''' Misc ''' from __future__ import absolute_import, division import sys import fractions import numpy as np from numpy import ( meshgrid, abs, amax, any, logical_and, arange, linspace, atleast_1d, asarray, ceil, floor, frexp, hypot, sqrt, arctan2, sin, cos, exp, log, log1p, mod, diff, finfo, inf, pi, interp, isscalar, zeros, ones, linalg, r_, sign, unique, hstack, vstack, nonzero, where, extract) from scipy.special import gammaln from scipy.integrate import trapz, simps import warnings from time import strftime, gmtime from numdifftools.extrapolation import dea3 # @UnusedImport from wafo.plotbackend import plotbackend from collections import Callable try: from wafo import c_library as clib # @UnresolvedImport except ImportError: warnings.warn('c_library not found. Check its compilation.') clib = None floatinfo = finfo(float) _TINY = np.finfo(float).tiny _EPS = np.finfo(float).eps __all__ = ['now', 'spaceline', 'narg_smallest', 'args_flat', 'is_numlike', 'JITImport', 'DotDict', 'Bunch', 'printf', 'sub_dict_select', 'parse_kwargs', 'detrendma', 'ecross', 'findcross', 'findextrema', 'findpeaks', 'findrfc', 'rfcfilter', 'findtp', 'findtc', 'findoutliers', 'common_shape', 'argsreduce', 'stirlerr', 'getshipchar', 'dea3', 'betaloge', 'gravity', 'nextpow2', 'discretize', 'polar2cart', 'cart2polar', 'meshgrid', 'ndgrid', 'trangood', 'tranproc', 'plot_histgrm', 'num2pistr', 'test_docstrings', 'lazywhere', 'piecewise', 'valarray'] def valarray(shape, value=np.NaN, typecode=None): """Return an array of all value. """ if typecode is None: typecode = bool out = ones(shape, dtype=typecode) * value if not isinstance(out, np.ndarray): out = asarray(out) return out def piecewise(condlist, funclist, xi=None, fill_value=0.0, args=(), **kw): """ Evaluate a piecewise-defined function. Given a set of conditions and corresponding functions, evaluate each function on the input data wherever its condition is true. Parameters ---------- condlist : list of bool arrays Each boolean array corresponds to a function in `funclist`. Wherever `condlist[i]` is True, `funclist[i](x0,x1,...,xn)` is used as the output value. Each boolean array in `condlist` selects a piece of `xi`, and should therefore be of the same shape as `xi`. The length of `condlist` must correspond to that of `funclist`. If one extra function is given, i.e. if ``len(funclist) - len(condlist) == 1``, then that extra function is the default value, used wherever all conditions are false. funclist : list of callables, f(*(xi + args), **kw), or scalars Each function is evaluated over `x` wherever its corresponding condition is True. It should take an array as input and give an array or a scalar value as output. If, instead of a callable, a scalar is provided then a constant function (``lambda x: scalar``) is assumed. xi : tuple input arguments to the functions in funclist, i.e., (x0, x1,...., xn) fill_value : scalar fill value for out of range values. Default 0. args : tuple, optional Any further arguments given here passed to the functions upon execution, i.e., if called ``piecewise(..., ..., args=(1, 'a'))``, then each function is called as ``f(x0, x1,..., xn, 1, 'a')``. kw : dict, optional Keyword arguments used in calling `piecewise` are passed to the functions upon execution, i.e., if called ``piecewise(..., ..., lambda=1)``, then each function is called as ``f(x0, x1,..., xn, lambda=1)``. Returns ------- out : ndarray The output is the same shape and type as x and is found by calling the functions in `funclist` on the appropriate portions of `x`, as defined by the boolean arrays in `condlist`. Portions not covered by any condition have undefined values. See Also -------- choose, select, where Notes ----- This is similar to choose or select, except that functions are evaluated on elements of `xi` that satisfy the corresponding condition from `condlist`. The result is:: |-- |funclist[0](x0[condlist[0]],x1[condlist[0]],...,xn[condlist[0]]) out = |funclist[1](x0[condlist[1]],x1[condlist[1]],...,xn[condlist[1]]) |... |funclist[n2](x0[condlist[n2]],x1[condlist[n2]],...,xn[condlist[n2]]) |-- Examples -------- Define the sigma function, which is -1 for ``x < 0`` and +1 for ``x >= 0``. >>> x = np.linspace(-2.5, 2.5, 6) >>> piecewise([x < 0, x >= 0], [-1, 1]) array([-1., -1., -1., 1., 1., 1.]) Define the absolute value, which is ``-x`` for ``x <0`` and ``x`` for ``x >= 0``. >>> piecewise([x < 0, x >= 0], [lambda x: -x, lambda x: x], xi=(x,)) array([ 2.5, 1.5, 0.5, 0.5, 1.5, 2.5]) Define the absolute value, which is ``-x*y`` for ``x*y <0`` and ``x*y`` for ``x*y >= 0`` >>> X, Y = np.meshgrid(x, x) >>> piecewise([X * Y < 0, ], [lambda x, y: -x * y, lambda x, y: x * y], ... xi=(X, Y)) array([[ 6.25, 3.75, 1.25, 1.25, 3.75, 6.25], [ 3.75, 2.25, 0.75, 0.75, 2.25, 3.75], [ 1.25, 0.75, 0.25, 0.25, 0.75, 1.25], [ 1.25, 0.75, 0.25, 0.25, 0.75, 1.25], [ 3.75, 2.25, 0.75, 0.75, 2.25, 3.75], [ 6.25, 3.75, 1.25, 1.25, 3.75, 6.25]]) """ def otherwise_condition(condlist): return ~np.logical_or.reduce(condlist, axis=0) def check_shapes(condlist, funclist): nc, nf = len(condlist), len(funclist) if nc not in [nf-1, nf]: raise ValueError("function list and condition list" + " must be the same length") check_shapes(condlist, funclist) condlist = np.broadcast_arrays(*condlist) if len(condlist) == len(funclist)-1: condlist.append(otherwise_condition(condlist)) if xi is None: arrays = () dtype = np.result_type(*funclist) shape = condlist[0].shape else: if not isinstance(xi, tuple): xi = (xi,) arrays = np.broadcast_arrays(*xi) dtype = np.result_type(*arrays) shape = arrays[0].shape out = valarray(shape, fill_value, dtype) for cond, func in zip(condlist, funclist): if isinstance(func, Callable): temp = tuple(np.extract(cond, arr) for arr in arrays) + args np.place(out, cond, func(*temp, **kw)) else: # func is a scalar value or a list np.putmask(out, cond, func) return out def lazywhere(cond, arrays, f, fillvalue=None, f2=None): """ np.where(cond, x, fillvalue) always evaluates x even where cond is False. This one only evaluates f(arr1[cond], arr2[cond], ...). For example, >>> a, b = np.array([1, 2, 3, 4]), np.array([5, 6, 7, 8]) >>> def f(a, b): ... return a*b >>> def f2(a, b): ... return np.ones(np.shape(a))*np.ones(np.shape(b)) >>> lazywhere(a > 2, (a, b), f, np.nan) array([ nan, nan, 21., 32.]) >>> lazywhere(a > 2, (a, b), f, f2=f2) array([ 1., 1., 21., 32.]) Notice it assumes that all `arrays` are of the same shape, or can be broadcasted together. """ if fillvalue is None: _assert(f2 is not None, "One of (fillvalue, f2) must be given.") fillvalue = np.nan else: _assert(f2 is None, "Only one of (fillvalue, f2) can be given.") arrays = np.broadcast_arrays(*arrays) temp = tuple(np.extract(cond, arr) for arr in arrays) out = valarray(np.shape(arrays[0]), value=fillvalue) np.place(out, cond, f(*temp)) if f2 is not None: temp = tuple(np.extract(~cond, arr) for arr in arrays) np.place(out, ~cond, f2(*temp)) return out def rotation_matrix(heading, pitch, roll): ''' Examples -------- >>> import numpy as np >>> rotation_matrix(heading=0, pitch=0, roll=0) array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]]) >>> np.all(np.abs(rotation_matrix(heading=180, pitch=0, roll=0)- ... np.array([[ -1.000000e+00, -1.224647e-16, 0.000000e+00], ... [ 1.224647e-16, -1.000000e+00, 0.000000e+00], ... [ -0.000000e+00, 0.000000e+00, 1.000000e+00]]))<1e-7) True >>> np.all(np.abs(rotation_matrix(heading=0, pitch=180, roll=0)- ... np.array([[ -1.000000e+00, 0.000000e+00, 1.224647e-16], ... [ -0.000000e+00, 1.000000e+00, 0.000000e+00], ... [ -1.224647e-16, -0.000000e+00, -1.000000e+00]]))<1e-7) True >>> np.all(np.abs(rotation_matrix(heading=0, pitch=0, roll=180)- ... np.array([[ 1.000000e+00, 0.000000e+00, 0.000000e+00], ... [ 0.000000e+00, -1.000000e+00, -1.224647e-16], ... [ -0.000000e+00, 1.224647e-16, -1.000000e+00]]))<1e-7) True ''' data = np.diag(np.ones(3)) # No transform if H=P=R=0 if heading != 0 or pitch != 0 or roll != 0: deg2rad = np.pi / 180 H = heading * deg2rad P = pitch * deg2rad R = roll * deg2rad # Convert to radians data.put(0, cos(H) * cos(P)) data.put(1, cos(H) * sin(P) * sin(R) - sin(H) * cos(R)) data.put(2, cos(H) * sin(P) * cos(R) + sin(H) * sin(R)) data.put(3, sin(H) * cos(P)) data.put(4, sin(H) * sin(P) * sin(R) + cos(H) * cos(R)) data.put(5, sin(H) * sin(P) * cos(R) - cos(H) * sin(R)) data.put(6, -sin(P)) data.put(7, cos(P) * sin(R)) data.put(8, cos(P) * cos(R)) return data def rotate(x, y, z, heading=0, pitch=0, roll=0): """ Example ------- >>> import numpy as np >>> x, y, z = 1, 1, 1 >>> np.allclose(rotate(x, y, z, heading=0, pitch=0, roll=0), ... (1.0, 1.0, 1.0)) True >>> np.allclose(rotate(x, y, z, heading=90, pitch=0, roll=0), ... (-1.0, 1.0, 1.0)) True >>> np.allclose(rotate(x, y, z, heading=0, pitch=90, roll=0), ... (1.0, 1.0, -1.0)) True >>> np.allclose(rotate(x, y, z, heading=0, pitch=0, roll=90), ... (1.0, -1.0, 1.0)) True """ rot_param = rotation_matrix(heading, pitch, roll).ravel() X = x * rot_param[0] + y * rot_param[1] + z * rot_param[2] Y = x * rot_param[3] + y * rot_param[4] + z * rot_param[5] Z = x * rot_param[6] + y * rot_param[7] + z * rot_param[8] return X, Y, Z def rotate_2d(x, y, angle_deg): ''' Rotate points in the xy cartesian plane counter clockwise Examples -------- >>> rotate_2d(x=1, y=0, angle_deg=0) (1.0, 0.0) >>> rotate_2d(x=1, y=0, angle_deg=90) (6.123233995736766e-17, 1.0) >>> rotate_2d(x=1, y=0, angle_deg=180) (-1.0, 1.2246467991473532e-16) >>> rotate_2d(x=1, y=0, angle_deg=360) (1.0, -2.4492935982947064e-16) ''' angle_rad = angle_deg * pi / 180 ch = cos(angle_rad) sh = sin(angle_rad) return ch * x - sh * y, sh * x + ch * y def now(show_seconds=True): ''' Return current date and time as a string ''' if show_seconds: return strftime("%a, %d %b %Y %H:%M:%S", gmtime()) return strftime("%a, %d %b %Y %H:%M", gmtime()) def _assert(cond, txt=''): if not cond: raise ValueError(txt) def spaceline(start_point, stop_point, num=10): '''Return `num` evenly spaced points between the start and stop points. Parameters ---------- start_point : vector, size=3 The starting point of the sequence. stop_point : vector, size=3 The end point of the sequence. num : int, optional Number of samples to generate. Default is 10. Returns ------- space_points : ndarray of shape n x 3 There are `num` equally spaced points in the closed interval ``[start, stop]``. See Also -------- linspace : similar to spaceline, but in 1D. arange : Similiar to `linspace`, but uses a step size (instead of the number of samples). logspace : Samples uniformly distributed in log space. Example ------- >>> import wafo.misc as pm >>> pm.spaceline((2,0,0), (3,0,0), num=5) array([[ 2. , 0. , 0. ], [ 2.25, 0. , 0. ], [ 2.5 , 0. , 0. ], [ 2.75, 0. , 0. ], [ 3. , 0. , 0. ]]) ''' num = int(num) e1, e2 = np.atleast_1d(start_point, stop_point) e2m1 = e2 - e1 length = np.sqrt((e2m1 ** 2).sum()) # length = sqrt((E2[0]-E1(1))^2 + (E2(2)-E1(2))^2 + (E2(3)-E1(3))^2) C = e2m1 / length delta = length / float(num - 1) return np.array([e1 + n * delta * C for n in range(num)]) def narg_smallest(arr, n=1): ''' Return the n smallest indices to the arr Examples -------- >>> import numpy as np >>> t = np.array([37, 11, 4, 23, 4, 6, 3, 2, 7, 4, 0]) >>> ix = narg_smallest(t, 3) >>> np.allclose(ix, ... [10, 7, 6]) True >>> np.allclose(t[ix], [0, 2, 3]) True ''' return np.array(arr).argsort()[:n] def args_flat(*args): ''' Return x,y,z positions as a N x 3 ndarray Parameters ---------- pos : array-like, shape N x 3 [x,y,z] positions or x,y,z : array-like [x,y,z] positions Returns ------ pos : ndarray, shape N x 3 [x,y,z] positions common_shape : None or tuple common shape of x, y and z variables if given as triple input. Examples -------- >>> x = [1,2,3] >>> pos, c_shape =args_flat(x,2,3) >>> pos array([[1, 2, 3], [2, 2, 3], [3, 2, 3]]) >>> c_shape (3,) >>> pos1, c_shape1 = args_flat([1,2,3]) >>> pos1 array([[1, 2, 3]]) >>> c_shape1 is None True >>> pos1, c_shape1 = args_flat(1,2,3) >>> pos1 array([[1, 2, 3]]) >>> c_shape1 () >>> pos1, c_shape1 = args_flat([1],2,3) >>> pos1 array([[1, 2, 3]]) >>> c_shape1 (1,) ''' nargin = len(args) _assert(nargin in [1, 3], 'Number of arguments must be 1 or 3!') if (nargin == 1): # pos pos = np.atleast_2d(args[0]) _assert((pos.shape[1] == 3) and (pos.ndim == 2), 'POS array must be of shape N x 3!') return pos, None x, y, z = np.broadcast_arrays(*args[:3]) c_shape = x.shape return np.vstack((x.ravel(), y.ravel(), z.ravel())).T, c_shape def index2sub(shape, index, order='C'): ''' Returns Multiple subscripts from linear index. Parameters ---------- shape : array-like shape of array index : linear index into array order : {'C','F'}, optional The order of the linear index. 'C' means C (row-major) order. 'F' means Fortran (column-major) order. By default, 'C' order is used. This function is used to determine the equivalent subscript values corresponding to a given single index into an array. Example ------- >>> shape = (3,3,4) >>> a = np.arange(np.prod(shape)).reshape(shape) >>> order = 'C' >>> a[1, 2, 3] 23 >>> i = sub2index(shape, 1, 2, 3, order=order) >>> a.ravel(order)[i] 23 >>> index2sub(shape, i, order=order) (1, 2, 3) See also -------- sub2index ''' return np.unravel_index(index, shape, order=order) def sub2index(shape, *subscripts, **kwds): ''' Returns linear index from multiple subscripts. Parameters ---------- shape : array-like shape of array *subscripts : subscripts into array order : {'C','F'}, optional The order of the linear index. 'C' means C (row-major) order. 