# Source code for filterpy.kalman.kalman_filter

```
# -*- coding: utf-8 -*-
# pylint: disable=invalid-name, too-many-arguments, too-many-branches,
# pylint: disable=too-many-locals, too-many-instance-attributes, too-many-lines
"""
This module implements the linear Kalman filter in both an object
oriented and procedural form. The KalmanFilter class implements
the filter by storing the various matrices in instance variables,
minimizing the amount of bookkeeping you have to do.
All Kalman filters operate with a predict->update cycle. The
predict step, implemented with the method or function predict(),
uses the state transition matrix F to predict the state in the next
time period (epoch). The state is stored as a gaussian (x, P), where
x is the state (column) vector, and P is its covariance. Covariance
matrix Q specifies the process covariance. In Bayesian terms, this
prediction is called the *prior*, which you can think of colloquially
as the estimate prior to incorporating the measurement.
The update step, implemented with the method or function `update()`,
incorporates the measurement z with covariance R, into the state
estimate (x, P). The class stores the system uncertainty in S,
the innovation (residual between prediction and measurement in
measurement space) in y, and the Kalman gain in k. The procedural
form returns these variables to you. In Bayesian terms this computes
the *posterior* - the estimate after the information from the
measurement is incorporated.
Whether you use the OO form or procedural form is up to you. If
matrices such as H, R, and F are changing each epoch, you'll probably
opt to use the procedural form. If they are unchanging, the OO
form is perhaps easier to use since you won't need to keep track
of these matrices. This is especially useful if you are implementing
banks of filters or comparing various KF designs for performance;
a trivial coding bug could lead to using the wrong sets of matrices.
This module also offers an implementation of the RTS smoother, and
other helper functions, such as log likelihood computations.
The Saver class allows you to easily save the state of the
KalmanFilter class after every update
This module expects NumPy arrays for all values that expect
arrays, although in a few cases, particularly method parameters,
it will accept types that convert to NumPy arrays, such as lists
of lists. These exceptions are documented in the method or function.
Examples
--------
The following example constructs a constant velocity kinematic
filter, filters noisy data, and plots the results. It also demonstrates
using the Saver class to save the state of the filter at each epoch.
.. code-block:: Python
import matplotlib.pyplot as plt
import numpy as np
from filterpy.kalman import KalmanFilter
from filterpy.common import Q_discrete_white_noise, Saver
r_std, q_std = 2., 0.003
cv = KalmanFilter(dim_x=2, dim_z=1)
cv.x = np.array([[0., 1.]]) # position, velocity
cv.F = np.array([[1, dt],[ [0, 1]])
cv.R = np.array([[r_std^^2]])
f.H = np.array([[1., 0.]])
f.P = np.diag([.1^^2, .03^^2)
f.Q = Q_discrete_white_noise(2, dt, q_std**2)
saver = Saver(cv)
for z in range(100):
cv.predict()
cv.update([z + randn() * r_std])
saver.save() # save the filter's state
saver.to_array()
plt.plot(saver.x[:, 0])
# plot all of the priors
plt.plot(saver.x_prior[:, 0])
# plot mahalanobis distance
plt.figure()
plt.plot(saver.mahalanobis)
This code implements the same filter using the procedural form
x = np.array([[0., 1.]]) # position, velocity
F = np.array([[1, dt],[ [0, 1]])
R = np.array([[r_std^^2]])
H = np.array([[1., 0.]])
P = np.diag([.1^^2, .03^^2)
Q = Q_discrete_white_noise(2, dt, q_std**2)
for z in range(100):
x, P = predict(x, P, F=F, Q=Q)
x, P = update(x, P, z=[z + randn() * r_std], R=R, H=H)
xs.append(x[0, 0])
plt.plot(xs)
For more examples see the test subdirectory, or refer to the
book cited below. In it I both teach Kalman filtering from basic
principles, and teach the use of this library in great detail.
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.
Copyright 2014-2018 Roger R Labbe Jr.
"""
from __future__ import absolute_import, division
from copy import deepcopy
from math import log, exp, sqrt
import sys
import warnings
import numpy as np
from numpy import dot, zeros, eye, isscalar, shape
import numpy.linalg as linalg
from filterpy.stats import logpdf
from filterpy.common import pretty_str, reshape_z
[docs]class KalmanFilter(object):
r""" Implements a Kalman filter. You are responsible for setting the
various state variables to reasonable values; the defaults will
not give you a functional filter.
For now the best documentation is my free book Kalman and Bayesian
Filters in Python [2]_. The test files in this directory also give you a
basic idea of use, albeit without much description.
