/usr/lib/python2.7/dist-packages/csb/numeric/__init__.py is in python-csb 1.2.3+dfsg-3.
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Low level numeric / math utility functions.
"""
import sys
import math
import numpy
EXP_MIN = -308
EXP_MAX = +709
LOG_MIN = 1e-308
LOG_MAX = 1e+308
## Euler-Mascheroni constant
EULER_MASCHERONI = 0.57721566490153286060651209008240243104215933593992
def log(x, x_min=LOG_MIN, x_max=LOG_MAX):
"""
Safe version of log, clips argument such that overflow does not occur.
@param x: input
@type x: numpy array or float or int
@param x_min: lower value for clipping
@type x_min: float
@param x_max: upper value for clipping
@type x_max: float
"""
x_min = max(x_min, LOG_MIN)
x_max = min(x_max, LOG_MAX)
return numpy.log(numpy.clip(x, x_min, x_max))
def exp(x, x_min=EXP_MIN, x_max=EXP_MAX):
"""
Safe version of exp, clips argument such that overflow does not occur.
@param x: input
@type x: numpy array or float or int
@param x_min: lower value for clipping
@type x_min: float
@param x_max: upper value for clipping
@type x_max: float
"""
x_min = max(x_min, EXP_MIN)
x_max = min(x_max, EXP_MAX)
return numpy.exp(numpy.clip(x, x_min, x_max))
def sign(x):
"""
Return the sign of the input.
"""
return numpy.sign(x)
def isreal(x, tol=1e-14):
"""
Check if input array has no imaginary part.
@param x: input array
@type x: numpy array
@param tol: tolerance to check for equality zero
@type tol: float
"""
return not hasattr(x, 'real') or abs(x.imag) < tol
def log_sum_exp(x, axis=0):
"""
Return the logarithm of the sum of exponentials.
@type x: Numpy array
"""
xmax = x.max(axis)
return log(exp(x - xmax).sum(axis)) + xmax
def log_sum_exp_accumulate(x, axis=0):
"""
Return the logarithm of the accumulated sums of exponentials.
@type x: Numpy array
"""
xmax = x.max(axis)
return log(numpy.add.accumulate(exp(x - xmax), axis)) + xmax
def radian2degree(x):
"""
Convert radians angles to torsion angles.
@param x: radian angle
@return: torsion angle of x
"""
x = x % (2 * numpy.pi)
numpy.putmask(x, x > numpy.pi, x - 2 * numpy.pi)
return x * 180. / numpy.pi
def degree2radian(x):
"""
Convert randian angles to torsion angles.
@param x: torsion angle
@return: radian angle of x
"""
numpy.putmask(x, x < 0., x + 360.)
return x * numpy.pi / 180.
def euler_angles(r):
"""
Calculate the euler angles from a three dimensional rotation matrix.
@param r: 3x3 Rotation matrix
"""
a = numpy.arctan2(r[2, 1], r[2, 0]) % (2 * numpy.pi)
b = numpy.arctan2((r[2, 0] + r[2, 1]) / (numpy.cos(a) + numpy.sin(a)), r[2, 2]) % (2 * numpy.pi)
c = numpy.arctan2(r[1, 2], -r[0, 2]) % (2 * numpy.pi)
return a, b, c
def euler(a, b, c):
"""
Calculate a three dimensional rotation matrix from the euler angles.
@param a: alpha, angle between the x-axis and the line of nodes
@param b: beta, angle between the z axis of the different coordinate systems
@param c: gamma, angle between the line of nodes and the X-axis
"""
from numpy import cos, sin, array
ca, cb, cc = cos(a), cos(b), cos(c)
sa, sb, sc = sin(a), sin(b), sin(c)
return array([[ cc * cb * ca - sc * sa, cc * cb * sa + sc * ca, -cc * sb],
[-sc * cb * ca - cc * sa, -sc * cb * sa + cc * ca, sc * sb],
[ sb * ca, sb * sa, cb ]])
def rotation_matrix(axis, angle):
"""
Calculate a three dimensional rotation matrix for a rotation around
the given angle and axis.
@type axis: (3,) numpy array
@param angle: angle in radians
@type angle: float
@rtype: (3,3) numpy.array
"""
axis = numpy.asfarray(axis) / norm(axis)
assert axis.shape == (3,)
c = math.cos(angle)
s = math.sin(angle)
r = (1.0 - c) * numpy.outer(axis, axis)
r.flat[[0,4,8]] += c
r.flat[[5,6,1]] += s * axis
r.flat[[7,2,3]] -= s * axis
return r
def axis_and_angle(r):
"""
Calculate axis and angle of rotation from a three dimensional
rotation matrix.