'F' means Fortran (column-major) order. By default, 'C' order is used. This function is used to determine the equivalent single index corresponding to a given set of subscript values into an array. Example ------- >>> shape = (3,3,4) >>> a = np.arange(np.prod(shape)).reshape(shape) >>> order = 'C' >>> i = sub2index(shape, 1, 2, 3, order=order) >>> a[1, 2, 3] 23 >>> a.ravel(order)[i] 23 >>> index2sub(shape, i, order=order) (1, 2, 3) See also -------- index2sub ''' return np.ravel_multi_index(subscripts, shape, **kwds) def is_numlike(obj): """return true if *obj* looks like a number Examples -------- >>> is_numlike(1) True >>> is_numlike('1') False """ try: obj + 1 except TypeError: return False 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 AttributeError as exc: 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) raise exc 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 >>> sorted(d.keys()) == ['test1', 'test2'] True >>> d.update(test1=2) >>> d.test1 2 ''' def __init__(self, **kwargs): self.__dict__.update(kwargs) def keys(self): return list(self.__dict__) 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} True 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) >>> opt == {'arg1': 2, 'arg2': 100} True >>> 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 numpy as np >>> import wafo.misc as wm >>> exp = np.exp; cos = np.cos; randn = np.random.randn >>> x = np.linspace(0,1,200) >>> noise = 0.1*randn(x.size) >>> noise = 0.1*np.sin(100*x) >>> y = exp(x)+cos(5*2*pi*x) + noise >>> y0 = wm.detrendma(y,20) >>> tr = y-y0 >>> np.allclose(tr[:5], ... [ 1.14134814, 1.14134814, 1.14134814, 1.14134814, 1.14134814]) True >>> y1 = wm.detrendma(y, 200) >>> np.allclose((y-y1), 1.7239972279640454) True import pylab as plt h = plt.plot(x, y, x, y0, 'r', x, exp(x), 'k', x, tr, 'm') plt.close('all') See also -------- Reconstruct """ _assert(0 < L, 'L must be positive') _assert(L == round(L), '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) >>> np.allclose(ind, [ 9, 25, 80, 97, 151, 168, 223, 239]) True >>> t0 = wm.ecross(t,x,ind,0.75) >>> np.allclose(t0, [0.84910514, 2.2933879 , 7.13205663, 8.57630119, ... 13.41484739, 14.85909194, 19.69776067, 21.14204343]) True 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, method='clib'): '''Return indices to zero up and downcrossings of a vector ''' if clib is not None and method == 'clib': 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, method='clib'): ''' 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 >>> np.allclose(findcross([0, 1, -1, 1], 0), [0, 1, 2]) True >>> v = 0.75 >>> t = np.linspace(0,7*np.pi,250) >>> x = np.sin(t) >>> ind = wm.findcross(x,v) # all crossings >>> np.allclose(ind, [ 9, 25, 80, 97, 151, 168, 223, 239]) True >>> ind2 = wm.findcross(x,v,'u') >>> np.allclose(ind2, [ 9, 80, 151, 223]) True >>> ind3 = wm.findcross(x,v,'d') >>> np.allclose(ind3, [ 25, 97, 168, 239]) True >>> ind4 = wm.findcross(x,v,'d', method='2') >>> np.allclose(ind4, [ 25, 97, 168, 239]) True t0 = plt.plot(t,x,'.',t[ind],x[ind],'r.', t, ones(t.shape)*v) 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, method) 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) >>> np.allclose(ind, [ 18, 53, 89, 125, 160, 196, 231]) True 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) >>> np.allclose(wm.findpeaks(data, n=8, min_h=5, min_p=0.3), ... [908, 694, 481]) True 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[:n - cnr[0]] 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] >>> ind1 = wm.findrfc(tp, 0.3) >>> np.allclose(ind1, [ 0, 9, 32, 53, 74, 95, 116, 137]) True >>> ind2 = wm.findrfc(tp, 0.3, method='2') >>> np.allclose(ind2, [ 0, 9, 32, 53, 74, 95, 116, 137]) True >>> ind3 = wm.findrfc(tp, 0.3, method='3') >>> np.allclose(ind2, [ 0, 9, 32, 53, 74, 95, 116, 137]) True a = plt.plot(t,x,'.',ti,tp,'r.') a = plt.plot(ti[ind1],tp[ind1]) plt.close('all') See also -------- rfcfilter, findtp. ''' y = atleast_1d(tp).ravel() t_start = int(y[0] > y[1]) # first is a max, ignore it y = y[t_start::] n = len(y) NC = np.floor(n / 2) - 1 if (NC < 1): return zeros(0, dtype=np.int) # No RFC cycles*/ if (y[0] > y[1] and y[1] > y[2] or y[0] < y[1] and y[1] < y[2]): warnings.warn('This is not a sequence of turningpoints, exit') return zeros(0, dtype=np.int) if clib is not None and method == 'clib': ind, ix = clib.findrfc(y, h) elif method == '2': ix = -1 ind = _findrfc2(y, h) else: ind, ix = _findrfc(y, h) return np.sort(ind[:ix]) + t_start def _findrfc(y, h): # TODO: merge rfcfilter and _findrfc t_start = 0 n = len(y) NC = np.floor(n / 2) - 1 ind = zeros(n, dtype=np.int) NC = np.int(NC) ix = 0 for i in range(NC): Tmi = t_start + 2 * i Tpl = t_start + 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 = t_start + 2 * j j -= 1 if (xminus >= xplus): if (y[2 * i + 1] - xminus >= h): ind[ix] = Tmi ix += 1 ind[ix] = (t_start + 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 = (t_start + 2 * j + 2) j += 1 else: if ((y[2 * i + 1] - xminus) >= h): ind[ix] = Tmi ix += 1 ind[ix] = (t_start + 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] = (t_start + 2 * i + 1) ix += 1 elif ((y[2 * i + 1] - xplus) >= h): ind[ix] = (t_start + 2 * i + 1) ix += 1 ind[ix] = Tpl ix += 1 # L180: # iy=i # /* for i */ return ind, ix def _raise_kind_error(kind): if kind in (-1, 0): raise NotImplementedError('kind = {} not yet implemented'.format(kind)) else: raise ValueError('kind = {}: not a valid value of kind'.format(kind)) def nt2cmat(nt, kind=1): """ Return cycle matrix from a counting distribution. Parameters ---------- NT: 2D array Counting distribution. [nxn] kind = 1: causes peaks to be projected upwards and troughs downwards to the closest discrete level (default). = 0: causes peaks and troughs to be projected to the closest discrete level. = -1: causes peaks to be projected downwards and the troughs upwards to the closest discrete level. Returns ------- cmat = Cycle matrix. [nxn] Example -------- >>> import numpy as np >>> cmat0 = np.round(np.triu(np.random.rand(4, 4), 1)*10) >>> cmat0 = np.array([[ 0., 5., 6., 9.], ... [ 0., 0., 1., 7.], ... [ 0., 0., 0., 4.], ... [ 0., 0., 0., 0.]]) >>> nt = cmat2nt(cmat0) >>> np.allclose(nt, ... [[ 0., 0., 0., 0.], ... [ 20., 15., 9., 0.], ... [ 28., 23., 16., 0.], ... [ 32., 27., 20., 0.]]) True >>> cmat = nt2cmat(nt) >>> np.allclose(cmat, [[ 0., 5., 6., 9.], ... [ 0., 0., 1., 7.], ... [ 0., 0., 0., 4.], ... [ 0., 0., 0., 0.]]) True See also -------- cmat2nt """ n = len(nt) # Number of discrete levels if kind == 1: I = np.r_[0:n-1] J = np.r_[1:n] c = nt[I+1][:, J-1] - nt[I][:, J-1] - nt[I+1][:, J] + nt[I][:, J] c2 = np.vstack((c, np.zeros((n-1)))) cmat = np.hstack((np.zeros((n, 1)), c2)) elif kind == 11: # same as def=1 but using for-loop cmat = np.zeros((n, n)) j = np.r_[1:n] for i in range(n-1): cmat[i, j] = nt[i+1, j-1] - nt[i, j-1] - nt[i+1, j] + nt[i, j] else: _raise_kind_error(kind) return cmat def cmat2nt(cmat, kind=1): """ CMAT2NT Calculates a counting distribution from a cycle matrix. Parameters ---------- cmat = Cycle matrix. [nxn] kind = 1: causes peaks to be projected upwards and troughs downwards to the closest discrete level (default). = 0: causes peaks and troughs to be projected to the closest discrete level. = -1: causes peaks to be projected downwards and the troughs upwards to the closest discrete level. Returns ------- NT: n x n array Counting distribution. Example ------- >>> import numpy as np >>> cmat0 = np.round(np.triu(np.random.rand(4, 4), 1)*10) >>> cmat0 = np.array([[ 0., 5., 6., 9.], ... [ 0., 0., 1., 7.], ... [ 0., 0., 0., 4.], ... [ 0., 0., 0., 0.]]) >>> nt = cmat2nt(cmat0, kind=11) >>> np.allclose(nt, ... [[ 0., 0., 0., 0.], ... [ 20., 15., 9., 0.], ... [ 28., 23., 16., 0.], ... [ 32., 27., 20., 0.]]) True >>> cmat = nt2cmat(nt, kind=11) >>> np.allclose(cmat, [[ 0., 5., 6., 9.], ... [ 0., 0., 1., 7.], ... [ 0., 0., 0., 4.], ... [ 0., 0., 0., 0.]]) True See also -------- nt2cmat """ n = len(cmat) # Number of discrete levels nt = zeros((n, n)) if kind == 1: csum = np.cumsum flip = np.fliplr nt[1:n, :n-1] = flip(csum(flip(csum(cmat[:-1, 1:], axis=0)), axis=1)) elif kind == 11: # same as def=1 but using for-loop # j = np.r_[1:n] for i in range(1, n): for j in range(n-1): nt[i, j] = np.sum(cmat[:i, j+1:n]) else: _raise_kind_error(kind) return nt def mctp2tc(f_Mm, utc, param, f_mM=None): """ MCTP2TC Calculates frequencies for the upcrossing troughs and crests using Markov chain of turning points. CALL: f_tc = mctp2tc(f_Mm,utc,param); where f_tc = the matrix with frequences of upcrossing troughs and crests, f_Mm = the frequency matrix for the Max2min cycles, utc = the reference level, param = a vector defining the discretization used to compute f_Mm, note that f_mM has to be computed on the same grid as f_mM. optional call: f_tc = mctp2tc(f_Mm,utc,param,f_mM) f_mM = the frequency matrix for the min2Max cycles. """ raise NotImplementedError('') # if f_mM is None: # f_mM = np.copy(f_Mm) # # u = np.linspace(*param) # udisc = np.fliplr(u) # ntc = np.sum(udisc >= utc) # n = len(f_Mm) # if ntc > n-1: # raise IndexError('index for mean-level out of range, stop') # if param[2]-1 < ntc or ntc < 2 : # raise ValueError('the reference level out of range, stop') # # # normalization of frequency matrices # # for i in range(n): # rowsum = np.sum(f_Mm[i]) # if rowsum!