind, n - 1] else: # adds index to the last value ind = r_[ind, n - 1] if h > 0.0: ind1 = findrfc(x[ind], h) ind = ind[ind1] if kind in ('mw', 'Mw'): # make sure that the first is a Max if wdef == 'Mw' # or make sure that the first is a min if wdef == 'mw' first_is_max = (x[ind[0]] > x[ind[1]]) remove_first = xor(first_is_max, kind.startswith('Mw')) if remove_first: ind = ind[1::] # make sure the number of minima and Maxima are according to the # wavedef. i.e., make sure Nm=length(ind) is odd if (mod(ind.size, 2)) != 1: ind = ind[:-1] return ind def findtc(x_in, v=None, kind=None): """ Return indices to troughs and crests of data. Parameters ---------- x : vector surface elevation. v : real scalar reference level (default v = mean of x). kind : string defines the type of wave. Possible options are 'dw', 'uw', 'tw', 'cw' or None. If None indices to all troughs and crests will be returned, otherwise only the paired ones will be returned according to the wavedefinition. Returns -------- tc_ind : vector of ints indices to the trough and crest turningpoints of sequence x. v_ind : vector of ints indices to the level v crossings of the original sequence x. (d,u) Example: -------- >>> import pylab as plt >>> import wafo.misc as wm >>> t = np.linspace(0,30,500).reshape((-1,1)) >>> x = np.hstack((t, np.cos(t))) >>> x1 = x[0:200,:] >>> itc, iv = wm.findtc(x1[:,1],0,'dw') >>> tc = x1[itc,:] >>> a = plt.plot(x1[:,0],x1[:,1],tc[:,0],tc[:,1],'ro') >>> plt.close('all') See also -------- findtp findcross, wavedef """ x = atleast_1d(x_in) if v is None: v = x.mean() v_ind = findcross(x, v, kind) n_c = v_ind.size if n_c <= 2: warnings.warn('There are no waves!') return zeros(0, dtype=np.int), zeros(0, dtype=np.int) # determine the number of trough2crest (or crest2trough) cycles is_even = mod(n_c + 1, 2) n_tc = int((n_c - 1 - is_even) / 2) # allocate variables before the loop increases the speed ind = zeros(n_c - 1, dtype=np.int) first_is_down_crossing = (x[v_ind[0]] > x[v_ind[0] + 1]) if first_is_down_crossing: for i in xrange(n_tc): # trough j = 2 * i ind[j] = x[v_ind[j] + 1:v_ind[j + 1] + 1].argmin() # crest ind[j + 1] = x[v_ind[j + 1] + 1:v_ind[j + 2] + 1].argmax() if (2 * n_tc + 1 < n_c) and (kind in (None, 'tw')): # trough ind[n_c - 2] = x[v_ind[n_c - 2] + 1:v_ind[n_c - 1]].argmin() else: # the first is a up-crossing for i in xrange(n_tc): # crest j = 2 * i ind[j] = x[v_ind[j] + 1:v_ind[j + 1] + 1].argmax() # trough ind[j + 1] = x[v_ind[j + 1] + 1:v_ind[j + 2] + 1].argmin() if (2 * n_tc + 1 < n_c) and (kind in (None, 'cw')): # crest ind[n_c - 2] = x[v_ind[n_c - 2] + 1:v_ind[n_c - 1]].argmax() return v_ind[:n_c - 1] + ind + 1, v_ind def findoutliers(x, zcrit=0.0, dcrit=None, ddcrit=None, verbose=False): """ Return indices to spurious points of data Parameters ---------- x : vector of data values. zcrit : real scalar critical distance between consecutive points. dcrit : real scalar critical distance of Dx used for determination of spurious points. (Default 1.5 standard deviation of x) ddcrit : real scalar critical distance of DDx used for determination of spurious points. (Default 1.5 standard deviation of x) Returns ------- inds : ndarray of integers indices to spurious points. indg : ndarray of integers indices to the rest of the points. Notes ----- Consecutive points less than zcrit apart are considered as spurious. The point immediately after and before are also removed. Jumps greater than dcrit in Dxn and greater than ddcrit in D^2xn are also considered as spurious. (All distances to be interpreted in the vertical direction.) Another good choice for dcrit and ddcrit are: dcrit = 5*dT and ddcrit = 9.81/2*dT**2 where dT is the timestep between points. Examples -------- >>> import numpy as np >>> import wafo.misc as wm >>> t = np.linspace(0,30,500).reshape((-1,1)) >>> xx = np.hstack((t, np.cos(t))) >>> dt = np.diff(xx[:2,0]) >>> dcrit = 5*dt >>> ddcrit = 9.81/2*dt*dt >>> zcrit = 0 >>> [inds, indg] = wm.findoutliers(xx[:,1],zcrit,dcrit,ddcrit,verbose=True) Found 0 spurious positive jumps of Dx Found 0 spurious negative jumps of Dx Found 0 spurious positive jumps of D^2x Found 0 spurious negative jumps of D^2x Found 0 consecutive equal values Found the total of 0 spurious points #waveplot(xx,'-',xx(inds,:),1,1,1) See also -------- waveplot, reconstruct """ # finding outliers findjumpsDx = True # find jumps in Dx # two point spikes and Spikes dcrit above/under the # previous and the following point are spurios. findSpikes = False # find spikes findDspikes = False # find double (two point) spikes findjumpsD2x = True # find jumps in D^2x findNaN = True # % find missing values xn = asarray(x).flatten() if xn.size < 2: raise ValueError('The vector must have more than 2 elements!') ind = zeros(0, dtype=int) # indg=[] indmiss = isnan(xn) if findNaN and indmiss.any(): ind, = nonzero(indmiss) if verbose: print('Found %d missing points' % ind.size) xn[indmiss] = 0. # %set NaN's to zero if dcrit is None: dcrit = 1.5 * xn.std() if verbose: print('dcrit is set to %g' % dcrit) if ddcrit is None: ddcrit = 1.5 * xn.std() if verbose: print('ddcrit is set to %g' % ddcrit) dxn = diff(xn) ddxn = diff(dxn) if findSpikes: # finding spurious spikes tmp, = nonzero((dxn[:-1] > dcrit) * (dxn[1::] < -dcrit) | (dxn[:-1] < -dcrit) * (dxn[1::] > dcrit)) if tmp.size > 0: tmp = tmp + 1 ind = hstack((ind, tmp)) if verbose: print('Found %d spurious spikes' % tmp.size) if findDspikes: # ,% finding spurious double (two point) spikes tmp, = nonzero((dxn[:-2] > dcrit) * (dxn[2::] < -dcrit) | (dxn[:-2] < -dcrit) * (dxn[2::] > dcrit)) if tmp.size > 0: tmp = tmp + 1 ind = hstack((ind, tmp, tmp + 1)) # %removing both points if verbose: print('Found %d spurious two point (double) spikes' % tmp.size) if findjumpsDx: # ,% finding spurious jumps in Dx tmp, = nonzero(dxn > dcrit) if verbose: print('Found %d spurious positive jumps of Dx' % tmp.size) if