In brief, you will first construct this object, specifying the size of
the state vector with dim_x and the size of the measurement vector that
you will be using with dim_z. These are mostly used to perform size checks
when you assign values to the various matrices. For example, if you
specified dim_z=2 and then try to assign a 3x3 matrix to R (the
measurement noise matrix you will get an assert exception because R
should be 2x2. (If for whatever reason you need to alter the size of
things midstream just use the underscore version of the matrices to
assign directly: your_filter._R = a_3x3_matrix.)
After construction the filter will have default matrices created for you,
but you must specify the values for each. Itâ€™s usually easiest to just
overwrite them rather than assign to each element yourself. This will be
clearer in the example below. All are of type numpy.array.
Examples
--------
Here is a filter that tracks position and velocity using a sensor that only
reads position.
First construct the object with the required dimensionality.
.. code::
from filterpy.kalman import KalmanFilter
f = KalmanFilter (dim_x=2, dim_z=1)
Assign the initial value for the state (position and velocity). You can do this
with a two dimensional array like so:
.. code::
f.x = np.array([[2.], # position
[0.]]) # velocity
or just use a one dimensional array, which I prefer doing.
.. code::
f.x = np.array([2., 0.])
Define the state transition matrix:
.. code::
f.F = np.array([[1.,1.],
[0.,1.]])
Define the measurement function:
.. code::
f.H = np.array([[1.,0.]])
Define the covariance matrix. Here I take advantage of the fact that
P already contains np.eye(dim_x), and just multiply by the uncertainty:
.. code::
f.P *= 1000.
I could have written:
.. code::
f.P = np.array([[1000., 0.],
[ 0., 1000.] ])
You decide which is more readable and understandable.
Now assign the measurement noise. Here the dimension is 1x1, so I can
use a scalar
.. code::
f.R = 5
I could have done this instead:
.. code::
f.R = np.array([[5.]])
Note that this must be a 2 dimensional array, as must all the matrices.
Finally, I will assign the process noise. Here I will take advantage of
another FilterPy library function:
.. code::
from filterpy.common import Q_discrete_white_noise
f.Q = Q_discrete_white_noise(dim=2, dt=0.1, var=0.13)
Now just perform the standard predict/update loop:
while some_condition_is_true:
.. code::
z = get_sensor_reading()
f.predict()
f.update(z)
do_something_with_estimate (f.x)
**Procedural Form**
This module also contains stand alone functions to perform Kalman filtering.
Use these if you are not a fan of objects.
**Example**
.. code::
while True:
z, R = read_sensor()
x, P = predict(x, P, F, Q)
x, P = update(x, P, z, R, H)
See my book Kalman and Bayesian Filters in Python [2]_.
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.
dim_u : int (optional)
size of the control input, if it is being used.
Default value of 0 indicates it is not used.
compute_log_likelihood : bool (default = True)
Computes log likelihood by default, but this can be a slow
computation, so if you never use it you can turn this computation
off.
Attributes
----------
x : numpy.array(dim_x, 1)
Current state estimate. Any call to update() or predict() updates
this variable.
P : numpy.array(dim_x, dim_x)
Current state covariance matrix. Any call to update() or predict()
updates this variable.
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.
z : numpy.array
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
F : numpy.array()
State Transition matrix
H : numpy.array(dim_z, 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
System uncertainty (P projected to measurement space). Read only.
SI : numpy.array
Inverse system uncertainty. Read only.
log_likelihood : float
log-likelihood of the last measurement. Read only.
likelihood : float
likelihood of last measurement. 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. 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: kf.inv = np.linalg.pinv
This is only used to invert self.S. If you know it is diagonal, you
might choose to set it to filterpy.common.inv_diagonal, which is
several times faster than numpy.linalg.inv for diagonal matrices.
alpha : float
Fading memory setting. 1.0 gives the normal Kalman filter, and
values slightly larger than 1.0 (such as 1.02) give a fading
memory effect - previous measurements have less influence on the
filter's estimates. This formulation of the Fading memory filter
(there are many) is due to Dan Simon [1]_.
