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3521 lines
104 KiB
Python
3521 lines
104 KiB
Python
'''
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Misc
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'''
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from __future__ import division
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import collections
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import sys
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import fractions
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import numpy as np
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from numpy import (
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meshgrid,
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abs, amax, any, logical_and, arange, linspace, atleast_1d,
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asarray, ceil, floor, frexp, hypot,
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sqrt, arctan2, sin, cos, exp, log, log1p, mod, diff,
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finfo, inf, pi, interp, isnan, isscalar, zeros, ones, linalg,
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r_, sign, unique, hstack, vstack, nonzero, where, extract)
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from scipy.special import gammaln, gamma, psi
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from scipy.integrate import trapz, simps
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import warnings
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from time import strftime, gmtime
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from plotbackend import plotbackend
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from collections import OrderedDict
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try:
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import c_library as clib # @UnresolvedImport
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except ImportError:
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warnings.warn('c_library not found. Check its compilation.')
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clib = None
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floatinfo = finfo(float)
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_TINY = np.finfo(float).tiny
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_EPS = np.finfo(float).eps
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__all__ = ['now', 'spaceline', 'narg_smallest', 'args_flat', 'is_numlike',
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'JITImport', 'DotDict', 'Bunch', 'printf', 'sub_dict_select',
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'parse_kwargs', 'detrendma', 'ecross', 'findcross', 'findextrema',
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'findpeaks', 'findrfc', 'rfcfilter', 'findtp', 'findtc',
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'findoutliers', 'common_shape', 'argsreduce', 'stirlerr',
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'getshipchar',
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'betaloge', 'gravity', 'nextpow2', 'discretize', 'polar2cart',
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'cart2polar', 'meshgrid', 'ndgrid', 'trangood', 'tranproc',
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'plot_histgrm', 'num2pistr', 'test_docstrings', 'lazywhere',
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'piecewise',
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'valarray']
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def valarray(shape, value=np.NaN, typecode=None):
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"""Return an array of all value.
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"""
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if typecode is None:
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typecode = bool
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out = ones(shape, dtype=typecode) * value
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if not isinstance(out, np.ndarray):
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out = asarray(out)
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return out
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def piecewise(xi, condlist, funclist, fill_value=0.0, args=(), **kw):
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"""
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Evaluate a piecewise-defined function.
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Given a set of conditions and corresponding functions, evaluate each
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function on the input data wherever its condition is true.
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Parameters
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----------
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xi : tuple
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input arguments to the functions in funclist, i.e., (x0, x1,...., xn)
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condlist : list of bool arrays
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Each boolean array corresponds to a function in `funclist`. Wherever
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`condlist[i]` is True, `funclist[i](x0,x1,...,xn)` is used as the
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output value. Each boolean array in `condlist` selects a piece of `xi`,
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and should therefore be of the same shape as `xi`.
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The length of `condlist` must correspond to that of `funclist`.
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If one extra function is given, i.e. if
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``len(funclist) - len(condlist) == 1``, then that extra function
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is the default value, used wherever all conditions are false.
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funclist : list of callables, f(*(xi + args), **kw), or scalars
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Each function is evaluated over `x` wherever its corresponding
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condition is True. It should take an array as input and give an array
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or a scalar value as output. If, instead of a callable,
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a scalar is provided then a constant function (``lambda x: scalar``) is
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assumed.
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fill_value : scalar
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fill value for out of range values. Default 0.
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args : tuple, optional
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Any further arguments given here passed to the functions
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upon execution, i.e., if called ``piecewise(..., ..., args=(1, 'a'))``,
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then each function is called as ``f(x0, x1,..., xn, 1, 'a')``.
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kw : dict, optional
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Keyword arguments used in calling `piecewise` are passed to the
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functions upon execution, i.e., if called
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``piecewise(..., ..., lambda=1)``, then each function is called as
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``f(x0, x1,..., xn, lambda=1)``.