tmp.size > 0: ind = hstack((ind, tmp + 1)) # removing the point after the jump tmp, = nonzero(dxn < -dcrit) if verbose: print('Found %d spurious negative jumps of Dx' % tmp.size) if tmp.size > 0: ind = hstack((ind, tmp)) # removing the point before the jump if findjumpsD2x: # ,% finding spurious jumps in D^2x tmp, = nonzero(ddxn > ddcrit) if tmp.size > 0: tmp = tmp + 1 ind = hstack((ind, tmp)) # removing the jump if verbose: print('Found %d spurious positive jumps of D^2x' % tmp.size) tmp, = nonzero(ddxn < -ddcrit) if tmp.size > 0: tmp = tmp + 1 ind = hstack((ind, tmp)) # removing the jump if verbose: print('Found %d spurious negative jumps of D^2x' % tmp.size) if zcrit >= 0.0: # finding consecutive values less than zcrit apart. indzeros = (abs(dxn) <= zcrit) indz, = nonzero(indzeros) if indz.size > 0: indz = indz + 1 # finding the beginning and end of consecutive equal values indtr, = nonzero((diff(indzeros))) indtr = indtr + 1 # indices to consecutive equal points # removing the point before + all equal points + the point after if True: ind = hstack((ind, indtr - 1, indz, indtr, indtr + 1)) else: # % removing all points + the point after ind = hstack((ind, indz, indtr, indtr + 1)) if verbose: if zcrit == 0.: print('Found %d consecutive equal values' % indz.size) else: print('Found %d consecutive values less than %g apart.' % (indz.size, zcrit)) indg = ones(xn.size, dtype=bool) if ind.size > 1: ind = unique(ind) indg[ind] = 0 indg, = nonzero(indg) if verbose: print('Found the total of %d spurious points' % ind.size) return ind, indg def common_shape(*args, ** kwds): ''' Return the common shape of a sequence of arrays Parameters ----------- *args : arraylike sequence of arrays **kwds : shape Returns ------- shape : tuple common shape of the elements of args. Raises ------ An error is raised if some of the arrays do not conform to the common shape according to the broadcasting rules in numpy. Examples -------- >>> import numpy as np >>> import wafo.misc as wm >>> A = np.ones((4,1)) >>> B = 2 >>> C = np.ones((1,5))*5 >>> wm.common_shape(A,B,C) (4, 5) >>> wm.common_shape(A,B,C,shape=(3,4,1)) (3, 4, 5) See also -------- broadcast, broadcast_arrays ''' args = map(asarray, args) shapes = [x.shape for x in args] shape = kwds.get('shape') if shape is not None: if not isinstance(shape, (list, tuple)): shape = (shape,) shapes.append(tuple(shape)) if len(set(shapes)) == 1: # Common case where nothing needs to be broadcasted. return tuple(shapes[0]) shapes = [list(s) for s in shapes] nds = [len(s) for s in shapes] biggest = max(nds) # Go through each array and prepend dimensions of length 1 to each of the # shapes in order to make the number of dimensions equal. for i in range(len(shapes)): diff = biggest - nds[i] if diff > 0: shapes[i] = [1] * diff + shapes[i] # Check each dimension for compatibility. A dimension length of 1 is # accepted as compatible with any other length. c_shape = [] for axis in range(biggest): lengths = [s[axis] for s in shapes] unique = set(lengths + [1]) if len(unique) > 2: # There must be at least two non-1 lengths for this axis. raise ValueError("shape mismatch: two or more arrays have " "incompatible dimensions on axis %r." % (axis,)) elif len(unique) == 2: # There is exactly one non-1 length. # The common shape will take this value. unique.remove(1) new_length = unique.pop() c_shape.append(new_length) else: # Every array has a length of 1 on this axis. Strides can be left # alone as nothing is broadcasted. c_shape.append(1) return tuple(c_shape) def argsreduce(condition, * args): """ Return the elements of each input array that satisfy some condition. Parameters ---------- condition : array_like An array whose nonzero or True entries indicate the elements of each input array to extract. The shape of 'condition' must match the common shape of the input arrays according to the broadcasting rules in numpy. arg1, arg2, arg3, ... : array_like one or more input arrays. Returns ------- narg1, narg2, narg3, ... : ndarray sequence of extracted copies of the input arrays converted to the same size as the nonzero values of condition. Example ------- >>> import wafo.misc as wm >>> import numpy as np >>> rand = np.random.random_sample >>> A = rand((4,5)) >>> B = 2 >>> C = rand((1,5)) >>> cond = np.ones(A.shape) >>> [A1,B1,C1] = wm.argsreduce(cond,A,B,C) >>> B1.shape (20,) >>> cond[2,:] = 0 >>> [A2,B2,C2] = wm.argsreduce(cond,A,B,C) >>> B2.shape (15,) See also -------- numpy.extract """ newargs = atleast_1d(*args) if not isinstance(newargs, list): newargs = [newargs, ] expand_arr = (condition == condition) return [extract(condition, arr1 * expand_arr) for arr1 in newargs] def stirlerr(n): ''' Return error of Stirling approximation, i.e., log(n!) - log( sqrt(2*pi*n)*(n/exp(1))**n ) Example ------- >>> import wafo.misc as wm >>> np.abs(wm.stirlerr(2)- 0.0413407)<1e-7 array([ True], dtype=bool) See also --------- binom Reference ----------- Catherine Loader (2000). Fast and Accurate Computation of Binomial Probabilities ''' S0 = 0.083333333333333333333 # /* 1/12 */ S1 = 0.00277777777777777777778 # /* 1/360 */ S2 = 0.00079365079365079365079365 # /* 1/1260 */ S3 = 0.000595238095238095238095238 # /* 1/1680 */ S4 = 0.0008417508417508417508417508 # /* 1/1188 */ n1 = atleast_1d(n) y = gammaln(n1 + 1) - log(sqrt(2 * pi * n1) * (n1 / exp(1)) ** n1) nn = n1 * n1 n500 = 500 < n1 y[n500] = (S0 - S1 / nn[n500]) / n1[n500] n80 = logical_and(80 < n1, n1 <= 500) if any(n80): y[n80] = (S0 - (S1 - S2 / nn[n80]) / nn[n80]) / n1[n80] n35 = logical_and(35 < n1, n1 <= 80) if any(n35): nn35 = nn[n35] y[n35] = (S0 - (S1 - (S2 - S3 / nn35) / nn35) / nn35) / n1[n35] n15 = logical_and(15 < n1, n1 <= 35) if any(n15): nn15 = nn[n15] y[n15] = ( S0 - (S1 - (S2 - (S3 - S4 / nn15) / nn15) / nn15) / nn15) / n1[n15] return y def getshipchar(value=None, property="max_deadweight", # @ReservedAssignment ** kwds): # @IgnorePep8 ''' Return ship characteristics from value of one ship-property Parameters ---------- value : scalar value to use in the estimation. property : string defining the ship property used in the estimation. Options are: 'max_deadweight','length','beam','draft','service_speed', 'propeller_diameter'. The length was found from statistics of 40 vessels of size 85 to 100000 tonn. An exponential curve through 0 was selected, and the factor and exponent that minimized the standard deviation of the relative error was selected. (The error returned is the same for any ship.) The servicespeed was found for ships above 1000 tonns only. The propeller diameter formula is from [1]_. Returns ------- sc : dict containing estimated mean values and standard-deviations of ship characteristics: max_deadweight [kkg], (weight of cargo, fuel etc.) length [m] beam [m] draught [m] service_speed [m/s] propeller_diameter [m] Example --------- >>> import wafo.misc as wm >>> sc = wm.getshipchar(10,'service_speed') >>> for key in sorted(sc): key, sc[key] ('beam', 29.0) ('beamSTD', 2.9000000000000004) ('draught', 9.6) ('draughtSTD', 2.112) ('length', 216.0) ('lengthSTD', 2.011309883194276) ('max_deadweight', 30969.0) ('max_deadweightSTD', 3096.9) ('propeller_diameter', 6.761165385916601) ('propeller_diameterSTD', 0.20267047566705432) ('service_speed', 10.0) ('service_speedSTD', 0) Other units: 1 ft = 0.3048 m and 1 knot = 0.5144 m/s Reference --------- .. [1] Gray and Greeley, (1978), "Source level model for propeller blade rate radiation for the world's merchant fleet", Bolt Beranek and Newman Technical Memorandum No. 458. ''' if value is None: names = kwds.keys() if len(names) != 1: raise ValueError('Only on keyword') property = names[0] # @ReservedAssignment value = kwds[property] value = np.atleast_1d(value) valid_props = dict(l='length', b='beam', d='draught', m='max_deadweigth', s='service_speed', p='propeller_diameter') prop = valid_props[property[0]] prop2max_dw = dict(length=lambda x: (x / 3.45) ** (2.5), beam=lambda x: ((x / 1.78) ** (1 / 0.27)), draught=lambda x: ((x / 0.8) ** (1 / 0.24)), service_speed=lambda x: ((x / 1.14) ** (1 / 0.21)), propeller_diameter=lambda x: (((x / 0.12) ** (4 / 3) / 3.45) ** (2.5))) max_deadweight = prop2max_dw.get(prop, lambda x: x)(value) propertySTD = prop + 'STD' length = round(3.45 * max_deadweight ** 0.40) length_err = length ** 0.13 beam = round(1.78 * max_deadweight ** 0.27 * 10) / 10 beam_err = beam * 0.10 draught = round(0.80 * max_deadweight ** 0.24 * 10) / 10 draught_err = draught * 0.22 # S = round(2/3*(L)**0.525) speed = round(1.14 * max_deadweight ** 0.21 * 10) / 10 speed_err = speed * 0.10 p_diam = 0.12 * length ** (3.0 / 4.0) p_diam_err = 0.12 * length_err ** (3.0 / 4.0) max_deadweight = round(max_deadweight) max_deadweightSTD = 0.1 * max_deadweight shipchar = OrderedDict(beam=beam, beamSTD=beam_err, draught=draught, draughtSTD=draught_err, length=length, lengthSTD=length_err, max_deadweight=max_deadweight, max_deadweightSTD=max_deadweightSTD, propeller_diameter=p_diam, propeller_diameterSTD=p_diam_err, service_speed=speed, service_speedSTD=speed_err) shipchar[propertySTD] = 0 return shipchar def binomln(z, w): ''' Natural Logarithm of binomial coefficient. CALL binomln(z,w) BINOMLN computes the natural logarithm of the binomial function for corresponding elements of Z and W. The arrays Z and W must be real and nonnegative. Both arrays must be the same size, or either can be scalar. BETALOGE is defined as: y = LOG(binom(Z,W)) = gammaln(Z)-gammaln(W)-gammaln(Z-W) and is obtained without computing BINOM(Z,W). Since the binom function can range over very large or very small values, its logarithm is sometimes more useful. This implementation is more accurate than the log(BINOM(Z,W) implementation for large arguments Example ------- >>> abs(binomln(3,2)- 1.09861229)<1e-7 array([ True], dtype=bool) See also -------- binom ''' # log(n!) = stirlerr(n) + log( sqrt(2*pi*n)*(n/exp(1))**n ) # y = gammaln(z+1)-gammaln(w+1)-gammaln(z-w+1) zpw = z - w return (stirlerr(z + 1) - stirlerr(w + 1) - 0.5 * log(2 * pi) - (w + 0.5) * log1p(w) + (z + 0.5) * log1p(z) - stirlerr(zpw + 1) - (zpw + 0.5) * log1p(zpw) + 1) def betaloge(z, w): ''' Natural Logarithm of beta function. CALL betaloge(z,w) BETALOGE computes the natural logarithm of the beta function for corresponding elements of Z and W. The arrays Z and W must be real and nonnegative. Both arrays must be the same size, or either can be scalar. BETALOGE is defined as: y = LOG(BETA(Z,W)) = gammaln(Z)+gammaln(W)-gammaln(Z+W) and is obtained without computing BETA(Z,W). Since the beta function can range over very large or very small values, its logarithm is sometimes more useful. This implementation is more accurate than the BETALN implementation for large arguments Example ------- >>> import wafo.misc as wm >>> abs(wm.betaloge(3,2)+2.48490665)<1e-7 array([ True], dtype=bool) See also -------- betaln, beta ''' # y = gammaln(z)+gammaln(w)-gammaln(z+w) zpw = z + w return (stirlerr(z) + stirlerr(w) + 0.5 * log(2 * pi) + (w - 0.5) * log(w) + (z - 0.5) * log(z) - stirlerr(zpw) - (zpw - 0.5) * log(zpw)) # stirlings approximation: # (-(zpw-0.5).*log(zpw) +(w-0.5).*log(w)+(z-0.5).*log(z) +0.5*log(2*pi)) # return y def gravity(phi=45): ''' Returns the constant acceleration of gravity GRAVITY calculates the acceleration of gravity using the international gravitational formulae [1]_: g = 9.78049*(1+0.0052884*sin(phir)**2-0.0000059*sin(2*phir)**2) where phir = phi*pi/180 Parameters ---------- phi : {float, int} latitude in degrees Returns -------- g : ndarray acceleration of gravity [m/s**2] Examples -------- >>> import wafo.misc as wm >>> import numpy as np >>> phi = np.linspace(0,45,5) >>> np.abs(wm.gravity(phi)-np.array([ 9.78049 , 9.78245014, 9.78803583, ... 