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pywafo/wafo/misc.py

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'''
Misc
'''
from __future__ import absolute_import, division
import sys
from wafo import numba_misc
import fractions
import numpy as np
from numpy import (
meshgrid,
amax, logical_and, arange, linspace, atleast_1d,
asarray, ceil, floor, frexp, hypot,
sqrt, arctan2, sin, cos, exp, log, log1p, mod, diff,
inf, pi, interp, isscalar, zeros, ones, linalg,
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
import numbers
try:
from wafo import c_library as clib # @UnresolvedImport
except ImportError:
warnings.warn('c_library not found. Check its compilation.')
clib = None
floatinfo = np.finfo(float)
_TINY = floatinfo.tiny
_EPS = floatinfo.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',
'lazyselect',
'piecewise',
'valarray', 'check_random_state']
def check_random_state(seed):
"""Turn seed into a np.random.RandomState instance
If seed is None (or np.random), return the RandomState singleton used
by np.random.
If seed is an int, return a new RandomState instance seeded with seed.
If seed is already a RandomState instance, return it.
Otherwise raise ValueError.
Example
-------
>>> check_random_state(seed=None)
<mtrand.RandomState object at ...
>>> check_random_state(seed=1)
<mtrand.RandomState object at ...
>>> check_random_state(seed=np.random.RandomState(1))
<mtrand.RandomState object at ...
check_random_state(seed=2.5)
"""
if seed is None or seed is np.random:
return np.random.mtrand._rand
if isinstance(seed, (numbers.Integral, np.integer)):
return np.random.RandomState(seed)
if isinstance(seed, np.random.RandomState):
return seed
msg = '{} cannot be used to seed a numpy.random.RandomState instance'
raise ValueError(msg.format(seed))
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 lazyselect(condlist, choicelist, arrays, default=0):
"""
Mimic `np.select(condlist, choicelist)`.
Notice it assumes that all `arrays` are of the same shape, or can be
broadcasted together.
All functions in `choicelist` must accept array arguments in the order
given in `arrays` and must return an array of the same shape as broadcasted
`arrays`.
Examples
--------
>>> x = np.arange(6)
>>> np.select([x <3, x > 3], [x**2, x**3], default=0)
array([ 0, 1, 4, 0, 64, 125])
>>> lazyselect([x < 3, x > 3], [lambda x: x**2, lambda x: x**3], (x,))
array([ 0., 1., 4., 0., 64., 125.])
>>> a = -np.ones_like(x)
>>> lazyselect([x < 3, x > 3],
... [lambda x, a: x**2, lambda x, a: a * x**3],
... (x, a))
array([ 0., 1., 4., 0., -64., -125.])
"""
arrays = np.broadcast_arrays(*arrays)
tcode = np.mintypecode([a.dtype.char for a in arrays])
out = valarray(np.shape(arrays[0]), value=default, typecode=tcode)
for index, cond in enumerate(condlist):
func = choicelist[index]
if np.all(cond is False):
continue
cond, _ = np.broadcast_arrays(cond, arrays[0])
temp = tuple(np.extract(cond, arr) for arr in arrays)
np.place(out, cond, func(*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.allclose(rotation_matrix(heading=180, pitch=0, roll=0),
... [[ -1., 0., 0.],
... [ 0., -1., 0.],
... [ 0., 0., 1.]])
True
>>> np.allclose(rotation_matrix(heading=0, pitch=180, roll=0),
... [[ -1., 0., 0.],
... [ 0., 1., 0.],
... [ 0., 0., -1.]])
True
>>> np.allclose(rotation_matrix(heading=0, pitch=0, roll=180),
... [[ 1., 0., 0.],
... [ 0., -1., 0.],
... [ 0., 0., -1.]])