@param r: 3x3 Rotation matrix
@return: axis unit vector as (3,) numpy.array and angle in radians as float
@rtype: tuple
"""
from scipy.linalg import logm
B = logm(r).real
a = numpy.array([B[1,2], -B[0,2], B[0,1]])
n = norm(a)
return a / n, n
def norm(x):
"""
Calculate the Eucledian norm of a d-dimensional vector.
@param x: vector (i.e. rank one array)
@return: length of vector
"""
return numpy.linalg.norm(x)
def reverse(array, axis=0):
"""
Reverse the order of elements in an array.
"""
from numpy import take, arange
return take(array, arange(array.shape[axis] - 1, -1, -1), axis)
def polar(x):
"""
Polar coordinate representation of a d-dimensional vector.
@param x: vector (i.e. rank one array)
@return: polar coordinates (radius and polar angles)
"""
(d,) = x.shape
phi = numpy.zeros(d)
for i in reversed(range(1, d)):
phi[i - 1] = numpy.arctan2(x[i] / numpy.cos(phi[i]), x[i - 1])
return numpy.array([norm(x)] + phi[:-1].tolist())
def from_polar(x):
"""
Reconstruct d-dimensional vector from polar coordinates.
@param x: vector (i.e. rank one array)
@return: position in d-dimensional space
"""
(d,) = x.shape
c = numpy.cos(x[1:])
s = numpy.sin(x[1:])
r = x[0]
x = numpy.zeros(d)
x[0] = r
for i in range(d - 1):
x[i + 1] = x[i] * s[i]
x[i] *= c[i]
return x
def polar3d(x):
"""
Polar coordinate representation of a three-dimensional vector.
@param x: vector (i.e. rank one array)
@return: polar coordinates (radius and polar angles)
"""
if x.shape != (3,):
raise ValueError(x)
r = norm(x)
theta = numpy.arccos(x[2] / r)
phi = numpy.arctan2(x[1], x[0])
return numpy.array([r, theta, phi])
def from_polar3d(x):
"""
Reconstruct 3-dimensional vector from polar coordinates.
@param x: vector (i.e. rank one array)
@return: position in 3-dimensional space
"""
assert x.shape == (3,)
r, theta, phi = x[:]
s = numpy.sin(theta)
c = numpy.cos(theta)
S = numpy.sin(phi)
C = numpy.cos(phi)
return r * numpy.array([s * C, s * S, c])
def dihedral_angle(a, b, c, d):
"""
Calculate the dihedral angle between 4 vectors,
representing 4 connected points. The angle is in range [-180, 180].
@param a: the four points that define the dihedral angle
@type a: array
@return: angle in [-180, 180]
"""
v = b - c
m = numpy.cross((a - b), v)
m /= norm(m)
n = numpy.cross((d - c), v)
n /= norm(n)
c = numpy.dot(m, n)
s = numpy.dot(numpy.cross(n, m), v) / norm(v)
angle = math.degrees(math.atan2(s, c))
if angle > 0:
return numpy.fmod(angle + 180, 360) - 180
else:
return numpy.fmod(angle - 180, 360) + 180
def psi(x):
"""
Digamma function
"""
from numpy import inf, log, sum, exp
coef = [-1. / 12., 1. / 120., -1. / 252., 1. / 240., -1. / 132.,
691. / 32760., -1. / 12.]
if x == 0.:
return -inf
elif x < 0.:
raise ValueError('not defined for negative values')
elif x < 6.:
return psi(x + 1) - 1. / x
else:
logx = log(x)
res = logx - 0.5 / x
res += sum([coef[i] * exp(-2 * (i + 1) * logx) for i in range(7)])
return res
def approx_psi(x):
from numpy import log, clip, where
if type(x) == numpy.ndarray:
y = 0. * x
y[where(x < 0.6)] = -EULER_MASCHERONI - 1. / clip(x[where(x < 0.6)], 1e-154, 1e308)
y[where(x >= 0.6)] = log(x[where(x >= 0.6)] - 0.5)
return y
else:
if x < 0.6:
return -EULER_MASCHERONI - 1 / clip(x, 1e-154, 1e308)
else:
return log(x - 0.5)
def d_approx_psi(x):
from numpy import clip, where
if type(x) == numpy.ndarray:
y = 0. * x
y[where(x < 0.6)] = 1. / clip(x[where(x < 0.6)], 1e-154, 1e308) ** 2
y[where(x >= 0.6)] = 1. / (x[where(x >= 0.6)] - 0.5)
return y
else:
if x < 0.6:
return 1 / clip(x, 1e-154, 1e308) ** 2
else:
return 1 / (x - 0.5)
def inv_psi(y, tol=1e-10, n_iter=100, psi=psi):
"""
Inverse digamma function
"""
from numpy import exp
from scipy.special import digamma
## initial value
if y < -5 / 3. - EULER_MASCHERONI:
x = -1 / (EULER_MASCHERONI + y)
else:
x = 0.5 + exp(y)
## Newton root finding
for dummy in range(n_iter):
z = digamma(x) - y
if abs(z) < tol:
break
x -= z / d_approx_psi(x)
return x
def log_trapezoidal(log_y, x=None):
"""
Compute the logarithm of the 1D integral of x, using trepezoidal approximation.
