# Source code for filterpy.kalman.EKF

```
# -*- coding: utf-8 -*-
# pylint: disable=invalid-name,too-many-instance-attributes, too-many-arguments
"""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, unicode_literals)
from copy import deepcopy
from math import log, exp, sqrt
import sys
import numpy as np
from numpy import dot, zeros, eye
import scipy.linalg as linalg
from filterpy.stats import logpdf
from filterpy.common import pretty_str, reshape_z
[docs]class ExtendedKalmanFilter(object):
""" Implements an extended Kalman filter (EKF). You are responsible for
setting the various state variables to reasonable values; the defaults
will not give you a functional filter.
You will have to set the following attributes after constructing this
object for the filter to perform properly. Please note that there are
various checks in place to ensure that you have made everything the
'correct' size. However, it is possible to provide incorrectly sized
arrays such that the linear algebra can not perform an operation.
It can also fail silently - you can end up with matrices of a size that
allows the linear algebra to work, but are the wrong shape for the problem
you are trying to solve.
Parameters
----------
dim_x : int
Number of state variables for the Kalman filter. For example, if
you are tracking the position and velocity of an object in two
dimensions, dim_x would be 4.
This is used to set the default size of P, Q, and u
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.
Attributes
----------
x : numpy.array(dim_x, 1)
State estimate vector
P : numpy.array(dim_x, dim_x)
Covariance matrix
x_prior : numpy.array(dim_x, 1)
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, 1)
Posterior (updated) state estimate. Read Only.
P_post : numpy.array(dim_x, dim_x)
Posterior (updated) state covariance matrix. Read Only.
R : numpy.array(dim_z, dim_z)
Measurement noise matrix
Q : numpy.array(dim_x, dim_x)
Process noise matrix
F : numpy.array()
State Transition matrix
H : numpy.array(dim_x, dim_x)
Measurement function
y : numpy.array
Residual of the update step. Read only.
K : numpy.array(dim_x, dim_z)
Kalman gain of the update step. Read only.
S : numpy.array
Systen uncertaintly projected to measurement space. Read only.
z : ndarray
Last measurement used in update(). Read only.
log_likelihood : float
log-likelihood of the last measurement. Read only.
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 innovation. E.g. 3 means measurement
was 3 standard deviations away from the predicted value.
Read only.
Examples
--------
See my book Kalman and Bayesian Filters in Python
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
"""
[docs] def __init__(self, dim_x, dim_z, dim_u=0):
self.dim_x = dim_x
self.dim_z = dim_z
self.dim_u = dim_u
self.x = zeros((dim_x, 1)) # state
self.P = eye(dim_x) # uncertainty covariance
self.B = 0 # control transition matrix
self.F = np.eye(dim_x) # state transition matrix
self.R = eye(dim_z) # state uncertainty
self.Q = eye(dim_x) # process uncertainty
self.y = zeros((dim_z, 1)) # residual
z = np.array([None]*self.dim_z)
self.z = reshape_z(z, self.dim_z, self.x.ndim)
# gain and residual are computed during the innovation step. We
# save them so that in case you want to inspect them for various
# purposes
self.K = np.zeros(self.x.shape) # kalman gain
self.y = zeros((dim_z, 1))
self.S = np.zeros((dim_z, dim_z)) # system uncertainty
self.SI = np.zeros((dim_z, dim_z)) # inverse system uncertainty
# identity matrix. Do not alter this.
self._I = np.eye(dim_x)
self._log_likelihood = log(sys.float_info.min)
self._likelihood = sys.float_info.min
self._mahalanobis = None
# 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_update(self, z, HJacobian, Hx, args=(), hx_args=(), u=0):
""" Performs the predict/update innovation of the extended Kalman
filter.
Parameters
----------
z : np.array
measurement for this step.
If `None`, only predict step is perfomed.
HJacobian : function
function which computes the Jacobian of the H matrix (measurement
function). Takes state variable (self.x) as input, along with the
optional arguments in args, and returns H.
Hx : function
function which takes as input the state variable (self.x) along
with the optional arguments in hx_args, and returns the measurement
that would correspond to that state.
args : tuple, optional, default (,)
arguments to be passed into HJacobian after the required state
variable.
hx_args : tuple, optional, default (,)
arguments to be passed into Hx after the required state
variable.
u : np.array or scalar
optional control vector input to the filter.