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Returns
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-------
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out : ndarray
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The output is the same shape and type as x and is found by
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calling the functions in `funclist` on the appropriate portions of `x`,
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as defined by the boolean arrays in `condlist`. Portions not covered
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by any condition have undefined values.
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See Also
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--------
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choose, select, where
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Notes
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-----
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This is similar to choose or select, except that functions are
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evaluated on elements of `xi` that satisfy the corresponding condition from
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`condlist`.
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The result is::
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|--
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|funclist[0](x0[condlist[0]],x1[condlist[0]],...,xn[condlist[0]])
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out = |funclist[1](x0[condlist[1]],x1[condlist[1]],...,xn[condlist[1]])
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|...
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|funclist[n2](x0[condlist[n2]],x1[condlist[n2]],...,xn[condlist[n2]])
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|--
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Examples
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--------
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Define the sigma function, which is -1 for ``x < 0`` and +1 for ``x >= 0``.
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>>> x = np.linspace(-2.5, 2.5, 6)
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>>> piecewise(x, [x < 0, x >= 0], [-1, 1])
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array([-1., -1., -1., 1., 1., 1.])
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Define the absolute value, which is ``-x`` for ``x <0`` and ``x`` for
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``x >= 0``.
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>>> piecewise((x,), [x < 0, x >= 0], [lambda x: -x, lambda x: x])
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array([ 2.5, 1.5, 0.5, 0.5, 1.5, 2.5])
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Define the absolute value, which is ``-x*y`` for ``x*y <0`` and ``x*y`` for
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``x*y >= 0``
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>>> X, Y = np.meshgrid(x, x)
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>>> piecewise((X, Y), [X * Y < 0, ],
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... [lambda x, y: -x * y, lambda x, y: x * y])
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array([[ 6.25, 3.75, 1.25, 1.25, 3.75, 6.25],
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[ 3.75, 2.25, 0.75, 0.75, 2.25, 3.75],
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[ 1.25, 0.75, 0.25, 0.25, 0.75, 1.25],
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[ 1.25, 0.75, 0.25, 0.25, 0.75, 1.25],
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[ 3.75, 2.25, 0.75, 0.75, 2.25, 3.75],
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[ 6.25, 3.75, 1.25, 1.25, 3.75, 6.25]])
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"""
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def otherwise_condition(condlist):
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return ~np.logical_or.reduce(condlist, axis=0)
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def check_shapes(condlist, funclist):
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nc, nf = len(condlist), len(funclist)
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if nc not in [nf-1, nf]:
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raise ValueError("function list and condition list" +
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" must be the same length")
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check_shapes(condlist, funclist)
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if not isinstance(xi, tuple):
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xi = (xi,)
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condlist = np.broadcast_arrays(*condlist)
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if len(condlist) == len(funclist)-1:
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condlist.append(otherwise_condition(condlist))
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arrays = np.broadcast_arrays(*xi)
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dtype = np.result_type(*arrays)
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out = valarray(arrays[0].shape, fill_value, dtype)
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for cond, func in zip(condlist, funclist):
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if isinstance(func, collections.Callable):
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temp = tuple(np.extract(cond, arr) for arr in arrays) + args
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np.place(out, cond, func(*temp, **kw))
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else: # func is a scalar value
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np.place(out, cond, func)
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return out
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def lazywhere(cond, arrays, f, fillvalue=None, f2=None):
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"""
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np.where(cond, x, fillvalue) always evaluates x even where cond is False.
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This one only evaluates f(arr1[cond], arr2[cond], ...).
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For example,
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>>> a, b = np.array([1, 2, 3, 4]), np.array([5, 6, 7, 8])
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>>> def f(a, b):
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... return a*b
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>>> lazywhere(a > 2, (a, b), f, np.nan)
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array([ nan, nan, 21., 32.])
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Notice it assumes that all `arrays` are of the same shape, or can be
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broadcasted together.
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"""
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if fillvalue is None:
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if f2 is None:
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raise ValueError("One of (fillvalue, f2) must be given.")