References
----------
.. [1] Dan Simon. "Optimal State Estimation." John Wiley & Sons.
p. 208-212. (2006)
.. [2] Roger Labbe. "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):
if dim_x < 1:
raise ValueError('dim_x must be 1 or greater')
if dim_z < 1:
raise ValueError('dim_z must be 1 or greater')
if dim_u < 0:
raise ValueError('dim_u must be 0 or greater')
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.Q = eye(dim_x) # process uncertainty
self.B = None # control transition matrix
self.F = eye(dim_x) # state transition matrix
self.H = zeros((dim_z, dim_x)) # Measurement function
self.R = eye(dim_z) # state uncertainty
self._alpha_sq = 1. # fading memory control
self.M = np.zeros((dim_z, dim_z)) # process-measurement cross correlation
self.z = np.array([[None]*self.dim_z]).T
# 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((dim_x, dim_z)) # 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)
# 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()
# Only computed only if requested via property
self._log_likelihood = log(sys.float_info.min)
self._likelihood = sys.float_info.min
self._mahalanobis = None
self.inv = np.linalg.inv
[docs] def predict(self, u=None, B=None, F=None, Q=None):
"""
Predict next state (prior) using the Kalman filter state propagation
equations.
Parameters
----------
u : np.array
Optional control vector. If not `None`, it is multiplied by B
to create the control input into the system.
B : np.array(dim_x, dim_z), or None
Optional control transition matrix; a value of None
will cause the filter to use `self.B`.
F : np.array(dim_x, dim_x), or None
Optional state transition matrix; a value of None
will cause the filter to use `self.F`.
Q : np.array(dim_x, dim_x), scalar, or None
Optional process noise matrix; a value of None will cause the
filter to use `self.Q`.
"""
if B is None:
B = self.B
if F is None:
F = self.F
if Q is None:
Q = self.Q
elif isscalar(Q):
Q = eye(self.dim_x) * Q
# x = Fx + Bu
if B is not None and u is not None:
self.x = dot(F, self.x) + dot(B, u)
else:
self.x = dot(F, self.x)
# P = FPF' + Q
self.P = self._alpha_sq * dot(dot(F, self.P), F.T) + Q
# save prior
self.x_prior = self.x.copy()
self.P_prior = self.P.copy()
[docs] def update(self, z, R=None, H=None):
"""
Add a new measurement (z) to the Kalman filter.
If z is None, nothing is computed. However, x_post and P_post are
updated with the prior (x_prior, P_prior), and self.z is set to None.
Parameters
----------
z : (dim_z, 1): array_like
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be convertible to a column vector.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
H : np.array, or None
Optionally provide H to override the measurement function for this
one call, otherwise self.H will be used.
"""
# set to None to force recompute
self._log_likelihood = None
self._likelihood = None
self._mahalanobis = None
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()
self.y = zeros((self.dim_z, 1))
return
z = reshape_z(z, self.dim_z, self.x.ndim)
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
if H is None:
H = self.H
# y = z - Hx
# error (residual) between measurement and prediction
self.y = z - dot(H, self.x)
# common subexpression for speed
PHT = dot(self.P, H.T)
# S = HPH' + R
# project system uncertainty into measurement space
self.S = dot(H, PHT) + R
self.SI = self.inv(self.S)
# K = PH'inv(S)
# map system uncertainty into kalman gain
self.K = dot(PHT, self.SI)
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self.x = self.x + dot(self.K, self.y)
# P = (I-KH)P(I-KH)' + KRK'
# This 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(dot(I_KH, self.P), I_KH.T) + dot(dot(self.K, R), self.K.T)
# save measurement and posterior state
self.z = deepcopy(z)
self.x_post = self.x.copy()
self.P_post = self.P.copy()
[docs] def predict_steadystate(self, u=0, B=None):
"""
Predict state (prior) using the Kalman filter state propagation
equations. Only x is updated, P is left unchanged. See
update_steadstate() for a longer explanation of when to use this
method.
Parameters
----------
u : np.array
Optional control vector. If non-zero, it is multiplied by B
to create the control input into the system.
B : np.array(dim_x, dim_z), or None
Optional control transition matrix; a value of None
will cause the filter to use `self.B`.
"""
if B is None:
B = self.B
# x = Fx + Bu
if B is not None:
self.x = dot(self.F, self.x) + dot(B, u)
else:
self.x = dot(self.F, self.x)
# save prior
self.x_prior = self.x.copy()
self.P_prior = self.P.copy()
[docs] def update_steadystate(self, z):
"""
Add a new measurement (z) to the Kalman filter without recomputing
the Kalman gain K, the state covariance P, or the system
uncertainty S.
You can use this for LTI systems since the Kalman gain and covariance
converge to a fixed value. Precompute these and assign them explicitly,
or run the Kalman filter using the normal predict()/update(0 cycle
until they converge.
The main advantage of this call is speed. We do significantly less
computation, notably avoiding a costly matrix inversion.
Use in conjunction with predict_steadystate(), otherwise P will grow
without bound.