Assumes x is monotonically increasing.
"""
if x is not None:
log_x_diff = log(x[1:] - x[:-1])
else:
log_x_diff = 0.
log_y_add = log_sum_exp(numpy.vstack((log_y[:-1], log_y[1:])), 0)
return log_sum_exp(log_y_add + log_x_diff) - log(2)
def log_midpoint_rule_2d(log_f, x, y):
x_delta = x[:-1] - x[1:]
y_delta = y[:-1] - y[1:]
z = numpy.array([log_f[:, 1:] , log_f[:, :-1]])
y_hat = log_sum_exp(z.reshape((2, -1)), 0)
y_hat = numpy.reshape(y_hat, (len(x), len(y) - 1))
y_hat += log(y_delta) - log(2)
return log_sum_exp(y_hat + log(x_delta)) - log(2.)
def log_trapezoidal_2d(log_f, x=None, y=None):
"""
Compute the logarithm of the 1D integral of x, using trepezoidal approximation.
Assumes x and y is monotonically increasing.
"""
int_y = numpy.array([log_trapezoidal(log_f[i, :], y) for i in range(len(y))])
return log_trapezoidal(int_y, x)
def trapezoidal(x, y):
return 0.5 * numpy.dot(x[1:] - x[:-1], y[1:] + y[:-1])
def trapezoidal_2d(f):
"""
Approximate the integral of f from a to b in two dimensions,
using trepezoidal approximation.
@param f: 2D numpy array of function values at equally spaces positions
@return: approximation of the definit integral
"""
I = f[0, 0] + f[-1, -1] + f[0, -1] + f[-1, 0]
I += 2 * (f[1:-1, (0, -1)].sum() + f[(0, -1), 1:-1].sum())
I += 4 * f[1:-1, 1:-1].sum()
return I / 4.
def simpson_2d(f):
"""
Approximate the integral of f from a to b in two dimensions,
using Composite Simpson's rule.
@param f: 2D numpy array of function values
@return: approximation of the definit integral
"""
n = int((f.shape[0] - 1) / 2)
i = 2 * numpy.arange(1, n + 1) - 1
j = 2 * numpy.arange(1, n)
I = f[0, 0] + f[-1, -1] + f[0, -1] + f[-1, 0]
I += 4 * (f[0, i].sum() + f[-1, i].sum() + f[0, j].sum() + f[-1, j].sum())
I += 4 * (f[i, 0].sum() + f[i, -1].sum() + f[j, 0].sum() + f[j, -1].sum())
I += 16 * f[i][:, i].sum() + 8 * (f[i][:, j].sum() + f[j][:, i].sum())
I += 4 * f[j][:, j].sum()
return I / 9.
def pad(x, s):
"""
Add layers of zeros around grid.
"""
s = numpy.array(s) - 1
y = numpy.zeros(numpy.array(x.shape) + s)
s /= 2
slices = [slice(s[i], -s[i]) for i in range(len(s))]
y[slices] = x
return y
def trim(x, s):
"""
Remove additional layers.
"""
s = numpy.array(s) - 1
s /= 2
slices = [slice(s[i], -s[i]) for i in range(len(s))]
return x[slices]
def zerofill(x, s):
y = numpy.zeros(s)
slices = [slice(-x.shape[i], None) for i in range(len(s))]
y[slices] = x
return y
def convolve(x, f):
from numpy import fft, all
sx = numpy.array(x.shape)
sf = numpy.array(f.shape)
if not all(sx >= sf): return convolve(f, x)
y = fft.ifftn(fft.fftn(x) * fft.fftn(f, sx)).real
slices = [slice(sf[i] - 1, sx[i]) for i in range(len(sf))]
return y[slices]
def correlate(x, y):
from numpy import fft
sx = numpy.array(x.shape)
sy = numpy.array(y.shape)
if (sx >= sy).sum():
slices = [slice(None, sx[i] - sy[i] + 1) for i in range(len(sx))]
X = fft.fftn(x)
Y = fft.fftn(zerofill(y, sx))
else:
sf = sx + sy - 1
slices = [slice(None, sf[i]) for i in range(len(sf))]
X = fft.fftn(x, sf)
Y = fft.fftn(zerofill(y, sf), sf)
return fft.ifftn(X.conjugate() * Y)[slices].real
def gower_matrix(X):
"""
Gower, J.C. (1966). Some distance properties of latent root
and vector methods used in multivariate analysis.