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else:
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fillvalue = np.nan
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else:
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if f2 is not None:
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raise ValueError("Only one of (fillvalue, f2) can be given.")
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arrays = np.broadcast_arrays(*arrays)
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temp = tuple(np.extract(cond, arr) for arr in arrays)
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out = valarray(np.shape(arrays[0]), value=fillvalue)
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np.place(out, cond, f(*temp))
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if f2 is not None:
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temp = tuple(np.extract(~cond, arr) for arr in arrays)
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np.place(out, ~cond, f2(*temp))
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return out
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def rotation_matrix(heading, pitch, roll):
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'''
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Examples
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>>> import numpy as np
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>>> rotation_matrix(heading=0, pitch=0, roll=0)
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array([[ 1., 0., 0.],
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[ 0., 1., 0.],
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[ 0., 0., 1.]])
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>>> np.all(np.abs(rotation_matrix(heading=180, pitch=0, roll=0)-
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... np.array([[ -1.000000e+00, -1.224647e-16, 0.000000e+00],
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... [ 1.224647e-16, -1.000000e+00, 0.000000e+00],
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... [ -0.000000e+00, 0.000000e+00, 1.000000e+00]]))<1e-7)
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True
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>>> np.all(np.abs(rotation_matrix(heading=0, pitch=180, roll=0)-
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... np.array([[ -1.000000e+00, 0.000000e+00, 1.224647e-16],
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... [ -0.000000e+00, 1.000000e+00, 0.000000e+00],
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... [ -1.224647e-16, -0.000000e+00, -1.000000e+00]]))<1e-7)
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True
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>>> np.all(np.abs(rotation_matrix(heading=0, pitch=0, roll=180)-
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... np.array([[ 1.000000e+00, 0.000000e+00, 0.000000e+00],
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... [ 0.000000e+00, -1.000000e+00, -1.224647e-16],
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... [ -0.000000e+00, 1.224647e-16, -1.000000e+00]]))<1e-7)
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True
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'''
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data = np.diag(np.ones(3)) # No transform if H=P=R=0
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if heading != 0 or pitch != 0 or roll != 0:
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deg2rad = np.pi / 180
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H = heading * deg2rad
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P = pitch * deg2rad
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R = roll * deg2rad # Convert to radians
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data.put(0, cos(H) * cos(P))
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data.put(1, cos(H) * sin(P) * sin(R) - sin(H) * cos(R))
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data.put(2, cos(H) * sin(P) * cos(R) + sin(H) * sin(R))
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data.put(3, sin(H) * cos(P))
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data.put(4, sin(H) * sin(P) * sin(R) + cos(H) * cos(R))
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data.put(5, sin(H) * sin(P) * cos(R) - cos(H) * sin(R))
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data.put(6, -sin(P))
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data.put(7, cos(P) * sin(R))
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data.put(8, cos(P) * cos(R))
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return data
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def rotate(x, y, z, heading=0, pitch=0, roll=0):
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rot_param = rotation_matrix(heading, pitch, roll).ravel()
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X = x * rot_param[0] + y * rot_param[1] + z * rot_param[2]
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Y = x * rot_param[3] + y * rot_param[4] + z * rot_param[5]
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Z = x * rot_param[6] + y * rot_param[7] + z * rot_param[8]
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return X, Y, Z
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def rotate_2d(x, y, angle_deg):
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'''
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Rotate points in the xy cartesian plane counter clockwise
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Examples
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--------
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>>> rotate_2d(x=1, y=0, angle_deg=0)
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(1.0, 0.0)
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>>> rotate_2d(x=1, y=0, angle_deg=90)
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(6.123233995736766e-17, 1.0)
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>>> rotate_2d(x=1, y=0, angle_deg=180)
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(-1.0, 1.2246467991473532e-16)
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>>> rotate_2d(x=1, y=0, angle_deg=360)
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(1.0, -2.4492935982947064e-16)
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'''
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angle_rad = angle_deg * pi / 180
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ch = cos(angle_rad)
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sh = sin(angle_rad)
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return ch * x - sh * y, sh * x + ch * y
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def now(show_seconds=True):
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'''
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Return current date and time as a string
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'''
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if show_seconds:
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return strftime("%a, %d %b %Y %H:%M:%S", gmtime())
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else:
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return strftime("%a, %d %b %Y %H:%M", gmtime())
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|
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def _assert(cond, txt=''):
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if not cond:
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raise ValueError(txt)
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|
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def spaceline(start_point, stop_point, num=10):
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'''Return `num` evenly spaced points between the start and stop points.