Parameters
----------
z : (dim_z, 1): array_like
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be convertible to a column vector.
Examples
--------
>>> cv = kinematic_kf(dim=3, order=2) # 3D const velocity filter
>>> # let filter converge on representative data, then save k and P
>>> for i in range(100):
>>> cv.predict()
>>> cv.update([i, i, i])
>>> saved_k = np.copy(cv.K)
>>> saved_P = np.copy(cv.P)
later on:
>>> cv = kinematic_kf(dim=3, order=2) # 3D const velocity filter
>>> cv.K = np.copy(saved_K)
>>> cv.P = np.copy(saved_P)
>>> for i in range(100):
>>> cv.predict_steadystate()
>>> cv.update_steadystate([i, i, i])
"""
# set to None to force recompute
self._log_likelihood = None
self._likelihood = None
self._mahalanobis = None
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()
self.y = zeros((self.dim_z, 1))
return
z = reshape_z(z, self.dim_z, self.x.ndim)
# y = z - Hx
# error (residual) between measurement and prediction
self.y = z - dot(self.H, self.x)
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self.x = self.x + dot(self.K, self.y)
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 batch_filter(self, zs, Fs=None, Qs=None, Hs=None,
Rs=None, Bs=None, us=None, update_first=False,
saver=None):
""" Batch processes a sequences of measurements.
Parameters
----------
zs : list-like
list of measurements at each time step `self.dt`. Missing
measurements must be represented by `None`.
Fs : None, list-like, default=None
optional value or list of values to use for the state transition
matrix F.
If Fs is None then self.F is used for all epochs.
Otherwise it must contain a list-like list of F's, one for
each epoch. This allows you to have varying F per epoch.
Qs : None, np.array or list-like, default=None
optional value or list of values to use for the process error
covariance Q.
If Qs is None then self.Q is used for all epochs.
Otherwise it must contain a list-like list of Q's, one for
each epoch. This allows you to have varying Q per epoch.
Hs : None, np.array or list-like, default=None
optional list of values to use for the measurement matrix H.
If Hs is None then self.H is used for all epochs.
If Hs contains a single matrix, then it is used as H for all
epochs.
Otherwise it must contain a list-like list of H's, one for
each epoch. This allows you to have varying H per epoch.
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.
Otherwise it must contain a list-like list of R's, one for
each epoch. This allows you to have varying R per epoch.
Bs : None, np.array or list-like, default=None
optional list of values to use for the control transition matrix B.
If Bs is None then self.B is used for all epochs.
Otherwise it must contain a list-like list of B's, one for
each epoch. This allows you to have varying B per epoch.
us : None, np.array or list-like, default=None
optional list of values to use for the control input vector;
If us is None then None is used for all epochs (equivalent to 0,
or no control input).
Otherwise it must contain a list-like list of u's, one for
each epoch.
update_first : bool, optional, default=False
controls whether the order of operations is update followed by
predict, or predict followed by update. Default is predict->update.
saver : filterpy.common.Saver, optional
filterpy.common.Saver object. If provided, saver.save() will be
called after every epoch
Returns
-------
means : np.array((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 : np.array((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`.
means_predictions : np.array((n,dim_x,1))
array of the state for each time step after the predictions. Each
entry is an np.array. In other words `means[k,:]` is the state at
step `k`.
covariance_predictions : np.array((n,dim_x,dim_x))
array of the covariances for each time step after the prediction.