9.79640552, 9.80629387]))<1.e-7 array([ True, True, True, True, True], dtype=bool) See also -------- wdensity References ---------- .. [1] Irgens, Fridtjov (1987) "Formelsamling i mekanikk: statikk, fasthetsl?re, dynamikk fluidmekanikk" tapir forlag, University of Trondheim, ISBN 82-519-0786-1, pp 19 ''' phir = phi * pi / 180. # change from degrees to radians return 9.78049 * (1. + 0.0052884 * sin(phir) ** 2. - 0.0000059 * sin(2 * phir) ** 2.) def dea3(v0, v1, v2): ''' Extrapolate a slowly convergent sequence Parameters ---------- v0, v1, v2 : array-like 3 values of a convergent sequence to extrapolate Returns ------- result : array-like extrapolated value abserr : array-like absolute error estimate Description ----------- DEA3 attempts to extrapolate nonlinearly to a better estimate of the sequence's limiting value, thus improving the rate of convergence. The routine is based on the epsilon algorithm of P. Wynn, see [1]_. Example ------- # integrate sin(x) from 0 to pi/2 >>> import numpy as np >>> import numdifftools as nd >>> Ei= np.zeros(3) >>> linfun = lambda k : np.linspace(0,np.pi/2.,2.**(k+5)+1) >>> for k in np.arange(3): ... x = linfun(k) ... Ei[k] = np.trapz(np.sin(x),x) >>> [En, err] = nd.dea3(Ei[0], Ei[1], Ei[2]) >>> truErr = Ei-1. >>> (truErr, err, En) (array([ -2.00805680e-04, -5.01999079e-05, -1.25498825e-05]), array([ 0.00020081]), array([ 1.])) See also -------- dea Reference --------- .. [1] C. Brezinski (1977) "Acceleration de la convergence en analyse numerique", "Lecture Notes in Math.", vol. 584, Springer-Verlag, New York, 1977. ''' E0, E1, E2 = np.atleast_1d(v0, v1, v2) abs = np.abs # @ReservedAssignment max = np.maximum # @ReservedAssignment delta2, delta1 = E2 - E1, E1 - E0 err2, err1 = abs(delta2), abs(delta1) tol2, tol1 = max(abs(E2), abs(E1)) * _EPS, max(abs(E1), abs(E0)) * _EPS with warnings.catch_warnings(): warnings.simplefilter("ignore") # ignore division by zero and overflow ss = 1.0 / delta2 - 1.0 / delta1 smallE2 = (abs(ss * E1) <= 1.0e-3).ravel() result = 1.0 * E2 abserr = err1 + err2 + abs(E2) * _EPS * 10.0 converged = (err1 <= tol1) & (err2 <= tol2).ravel() | smallE2 k4, = (1 - converged).nonzero() if k4.size > 0: result[k4] = E1[k4] + 1.0 / ss[k4] abserr[k4] = err1[k4] + err2[k4] + abs(result[k4] - E2[k4]) return result, abserr def nextpow2(x): ''' Return next higher power of 2 Example ------- >>> import wafo.misc as wm >>> wm.nextpow2(10) 4 >>> wm.nextpow2(np.arange(5)) 3 ''' t = isscalar(x) or len(x) if (t > 1): f, n = frexp(t) else: f, n = frexp(abs(x)) if (f == 0.5): n = n - 1 return n def discretize(fun, a, b, tol=0.005, n=5, method='linear'): ''' Automatic discretization of function Parameters ---------- fun : callable function to discretize a,b : real scalars evaluation limits tol : real, scalar absoute error tolerance n : scalar integer number of values method : string defining method of gridding, options are 'linear' and 'adaptive' Returns ------- x : discretized values y : fun(x) Example ------- >>> import wafo.misc as wm >>> import numpy as np >>> import pylab as plt >>> x,y = wm.discretize(np.cos, 0, np.pi) >>> xa,ya = wm.discretize(np.cos, 0, np.pi, method='adaptive') >>> t = plt.plot(x, y, xa, ya, 'r.') plt.show() >>> plt.close('all') ''' if method.startswith('a'): return _discretize_adaptive(fun, a, b, tol, n) else: return _discretize_linear(fun, a, b, tol, n) def _discretize_linear(fun, a, b, tol=0.005, n=5): ''' Automatic discretization of function, linear gridding ''' x = linspace(a, b, n) y = fun(x) err0 = inf err = 10000 nmax = 2 ** 20 while (err != err0 and err > tol and n < nmax): err0 = err x0 = x y0 = y n = 2 * (n - 1) + 1 x = linspace(a, b, n) y = fun(x) y00 = interp(x, x0, y0) err = 0.5 * amax(abs((y00 - y) / (abs(y00 + y) + _TINY))) return x, y def _discretize_adaptive(fun, a, b, tol=0.005, n=5): ''' Automatic discretization of function, adaptive gridding. ''' n += (mod(n, 2) == 0) # make sure n is odd x = linspace(a, b, n) fx = fun(x) n2 = (n - 1) / 2 erri = hstack((zeros((n2, 1)), ones((n2, 1)))).ravel() err = erri.max() err0 = inf # while (err != err0 and err > tol and n < nmax): for j in range(50): if err != err0 and np.any(erri > tol): err0 = err # find top errors I, = where(erri > tol) # double the sample rate in intervals with the most error y = (vstack(((x[I] + x[I - 1]) / 2, (x[I + 1] + x[I]) / 2)).T).ravel() fy = fun(y) fy0 = interp(y, x, fx) erri = 0.5 * (abs((fy0 - fy) / (abs(fy0 + fy) + _TINY))) err = erri.max() x = hstack((x, y)) I = x.argsort() x = x[I] erri = hstack((zeros(len(fx)), erri))[I] fx = hstack((fx, fy))[I] else: break else: warnings.warn('Recursion level limit reached j=%d' % j) return x, fx def polar2cart(theta, rho, z=None): ''' Transform polar coordinates into 2D cartesian coordinates. Returns ------- x, y : array-like Cartesian coordinates, x = rho*cos(theta), y = rho*sin(theta) See also -------- cart2polar ''' x, y = rho * cos(theta), rho * sin(theta) if z is None: return x, y else: return x, y, z pol2cart = polar2cart def cart2polar(x, y, z=None): ''' Transform 2D cartesian coordinates into polar coordinates. Returns ------- theta : array-like radial angle, arctan2(y,x) rho : array-like radial distance, sqrt(x**2+y**2) See also -------- polar2cart ''' t, r = arctan2(y, x), hypot(x, y) if z is None: return t, r else: return t, r, z cart2pol = cart2polar def ndgrid(*args, **kwargs): """ Same as calling meshgrid with indexing='ij' (see meshgrid for documentation). """ kwargs['indexing'] = 'ij' return meshgrid(*args, ** kwargs) def trangood(x, f, min_n=None, min_x=None, max_x=None, max_n=inf): """ Make sure transformation is efficient. Parameters ------------ x, f : array_like input transform function, (x,f(x)). min_n : scalar, int minimum number of points in the good transform. (Default x.shape[0]) min_x : scalar, real minimum x value to transform. (Default min(x)) max_x : scalar, real maximum x value to transform. (Default max(x)) max_n : scalar, int maximum