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Parameters
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----------
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start_point : vector, size=3
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The starting point of the sequence.
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stop_point : vector, size=3
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The end point of the sequence.
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num : int, optional
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Number of samples to generate. Default is 10.
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Returns
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-------
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space_points : ndarray of shape n x 3
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There are `num` equally spaced points in the closed interval
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``[start, stop]``.
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See Also
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--------
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linspace : similar to spaceline, but in 1D.
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arange : Similiar to `linspace`, but uses a step size (instead of the
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number of samples).
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logspace : Samples uniformly distributed in log space.
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Example
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-------
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>>> import wafo.misc as pm
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>>> pm.spaceline((2,0,0), (3,0,0), num=5)
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array([[ 2. , 0. , 0. ],
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[ 2.25, 0. , 0. ],
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[ 2.5 , 0. , 0. ],
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[ 2.75, 0. , 0. ],
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[ 3. , 0. , 0. ]])
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'''
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num = int(num)
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e1, e2 = np.atleast_1d(start_point, stop_point)
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e2m1 = e2 - e1
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length = np.sqrt((e2m1 ** 2).sum())
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# length = sqrt((E2[0]-E1(1))^2 + (E2(2)-E1(2))^2 + (E2(3)-E1(3))^2)
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C = e2m1 / length
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delta = length / float(num - 1)
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return np.array([e1 + n * delta * C for n in range(num)])
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|
|
|
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def narg_smallest(n, arr):
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''' Return the n smallest indicis to the arr
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'''
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return np.array(arr).argsort()[:n]
|
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|
|
|
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def args_flat(*args):
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'''
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Return x,y,z positions as a N x 3 ndarray
|
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|
Parameters
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----------
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pos : array-like, shape N x 3
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[x,y,z] positions
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or
|
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x,y,z : array-like
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[x,y,z] positions
|
|
|
|
Returns
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|
------
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pos : ndarray, shape N x 3
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[x,y,z] positions
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common_shape : None or tuple
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common shape of x, y and z variables if given as triple input.
|
|
|
|
Examples
|
|
--------
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>>> x = [1,2,3]
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>>> pos, c_shape =args_flat(x,2,3)
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>>> pos
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array([[1, 2, 3],
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[2, 2, 3],
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[3, 2, 3]])
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>>> c_shape
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(3,)
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>>> pos1, c_shape1 = args_flat([1,2,3])
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>>> pos1
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array([[1, 2, 3]])
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>>> c_shape1 is None
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True
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>>> pos1, c_shape1 = args_flat(1,2,3)
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>>> pos1
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array([[1, 2, 3]])
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>>> c_shape1
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()
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>>> pos1, c_shape1 = args_flat([1],2,3)
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>>> pos1
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array([[1, 2, 3]])
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>>> c_shape1
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(1,)
|
|
|
|
'''
|
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nargin = len(args)
|
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|
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if (nargin == 1): # pos
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pos = np.atleast_2d(args[0])
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_assert((pos.shape[1] == 3) and (pos.ndim == 2),
|
|
'POS array must be of shape N x 3!')