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. This requires
# that F be recomputed for each epoch. The output is then smoothed
# with an RTS smoother.
zs = [t + random.randn()*4 for t in range (40)]
Fs = [np.array([[1., dt], [0, 1]] for dt in dts]
(mu, cov, _, _) = kf.batch_filter(zs, Fs=Fs)
(xs, Ps, Ks) = kf.rts_smoother(mu, cov, Fs=Fs)
"""
#pylint: disable=too-many-statements
n = np.size(zs, 0)
if Fs is None:
Fs = [self.F] * n
if Qs is None:
Qs = [self.Q] * n
if Hs is None:
Hs = [self.H] * n
if Rs is None:
Rs = [self.R] * n
if Bs is None:
Bs = [self.B] * n
if us is None:
us = [0] * n
# mean estimates from Kalman Filter
if self.x.ndim == 1:
means = zeros((n, self.dim_x))
means_p = zeros((n, self.dim_x))
else:
means = zeros((n, self.dim_x, 1))
means_p = zeros((n, self.dim_x, 1))
# state covariances from Kalman Filter
covariances = zeros((n, self.dim_x, self.dim_x))
covariances_p = zeros((n, self.dim_x, self.dim_x))
if update_first:
for i, (z, F, Q, H, R, B, u) in enumerate(zip(zs, Fs, Qs, Hs, Rs, Bs, us)):
self.update(z, R=R, H=H)
means[i, :] = self.x
covariances[i, :, :] = self.P
self.predict(u=u, B=B, F=F, Q=Q)
means_p[i, :] = self.x
covariances_p[i, :, :] = self.P
if saver is not None:
saver.save()
else:
for i, (z, F, Q, H, R, B, u) in enumerate(zip(zs, Fs, Qs, Hs, Rs, Bs, us)):
self.predict(u=u, B=B, F=F, Q=Q)
means_p[i, :] = self.x
covariances_p[i, :, :] = self.P
self.update(z, R=R, H=H)
means[i, :] = self.x
covariances[i, :, :] = self.P
if saver is not None:
saver.save()
return (means, covariances, means_p, covariances_p)
[docs] def rts_smoother(self, Xs, Ps, Fs=None, Qs=None, inv=np.linalg.inv):
"""
Runs the Rauch-Tung-Striebal Kalman smoother on a set of
means and covariances computed by a Kalman filter. The usual input
would come from the output of `KalmanFilter.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.
Fs : list-like collection of numpy.array, optional
State transition matrix of the Kalman filter at each time step.
Optional, if not provided the filter's self.F will be used
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
inv : function, default numpy.linalg.inv
If you prefer another inverse function, such as the Moore-Penrose
pseudo inverse, set it to that instead: kf.inv = np.linalg.pinv
Returns
-------
x : numpy.ndarray
smoothed means
P : numpy.ndarray
smoothed state covariances
K : numpy.ndarray
smoother gain at each step
Pp : numpy.ndarray
Predicted state covariances
Examples
--------
.. code-block:: Python
zs = [t + random.randn()*4 for t in range (40)]
(mu, cov, _, _) = kalman.batch_filter(zs)
(x, P, K, Pp) = rts_smoother(mu, cov, kf.F, kf.Q)
"""
if len(Xs) != len(Ps):
raise ValueError('length of Xs and Ps must be the same')
n = Xs.shape[0]
dim_x = Xs.shape[1]
if Fs is None:
Fs = [self.F] * n
if Qs is None:
Qs = [self.Q] * n
# smoother gain
K = zeros((n, dim_x, dim_x))
x, P, Pp = Xs.copy(), Ps.copy(), Ps.copy()
for k in range(n-2, -1, -1):
Pp[k] = dot(dot(Fs[k+1], P[k]), Fs[k+1].T) + Qs[k+1]
#pylint: disable=bad-whitespace
K[k] = dot(dot(P[k], Fs[k+1].T), inv(Pp[k]))
x[k] += dot(K[k], x[k+1] - dot(Fs[k+1], x[k]))
P[k] += dot(dot(K[k], P[k+1] - Pp[k]), K[k].T)
return (x, P, K, Pp)
[docs] def get_prediction(self, u=0):
"""
Predicts the next state of the filter and returns it without
altering the state of the filter.
Parameters
----------
u : np.array
optional control input
Returns
-------
(x, P) : tuple
State vector and covariance array of the prediction.
"""
x = dot(self.F, self.x) + dot(self.B, u)
P = self._alpha_sq * dot(dot(self.F, self.P), self.F.T) + self.Q
return (x, P)
[docs] def get_update(self, z=None):
"""
Computes the new estimate based on measurement `z` and returns it
without altering the state of the filter.
Parameters
----------
z : (dim_z, 1): array_like
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be convertible to a column vector.
Returns
-------
(x, P) : tuple
State vector and covariance array of the update.
"""
if z is None:
return self.x, self.P
z = reshape_z(z, self.dim_z, self.x.ndim)
R = self.R
H = self.H
P = self.P
x = self.x
# error (residual) between measurement and prediction
y = z - dot(H, x)
# common subexpression for speed
PHT = dot(P, H.T)
# project system uncertainty into measurement space
S = dot(H, PHT) + R
# map system uncertainty into kalman gain
K = dot(PHT, self.inv(S))
# predict new x with residual scaled by the kalman gain
x = x + dot(K, y)
# P = (I-KH)P(I-KH)' + KRK'
I_KH = self._I - dot(K, H)
P = dot(dot(I_KH, P), I_KH.T) + dot(dot(K, R), K.T)
return x, P
[docs] def residual_of(self, z):
"""
Returns the residual for the given measurement (z). Does not alter
the state of the filter.
