# Source code for filterpy.kalman.UKF

```
# -*- coding: utf-8 -*-
# pylint: disable=invalid-name
"""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 (absolute_import, division)
from copy import deepcopy
from math import log, exp, sqrt
import sys
import numpy as np
from numpy import eye, zeros, dot, isscalar, outer
from scipy.linalg import cholesky
from filterpy.kalman import unscented_transform
from filterpy.stats import logpdf
from filterpy.common import pretty_str
[docs]class UnscentedKalmanFilter(object):
# pylint: disable=too-many-instance-attributes
# pylint: disable=invalid-name
r"""
Implements the Scaled Unscented Kalman filter (UKF) as defined by
Simon Julier in [1], using the formulation provided by Wan and Merle
in [2]. This filter scales the sigma points to avoid strong nonlinearities.
Parameters
----------
dim_x : int
Number of state variables for the filter. For example, if
you are tracking the position and velocity of an object in two
dimensions, dim_x would be 4.
dim_z : int
Number of of measurement inputs. For example, if the sensor
provides you with position in (x,y), dim_z would be 2.
This is for convience, so everything is sized correctly on
creation. If you are using multiple sensors the size of `z` can
change based on the sensor. Just provide the appropriate hx function
dt : float
Time between steps in seconds.
hx : function(x)
Measurement function. Converts state vector x into a measurement
vector of shape (dim_z).
fx : function(x,dt)
function that returns the state x transformed by the
state transistion function. dt is the time step in seconds.
points : class
Class which computes the sigma points and weights for a UKF
algorithm. You can vary the UKF implementation by changing this
class. For example, MerweScaledSigmaPoints implements the alpha,
beta, kappa parameterization of Van der Merwe, and
JulierSigmaPoints implements Julier's original kappa
parameterization. See either of those for the required
signature of this class if you want to implement your own.
sqrt_fn : callable(ndarray), default=None (implies 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 as far as this class is concerned.
x_mean_fn : callable (sigma_points, weights), optional
Function that computes the mean of the provided sigma points
and weights. Use this if your state variable contains nonlinear
values such as angles which cannot be summed.
.. code-block:: Python
def state_mean(sigmas, Wm):
x = np.zeros(3)
sum_sin, sum_cos = 0., 0.
for i in range(len(sigmas)):
s = sigmas[i]
x[0] += s[0] * Wm[i]
x[1] += s[1] * Wm[i]
sum_sin += sin(s[2])*Wm[i]
sum_cos += cos(s[2])*Wm[i]
x[2] = atan2(sum_sin, sum_cos)
return x
z_mean_fn : callable (sigma_points, weights), optional
Same as x_mean_fn, except it is called for sigma points which
form the measurements after being passed through hx().
residual_x : callable (x, y), optional
residual_z : callable (x, y), optional
Function that computes the residual (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. One is for the state variable,
the other is for the measurement state.
.. code-block:: Python
def residual(a, b):
y = a[0] - b[0]
if y > np.pi:
y -= 2*np.pi
if y < -np.pi:
y = 2*np.pi
return y
Attributes
----------
x : numpy.array(dim_x)
state estimate vector
P : numpy.array(dim_x, dim_x)
covariance estimate matrix
x_prior : numpy.array(dim_x)
Prior (predicted) state estimate. The *_prior and *_post attributes
are for convienence; they store the prior and posterior of the
current epoch. Read Only.
P_prior : numpy.array(dim_x, dim_x)
Prior (predicted) state covariance matrix. Read Only.
x_post : numpy.array(dim_x)
Posterior (updated) state estimate. Read Only.
P_post : numpy.array(dim_x, dim_x)
Posterior (updated) state covariance matrix. Read Only.
z : ndarray
Last measurement used in update(). Read only.
R : numpy.array(dim_z, dim_z)
measurement noise matrix
Q : numpy.array(dim_x, dim_x)
process noise matrix
K : numpy.array
Kalman gain
y : numpy.array
innovation residual
log_likelihood : scalar
Log likelihood of last measurement update.
likelihood : float
likelihood of last measurment. Read only.
Computed from the log-likelihood. The log-likelihood can be very
small, meaning a large negative value such as -28000. Taking the
exp() of that results in 0.0, which can break typical algorithms
which multiply by this value, so by default we always return a
number >= sys.float_info.min.
mahalanobis : float
mahalanobis distance of the measurement. Read only.
inv : function, default numpy.linalg.inv
If you prefer another inverse function, such as the Moore-Penrose
pseudo inverse, set it to that instead:
.. code-block:: Python
kf.inv = np.linalg.pinv
Examples
--------
Simple example of a linear order 1 kinematic filter in 2D. There is no
need to use a UKF for this example, but it is easy to read.