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return pos, None
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elif nargin == 3:
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x, y, z = np.broadcast_arrays(*args[:3])
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c_shape = x.shape
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return np.vstack((x.ravel(), y.ravel(), z.ravel())).T, c_shape
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else:
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raise ValueError('Number of arguments must be 1 or 3!')
|
|
|
|
|
|
def index2sub(shape, index, order='C'):
|
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'''
|
|
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)
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|
>>> a = np.arange(np.prod(shape)).reshape(shape)
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>>> order = 'C'
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|
>>> a[1, 2, 3]
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23
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>>> i = sub2index(shape, 1, 2, 3, order=order)
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|
>>> a.ravel(order)[i]
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|
23
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>>> index2sub(shape, i, order=order)
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(1, 2, 3)
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|
|
|
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
|
|
<http://lists.gnu.org/archive/html/octave-maintainers/2011-09/pdfK0uKOST642.pdf>
|
|
'''
|
|
|
|
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)|<pi
|
|
h = exp(e3 + e8 - e6 - e7) * \
|
|
hyp2f1_taylor(a, b, a + b - c, 1 - z, 1e-15) \
|
|
+ (1 - z) ** (c - a - b) * exp(e3 + e9 - e1 - e2) \
|
|
* hyp2f1_taylor(c - a, c - b, c - a - b + 1, 1 - z, 1e-15)
|
|
elif abs(z / (z - 1)) <= rho:
|
|
h = (1 - z) ** (-a) \
|
|
* hyp2f1_taylor(a, c - b, c, (z / (z - 1)), 1e-15)
|
|
elif abs(1 / z) <= rho: # % Require that |arg(z)|<pi and |arg(1-z)|<pi
|
|
h = (-z + 0j) ** (-a) * exp(e3 + e4 - e2 - e6) \
|
|
* hyp2f1_taylor(a, a - c + 1, a - b + 1, 1. / z, 1e-15) \
|
|
+ (-z + 0j) ** (-b) * exp(e3 + e5 - e1 - e7) \
|
|
* hyp2f1_taylor(b - c + 1, b, b - a + 1, (1. / z), 1e-15)
|
|
elif abs(1. / (1 - z)) <= rho: # % Require that |arg(1-z)|<pi
|
|
h = (1 - z) ** (-a) * exp(e3 + e4 - e2 - e6) \
|
|
* hyp2f1_taylor(a, c - b, a - b + 1, (1. / (1 - z)), 1e-15)\
|
|
+ (1 - z) ** (-b) * exp(e3 + e5 - e1 - e7) \
|
|
* hyp2f1_taylor(b, c - a, b - a + 1, (1. / (1 - z)), 1e-15)
|
|
elif abs(1 - 1 / z) < rho: # % Require that |arg(z)|<pi and |arg(1-z)|<pi
|
|
h = z ** (-a) * exp(e3 + e8 - e6 - e7) \
|
|
* hyp2f1_taylor(a, a - c + 1, a + b - c + 1, (1 - 1 / z), 1e-15) \
|
|
+ z ** (a - c) * (1 - z) ** (c - a - b) * exp(e3 + e9 - e1 - e2) \
|
|
* hyp2f1_taylor(c - a, 1 - a, c - a - b + 1, (1 - 1 / z), 1e-15)
|
|
else:
|
|
warnings.warn('Another method is needed')
|
|
return h
|
|
|
|
|
|
def hyp2f1_wrong(a, b, c, z, tol=1e-13, itermax=500):
|
|
ajm1 = 0
|
|
bjm1 = 1
|
|
cjm1 = 1
|
|
xjm1 = np.ones(np.shape(c + a * b * z))
|
|
xjm2 = 2 * np.ones(xjm1.shape)
|
|
|
|
for j in range(1, itermax):
|
|
aj = (ajm1 + bjm1) * j * (c + j - 1)
|
|
bj = bjm1 * (a + j - 1) * (b + j - 1) * z
|
|
cj = cjm1 * j * (c + j - 1)
|
|
if np.any((aj == np.inf) | (bj == np.inf) | (cj == np.inf)):
|
|
break
|
|
xj = (aj + bj) / cj
|
|
h, err = dea3(xjm2, xjm1, xj)
|
|
if np.all(err <= tol * np.abs(h)) and j > 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("<pre>")
|
|
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("</pre>")
|
|
|
|
|
|
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()
|