=0: # f_Mm[i] = f_Mm[i] /rowsum # # P = np.fliplr(f_Mm) # # Ph = np.rot90(np.fliplr(f_mM), -1) # for i in range(n): # rowsum = np.sum(Ph[i]) # if rowsum!=0: # Ph[i] = Ph[i] / rowsum # # Ph = np.fliplr(Ph) # # F = np.zeros((n, n)) # F[:ntc-1,:(n-ntc)] = f_mM[:ntc-1, :(n-ntc)] # F = cmat2nt(F) # # for i in range(1, ntc): # for j in range(ntc, n-1): # # if i 1e-10 # if dim_p == 1: # tempp[0] = (Ap/(1-Bp*Ap)*e); # else: # tempp = Ap*((I-Bp*Ap)\e) # # end # # end # # end # # if j>ntc # # Am=P(ntc:j-1,ntc+1:j); Bm=Ph(ntc+1:j,ntc:j-1); # dim_m=j-ntc; # tempm=zeros(dim_m,1); # Im=eye(size(Am)); # if j==n-1 # em=P(ntc:j-1,n); # else # em=sum(P(ntc:j-1,j+1:n),2); # end # if max(abs(em))>1e-10 # if dim_m==1 # tempm(1,1)=(Bm/(1-Am*Bm)*em); # else # tempm=Bm*((Im-Am*Bm)\em); # end # end # end # # if (j>ntc) && (intc) && (i==ntc) # F(i,n-j+1)=F(i,n-j+1)+ones(1,i-1)*freqPVL(1:i-1,n-ntc:-1:n-j+1)*tempm; # for k=ntc:n # F(k,n-j+1)=F(ntc,n-j+1); # end # end # end # end # # return nt2cmat(F); # fmax=max(max(F)); # contour (u,u,flipud(F),... # fmax*[0.005 0.01 0.02 0.05 0.1 0.2 0.4 0.6 0.8]) # axis([param(1) param(2) param(1) param(2)]) # title('Crest-trough density') # ylabel('crest'), xlabel('trough') # axis('square') # if mlver>1, commers, end 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) ''' def _get_PMm(AA1, MA, nA): 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) return PMm if fMm is None: fmM = np.atleast_1d(fmM) fMm = fmM 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 = _get_PMm(AA1, MA, nA) 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 _findrfc2(y, h, method=0): n = len(y) t = zeros(n, dtype=np.int) j = 0 t0 = 0 y0 = y[t0] z0 = 0 def a_le_b(a, b): return a <= b def a_lt_b(a, b): return a < b if method == 0: cmpfun1 = a_le_b cmpfun2 = a_lt_b else: cmpfun1 = a_lt_b cmpfun2 = a_le_b # 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 t[:j + 1] 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() ix = _findrfc2(y, h, method) return y[ix] 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,:] >>> np.allclose(itp, [ 5, 18, 24, 38, 46, 57, 70, 76, 91, 98, 99]) True >>> np.allclose(itph, 91) True a = plt.plot(x1[:,0],x1[:,1], tp[:,0],tp[:,1],'ro', tph[:,0],tph[:,1],'k.') plt.close('all') 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,:] >>> np.allclose(itc, [ 52, 105]) True >>> itc, iv = wm.findtc(x1[:,1],0,'uw') >>> np.allclose(itc, [ 105, 157]) True 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 range(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 range(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], verbose=True) Found 0 missing points dcrit is set to 1.05693 ddcrit is set to 1.05693 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 """ def _find_nans(xn): i_missing = np.flatnonzero(np.isnan(xn)) if verbose: print('Found %d missing points' % i_missing.size) return i_missing def _find_spurious_jumps(dxn, dcrit, name='Dx'): i_p = np.flatnonzero(dxn > dcrit) if i_p.size > 0: i_p += 1 # the point after the jump if verbose: print('Found {0:d} spurious positive jumps of {1}'.format(i_p.size, name)) i_n = np.flatnonzero(dxn < -dcrit) # the point before the jump if verbose: print('Found {0:d} spurious negative jumps of {1}'.format(i_n.size, name)) if i_n.size > 0: return hstack((i_p, i_n)) return i_p def _find_consecutive_equal_values(dxn, zcrit): mask_small = (abs(dxn) <= zcrit) i_small = np.flatnonzero(mask_small) if verbose: if zcrit == 0.: print('Found %d consecutive equal values' % i_small.size) else: print('Found %d consecutive values less than %g apart.' % (i_small.size, zcrit)) if i_small.size > 0: i_small += 1 # finding the beginning and end of consecutive equal values i_step = np.flatnonzero((diff(mask_small))) + 1 # indices to consecutive equal points # removing the point before + all equal points + the point after return hstack((i_step - 1, i_small, i_step, i_step + 1)) return i_small xn = asarray(x).flatten() _assert(2 < xn.size, 'The vector must have more than 2 elements!') i_missing = _find_nans(xn) if np.any(i_missing): xn[i_missing] = 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) ind = np.hstack((_find_spurious_jumps(dxn, dcrit, name='Dx'), _find_spurious_jumps(ddxn, ddcrit, name='D^2x'), _find_consecutive_equal_values(dxn, 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 ''' shape = kwds.get('shape') x0 = 1 if shape is None else np.ones(shape) x1 = np.broadcast(x0, *args) return tuple(x1.