# -*- coding: utf-8 -*-
# pylint: disable=invalid-name, too-many-instance-attributes
"""Copyright 2015 Roger R Labbe Jr.
FilterPy library.
http://github.com/rlabbe/filterpy
Documentation at:
https://filterpy.readthedocs.org
Supporting book at:
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
This is licensed under an MIT license. See the readme.MD file
for more information.
"""
from __future__ import division
import numpy as np
from scipy.linalg import cholesky
from filterpy.common import pretty_str
[docs]class MerweScaledSigmaPoints(object):
"""
Generates sigma points and weights according to Van der Merwe's
2004 dissertation[1] for the UnscentedKalmanFilter class.. It
parametizes the sigma points using alpha, beta, kappa terms, and
is the version seen in most publications.
Unless you know better, this should be your default choice.
Parameters
----------
n : int
Dimensionality of the state. 2n+1 weights will be generated.
alpha : float
Determins the spread of the sigma points around the mean.
Usually a small positive value (1e-3) according to [3].
beta : float
Incorporates prior knowledge of the distribution of the mean. For
Gaussian x beta=2 is optimal, according to [3].
kappa : float, default=0.0
Secondary scaling parameter usually set to 0 according to [4],
or to 3-n according to [5].
sqrt_method : function(ndarray), default=scipy.linalg.cholesky
Defines how we compute the square root of a matrix, which has
no unique answer. Cholesky is the default choice due to its
speed. Typically your alternative choice will be
scipy.linalg.sqrtm. Different choices affect how the sigma points
are arranged relative to the eigenvectors of the covariance matrix.
Usually this will not matter to you; if so the default cholesky()
yields maximal performance. As of van der Merwe's dissertation of
2004 [6] this was not a well reseached area so I have no advice
to give you.
If your method returns a triangular matrix it must be upper
triangular. Do not use numpy.linalg.cholesky - for historical
reasons it returns a lower triangular matrix. The SciPy version
does the right thing.
subtract : callable (x, y), optional
Function that computes the difference between x and y.
You will have to supply this if your state variable cannot support
subtraction, such as angles (359-1 degreees is 2, not 358). x and y
are state vectors, not scalars.
Attributes
----------
Wm : np.array
weight for each sigma point for the mean
Wc : np.array
weight for each sigma point for the covariance
Examples
--------
See my book Kalman and Bayesian Filters in Python
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
References
----------
.. [1] R. Van der Merwe "Sigma-Point Kalman Filters for Probabilitic
Inference in Dynamic State-Space Models" (Doctoral dissertation)
"""
[docs] def __init__(self, n, alpha, beta, kappa, sqrt_method=None, subtract=None):
#pylint: disable=too-many-arguments
self.n = n
self.alpha = alpha
self.beta = beta
self.kappa = kappa
if sqrt_method is None:
self.sqrt = cholesky
else:
self.sqrt = sqrt_method
if subtract is None:
self.subtract = np.subtract
else:
self.subtract = subtract
self._compute_weights()
[docs] def num_sigmas(self):
""" Number of sigma points for each variable in the state x"""
return 2*self.n + 1
[docs] def sigma_points(self, x, P):
""" Computes the sigma points for an unscented Kalman filter
given the mean (x) and covariance(P) of the filter.
Returns tuple of the sigma points and weights.
Works with both scalar and array inputs:
sigma_points (5, 9, 2) # mean 5, covariance 9
sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I
Parameters
----------
x : An array-like object of the means of length n
Can be a scalar if 1D.
examples: 1, [1,2], np.array([1,2])
P : scalar, or np.array
Covariance of the filter. If scalar, is treated as eye(n)*P.
Returns
-------
sigmas : np.array, of size (n, 2n+1)
Two dimensional array of sigma points. Each column contains all of
the sigmas for one dimension in the problem space.
Ordered by Xi_0, Xi_{1..n}, Xi_{n+1..2n}
"""
if self.n != np.size(x):
raise ValueError("expected size(x) {}, but size is {}".format(
self.n, np.size(x)))
n = self.n
if np.isscalar(x):
x = np.asarray([x])
if np.isscalar(P):
P = np.eye(n)*P
else:
P = np.atleast_2d(P)
lambda_ = self.alpha**2 * (n + self.kappa) - n
U = self.sqrt((lambda_ + n)*P)
sigmas = np.zeros((2*n+1, n))
sigmas[0] = x
for k in range(n):
# pylint: disable=bad-whitespace
sigmas[k+1] = self.subtract(x, -U[k])
sigmas[n+k+1] = self.subtract(x, U[k])
return sigmas
def _compute_weights(self):
""" Computes the weights for the scaled unscented Kalman filter.