"""
return z - dot(self.H, self.x_prior)
[docs] def measurement_of_state(self, x):
"""
Helper function that converts a state into a measurement.
Parameters
----------
x : np.array
kalman state vector
Returns
-------
z : (dim_z, 1): array_like
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be convertible to a column vector.
"""
return dot(self.H, x)
@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
@property
def alpha(self):
"""
Fading memory setting. 1.0 gives the normal Kalman filter, and
values slightly larger than 1.0 (such as 1.02) give a fading
memory effect - previous measurements have less influence on the
filter's estimates. This formulation of the Fading memory filter
(there are many) is due to Dan Simon [1]_.
"""
return self._alpha_sq**.5
[docs] def log_likelihood_of(self, z):
"""
log likelihood of the measurement `z`. This should only be called
after a call to update(). Calling after predict() will yield an
incorrect result."""
if z is None:
return log(sys.float_info.min)
return logpdf(z, dot(self.H, self.x), self.S)
@alpha.setter
def alpha(self, value):
if not np.isscalar(value) or value < 1:
raise ValueError('alpha must be a float greater than 1')
self._alpha_sq = value**2
def __repr__(self):
return '\n'.join([
'KalmanFilter object',
pretty_str('dim_x', self.dim_x),
pretty_str('dim_z', self.dim_z),
pretty_str('dim_u', self.dim_u),
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('x_post', self.x_post),
pretty_str('P_post', self.P_post),
pretty_str('F', self.F),
pretty_str('Q', self.Q),
pretty_str('R', self.R),
pretty_str('H', self.H),
pretty_str('K', self.K),
pretty_str('y', self.y),
pretty_str('S', self.S),
pretty_str('SI', self.SI),
pretty_str('M', self.M),
pretty_str('B', self.B),
pretty_str('z', self.z),
pretty_str('log-likelihood', self.log_likelihood),
pretty_str('likelihood', self.likelihood),
pretty_str('mahalanobis', self.mahalanobis),
pretty_str('alpha', self.alpha),
pretty_str('inv', self.inv)
])
[docs] def test_matrix_dimensions(self, z=None, H=None, R=None, F=None, Q=None):
"""
Performs a series of asserts to check that the size of everything
is what it should be. This can help you debug problems in your design.
If you pass in H, R, F, Q those will be used instead of this object's
value for those matrices.
Testing `z` (the measurement) is problamatic. x is a vector, and can be
implemented as either a 1D array or as a nx1 column vector. Thus Hx
can be of different shapes. Then, if Hx is a single value, it can
be either a 1D array or 2D vector. If either is true, z can reasonably
be a scalar (either '3' or np.array('3') are scalars under this
definition), a 1D, 1 element array, or a 2D, 1 element array. You are
allowed to pass in any combination that works.
"""
if H is None:
H = self.H
if R is None:
R = self.R
if F is None:
F = self.F
if Q is None:
Q = self.Q
x = self.x
P = self.P
assert x.ndim == 1 or x.ndim == 2, \
"x must have one or two dimensions, but has {}".format(x.ndim)
if x.ndim == 1:
assert x.shape[0] == self.dim_x, \
"Shape of x must be ({},{}), but is {}".format(
self.dim_x, 1, x.shape)
else:
assert x.shape == (self.dim_x, 1), \
"Shape of x must be ({},{}), but is {}".format(
self.dim_x, 1, x.shape)
assert P.shape == (self.dim_x, self.dim_x), \
"Shape of P must be ({},{}), but is {}".format(
self.dim_x, self.dim_x, P.shape)
assert Q.shape == (self.dim_x, self.dim_x), \
"Shape of P must be ({},{}), but is {}".format(
self.dim_x, self.dim_x, P.shape)
assert F.shape == (self.dim_x, self.dim_x), \
"Shape of F must be ({},{}), but is {}".format(
self.dim_x, self.dim_x, F.shape)
assert np.ndim(H) == 2, \
"Shape of H must be (dim_z, {}), but is {}".format(
P.shape[0], shape(H))
assert H.shape[1] == P.shape[0], \
"Shape of H must be (dim_z, {}), but is {}".format(
P.shape[0], H.shape)
# shape of R must be the same as HPH'
hph_shape = (H.shape[0], H.shape[0])
r_shape = shape(R)
if H.shape[0] == 1:
# r can be scalar, 1D, or 2D in this case
assert r_shape == () or r_shape == (1,) or r_shape == (1, 1), \
"R must be scalar or one element array, but is shaped {}".format(
r_shape)
else:
assert r_shape == hph_shape, \
"shape of R should be {} but it is {}".format(hph_shape, r_shape)
if z is not None:
z_shape = shape(z)
else:
z_shape = (self.dim_z, 1)
# H@x must have shape of z
Hx = dot(H, x)
if z_shape == (): # scalar or np.array(scalar)
assert Hx.ndim == 1 or shape(Hx) == (1, 1), \
"shape of z should be {}, not {} for the given H".format(
shape(Hx), z_shape)
elif shape(Hx) == (1,):
assert z_shape[0] == 1, 'Shape of z must be {} for the given H'.format(shape(Hx))
else:
assert (z_shape == shape(Hx) or
(len(z_shape) == 1 and shape(Hx) == (z_shape[0], 1))), \
"shape of z should be {}, not {} for the given H".format(
shape(Hx), z_shape)
if np.ndim(Hx) > 1 and shape(Hx) != (1, 1):
assert shape(Hx) == z_shape, \
'shape of z should be {} for the given H, but it is {}'.format(
shape(Hx), z_shape)
[docs]def update(x, P, z, R, H=None, return_all=False):
"""
Add a new measurement (z) to the Kalman filter. If z is None, nothing
is changed.