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
--------
>>> np.allclose(rotate_2d(x=1, y=0, angle_deg=0), (1.0, 0.0))
True
>>> np.allclose(rotate_2d(x=1, y=0, angle_deg=90), (0, 1.0))
True
>>> np.allclose(rotate_2d(x=1, y=0, angle_deg=180), (-1.0, 0))
True
>>> np.allclose(rotate_2d(x=1, y=0, angle_deg=360), (1.0, 0))
True
'''
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):
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]
return numba_misc.findcross(xn)
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 kind=='dw' or kind=='tw'
# or make sure the first is a level v up-crossing
# if kind=='uw' or kind=='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 kind is dw or uw
# and even if kind is 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!'
' ({})'.format(kind))
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)
"""
return numba_misc.findrfc_astm(tp)
# 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=0)
>>> np.allclose(ind2, [ 0, 9, 32, 53, 74, 95, 116, 137, 146])
True
>>> ind3 = wm.findrfc(tp, 0.3, method=1)
>>> np.allclose(ind3, [ 0, 9, 32, 53, 74, 95, 116, 137, 146])
True
>>> ind3 = wm.findrfc(tp, 0.3, method=2)
>>> np.allclose(ind3, [ 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)
else:
ind = numba_misc.findrfc(y, h, method)
ix = len(ind)
return np.sort(ind[:ix]) + t_start
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.
Parameters
----------
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.
f_mM = the frequency matrix for the min2Max cycles.
Returns
-------
f_tc = the matrix with frequences of upcrossing troughs and crests,
"""
def _check_ntc(ntc, n):
if ntc > n - 1:
raise IndexError('index for mean-level out of range, stop')
def _check_discretization(param, ntc):
if param[2] - 1 < ntc or ntc < 2:
raise ValueError('the reference level out of range, stop')
def _normalize_rows(arr):
n = len(arr)
for i in range(n):
rowsum = np.sum(arr[i])
if rowsum != 0:
arr[i] = arr[i] / rowsum
return arr
def _make_tempp(P, Ph, i, ntc):
Ap = P[i:ntc - 1, i + 1:ntc]
Bp = Ph[i + 1:ntc, i:ntc - 1]
dim_p = ntc - i
tempp = zeros((dim_p, 1))
I = np.eye(np.shape(Ap))
if i == 2:
e = Ph[i + 1:ntc, 0]
else:
e = np.sum(Ph[i + 1:ntc, 1:i - 1], axis=1)
if max(abs(e)) > 1e-10:
if dim_p == 1:
tempp[0] = (Ap / (1 - Bp * Ap) * e)
else:
rh = I - np.dot(Bp, Ap)
tempp = np.dot(Ap, linalg.solve(rh, e))
# end
# end
return tempp
def _make_tempm(P, Ph, 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 = np.eye(np.shape(Am))
if j == n - 1:
em = P[ntc:j - 1, n]
else:
em = np.sum(P[ntc:j - 1, j + 1:n], axis=1)
# end
if max(abs(em)) > 1e-10:
if dim_m == 1:
tempm[0, 0] = (Bm / (1 - Am * Bm) * em)
else:
rh = Im - np.dot(Am, Bm)
tempm = np.dot(Bm, linalg.lstsq(rh, em)[0])
# end
# end
return tempm
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)
_check_ntc(ntc, n)
_check_discretization(param, ntc)
# normalization of frequency matrices
f_Mm = _normalize_rows(f_Mm)
P = np.fliplr(f_Mm)
Ph = np.rot90(np.fliplr(f_mM), -1)
Ph = _normalize_rows(Ph)
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-1, n - 1):
if i < ntc-1:
tempp = _make_tempp(P, Ph, i, ntc)
b = np.dot(np.dot(tempp.T, f_mM[i:ntc - 1, n - j:-1:-1]),
ones((n - j, 1)))
# end
if j > ntc-1:
tempm = _make_tempm(P,Ph, j, ntc)
c = np.dot(np.dot(ones((1, i - 1)),
f_mM[:i - 1, n - ntc:n - j + 1:-1]),
tempm)
# end
if (j > ntc-1) and (i < ntc-1):
a = np.dot(np.dot(tempp.T,
f_mM[i:ntc - 1, n - ntc:-1:n - j + 1]),
tempm)
F[i, n - j + 1] = F[i, n - j + 1] + a + b + c
# end
if (j == ntc-1) and (i < ntc-1):
F[i, n - j + 1] = F[i, n - j + 1] + b
for k in range(ntc):
F[i, n - k + 1] = F[i, n - ntc + 1]
# end
# end
if (j > ntc-1) and (i == ntc-1):