number of points in the good transform (default inf) Returns ------- x, f : array_like the good transform function. TRANGOOD interpolates f linearly and optionally extrapolate it linearly outside the range of x with X uniformly spaced. See also --------- tranproc, numpy.interp """ xo, fo = atleast_1d(x, f) # n = xo.size if (xo.ndim != 1): raise ValueError('x must be a vector.') if (fo.ndim != 1): raise ValueError('f must be a vector.') i = xo.argsort() xo = xo[i] fo = fo[i] del i dx = diff(xo) if (any(dx <= 0)): raise ValueError('Duplicate x-values not allowed.') nf = fo.shape[0] if max_x is None: max_x = xo[-1] if min_x is None: min_x = xo[0] if min_n is None: min_n = nf if (min_n < 2): min_n = 2 if (max_n < 2): max_n = 2 ddx = diff(dx) xn = xo[-1] x0 = xo[0] L = float(xn - x0) if ((nf < min_n) or (max_n < nf) or any(abs(ddx) > 10 * _EPS * (L))): # pab 07.01.2001: Always choose the stepsize df so that # it is an exactly representable number. # This is important when calculating numerical derivatives and is # accomplished by the following. dx = L / (min(min_n, max_n) - 1) dx = (dx + 2.) - 2. xi = arange(x0, xn + dx / 2., dx) # New call pab 11.11.2000: This is much quicker fo = interp(xi, xo, fo) xo = xi # x is now uniformly spaced dx = xo[1] - xo[0] # Extrapolate linearly outside the range of ff if (min_x < xo[0]): x1 = dx * arange(floor((min_x - xo[0]) / dx), -2) f2 = fo[0] + x1 * (fo[1] - fo[0]) / (xo[1] - xo[0]) fo = hstack((f2, fo)) xo = hstack((x1 + xo[0], xo)) if (max_x > xo[-1]): x1 = dx * arange(1, ceil((max_x - xo[-1]) / dx) + 1) f2 = f[-1] + x1 * (f[-1] - f[-2]) / (xo[-1] - xo[-2]) fo = hstack((fo, f2)) xo = hstack((xo, x1 + xo[-1])) return xo, fo def tranproc(x, f, x0, *xi): """ Transforms process X and up to four derivatives using the transformation f. Parameters ---------- x,f : array-like [x,f(x)], transform function, y = f(x). x0, x1,...,xn : vectors where xi is the i'th time derivative of x0. 0<=N<=4. Returns ------- y0, y1,...,yn : vectors where yi is the i'th time derivative of y0 = f(x0). By the basic rules of derivation: Y1 = f'(X0)*X1 Y2 = f''(X0)*X1^2 + f'(X0)*X2 Y3 = f'''(X0)*X1^3 + f'(X0)*X3 + 3*f''(X0)*X1*X2 Y4 = f''''(X0)*X1^4 + f'(X0)*X4 + 6*f'''(X0)*X1^2*X2 + f''(X0)*(3*X2^2 + 4*X1*X3) The derivation of f is performed numerically with a central difference method with linear extrapolation towards the beginning and end of f, respectively. Example -------- Derivative of g and the transformed Gaussian model. >>> import pylab as plt >>> import wafo.misc as wm >>> import wafo.transform.models as wtm >>> tr = wtm.TrHermite() >>> x = linspace(-5,5,501) >>> g = tr(x) >>> gder = wm.tranproc(x, g, x, ones(g.shape[0])) >>> h = plt.plot(x, g, x, gder[1]) plt.plot(x,pdfnorm(g)*gder[1],x,pdfnorm(x)) plt.legend('Transformed model','Gaussian model') >>> plt.close('all') See also -------- trangood. """ xo, fo, x0 = atleast_1d(x, f, x0) xi = atleast_1d(*xi) if not isinstance(xi, list): xi = [xi, ] N = len(xi) # N = number of derivatives nmax = ceil((xo.ptp()) * 10 ** (7. / max(N, 1))) xo, fo = trangood(xo, fo, min_x=min(x0), max_x=max(x0), max_n=nmax) n = f.shape[0] # y = x0.copy() xu = (n - 1) * (x0 - xo[0]) / (xo[-1] - xo[0]) fi = asarray(floor(xu), dtype=int) fi = where(fi == n - 1, fi - 1, fi) xu = xu - fi y0 = fo[fi] + (fo[fi + 1] - fo[fi]) * xu y = y0 if N > 0: y = [y0] hn = xo[1] - xo[0] if hn ** N < sqrt(_EPS): msg = ('Numerical problems may occur for the derivatives in ' + 'tranproc.\n' + 'The sampling of the transformation may be too small.') warnings.warn(msg) # Transform X with the derivatives of f. fxder = zeros((N, x0.size)) fder = vstack((xo, fo)) for k in range(N): # Derivation of f(x) using a difference method. n = fder.shape[-1] fder = vstack([(fder[0, 0:n - 1] + fder[0, 1:n]) / 2, diff(fder[1, :]) / hn]) fxder[k] = tranproc(fder[0], fder[1], x0) # Calculate the transforms of the derivatives of X. # First time derivative of y: y1 = f'(x)*x1 y1 = fxder[0] * xi[0] y.append(y1) if N > 1: # Second time derivative of y: # y2 = f''(x)*x1.^2+f'(x)*x2 y2 = fxder[1] * xi[0] ** 2. + fxder[0] * xi[1] y.append(y2) if N > 2: # Third time derivative of y: # y3 = f'''(x)*x1.^3+f'(x)*x3 +3*f''(x)*x1*x2 y3 = fxder[2] * xi[0] ** 3 + fxder[0] * xi[2] + \ 3 * fxder[1] * xi[0] * xi[1] y.append(y3) if N > 3: # Fourth time derivative of y: # y4 = f''''(x)*x1.^4+f'(x)*x4 # +6*f'''(x)*x1^2*x2+f''(x)*(3*x2^2+4x1*x3) y4 = (fxder[3] * xi[0] ** 4. + fxder[0] * xi[3] + 6. * fxder[2] * xi[0] ** 2. * xi[1] + fxder[1] * (3. * xi[1] ** 2. + 4. * xi[0] * xi[1])) y.append(y4) if N > 4: warnings.warn('Transformation of derivatives of ' + 'order>4 not supported.') return y # y0,y1,y2,y3,y4 def good_bins(data=None, range=None, num_bins=None, # @ReservedAssignment num_data=None, odd=False, loose=True): ''' Return good bins for histogram Parameters ---------- data : array-like the data range : (float, float) minimum and maximum range of bins (default data.min(), data.max()) num_bins : scalar integer approximate number of bins wanted (default depending on num_data=len(data)) odd : bool placement of bins (0 or 1) (default 0) loose : bool if True add extra space to min and max if False the bins are made tight to the min and max Example ------- >>> import wafo.misc as wm >>> wm.good_bins(range=(0,5), num_bins=6) array([-1., 0., 1., 2., 3., 4., 5., 6.]) >>> wm.good_bins(range=(0,5), num_bins=6, loose=False) array([ 0., 1., 2., 3., 4., 5.]) >>> wm.good_bins(range=(0,5), num_bins=6, odd=True) array([-1.5, -0.5, 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5]) >>> wm.good_bins(range=(0,5), num_bins=6, odd=True, loose=False) array([-0.5, 0.5, 1.5, 2.5, 3.5, 4.5, 5.5]) ''' if data is not None: x = np.atleast_1d(data) num_data = len(x) mn, mx = range if range else (x.min(), x.max()) if num_bins is None: num_bins = np.ceil(4 * np.sqrt(np.sqrt(num_data))) d = float(mx - mn) / num_bins * 2 e = np.floor(np.log(d) / np.log(10)) m = np.floor(d / 10 ** e) if m > 5: m = 5 elif m > 2: m = 2 d = m * 10 ** e mn = (np.floor(mn / d) - loose) * d - odd * d / 2 mx = (np.ceil(mx / d) + loose) * d + odd * d / 2 limits = np.arange(mn, mx + d / 2, d) return limits def plot_histgrm(data, bins=None, range=None, # @ReservedAssignment normed=False, weights=None, lintype='b-'): ''' Plot histogram Parameters ----------- data : array-like the data bins : int or sequence of scalars, optional If an int, it defines the number of equal-width bins in the given range (4 * sqrt(sqrt(len(data)), by default). If 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 ``(data.min(), data.max())``. Values outside the range are ignored. 