This can handle either the multidimensional or unidimensional case. If
all parameters are floats instead of arrays the filter will still work,
and return floats for x, P as the result.
update(1, 2, 1, 1, 1) # univariate
update(x, P, 1
Parameters
----------
x : numpy.array(dim_x, 1), or float
State estimate vector
P : numpy.array(dim_x, dim_x), or float
Covariance matrix
z : (dim_z, 1): array_like
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be convertible to a column vector.
R : numpy.array(dim_z, dim_z), or float
Measurement noise matrix
H : numpy.array(dim_x, dim_x), or float, optional
Measurement function. If not provided, a value of 1 is assumed.
return_all : bool, default False
If true, y, K, S, and log_likelihood are returned, otherwise
only x and P are returned.
Returns
-------
x : numpy.array
Posterior state estimate vector
P : numpy.array
Posterior covariance matrix
y : numpy.array or scalar
Residua. Difference between measurement and state in measurement space
K : numpy.array
Kalman gain
S : numpy.array
System uncertainty in measurement space
log_likelihood : float
log likelihood of the measurement
"""
#pylint: disable=bare-except
if z is None:
if return_all:
return x, P, None, None, None, None
return x, P
if H is None:
H = np.array([1])
if np.isscalar(H):
H = np.array([H])
Hx = np.atleast_1d(dot(H, x))
z = reshape_z(z, Hx.shape[0], x.ndim)
# error (residual) between measurement and prediction
y = z - Hx
# project system uncertainty into measurement space
S = dot(dot(H, P), H.T) + R
# map system uncertainty into kalman gain
try:
K = dot(dot(P, H.T), linalg.inv(S))
except:
# can't invert a 1D array, annoyingly
K = dot(dot(P, H.T), 1./S)
# predict new x with residual scaled by the kalman gain
x = x + dot(K, y)
# P = (I-KH)P(I-KH)' + KRK'
KH = dot(K, H)
try:
I_KH = np.eye(KH.shape[0]) - KH
except:
I_KH = np.array([1 - KH])
P = dot(dot(I_KH, P), I_KH.T) + dot(dot(K, R), K.T)
if return_all:
# compute log likelihood
log_likelihood = logpdf(z, dot(H, x), S)
return x, P, y, K, S, log_likelihood
return x, P
def update_steadystate(x, z, K, H=None):
"""
Add a new measurement (z) to the Kalman filter. If z is None, nothing
is changed.
Parameters
----------
x : numpy.array(dim_x, 1), or float
State estimate vector
z : (dim_z, 1): array_like
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be convertible to a column vector.
K : numpy.array, or float
Kalman gain matrix
H : numpy.array(dim_x, dim_x), or float, optional
Measurement function. If not provided, a value of 1 is assumed.
Returns
-------
x : numpy.array
Posterior state estimate vector
Examples
--------
This can handle either the multidimensional or unidimensional case. If
all parameters are floats instead of arrays the filter will still work,
and return floats for x, P as the result.
>>> update_steadystate(1, 2, 1) # univariate
>>> update_steadystate(x, P, z, H)
"""
if z is None:
return x
if H is None:
H = np.array([1])
if np.isscalar(H):
H = np.array([H])
Hx = np.atleast_1d(dot(H, x))
z = reshape_z(z, Hx.shape[0], x.ndim)
# error (residual) between measurement and prediction
y = z - Hx
# estimate new x with residual scaled by the kalman gain
return x + dot(K, y)
[docs]def predict(x, P, F=1, Q=0, u=0, B=1, alpha=1.):
"""
Predict next state (prior) using the Kalman filter state propagation
equations.