>>> def fx(x, dt):
>>> # state transition function - predict next state based
>>> # on constant velocity model x = vt + x_0
>>> F = np.array([[1, dt, 0, 0],
>>> [0, 1, 0, 0],
>>> [0, 0, 1, dt],
>>> [0, 0, 0, 1]], dtype=float)
>>> return np.dot(F, x)
>>>
>>> def hx(x):
>>> # measurement function - convert state into a measurement
>>> # where measurements are [x_pos, y_pos]
>>> return np.array([x[0], x[2]])
>>>
>>> dt = 0.1
>>> # create sigma points to use in the filter. This is standard for Gaussian processes
>>> points = MerweScaledSigmaPoints(4, alpha=.1, beta=2., kappa=-1)
>>>
>>> kf = UnscentedKalmanFilter(dim_x=4, dim_z=2, dt=dt, fx=fx, hx=hx, points=points)
>>> kf.x = np.array([-1., 1., -1., 1]) # initial state
>>> kf.P *= 0.2 # initial uncertainty
>>> z_std = 0.1
>>> kf.R = np.diag([z_std**2, z_std**2]) # 1 standard
>>> kf.Q = Q_discrete_white_noise(dim=2, dt=dt, var=0.01**2, block_size=2)
>>>
>>> zs = [[i+randn()*z_std, i+randn()*z_std] for i in range(50)] # measurements
>>> for z in zs:
>>> kf.predict()
>>> kf.update(z)
>>> print(kf.x, 'log-likelihood', kf.log_likelihood)
For in depth explanations see my book Kalman and Bayesian Filters in Python
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
Also see the filterpy/kalman/tests subdirectory for test code that
may be illuminating.
References
----------
.. [1] Julier, Simon J. "The scaled unscented transformation,"
American Control Converence, 2002, pp 4555-4559, vol 6.
Online copy:
https://www.cs.unc.edu/~welch/kalman/media/pdf/ACC02-IEEE1357.PDF
.. [2] E. A. Wan and R. Van der Merwe, “The unscented Kalman filter for
nonlinear estimation,” in Proc. Symp. Adaptive Syst. Signal
Process., Commun. Contr., Lake Louise, AB, Canada, Oct. 2000.
Online Copy:
https://www.seas.harvard.edu/courses/cs281/papers/unscented.pdf
.. [3] S. Julier, J. Uhlmann, and H. Durrant-Whyte. "A new method for
the nonlinear transformation of means and covariances in filters
and estimators," IEEE Transactions on Automatic Control, 45(3),
pp. 477-482 (March 2000).
.. [4] E. A. Wan and R. Van der Merwe, “The Unscented Kalman filter for
Nonlinear Estimation,” in Proc. Symp. Adaptive Syst. Signal
Process., Commun. Contr., Lake Louise, AB, Canada, Oct. 2000.
https://www.seas.harvard.edu/courses/cs281/papers/unscented.pdf
.. [5] Wan, Merle "The Unscented Kalman Filter," chapter in *Kalman
Filtering and Neural Networks*, John Wiley & Sons, Inc., 2001.
.. [6] R. Van der Merwe "Sigma-Point Kalman Filters for Probabilitic
Inference in Dynamic State-Space Models" (Doctoral dissertation)
"""
[docs] def __init__(self, dim_x, dim_z, dt, hx, fx, points,
sqrt_fn=None, x_mean_fn=None, z_mean_fn=None,
residual_x=None,
residual_z=None):
"""
Create a Kalman filter. You are responsible for setting the
various state variables to reasonable values; the defaults below will
not give you a functional filter.
"""
#pylint: disable=too-many-arguments
self.x = zeros(dim_x)
self.P = eye(dim_x)
self.x_prior = np.copy(self.x)
self.P_prior = np.copy(self.P)
self.Q = eye(dim_x)
self.R = eye(dim_z)
self._dim_x = dim_x
self._dim_z = dim_z
self.points_fn = points
self._dt = dt
self._num_sigmas = points.num_sigmas()
self.hx = hx
self.fx = fx
self.x_mean = x_mean_fn
self.z_mean = z_mean_fn
# Only computed only if requested via property
self._log_likelihood = log(sys.float_info.min)
self._likelihood = sys.float_info.min
self._mahalanobis = None
if sqrt_fn is None:
self.msqrt = cholesky
else:
self.msqrt = sqrt_fn
# weights for the means and covariances.
self.Wm, self.Wc = points.Wm, points.Wc
if residual_x is None:
self.residual_x = np.subtract
else:
self.residual_x = residual_x
if residual_z is None:
self.residual_z = np.subtract
else:
self.residual_z = residual_z
# sigma points transformed through f(x) and h(x)
# variables for efficiency so we don't recreate every update
self.sigmas_f = zeros((self._num_sigmas, self._dim_x))
self.sigmas_h = zeros((self._num_sigmas, self._dim_z))
self.K = np.zeros((dim_x, dim_z)) # Kalman gain
self.y = np.zeros((dim_z)) # residual
self.z = np.array([[None]*dim_z]).T # measurement
self.S = np.zeros((dim_z, dim_z)) # system uncertainty
self.SI = np.zeros((dim_z, dim_z)) # inverse system uncertainty
self.inv = np.linalg.inv
# these will always be a copy of x,P after predict() is called
self.x_prior = self.x.copy()
self.P_prior = self.P.copy()
# these will always be a copy of x,P after update() is called
self.x_post = self.x.copy()
self.P_post = self.P.copy()
[docs] def predict(self, dt=None, UT=None, fx=None, **fx_args):
r"""
Performs the predict step of the UKF. On return, self.x and
self.P contain the predicted state (x) and covariance (P). '
Important: this MUST be called before update() is called for the first
time.
Parameters
----------
dt : double, optional
If specified, the time step to be used for this prediction.
self._dt is used if this is not provided.
fx : callable f(x, **fx_args), optional
State transition function. If not provided, the default
function passed in during construction will be used.
UT : function(sigmas, Wm, Wc, noise_cov), optional
Optional function to compute the unscented transform for the sigma
points passed through hx. Typically the default function will
work - you can use x_mean_fn and z_mean_fn to alter the behavior
of the unscented transform.
**fx_args : keyword arguments
optional keyword arguments to be passed into f(x).
"""
if dt is None:
dt = self._dt
if UT is None:
UT = unscented_transform
# calculate sigma points for given mean and covariance
self.compute_process_sigmas(dt, fx, **fx_args)
#and pass sigmas through the unscented transform to compute prior
self.x, self.P = UT(self.sigmas_f, self.Wm, self.Wc, self.Q,
self.x_mean, self.residual_x)
# save prior
self.x_prior = np.copy(self.x)
self.P_prior = np.copy(self.P)
[docs] def update(self, z, R=None, UT=None, hx=None, **hx_args):
"""
Update the UKF with the given measurements. On return,
self.x and self.P contain the new mean and covariance of the filter.
Parameters
----------
z : numpy.array of shape (dim_z)
measurement vector
R : numpy.array((dim_z, dim_z)), optional
Measurement noise. If provided, overrides self.R for
this function call.
UT : function(sigmas, Wm, Wc, noise_cov), optional
Optional function to compute the unscented transform for the sigma
points passed through hx. Typically the default function will
work - you can use x_mean_fn and z_mean_fn to alter the behavior
of the unscented transform.
**hx_args : keyword argument
arguments to be passed into h(x) after x -> h(x, **hx_args)
"""
if z is None:
self.z = np.array([[None]*self._dim_z]).T
self.x_post = self.x.copy()
self.P_post = self.P.copy()
return
if hx is None:
hx = self.hx
if UT is None:
UT = unscented_transform
if R is None:
R = self.R
elif isscalar(R):
R = eye(self._dim_z) * R
# pass prior sigmas through h(x) to get measurement sigmas
# the shape of sigmas_h will vary if the shape of z varies, so
# recreate each time
sigmas_h = []
for s in self.sigmas_f:
sigmas_h.append(hx(s, **hx_args))
self.sigmas_h = np.atleast_2d(sigmas_h)
# mean and covariance of prediction passed through unscented transform
zp, self.S = UT(self.sigmas_h, self.Wm, self.Wc, R, self.z_mean, self.residual_z)
self.SI = self.inv(self.S)
# compute cross variance of the state and the measurements
Pxz = self.cross_variance(self.x, zp, self.sigmas_f, self.sigmas_h)
self.K = dot(Pxz, self.SI) # Kalman gain
self.y = self.residual_z(z, zp) # residual
# update Gaussian state estimate (x, P)
self.x = self.x + dot(self.K, self.y)
self.P = self.P - dot(self.K, dot(self.S, self.K.T))
# save measurement and posterior state
self.z = deepcopy(z)
self.x_post = self.x.copy()
self.P_post = self.P.copy()
# set to None to force recompute
self._log_likelihood = None
self._likelihood = None
self._mahalanobis = None
[docs] def cross_variance(self, x, z, sigmas_f, sigmas_h):
"""
Compute cross variance of the state `x` and measurement `z`.