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.allclose(wm.stirlerr(2), 0.0413407) True >>> np.allclose(wm.stirlerr(5), 0.01664469) True >>> np.allclose(wm.stirlerr(8), 0.01041127) True >>> np.allclose(wm.stirlerr(12), 0.00694284) True >>> np.allclose(wm.stirlerr(25), 0.00333316) True >>> np.allclose(wm.stirlerr(70), 0.00119047) True >>> np.allclose(wm.stirlerr(100), 0.00083333) True See also --------- binom Reference ----------- Catherine Loader (2000). Fast and Accurate Computation of Binomial Probabilities ''' S0 = 0.083333333333333333333 # /* 1/12 */ S1 = 0.00277777777777777777778 # /* 1/360 */ S2 = 0.00079365079365079365079365 # /* 1/1260 */ S3 = 0.000595238095238095238095238 # /* 1/1680 */ S4 = 0.0008417508417508417508417508 # /* 1/1188 */ n1 = atleast_1d(n) y = gammaln(n1 + 1) - log(sqrt(2 * pi * n1) * (n1 / exp(1)) ** n1) nn = n1 * n1 n500 = 500 < n1 y[n500] = (S0 - S1 / nn[n500]) / n1[n500] n80 = logical_and(80 < n1, n1 <= 500) if any(n80): y[n80] = (S0 - (S1 - S2 / nn[n80]) / nn[n80]) / n1[n80] n35 = logical_and(35 < n1, n1 <= 80) if any(n35): nn35 = nn[n35] y[n35] = (S0 - (S1 - (S2 - S3 / nn35) / nn35) / nn35) / n1[n35] n15 = logical_and(15 < n1, n1 <= 35) if any(n15): nn15 = nn[n15] y[n15] = ( S0 - (S1 - (S2 - (S3 - S4 / nn15) / nn15) / nn15) / nn15) / n1[n15] return y def getshipchar(value=None, property="max_deadweight", # @ReservedAssignment **kwds): ''' 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 >>> true_sc = {'service_speedSTD': 0, ... 'lengthSTD': 2.0113098831942762, ... 'draught': 9.5999999999999996, ... 'propeller_diameterSTD': 0.20267047566705432, ... 'max_deadweightSTD': 3096.9000000000001, ... 'beam': 29.0, 'length': 216.0, ... 'beamSTD': 2.9000000000000004, ... 'service_speed': 10.0, ... 'draughtSTD': 2.1120000000000001, ... 'max_deadweight': 30969.0, ... 'propeller_diameter': 6.761165385916601} >>> wm.getshipchar(10,'service_speed') == true_sc True >>> sc = wm.getshipchar(service_speed=10) >>> sc == true_sc True 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 = list(kwds) _assert(len(names) == 1, 'Only one keyword allowed!') property = names[0] # @ReservedAssignment value = kwds[property] value = np.array(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 = np.round(3.45 * max_deadweight ** 0.40) length_err = length ** 0.13 beam = np.round(1.78 * max_deadweight ** 0.27 * 10) / 10 beam_err = beam * 0.10 draught = np.round(0.80 * max_deadweight ** 0.24 * 10) / 10 draught_err = draught * 0.22 # S = round(2/3*(L)**0.525) speed = np.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 = np.round(max_deadweight) max_deadweightSTD = 0.1 * max_deadweight shipchar = dict(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 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') >>> np.allclose(xa[:5], ... [ 0. , 0.19634954, 0.39269908, 0.58904862, 0.78539816]) True 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) Examples -------- >>> np.allclose(polar2cart(0, 1, 1), (1, 0, 1)) True >>> np.allclose(polar2cart(0, 1), (1, 0)) True See also -------- cart2polar ''' x, y = rho * cos(theta), rho * sin(theta) if z is None: return x, y 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) Examples -------- >>> np.allclose(cart2polar(1, 0, 1), (0, 1, 1)) True >>> np.allclose(cart2polar(1, 0), (0, 1)) True See also -------- polar2cart ''' t, r = arctan2(y, x), hypot(x, y) if z is None: return t, r return t, r, z cart2pol = cart2polar def ndgrid(*args, **kwargs): """ Same as calling meshgrid with indexing='ij' (see meshgrid for documentation). Example ------- >>> x, y = ndgrid([1,2,3],[4,5,6]) >>> np.allclose(x, [[1, 1, 1], ... [2, 2, 2], ... [3, 3, 3]]) True >>> np.allclose(y, [[4, 5, 6], ... [4, 5, 6], ... [4, 5, 6]]) True """ 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) _assert(xo.ndim == 1, 'x must be a vector.') _assert(fo.ndim == 1, 'f must be a vector.') i = xo.argsort() xo, fo = xo[i], fo[i] del i dx = diff(xo) _assert(all(dx > 0), 'Duplicate x-values not allowed.') nf = fo.shape[0] max_x = xo[-1] if max_x is None else max_x min_x = xo[0] if min_x is None else min_x min_n = nf if min_n is None else min_n min_n = max(min_n, 2) max_n = max(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])) >>> np.allclose(gder[1][:5], ... [ 1.09938766, 1.39779849, 1.39538745, 1.39298656, 1.39059575]) True 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. """ def _default_step(xo, N): 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) return hn def _diff(xo, fo, x0, N): hn = _default_step(xo, N) # Transform X with the derivatives of f. fder = vstack((xo, fo)) fxder = zeros((N, x0.size)) 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) return fxder def _der_1(fxder, xi): """First time derivative of y: y1 = f'(x)*x1""" return fxder[0] * xi[0] def _der_2(fxder, xi): """Second time derivative of y: y2 = f''(x)*x1.