Biometrika 53: 325-338
@param X: ensemble coordinates
@type X: (m,n,k) numpy.array
@return: symmetric dissimilarity matrix
@rtype: (n,n) numpy.array
"""
X = numpy.asarray(X)
B = sum(numpy.dot(x, x.T) for x in X) / float(len(X))
b = B.mean(1)
bb = b.mean()
return (B - numpy.add.outer(b, b)) + bb
class MatrixInitError(Exception):
pass
class InvertibleMatrix(object):
"""
Matrix object which is intended to save time in MCMC sampling algorithms involving
repeated integration of Hamiltonian equations of motion and frequent draws from
multivariate normal distributions involving mass matrices as covariance matrices.
It can be initialized either with the matrix one wants to use or its inverse.
The main feature compared to standard numpy matrices / arrays is that it has also
a property "inverse", which gives the inverse matrix. If the matrix (its inverse) is
changed, the inverse (regular matrix) is calculated only when needed. This avoids costly
matrix inversions.
@param matrix: matrix-like object with whose values the Matrix object is supposed to
hold
@type matrix: invertible (n,n)-shaped numpy.ndarray
@param inverse_matrix: matrix-like object with whose inverse the Matrix object is supposed
to hold
@type inverse_matrix: invertible (n,n)-shaped numpy.ndarray
"""
def __init__(self, matrix=None, inverse_matrix=None):
if (matrix is None and inverse_matrix is None):
raise MatrixInitError("At least one matrix argument has to be specified")
self._matrix = None
self._inverse_matrix = None
self._matrix_updated = False
self._inverse_matrix_updated = False
self._is_unity_multiple = False
if matrix is not None and inverse_matrix is not None:
if type(matrix) != numpy.ndarray or type(inverse_matrix) != numpy.ndarray:
raise TypeError("Arguments have to be of type numpy.ndarray!")
matrix = matrix.copy()
inverse_matrix = inverse_matrix.copy()
self._check_equal_shape(matrix, inverse_matrix)
self._matrix = matrix
self._inverse_matrix = inverse_matrix
self._is_unity_multiple = self._check_unity_multiple(self._matrix)
self._matrix_updated = True
self._inverse_matrix_updated = True
else:
if matrix is not None:
if type(matrix) != numpy.ndarray:
raise TypeError("Arguments have to be of type numpy.ndarray!")
matrix = matrix.copy()
self._check_square(matrix)
self._matrix = matrix
self._matrix_updated = True
self._inverse_matrix_updated = False
self._is_unity_multiple = self._check_unity_multiple(self._matrix)
else:
if type(inverse_matrix) != numpy.ndarray:
raise TypeError("Arguments have to be of type numpy.ndarray!")
inverse_matrix = inverse_matrix.copy()
self._check_square(inverse_matrix)
self._inverse_matrix = inverse_matrix
self._matrix_updated = False
self._inverse_matrix_updated = True
self._is_unity_multiple = self._check_unity_multiple(self._inverse_matrix)
def _check_diagonal(self, matrix):
i, j = numpy.nonzero(matrix)
return numpy.array_equal(i, j)
def _check_unity_multiple(self, matrix):
diagonal_elements_equal = numpy.all([matrix[i][i] == matrix[i+1][i+1]
for i in range(len(matrix) - 1)])
return self._check_diagonal(matrix) and diagonal_elements_equal
def _check_square(self, matrix):
if matrix.shape[0] != matrix.shape[1]:
raise ValueError("Matrix " + matrix.__name__ + " must be a square matrix!")
def _check_equal_shape(self, matrix1, matrix2):
if not numpy.all(matrix1.shape == matrix2.shape):
raise ValueError("Matrices " + matrix1.__name__ + " and " + matrix2.__name__ +
" must have equal shape!")
def __getitem__(self, loc):
if self._matrix_updated == False:
if self.is_unity_multiple:
self._matrix = numpy.diag(1. / self._inverse_matrix.diagonal())
else:
self._matrix = numpy.linalg.inv(self._inverse_matrix)
self._matrix_updated = True
return self._matrix[loc]
def __setitem__(self, i, value):
if type(value) != numpy.ndarray:
raise TypeError("Arguments have to be of type numpy.ndarray!")
self._matrix[i] = numpy.array(value)
self._matrix_updated = True
self._inverse_matrix_updated = False
self._is_unity_multiple = self._check_unity_multiple(self._matrix)
def __array__(self, dtype=None):
return self._matrix
def __mul__(self, value):
if type(value) in (float, int):
v = float(value)
if self._matrix_updated:
return InvertibleMatrix(v * self._matrix)
else:
return InvertibleMatrix(inverse_matrix = self._inverse_matrix / v)
else:
raise ValueError("Only float and int are supported for multiplication!")