"""
Pxz = zeros((sigmas_f.shape[1], sigmas_h.shape[1]))
N = sigmas_f.shape[0]
for i in range(N):
dx = self.residual_x(sigmas_f[i], x)
dz = self.residual_z(sigmas_h[i], z)
Pxz += self.Wc[i] * outer(dx, dz)
return Pxz
[docs] def compute_process_sigmas(self, dt, fx=None, **fx_args):
"""
computes the values of sigmas_f. Normally a user would not call
this, but it is useful if you need to call update more than once
between calls to predict (to update for multiple simultaneous
measurements), so the sigmas correctly reflect the updated state
x, P.
"""
if fx is None:
fx = self.fx
# calculate sigma points for given mean and covariance
sigmas = self.points_fn.sigma_points(self.x, self.P)
for i, s in enumerate(sigmas):
self.sigmas_f[i] = fx(s, dt, **fx_args)
[docs] def batch_filter(self, zs, Rs=None, dts=None, UT=None, saver=None):
"""
Performs the UKF filter over the list of measurement in `zs`.
Parameters
----------
zs : list-like
list of measurements at each time step `self._dt` Missing
measurements must be represented by 'None'.
Rs : None, np.array or list-like, default=None
optional list of values to use for the measurement error
covariance R.
If Rs is None then self.R is used for all epochs.
If it is a list of matrices or a 3D array where
len(Rs) == len(zs), then it is treated as a list of R values, one
per epoch. This allows you to have varying R per epoch.
dts : None, scalar or list-like, default=None
optional value or list of delta time to be passed into predict.
If dtss is None then self.dt is used for all epochs.
If it is a list where len(dts) == len(zs), then it is treated as a
list of dt values, one per epoch. This allows you to have varying
epoch durations.
UT : function(sigmas, Wm, Wc, noise_cov), optional
Optional function to compute the unscented transform for the sigma
points passed through hx. Typically the default function will
work - you can use x_mean_fn and z_mean_fn to alter the behavior
of the unscented transform.
saver : filterpy.common.Saver, optional
filterpy.common.Saver object. If provided, saver.save() will be
called after every epoch
Returns
-------
means: ndarray((n,dim_x,1))
array of the state for each time step after the update. Each entry
is an np.array. In other words `means[k,:]` is the state at step
`k`.
covariance: ndarray((n,dim_x,dim_x))
array of the covariances for each time step after the update.
In other words `covariance[k,:,:]` is the covariance at step `k`.
Examples
--------
.. code-block:: Python
# this example demonstrates tracking a measurement where the time
# between measurement varies, as stored in dts The output is then smoothed
# with an RTS smoother.
zs = [t + random.randn()*4 for t in range (40)]
(mu, cov, _, _) = ukf.batch_filter(zs, dts=dts)
(xs, Ps, Ks) = ukf.rts_smoother(mu, cov)
"""
#pylint: disable=too-many-arguments
try:
z = zs[0]
except TypeError:
raise TypeError('zs must be list-like')
if self._dim_z == 1:
if not(isscalar(z) or (z.ndim == 1 and len(z) == 1)):
raise TypeError('zs must be a list of scalars or 1D, 1 element arrays')
else:
if len(z) != self._dim_z:
raise TypeError(
'each element in zs must be a 1D array of length {}'.format(self._dim_z))
z_n = np.size(zs, 0)
if Rs is None:
Rs = [self.R] * z_n
if dts is None:
dts = [self._dt] * z_n
# mean estimates from Kalman Filter
if self.x.ndim == 1:
means = zeros((z_n, self._dim_x))
else:
means = zeros((z_n, self._dim_x, 1))
# state covariances from Kalman Filter
covariances = zeros((z_n, self._dim_x, self._dim_x))
for i, (z, r, dt) in enumerate(zip(zs, Rs, dts)):
self.predict(dt=dt, UT=UT)
self.update(z, r, UT=UT)
means[i, :] = self.x
covariances[i, :, :] = self.P
if saver is not None:
saver.save()
return (means, covariances)
[docs] def rts_smoother(self, Xs, Ps, Qs=None, dts=None, UT=None):
"""
Runs the Rauch-Tung-Striebal Kalman smoother on a set of
means and covariances computed by the UKF. The usual input
would come from the output of `batch_filter()`.
Parameters
----------
Xs : numpy.array
array of the means (state variable x) of the output of a Kalman
filter.
Ps : numpy.array
array of the covariances of the output of a kalman filter.