F[i, n - j + 1] = F[i, n - j + 1] + c
for k in range(ntc-1, n):
F[k, n - j + 1] = F[ntc-1, n - j + 1]
# end
# end
# end
# end
# 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
return nt2cmat(F)
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
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(np.abs(e)) > 1e-6 and
max(np.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 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
"""
y = atleast_1d(x).ravel()
ix = numba_misc.findrfc(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 ind[0] != 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 = np.r_[0, ind, n - 1]
else: # adds index to the last value
ind = np.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:
f1, f2 = np.argmin, np.argmax
else:
f1, f2 = np.argmax, np.argmin
for i in range(n_tc):
# trough or crest
j = 2 * i
ind[j] = f1(x[v_ind[j] + 1:v_ind[j + 1] + 1])
# crest or trough
ind[j + 1] = f2(x[v_ind[j + 1] + 1:v_ind[j + 2] + 1])
if (2 * n_tc + 1 < n_c) and (kind in (None, 'tw', 'cw')):
# trough or crest
ind[n_c - 2] = f1(x[v_ind[n_c - 2] + 1:v_ind[n_c - 1] + 1])
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 = (np.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' % np.size(ind))
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)
return tuple(np.broadcast(x0, *args).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
<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 np.any(n80):
y[n80] = (S0 - (S1 - S2 / nn[n80]) / nn[n80]) / n1[n80]
n35 = logical_and(35 < n1, n1 <= 80)
if np.any(n35):
nn35 = nn[n35]
y[n35] = (S0 - (S1 - (S2 - S3 / nn35) / nn35) / nn35) / n1[n35]
n15 = logical_and(15 < n1, n1 <= 35)
if np.any(n15):
nn15 = nn[n15]
y[n15] = (
S0 - (S1 - (S2 - (S3 - S4 / nn15) / nn15) / nn15) / nn15) / n1[n15]
return y
def _get_max_deadweight(**ship_property):
names = list(ship_property)
_assert(len(ship_property) == 1, 'Only one ship property allowed!')
name = names[0]
value = np.array(ship_property[name])
valid_props = dict(le='length', be='beam', dr='draught',
ma='max_deadweigth',
se='service_speed', pr='propeller_diameter')
prop = valid_props[name[:2]]
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=lambda x: x
)
max_deadweight = prop2max_dw.get(prop, lambda x: x)(value)
return max_deadweight, prop
def getshipchar(**ship_property):
'''
Return ship characteristics from value of one ship-property
Parameters
----------
**ship_property : scalar
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(service_speed=10) == 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.
'''
max_deadweight, prop = _get_max_deadweight(**ship_property)
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
-------
>>> np.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
>>> np.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(np.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
num_tries = 0
while (num_tries < 5 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(np.abs((y00 - y) / (np.abs(y00 + y) + _TINY)))
num_tries += int(abs(err - err0) <= tol/2)
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
num_tries = 0
for j in range(50):
if num_tries < 5 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 * (np.abs((fy0 - fy) / (np.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]
num_tries += int(abs(err - err0) <= tol/2)
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 np.any(np.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 np.abs(den) == 1 else '/%d' % den
def _numerator_text(num):
if np.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 |
doctest.ELLIPSIS)
if __name__ == "__main__":
test_docstrings()
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