"""
n = self.n
lambda_ = self.alpha**2 * (n +self.kappa) - n
c = .5 / (n + lambda_)
self.Wc = np.full(2*n + 1, c)
self.Wm = np.full(2*n + 1, c)
self.Wc[0] = lambda_ / (n + lambda_) + (1 - self.alpha**2 + self.beta)
self.Wm[0] = lambda_ / (n + lambda_)
def __repr__(self):
return '\n'.join([
'MerweScaledSigmaPoints object',
pretty_str('n', self.n),
pretty_str('alpha', self.alpha),
pretty_str('beta', self.beta),
pretty_str('kappa', self.kappa),
pretty_str('Wm', self.Wm),
pretty_str('Wc', self.Wc),
pretty_str('subtract', self.subtract),
pretty_str('sqrt', self.sqrt)
])
[docs]class JulierSigmaPoints(object):
"""
Generates sigma points and weights according to Simon J. Julier
and Jeffery K. Uhlmann's original paper[1]. It parametizes the sigma
points using kappa.
Parameters
----------
n : int
Dimensionality of the state. 2n+1 weights will be generated.
kappa : float, default=0.
Scaling factor that can reduce high order errors. kappa=0 gives
the standard unscented filter. According to [Julier], if you set
kappa to 3-dim_x for a Gaussian x you will minimize the fourth
order errors in x and P.
sqrt_method : function(ndarray), default=scipy.linalg.cholesky
Defines how we compute the square root of a matrix, which has
no unique answer. Cholesky is the default choice due to its
speed. Typically your alternative choice will be
scipy.linalg.sqrtm. Different choices affect how the sigma points
are arranged relative to the eigenvectors of the covariance matrix.
Usually this will not matter to you; if so the default cholesky()
yields maximal performance. As of van der Merwe's dissertation of
2004 [6] this was not a well reseached area so I have no advice
to give you.
If your method returns a triangular matrix it must be upper
triangular. Do not use numpy.linalg.cholesky - for historical
reasons it returns a lower triangular matrix. The SciPy version
does the right thing.
subtract : callable (x, y), optional
Function that computes the difference between x and y.
You will have to supply this if your state variable cannot support
subtraction, such as angles (359-1 degreees is 2, not 358). x and y
Attributes
----------
Wm : np.array
weight for each sigma point for the mean
Wc : np.array
weight for each sigma point for the covariance
References
----------
.. [1] Julier, Simon J.; Uhlmann, Jeffrey "A New Extension of the Kalman
Filter to Nonlinear Systems". Proc. SPIE 3068, Signal Processing,
Sensor Fusion, and Target Recognition VI, 182 (July 28, 1997)
"""
[docs] def __init__(self, n, kappa=0., sqrt_method=None, subtract=None):
self.n = n
self.kappa = kappa
if sqrt_method is None:
self.sqrt = cholesky
else:
self.sqrt = sqrt_method
if subtract is None:
self.subtract = np.subtract
else:
self.subtract = subtract
self._compute_weights()
[docs] def num_sigmas(self):
""" Number of sigma points for each variable in the state x"""
return 2*self.n + 1
[docs] def sigma_points(self, x, P):
r""" Computes the sigma points for an unscented Kalman filter
given the mean (x) and covariance(P) of the filter.
kappa is an arbitrary constant. Returns sigma points.
Works with both scalar and array inputs:
sigma_points (5, 9, 2) # mean 5, covariance 9
sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I
Parameters
----------
x : array-like object of the means of length n
Can be a scalar if 1D.
examples: 1, [1,2], np.array([1,2])
P : scalar, or np.array
Covariance of the filter. If scalar, is treated as eye(n)*P.
kappa : float
Scaling factor.
Returns
-------
sigmas : np.array, of size (n, 2n+1)
2D array of sigma points :math:`\chi`. Each column contains all of
the sigmas for one dimension in the problem space. They
are ordered as:
.. math::
:nowrap:
\begin{eqnarray}
\chi[0] = &x \\
\chi[1..n] = &x + [\sqrt{(n+\kappa)P}]_k \\
\chi[n+1..2n] = &x - [\sqrt{(n+\kappa)P}]_k
\end{eqnarray}
"""
if self.n != np.size(x):
raise ValueError("expected size(x) {}, but size is {}".format(
self.n, np.size(x)))
n = self.n
if np.isscalar(x):
x = np.asarray([x])
n = np.size(x) # dimension of problem
if np.isscalar(P):
P = np.eye(n) * P
else:
P = np.atleast_2d(P)
sigmas = np.zeros((2*n+1, n))
# implements U'*U = (n+kappa)*P. Returns lower triangular matrix.
# Take transpose so we can access with U[i]
U = self.sqrt((n + self.kappa) * P)
sigmas[0] = x
for k in range(n):
# pylint: disable=bad-whitespace
sigmas[k+1] = self.subtract(x, -U[k])
sigmas[n+k+1] = self.subtract(x, U[k])
return sigmas
def _compute_weights(self):
""" Computes the weights for the unscented Kalman filter. In this
formulation the weights for the mean and covariance are the same.