^2+f'(x)*x2""" return fxder[1] * xi[0] ** 2. + fxder[0] * xi[1] def _der_3(fxder, xi): """Third time derivative of y: y3 = f'''(x)*x1.^3+f'(x)*x3 +3*f''(x)*x1*x2 """ return (fxder[2] * xi[0] ** 3 + fxder[0] * xi[2] + 3 * fxder[1] * xi[0] * xi[1]) def _der_4(fxder, xi): """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) """ return (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])) 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] 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 > 4: warnings.warn('Transformation of derivatives of order>4 is ' + 'not supported.') N = 4 if N > 0: y = [y0] fxder = _diff(xo, fo, x0, N) # Calculate the transforms of the derivatives of X. dfuns = [_der_1, _der_2, _der_3, _der_4] for dfun in dfuns[:N]: y.append(dfun(fxder, xi)) return y 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]) ''' def _default_range(range_, x): return range_ if range_ else (x.min(), x.max()) def _default_bins(num_bins, x): if num_bins is None: num_bins = np.ceil(4 * np.sqrt(np.sqrt(len(x)))) return num_bins def _default_step(mn, mx, num_bins): d = float(mx - mn) / num_bins * 2 e = np.floor(np.log(d) / np.log(10)) m = np.clip(np.floor(d / 10 ** e), a_min=0, a_max=5) if 2 < m < 5: m = 2 return m * 10 ** e if data is not None: data = np.atleast_1d(data) mn, mx = _default_range(range, data) num_bins = _default_bins(num_bins, data) d = _default_step(mn, mx, num_bins) 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 _make_bars(limits, bin_): 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 xx, yy def _histogram(data, bins=None, range=None, normed=False, weights=None, density=None): """ Example ------- >>> import numpy as np >>> data = np.linspace(0, 10) >>> xx, yy, limits = _histogram(data) >>> len(limits) 12 >>> xx, yy, limits = _histogram(data, bins=[0, 5, 11]) >>> np.allclose(xx, [ 0, 0, 5, 5, 5, 11, 11]) True >>> np.allclose(yy, [ 0., 25., 25., 0., 25., 25., 0.]) True >>> np.allclose(limits, [[ 0], [ 5], [11]]) True """ 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, range=range, normed=normed, weights=weights, density=density) xx, yy = _make_bars(limits, bin_) return xx, yy, limits def plot_histgrm(data, bins=None, range=None, # @ReservedAssignment normed=False, weights=None, density=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) >>> R = np.linspace(0,10) >>> bins = good_bins(R) >>> len(bins) 13 h0 = wm.plot_histgrm(R, bins, 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 ''' xx, yy, limits = _histogram(data, bins, range, normed, weights, density) return plotbackend.plot(xx, yy, lintype, limits, limits * 0) def num2pistr(x, n=3, numerator_max=10, denominator_max=10): ''' 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 >>> wm.num2pistr(-np.pi/4)=='-\\pi/4' True >>> wm.num2pistr(-np.pi)=='-\\pi' True >>> wm.num2pistr(-1/4)=='-0.25' True ''' def _denominator_text(den): return '' if abs(den) == 1 else '/%d' % den def _numerator_text(num): if abs(num) == 1: return '-' if num == -1 else '' return '{:d}'.format(num) frac = fractions.Fraction.from_float(x / pi).limit_denominator(int(1e+13)) num, den = frac.numerator, frac.denominator if (den < denominator_max) and (num < numerator_max) and (num != 0): return r'{0:s}\pi{1:s}'.format(_numerator_text(num), _denominator_text(den)) fmt = '{:0.' + '{:d}'.format(n) + 'g}' return fmt.format(x) def fourier(data, t=None, period=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. period : 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, period=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 t = np.arange(n) if t is None else np.atleast_1d(t) n = len(t) if n is None else n m = n if m is None else m period = t[-1] - t[0] if period is None else period intfun = trapz if method.startswith('trapz') else 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 / period 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 test_docstrings(): # np.set_printoptions(precision=6) import doctest print('Testing docstrings in %s' % __file__) doctest.testmod(optionflags=doctest.NORMALIZE_WHITESPACE) if __name__ == "__main__": test_docstrings()