__rmul__ = __mul__
def __truediv__(self, value):
if type(value) in (float, int):
v = float(value)
if self._matrix_updated:
return InvertibleMatrix(self._matrix / v)
else:
return InvertibleMatrix(inverse_matrix=self._inverse_matrix * v)
else:
raise ValueError("Only float and int are supported for division!")
__div__ = __truediv__
def __imul__(self, value):
if type(value) in (float, int):
if self._matrix_updated:
self._matrix *= float(value)
self._deprecate_inverse_matrix()
else:
self._inverse_matrix /= float(value)
self._inverse_matrix_updated = True
self._deprecate_matrix()
return self
else:
raise ValueError("Only float and int are supported for multiplication!")
def __itruediv__(self, value):
if type(value) in (float, int):
if self._matrix_updated:
self._matrix /= float(value)
self._matrix_updated = True
self._deprecate_inverse_matrix()
else:
self._inverse_matrix *= float(value)
self._inverse_matrix_updated = True
self._deprecate_matrix()
return self
else:
raise ValueError("Only float and int are supported for division!")
__idiv__ = __itruediv__
def __eq__(self, other):
if self._matrix is not None and other._matrix is not None:
return self._matrix == other._matrix
if self._inverse_matrix is not None and other._inverse_matrix is not None:
return self._inverse_matrix == other._inverse_matrix
def __str__(self):
if self._matrix is not None and self._inverse_matrix is not None:
return "csb.numeric.InvertibleMatrix object holding the following numpy matrix:\n"\
+self._matrix.__str__() +"\n and its inverse:\n"+self._inverse_matrix.__str__()
else:
if self._matrix is None:
return "csb.numeric.InvertibleMatrix object holding only the inverse matrix:\n"\
+self._inverse_matrix.__str__()
else:
return "csb.numeric.InvertibleMatrix object holding the matrix:\n"\
+self._matrix.__str__()
def __len__(self):
if self._matrix is not None:
return len(self._matrix)
else:
return len(self._inverse_matrix)
def _deprecate_matrix(self):
self._matrix_updated = False
def _deprecate_inverse_matrix(self):
self._inverse_matrix_updated = False
def _update_matrix(self, matrix=None):
"""
Updates the _matrix field given a new matrix or by setting
it to the inverse of the _inverse_matrix field.
@param matrix: matrix-like object which the Matrix object
is supposed to represent
@type matrix: (n,n)-shaped numpy.ndarray or list
"""
if matrix is None:
self._matrix = numpy.linalg.inv(self._inverse_matrix)
else:
self._matrix = matrix
self._matrix_updated = True
self._deprecate_inverse_matrix()
def _update_inverse_matrix(self, inverse=None):
"""
Updates the __inverse_matrix field given a new matrix or by setting
it to the inverse of the _matrix field.
@param inverse: matrix-like object which the Matrix object
is supposed to represent
@type inverse: (n,n)-shaped numpy.ndarray or list
"""
if inverse is None:
if self.is_unity_multiple:
self._inverse_matrix = numpy.diag(1. / self._matrix.diagonal())
else:
self._inverse_matrix = numpy.linalg.inv(self._matrix)
else:
self._inverse_matrix = inverse
self._inverse_matrix_updated = True
@property
def inverse(self):
if self._inverse_matrix_updated == False:
self._update_inverse_matrix()
return self._inverse_matrix.copy()
@inverse.setter
def inverse(self, value):
if type(value) != numpy.ndarray:
raise TypeError("Arguments have to be of type numpy.ndarray!")
self._check_equal_shape(value, self._matrix)
self._update_inverse_matrix(numpy.array(value))
self._deprecate_matrix()
self._is_unity_multiple = self._check_unity_multiple(self._inverse_matrix)
@property
def is_unity_multiple(self):
"""
This property can be used to save computation time when drawing
from multivariate normal distributions with the covariance matrix
given by an instance of this object. By probing this property,
one can instead draw from normal distributions and rescale the samples
afterwards to avoid costly matrix inversions
"""
return self._is_unity_multiple
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