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. weights : array_like, optional An array of weights, of the same shape as `data`. Each value in `data` 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 lintype : specify color and lintype, see PLOT for possibilities. Returns ------- h : list of plot-objects Example ------- >>> import pylab as plt >>> import wafo.misc as wm >>> import wafo.stats as ws >>> R = ws.weibull_min.rvs(2,loc=0,scale=2, size=100) >>> h0 = wm.plot_histgrm(R, 20, normed=True) >>> x = np.linspace(-3,16,200) >>> h1 = plt.plot(x,ws.weibull_min.pdf(x,2,0,2),'r') >>> plt.close('all') See also -------- wafo.misc.good_bins numpy.histogram ''' x = np.atleast_1d(data) if bins is None: bins = np.ceil(4 * np.sqrt(np.sqrt(len(x)))) bin_, limits = np.histogram( data, bins=bins, normed=normed, weights=weights) limits.shape = (-1, 1) xx = limits.repeat(3, axis=1) xx.shape = (-1,) xx = xx[1:-1] bin_.shape = (-1, 1) yy = bin_.repeat(3, axis=1) # yy[0,0] = 0.0 # pdf yy[:, 0] = 0.0 # histogram yy.shape = (-1,) yy = np.hstack((yy, 0.0)) return plotbackend.plot(xx, yy, lintype, limits, limits * 0) def num2pistr(x, n=3): ''' Convert a scalar to a text string in fractions of pi if the numerator is less than 10 and not equal 0 and if the denominator is less than 10. Parameters ---------- x = a scalar n = maximum digits of precision. (default 3) Returns ------- xtxt = a text string in fractions of pi Example >>> import wafo.misc as wm >>> wm.num2pistr(np.pi*3/4)=='3\\pi/4' True ''' frac = fractions.Fraction.from_float(x / pi).limit_denominator(10000000) num = frac.numerator den = frac.denominator if (den < 10) and (num < 10) and (num != 0): dtxt = '' if abs(den) == 1 else '/%d' % den if abs(num) == 1: # % numerator ntxt = '-' if num == -1 else '' else: ntxt = '%d' % num xtxt = ntxt + r'\pi' + dtxt else: format = '%0.' + '%dg' % n # @ReservedAssignment xtxt = format % x return xtxt def fourier(data, t=None, T=None, m=None, n=None, method='trapz'): ''' Returns Fourier coefficients. Parameters ---------- data : array-like vector or matrix of row vectors with data points shape p x n. t : array-like vector with n values indexed from 1 to N. T : real scalar, (default T = t[-1]-t[0]) primitive period of signal, i.e., smallest period. m : scalar integer defines no of harmonics desired (default M = N) n : scalar integer no of data points (default len(t)) method : string integration method used Returns ------- a,b = Fourier coefficients size m x p FOURIER finds the coefficients for a Fourier series representation of the signal x(t) (given in digital form). It is assumed the signal is periodic over T. N is the number of data points, and M-1 is the number of coefficients. The signal can be estimated by using M-1 harmonics by: M-1 x[i] = 0.5*a[0] + sum (a[n]*c[n,i] + b[n]*s[n,i]) n=1 where c[n,i] = cos(2*pi*(n-1)*t[i]/T) s[n,i] = sin(2*pi*(n-1)*t[i]/T) Note that a[0] is the "dc value". Remaining values are a[1], a[2], ... , a[M-1]. Example ------- >>> import wafo.misc as wm >>> import numpy as np >>> T = 2*np.pi >>> t = np.linspace(0,4*T) >>> x = np.sin(t) >>> a, b = wm.fourier(x, t, T=T, m=5) >>> np.abs(a.ravel())<1e-12 array([ True, True, True, True, True], dtype=bool) >>> np.abs(b.ravel()-np.array([ 0., 4., 0., 0., 0.]))<1e-12 array([ True, True, True, True, True], dtype=bool) See also -------- fft ''' x = np.atleast_2d(data) p, n = x.shape if t is None: t = np.arange(n) else: t = np.atleast_1d(t) n = len(t) if n is None else n m = n if n is None else m T = t[-1] - t[0] if T is None else T if method.startswith('trapz'): intfun = trapz elif method.startswith('simp'): intfun = simps # Define the vectors for computing the Fourier coefficients t.shape = (1, -1) a = zeros((m, p)) b = zeros((m, p)) a[0] = intfun(x, t, axis=-1) # Compute M-1 more coefficients tmp = 2 * pi * t / T # tmp = 2*pi*(0:N-1).'/(N-1); for i in range(1, m): a[i] = intfun(x * cos(i * tmp), t, axis=-1) b[i] = intfun(x * sin(i * tmp), t, axis=-1) a = a / pi b = b / pi # Alternative: faster for large M, but gives different results than above. # nper = diff(t([1 end]))/T; %No of periods given # if nper == round(nper): # N1 = n/nper # else: # N1 = n # # # # Fourier coefficients by fft # Fcof1 = 2*ifft(x(1:N1,:),[],1); # Pcor = [1; exp(sqrt(-1)*(1:M-1).'*t(1))] # correction term to get # # the correct integration limits # Fcof = Fcof1(1:M,:).