Qs: list-like collection of numpy.array, optional
Process noise of the Kalman filter at each time step. Optional,
if not provided the filter's self.Q will be used
dt : optional, float or array-like of float
If provided, specifies the time step of each step of the filter.
If float, then the same time step is used for all steps. If
an array, then each element k contains the time at step k.
Units are seconds.
UT : function(sigmas, Wm, Wc, noise_cov), optional
Optional function to compute the unscented transform for the sigma
points passed through hx. Typically the default function will
work - you can use x_mean_fn and z_mean_fn to alter the behavior
of the unscented transform.
Returns
-------
x : numpy.ndarray
smoothed means
P : numpy.ndarray
smoothed state covariances
K : numpy.ndarray
smoother gain at each step
Examples
--------
.. code-block:: Python
zs = [t + random.randn()*4 for t in range (40)]
(mu, cov, _, _) = kalman.batch_filter(zs)
(x, P, K) = rts_smoother(mu, cov, fk.F, fk.Q)
"""
#pylint: disable=too-many-locals, too-many-arguments
if len(Xs) != len(Ps):
raise ValueError('Xs and Ps must have the same length')
n, dim_x = Xs.shape
if dts is None:
dts = [self._dt] * n
elif isscalar(dts):
dts = [dts] * n
if Qs is None:
Qs = [self.Q] * n
if UT is None:
UT = unscented_transform
# smoother gain
Ks = zeros((n, dim_x, dim_x))
num_sigmas = self._num_sigmas
xs, ps = Xs.copy(), Ps.copy()
sigmas_f = zeros((num_sigmas, dim_x))
for k in reversed(range(n-1)):
# create sigma points from state estimate, pass through state func
sigmas = self.points_fn.sigma_points(xs[k], ps[k])
for i in range(num_sigmas):
sigmas_f[i] = self.fx(sigmas[i], dts[k])
xb, Pb = UT(
sigmas_f, self.Wm, self.Wc, self.Q,
self.x_mean, self.residual_x)
# compute cross variance
Pxb = 0
for i in range(num_sigmas):
y = self.residual_x(sigmas_f[i], xb)
z = self.residual_x(sigmas[i], Xs[k])
Pxb += self.Wc[i] * outer(z, y)
# compute gain
K = dot(Pxb, self.inv(Pb))
# update the smoothed estimates
xs[k] += dot(K, self.residual_x(xs[k+1], xb))
ps[k] += dot(K, ps[k+1] - Pb).dot(K.T)
Ks[k] = K
return (xs, ps, Ks)
@property
def log_likelihood(self):
"""
log-likelihood of the last measurement.
"""
if self._log_likelihood is None:
self._log_likelihood = logpdf(x=self.y, cov=self.S)
return self._log_likelihood
@property
def likelihood(self):
"""
Computed from the log-likelihood. The log-likelihood can be very
small, meaning a large negative value such as -28000. Taking the
exp() of that results in 0.0, which can break typical algorithms
which multiply by this value, so by default we always return a
number >= sys.float_info.min.
"""
if self._likelihood is None:
self._likelihood = exp(self.log_likelihood)
if self._likelihood == 0:
self._likelihood = sys.float_info.min
return self._likelihood
@property
def mahalanobis(self):
""""
Mahalanobis distance of measurement. E.g. 3 means measurement
was 3 standard deviations away from the predicted value.
Returns
-------
mahalanobis : float
"""
if self._mahalanobis is None:
self._mahalanobis = sqrt(float(dot(dot(self.y.T, self.SI), self.y)))
return self._mahalanobis
def __repr__(self):
return '\n'.join([
'UnscentedKalmanFilter object',
pretty_str('x', self.x),
pretty_str('P', self.P),
pretty_str('x_prior', self.x_prior),
pretty_str('P_prior', self.P_prior),
pretty_str('Q', self.Q),
pretty_str('R', self.R),
pretty_str('S', self.S),
pretty_str('K', self.K),
pretty_str('y', self.y),
pretty_str('log-likelihood', self.log_likelihood),
pretty_str('likelihood', self.likelihood),
pretty_str('mahalanobis', self.mahalanobis),
pretty_str('sigmas_f', self.sigmas_f),
pretty_str('h', self.sigmas_h),
pretty_str('Wm', self.Wm),
pretty_str('Wc', self.Wc),
pretty_str('residual_x', self.residual_x),
pretty_str('residual_z', self.residual_z),
pretty_str('msqrt', self.msqrt),
pretty_str('hx', self.hx),
pretty_str('fx', self.fx),
pretty_str('x_mean', self.x_mean),
pretty_str('z_mean', self.z_mean)
])
```