"""
n = self.n
k = self.kappa
self.Wm = np.full(2*n+1, .5 / (n + k))
self.Wm[0] = k / (n+k)
self.Wc = self.Wm
def __repr__(self):
return '\n'.join([
'JulierSigmaPoints object',
pretty_str('n', self.n),
pretty_str('kappa', self.kappa),
pretty_str('Wm', self.Wm),
pretty_str('Wc', self.Wc),
pretty_str('subtract', self.subtract),
pretty_str('sqrt', self.sqrt)
])
[docs]class SimplexSigmaPoints(object):
"""
Generates sigma points and weights according to the simplex
method presented in [1].
Parameters
----------
n : int
Dimensionality of the state. n+1 weights will be generated.
sqrt_method : function(ndarray), default=scipy.linalg.cholesky
Defines how we compute the square root of a matrix, which has
no unique answer. Cholesky is the default choice due to its
speed. Typically your alternative choice will be
scipy.linalg.sqrtm
If your method returns a triangular matrix it must be upper
triangular. Do not use numpy.linalg.cholesky - for historical
reasons it returns a lower triangular matrix. The SciPy version
does the right thing.
subtract : callable (x, y), optional
Function that computes the difference between x and y.
You will have to supply this if your state variable cannot support
subtraction, such as angles (359-1 degreees is 2, not 358). x and y
are state vectors, not scalars.
Attributes
----------
Wm : np.array
weight for each sigma point for the mean
Wc : np.array
weight for each sigma point for the covariance
References
----------
.. [1] Phillippe Moireau and Dominique Chapelle "Reduced-Order
Unscented Kalman Filtering with Application to Parameter
Identification in Large-Dimensional Systems"
DOI: 10.1051/cocv/2010006
"""
[docs] def __init__(self, n, alpha=1, sqrt_method=None, subtract=None):
self.n = n
self.alpha = alpha
if sqrt_method is None:
self.sqrt = cholesky
else:
self.sqrt = sqrt_method
if subtract is None:
self.subtract = np.subtract
else:
self.subtract = subtract
self._compute_weights()
[docs] def num_sigmas(self):
""" Number of sigma points for each variable in the state x"""
return self.n + 1
[docs] def sigma_points(self, x, P):
"""
Computes the implex sigma points for an unscented Kalman filter
given the mean (x) and covariance(P) of the filter.
Returns tuple of the sigma points and weights.
Works with both scalar and array inputs:
sigma_points (5, 9, 2) # mean 5, covariance 9
sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I
Parameters
----------
x : An array-like object of the means of length n
Can be a scalar if 1D.
examples: 1, [1,2], np.array([1,2])
P : scalar, or np.array
Covariance of the filter. If scalar, is treated as eye(n)*P.
Returns
-------
sigmas : np.array, of size (n, n+1)
Two dimensional array of sigma points. Each column contains all of
the sigmas for one dimension in the problem space.
Ordered by Xi_0, Xi_{1..n}
"""
if self.n != np.size(x):
raise ValueError("expected size(x) {}, but size is {}".format(
self.n, np.size(x)))
n = self.n
if np.isscalar(x):
x = np.asarray([x])
x = x.reshape(-1, 1)
if np.isscalar(P):
P = np.eye(n) * P
else:
P = np.atleast_2d(P)
U = self.sqrt(P)
lambda_ = n / (n + 1)
Istar = np.array([[-1/np.sqrt(2*lambda_), 1/np.sqrt(2*lambda_)]])
for d in range(2, n+1):
row = np.ones((1, Istar.shape[1] + 1)) * 1. / np.sqrt(lambda_*d*(d + 1))
row[0, -1] = -d / np.sqrt(lambda_ * d * (d + 1))
Istar = np.r_[np.c_[Istar, np.zeros((Istar.shape[0]))], row]
I = np.sqrt(n)*Istar
scaled_unitary = U.dot(I)
sigmas = self.subtract(x, -scaled_unitary)
return sigmas.T
def _compute_weights(self):
""" Computes the weights for the scaled unscented Kalman filter. """
n = self.n
c = 1. / (n + 1)
self.Wm = np.full(n + 1, c)
self.Wc = self.Wm
def __repr__(self):
return '\n'.join([
'SimplexSigmaPoints object',
pretty_str('n', self.n),
pretty_str('alpha', self.alpha),
pretty_str('Wm', self.Wm),
pretty_str('Wc', self.Wc),
pretty_str('subtract', self.subtract),
pretty_str('sqrt', self.sqrt)
])