*Pcor(:,ones(1,P)); # a = real(Fcof(1:M,:)); # b = imag(Fcof(1:M,:)); return a, b def hyp2f1_taylor(a, b, c, z, tol=1e-13, itermax=500): a, b, c, z = np.broadcast_arrays(*np.atleast_1d(a, b, c, z)) shape = a.shape ak, bk, ck, zk = [d.ravel() for d in (a, b, c, z)] ajm1 = np.ones(ak.shape) bjm2 = 0.5 * np.ones(ak.shape) bjm1 = np.ones(ak.shape) hout = np.zeros(ak.shape) k0 = np.arange(len(ak)) for j in range(0, itermax): aj = ajm1 * (ak + j) * (bk + j) / (ck + j) * zk / (j + 1) bj = bjm1 + aj h, err = dea3(bjm2, bjm1, bj) k = np.flatnonzero(err > tol * np.abs(h)) hout[k0] = h if len(k) == 0: break k0 = k0[k] ak, bk, ck, zk = ak[k], bk[k], ck[k], zk[k] ajm1 = aj[k] bjm2 = bjm1[k] bjm1 = bj[k] else: warnings.warn(('Reached %d limit! \n' + '#%d values did not converge! Max error=%g') % (j, len(k), np.max(err))) return hout.reshape(shape) def hyp2f1(a, b, c, z, rho=0.5): e1 = gammaln(a) e2 = gammaln(b) e3 = gammaln(c) e4 = gammaln(b - a) e5 = gammaln(a - b) e6 = gammaln(c - a) e7 = gammaln(c - b) e8 = gammaln(c - a - b) e9 = gammaln(a + b - c) _cmab = c - a - b # ~(np.round(cmab) == cmab & cmab <= 0) if abs(z) <= rho: h = hyp2f1_taylor(a, b, c, z, 1e-15) elif abs(1 - z) <= rho: # % Require that |arg(1-z)| 10: break xjm2 = xjm1 xjm1 = xj else: warnings.warn('Reached %d limit' % j) return h def hygfz(A, B, C, Z): ''' Return hypergeometric function for a complex argument, F(a,b,c,z) Parameters ---------- a, b, c: parameters where c <> 0,-1,-2,... z :--- Complex argument ''' X = np.real(Z) Y = np.imag(Z) EPS = 1.0e-15 L0 = C == np.round(C) and C < 0.0e0 L1 = abs(1.0 - X) < EPS and Y == 0.0 and C - A - B <= 0.0 L2 = abs(Z + 1.0) < EPS and abs(C - A + B - 1.0) < EPS L3 = A == np.round(A) and A < 0.0 L4 = B == np.round(B) and B < 0.0 L5 = C - A == np.round(C - A) and C - A <= 0.0 L6 = C - B == np.round(C - B) and C - B <= 0.0 AA = A BB = B A0 = abs(Z) if (A0 > 0.95): EPS = 1.0e-8 PI = 3.141592653589793 EL = .5772156649015329 if (L0 or L1): # 'The hypergeometric series is divergent' return np.inf NM = 0 if (A0 == 0.0 or A == 0.0 or B == 0.0): ZHF = 1.0 elif (Z == 1.0 and C - A - B > 0.0): GC = gamma(C) GCAB = gamma(C - A - B) GCA = gamma(C - A) GCB = gamma(C - B) ZHF = GC * GCAB / (GCA * GCB) elif L2: G0 = sqrt(PI) * 2.0 ** (-A) G1 = gamma(C) G2 = gamma(1.0 + A / 2.0 - B) G3 = gamma(0.5 + 0.5 * A) ZHF = G0 * G1 / (G2 * G3) elif L3 or L4: if (L3): NM = int(np.round(abs(A))) if (L4): NM = int(np.round(abs(B))) ZHF = 1.0 ZR = 1.0 for K in range(NM): ZR = ZR * (A + K) * (B + K) / ((K + 1.) * (C + K)) * Z ZHF = ZHF + ZR elif L5 or L6: if (L5): NM = np.round(abs(C - A)) if (L6): NM = np.round(abs(C - B)) ZHF = 1.0 + 0j ZR = 1.0 + 0j for K in range(NM): ZR *= (C - A + K) * (C - B + K) / ((K + 1.) * (C + K)) * Z ZHF = ZHF + ZR ZHF = (1.0 - Z) ** (C - A - B) * ZHF elif (A0 <= 1.0): if (X < 0.0): Z1 = Z / (Z - 1.0) if (C > A and B < A and B > 0.0): A = BB B = AA ZC0 = 1.0 / ((1.0 - Z) ** A) ZHF = 1.0 + 0j ZR0 = 1.0 + 0j ZW = 0 for K in range(500): ZR0 *= (A + K) * (C - B + K) / ((K + 1.0) * (C + K)) * Z1 ZHF += ZR0 if (abs(ZHF - ZW) < abs(ZHF) * EPS): break ZW = ZHF ZHF = ZC0 * ZHF elif (A0 >= 0.90): ZW = 0.0 GM = 0.0 MCAB = np.round(C - A - B) if (abs(C - A - B - MCAB) < EPS): M = int(np.round(C - A - B)) GA = gamma(A) GB = gamma(B) GC = gamma(C) GAM = gamma(A + M) GBM = gamma(B + M) PA = psi(A) PB = psi(B) if (M != 0): GM = 1.0 for j in range(1, abs(M)): GM *= j RM = 1.0 for j in range(1, abs(M) + 1): # DO 35 J=1,abs(M) RM *= j ZF0 = 1.0 ZR0 = 1.0 ZR1 = 1.0 SP0 = 0.0 SP = 0.0 if (M >= 0): ZC0 = GM * GC / (GAM * GBM) ZC1 = -GC * (Z - 1.0) ** M / (GA * GB * RM) for K in range(1, M): ZR0 = ZR0 * \ (A + K - 1.) * (B + K - 1.) / \ (K * (K - M)) * (1. - Z) ZF0 = ZF0 + ZR0 for K in range(M): SP0 = SP0 + 1.0 / \ (A + K) + 1.0 / (B + K) - 1. / (K + 1.) ZF1 = PA + PB + SP0 + 2.0 * EL + np.log(1.0 - Z) for K in range(1, 501): SP = SP + \ (1.0 - A) / (K * (A + K - 1.0)) + ( 1.0 - B) / (K * (B + K - 1.0)) SM = 0.0 for J in range(1, M): SM += (1.0 - A) / ( (J + K) * (A + J + K - 1.0)) + \ 1.0 / (B + J + K - 1.0) ZP = PA + PB + 2.0 * EL + SP + SM + np.log(1.0 - Z) ZR1 = ZR1 * \ (A + M + K - 1.0) * (B + M + K - 1.0) / ( K * (M + K)) * (1.0 - Z) ZF1 = ZF1 + ZR1 * ZP if (abs(ZF1 - ZW) < abs(ZF1) * EPS): break ZW = ZF1 ZHF = ZF0 * ZC0 + ZF1 * ZC1 elif (M < 0): M = -M ZC0 = GM * GC / (GA * GB * (1.0 - Z) ** M) ZC1 = -(-1) ** M * GC / (GAM * GBM * RM) for K in range(1, M): ZR0 = ZR0 * \ (A - M + K - 1.0) * (B - M + K - 1.0) / ( K * (K - M)) * (1.0 - Z) ZF0 = ZF0 + ZR0 for K in range(1, M + 1): SP0 = SP0 + 1.0 / K ZF1 = PA + PB - SP0 + 2.0 * EL + np.log(1.0 - Z) for K in range(1, 501): SP = SP + \ (1.0 - A) / (K * (A + K - 1.0)) + ( 1.0 - B) / (K * (B + K - 1.0)) SM = 0.0 for J in range(1, M + 1): SM = SM + 1.0 / (J + K) ZP = PA + PB + 2.0 * EL + SP - SM + np.log(1.0 - Z) ZR1 = ZR1 * \ (A + K - 1.) * (B + K - 1.) / \ (K * (M + K)) * (1. - Z) ZF1 = ZF1 + ZR1 * ZP if (abs(ZF1 - ZW) < abs(ZF1) * EPS): break ZW = ZF1 ZHF = ZF0 * ZC0 + ZF1 * ZC1 else: GA = gamma(A) GB = gamma(B) GC = gamma(C) GCA = gamma(C - A) GCB = gamma(C - B) GCAB = gamma(C - A - B) GABC = gamma(A + B - C) ZC0 = GC * GCAB / (GCA * GCB) ZC1 = GC * GABC / (GA * GB) * (1.0 - Z) ** (C - A - B) ZHF = 0 + 0j ZR0 = ZC0 ZR1 = ZC1 for K in range(1, 501): ZR0 = ZR0 * \ (A + K - 1.) * (B + K - 1.) / \ (K * (A + B - C + K)) * (1. - Z) ZR1 = ZR1 * \ (C - A + K - 1.0) * (C - B + K - 1.0) / ( K * (C - A - B + K)) * (1.0 - Z) ZHF = ZHF + ZR0 + ZR1 if (abs(ZHF - ZW) < abs(ZHF) * EPS): break ZW = ZHF ZHF = ZHF + ZC0 + ZC1 else: ZW = 0.0 Z00 = 1.0 # + 0j if (C - A < A and C - B < B): Z00 = (1.0 - Z) ** (C - A - B) A = C - A B = C - B ZHF = 1.0 ZR = 1.0 for K in range(1, 501): ZR = ZR * \ (A + K - 1.0) * (B + K - 1.0) / (K * (C + K - 1.0)) * Z ZHF = ZHF + ZR if (abs(ZHF - ZW) <= abs(ZHF) * EPS): break ZW = ZHF ZHF = Z00 * ZHF elif (A0 > 1.0): MAB = np.round(A - B) if (abs(A - B - MAB) < EPS and A0 <= 1.1): B = B + EPS if (abs(A - B - MAB) > EPS): GA = gamma(A) GB = gamma(B) GC = gamma(C) GAB = gamma(A - B) GBA = gamma(B - A) GCA = gamma(C - A) GCB = gamma(C - B) ZC0 = GC * GBA / (GCA * GB * (-Z) ** A) ZC1 = GC * GAB / (GCB * GA * (-Z) ** B) ZR0 = ZC0 ZR1 = ZC1 ZHF = 0.0 + 0j for K in range(1, 501): ZR0 = ZR0 * (A + K - 1.0) * (A - C + K) / ((A - B + K) * K * Z) ZR1 = ZR1 * (B + K - 1.0) * (B - C + K) / ((B - A + K) * K * Z) ZHF = ZHF + ZR0 + ZR1 if (abs((ZHF - ZW) / ZHF) <= EPS): break ZW = ZHF ZHF = ZHF + ZC0 + ZC1 else: if (A - B < 0.0): A = BB B = AA CA = C - A CB = C - B NCA = np.round(CA) NCB = np.round(CB) if (abs(CA - NCA) < EPS or abs(CB - NCB) < EPS): C = C + EPS GA = gamma(A) GC = gamma(C) GCB = gamma(C - B) PA = psi(A) PCA = psi(C - A) PAC = psi(A - C) MAB = np.round(A - B + EPS) ZC0 = GC / (GA * (-Z) ** B) GM = gamma(A - B) ZF0 = GM / GCB * ZC0 ZR = ZC0 for K in range(1, MAB): ZR = ZR * (B + K - 1.0) / (K * Z) T0 = A - B - K G0 = gamma(T0) GCBK = gamma(C - B - K) ZF0 = ZF0 + ZR * G0 / GCBK if (MAB == 0): ZF0 = 0.0 + 0j ZC1 = GC / (GA * GCB * (-Z) ** A) SP = -2.0 * EL - PA - PCA for J in range(1, MAB + 1): SP = SP + 1.0 / J ZP0 = SP + np.log(-Z) SQ = 1.0 for J in range(1, MAB + 1): SQ = SQ * (B + J - 1.0) * (B - C + J) / J ZF1 = (SQ * ZP0) * ZC1 ZR = ZC1 RK1 = 1.0 SJ1 = 0.0 W0 = 0.0 for K in range(1, 10001): ZR = ZR / Z RK1 = RK1 * (B + K - 1.0) * (B - C + K) / (K * K) RK2 = RK1 for J in range(K + 1, K + MAB + 1): RK2 = RK2 * (B + J - 1.0) * (B - C + J) / J SJ1 = SJ1 + \ (A - 1.0) / (K * (A + K - 1.0)) + \ (A - C - 1.0) / (K * (A - C + K - 1.0)) SJ2 = SJ1 for J in range(K + 1, K + MAB + 1): SJ2 = SJ2 + 1.0 / J ZP = -2.0 * EL - PA - PAC + SJ2 - 1.0 / \ (K + A - C) - PI / np.tan(PI * (K + A - C)) + np.log(-Z) ZF1 = ZF1 + RK2 * ZR * ZP WS = abs(ZF1) if (abs((WS - W0) / WS) < EPS): break W0 = WS ZHF = ZF0 + ZF1 A = AA B = BB if (K > 150): warnings.warn('Warning! You should check the accuracy') return ZHF # def hypgf(a, b, c, x, abseps=0, releps=1e-13, kmax=10000): # '''HYPGF Hypergeometric function F(a,b,c,x) # # CALL: [y ,abserr] = hypgf(a,b,c,x,abseps,releps) # # y = F(a,b,c,x) # abserr = absolute error estimate # a,b,c,x = input parameters # abseps = requested absolute error # releps = requested relative error # # HYPGF calculates one solution to Gauss's hypergeometric differential # equation: # # x*(1-x)Y''(x)+[c-(a+b+1)*x]*Y'(x)-a*b*Y(x) = 0 # where # F(a,b,c,x) = Y1(x) = 1 + a*b*x/c + a*(a+1)*b*(b+1)*x^2/(c*(c+1))+.... # # # Many elementary functions are special cases of F(a,b,c,x): # 1/(1-x) = F(1,1,1,x) = F(1,b,b,x) = F(a,1,a,x) # (1+x)^n = F(-n,b,b,-x) # atan(x) = x*F(.5,1,1.5,-x^2) # asin(x) = x*F(.5,.5,1.5,x^2) # log(x) = x*F(1,1,2,-x) # log(1+x)-log(1-x) = 2*x*F(.5,1,1.5,x^2) # # NOTE: only real x, abs(x) < 1 and c~=0,-1,-2,... are allowed. # # Examples: # x = linspace(-.99,.99)'; # [Sn1,err1] = hypgf(1,1,1,x) # plot(x,abs(Sn1-1./(1-x)),'b',x,err1,'r'),set(gca,'yscale','log') # [Sn2,err2] = hypgf(.5,.5,1.5,x.^2); # plot(x,abs(x.*Sn2-asin(x)),'b',x,abs(x.*err2),'r') # set(gca,'yscale','log') # # # Reference: # --------- # Kreyszig, Erwin (1988) # Advanced engineering mathematics # John Wiley & Sons, sixth edition, pp 204. # ''' # csize = common_shape(x, a, b, c) # kmin = 2 # fsum = np.zeros(csize) # delta = np.zeros(csize) # err = np.zeros(csize) # # ok = ~((np.round(c) == c & c <= 0) | np.abs(x) > 1) # if np.any(~ok): # warnings.warn('HYPGF', 'Illegal input: c = 0,-1,-2,... or abs(x)>1') # fsum[~ok] = np.NaN # err[~ok] = np.NaN # # k0=find(ok & abs(x)==1); # if any(k0) # cmab = c(k0)-a(k0)-b(k0); # fsum(k0) = exp(gammaln(c(k0))+gammaln(cmab)-... # gammaln(c(k0)-a(k0))-gammaln(c(k0)-b(k0))); # err(k0) = eps; # k00 = find(real(cmab)<=0); # if any(k00) # err(k0(k00)) = nan; # fsum(k0(k00)) = nan; # end # end # k=find(ok & abs(x)<1); # if any(k), # delta(k) = ones(size(k)); # fsum(k) = delta(k); # # k1 = k; # E = cell(1,3); # E{3} = fsum(k); # converge = 'n'; # for ix=0:Kmax-1, # delta(k1) = delta(k1).*((a(k1)+ix)./(ix+1)).*((b(k1)+ix)./(c(k1)+ ix)).*x(k1); @IgnorePep8 # fsum(k1) = fsum(k1)+delta(k1); # # E(1:2) = E(2:3); # E{3} = fsum(k1); # # if ix>Kmin # if useDEA, # [Sn, err(k1)] = dea3(E{:}); # k00 = find((abs(err(k1))) <= max(absEps,abs(relEps.*fsum(k1)))); # if any(k00) # fsum(k1(k00)) = Sn(k00); # end # if (ix==Kmax-1) # fsum(k1) = Sn; # end # k0 = (find((abs(err(k1))) > max(absEps,abs(relEps.*fsum(k1))))); # if any(k0),% compute more terms # %nk=length(k0);%# of values we have to compute again # E{2} = E{2}(k0); # E{3} = E{3}(k0); # else # converge='y'; # break; # end # else # err(k1) = 10*abs(delta(k1)); # k0 = (find((abs(err(k1))) > max(absEps,abs(relEps.* ... # fsum(k1))))); # if any(k0),% compute more terms # %nk=length(k0);%# of values we have to compute again # else # converge='y'; # break; # end # end # k1 = k1(k0); # end # end # if ~strncmpi(converge,'y',1) # disp(sprintf('#%d values did not converge',length(k1))) # end # end # %ix # return def _test_find_cross(): t = findcross([0, 0, 1, -1, 1], 0) # @UnusedVariable def real_main0(): x = np.arange(10000) y = np.arange(100).reshape(-1, 1) np.broadcast_arrays(x, y, x, x, x, x, x, x, x, x) def real_main(): x = np.arange(100000) y = np.arange(100).reshape(-1, 1) common_shape(x, y, x, x, x, x, x, x, x, x) def profile_main1(): # This is the main function for profiling # We've renamed our original main() above to real_main() import cProfile import pstats prof = cProfile.Profile() prof = prof.runctx("real_main()", globals(), locals()) print("
")
    stats = pstats.Stats(prof)
    stats.sort_stats("time")  # Or cumulative
    stats.print_stats(80)  # 80 = how many to print
    # The rest is optional.
    # stats.print_callees()
    # stats.print_callers()
    print("
") main = profile_main1 def test_docstrings(): # np.set_printoptions(precision=6) import doctest print('Testing docstrings in %s' % __file__) doctest.testmod(optionflags=doctest.NORMALIZE_WHITESPACE) def test_hyp2f1(): # 1/(1-x) = F(1,1,1,x) = F(1,b,b,x) = F(a,1,a,x) # (1+x)^n = F(-n,b,b,-x) # atan(x) = x*F(.5,1,1.5,-x^2) # asin(x) = x*F(.5,.5,1.5,x^2) # log(x) = x*F(1,1,2,-x) # log(1+x)-log(1-x) = 2*x*F(.5,1,1.5,x^2) x = linspace(0., .7, 20) y = hyp2f1_taylor(-1, -4, 1, .9) _y2 = hygfz(-1, -4, 1, .9) _y3 = hygfz(5, -300, 10, 0.5) _y4 = hyp2f1_taylor(5, -300, 10, 0.5) # y = hyp2f1(0.1, 0.2, 0.3, 0.5) # y = hyp2f1(1, 1.5, 3, -4 +3j) # y = hyp2f1(5, 7.5, 2.5, 5) # fun = lambda x : 1./(1-x) # x = .99 # y = hyp2f1(1,1,1,x) # print(y-fun(x)) # plt = plotbackend plt.interactive(False) plt.semilogy(x, np.abs(y - 1. / (1 - x)) + 1e-20, 'r') plt.show() if __name__ == "__main__": test_docstrings() # test_hyp2f1()