"""
#pylint: disable=too-many-locals
if not isinstance(args, tuple):
args = (args,)
if not isinstance(hx_args, tuple):
hx_args = (hx_args,)
if np.isscalar(z) and self.dim_z == 1:
z = np.asarray([z], float)
F = self.F
B = self.B
P = self.P
Q = self.Q
R = self.R
x = self.x
H = HJacobian(x, *args)
# predict step
x = dot(F, x) + dot(B, u)
P = dot(F, P).dot(F.T) + Q
# save prior
self.x_prior = np.copy(self.x)
self.P_prior = np.copy(self.P)
# update step
PHT = dot(P, H.T)
self.S = dot(H, PHT) + R
self.SI = linalg.inv(self.S)
self.K = dot(PHT, self.SI)
self.y = z - Hx(x, *hx_args)
self.x = x + dot(self.K, self.y)
I_KH = self._I - dot(self.K, H)
self.P = dot(I_KH, P).dot(I_KH.T) + dot(self.K, R).dot(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 update(self, z, HJacobian, Hx, R=None, args=(), hx_args=(),
residual=np.subtract):
""" Performs the update innovation of the extended Kalman filter.
Parameters
----------
z : np.array
measurement for this step.
If `None`, posterior is not computed
HJacobian : function
function which computes the Jacobian of the H matrix (measurement
function). Takes state variable (self.x) as input, returns H.
Hx : function
function which takes as input the state variable (self.x) along
with the optional arguments in hx_args, and returns the measurement
that would correspond to that state.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
args : tuple, optional, default (,)
arguments to be passed into HJacobian after the required state
variable. for robot localization you might need to pass in
information about the map and time of day, so you might have
`args=(map_data, time)`, where the signature of HCacobian will
be `def HJacobian(x, map, t)`
hx_args : tuple, optional, default (,)
arguments to be passed into Hx function after the required state
variable.
residual : function (z, z2), optional
Optional function that computes the residual (difference) between
the two measurement vectors. If you do not provide this, then the
built in minus operator will be used. You will normally want to use
the built in unless your residual computation is nonlinear (for
example, if they are angles)
"""
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 not isinstance(args, tuple):
args = (args,)
if not isinstance(hx_args, tuple):
hx_args = (hx_args,)
if R is None:
R = self.R
elif np.isscalar(R):
R = eye(self.dim_z) * R
if np.isscalar(z) and self.dim_z == 1:
z = np.asarray([z], float)
H = HJacobian(self.x, *args)
PHT = dot(self.P, H.T)
self.S = dot(H, PHT) + R
self.K = PHT.dot(linalg.inv(self.S))
hx = Hx(self.x, *hx_args)
self.y = residual(z, hx)
self.x = self.x + dot(self.K, self.y)
# P = (I-KH)P(I-KH)' + KRK' is more numerically stable
# and works for non-optimal K vs the equation
# P = (I-KH)P usually seen in the literature.
I_KH = self._I - dot(self.K, H)
self.P = dot(I_KH, self.P).dot(I_KH.T) + dot(self.K, R).dot(self.K.T)
# set to None to force recompute
self._log_likelihood = None
self._likelihood = None
self._mahalanobis = None
# save measurement and posterior state
self.z = deepcopy(z)
self.x_post = self.x.copy()
self.P_post = self.P.copy()
[docs] def predict_x(self, u=0):
"""
Predicts the next state of X. If you need to
compute the next state yourself, override this function. You would
need to do this, for example, if the usual Taylor expansion to
generate F is not providing accurate results for you.
"""
self.x = dot(self.F, self.x) + dot(self.B, u)
[docs] def predict(self, u=0):
"""
Predict next state (prior) using the Kalman filter state propagation
equations.
Parameters
----------
u : np.array
Optional control vector. If non-zero, it is multiplied by B
to create the control input into the system.
"""
self.predict_x(u)
self.P = dot(self.F, self.P).dot(self.F.T) + self.Q
# save prior
self.x_prior = np.copy(self.x)
self.P_prior = np.copy(self.P)
@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 innovation. 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([
'KalmanFilter 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('F', self.F),
pretty_str('Q', self.Q),
pretty_str('R', self.R),
pretty_str('K', self.K),
pretty_str('y', self.y),
pretty_str('S', self.S),
pretty_str('likelihood', self.likelihood),
pretty_str('log-likelihood', self.log_likelihood),
pretty_str('mahalanobis', self.mahalanobis)
])
```