Parameters
----------
x : numpy.array
State estimate vector
P : numpy.array
Covariance matrix
F : numpy.array()
State Transition matrix
Q : numpy.array, Optional
Process noise matrix
u : numpy.array, Optional, default 0.
Control vector. If non-zero, it is multiplied by B
to create the control input into the system.
B : numpy.array, optional, default 0.
Control transition matrix.
alpha : float, Optional, default=1.0
Fading memory setting. 1.0 gives the normal Kalman filter, and
values slightly larger than 1.0 (such as 1.02) give a fading
memory effect - previous measurements have less influence on the
filter's estimates. This formulation of the Fading memory filter
(there are many) is due to Dan Simon
Returns
-------
x : numpy.array
Prior state estimate vector
P : numpy.array
Prior covariance matrix
"""
if np.isscalar(F):
F = np.array(F)
x = dot(F, x) + dot(B, u)
P = (alpha * alpha) * dot(dot(F, P), F.T) + Q
return x, P
def predict_steadystate(x, F=1, u=0, B=1):
"""
Predict next state (prior) using the Kalman filter state propagation
equations. This steady state form only computes x, assuming that the
covariance is constant.
Parameters
----------
x : numpy.array
State estimate vector
P : numpy.array
Covariance matrix
F : numpy.array()
State Transition matrix
u : numpy.array, Optional, default 0.
Control vector. If non-zero, it is multiplied by B
to create the control input into the system.
B : numpy.array, optional, default 0.
Control transition matrix.
Returns
-------
x : numpy.array
Prior state estimate vector
"""
if np.isscalar(F):
F = np.array(F)
x = dot(F, x) + dot(B, u)
return x
[docs]def batch_filter(x, P, zs, Fs, Qs, Hs, Rs, Bs=None, us=None,
update_first=False, saver=None):
"""
Batch processes a sequences of measurements.
Parameters
----------
zs : list-like
list of measurements at each time step. Missing measurements must be
represented by None.
Fs : list-like
list of values to use for the state transition matrix matrix.
Qs : list-like
list of values to use for the process error
covariance.
Hs : list-like
list of values to use for the measurement matrix.
Rs : list-like
list of values to use for the measurement error
covariance.
Bs : list-like, optional
list of values to use for the control transition matrix;
a value of None in any position will cause the filter
to use `self.B` for that time step.
us : list-like, optional
list of values to use for the control input vector;
a value of None in any position will cause the filter to use
0 for that time step.
update_first : bool, optional
controls whether the order of operations is update followed by
predict, or predict followed by update. Default is predict->update.
saver : filterpy.common.Saver, optional
filterpy.common.Saver object. If provided, saver.save() will be
called after every epoch
Returns
-------
means : np.array((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 : np.array((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`.
means_predictions : np.array((n,dim_x,1))
array of the state for each time step after the predictions. Each
entry is an np.array. In other words `means[k,:]` is the state at
step `k`.
covariance_predictions : np.array((n,dim_x,dim_x))
array of the covariances for each time step after the prediction.
In other words `covariance[k,:,:]` is the covariance at step `k`.
Examples
--------
.. code-block:: Python
zs = [t + random.randn()*4 for t in range (40)]
Fs = [kf.F for t in range (40)]
Hs = [kf.H for t in range (40)]
(mu, cov, _, _) = kf.batch_filter(zs, Rs=R_list, Fs=Fs, Hs=Hs, Qs=None,
Bs=None, us=None, update_first=False)
(xs, Ps, Ks) = kf.rts_smoother(mu, cov, Fs=Fs, Qs=None)
"""
n = np.size(zs, 0)
dim_x = x.shape[0]
# mean estimates from Kalman Filter
if x.ndim == 1:
means = zeros((n, dim_x))
means_p = zeros((n, dim_x))
else:
means = zeros((n, dim_x, 1))
means_p = zeros((n, dim_x, 1))
# state covariances from Kalman Filter
covariances = zeros((n, dim_x, dim_x))
covariances_p = zeros((n, dim_x, dim_x))
if us is None:
us = [0.] * n
Bs = [0.] * n
if update_first:
for i, (z, F, Q, H, R, B, u) in enumerate(zip(zs, Fs, Qs, Hs, Rs, Bs, us)):
x, P = update(x, P, z, R=R, H=H)
means[i, :] = x
covariances[i, :, :] = P
x, P = predict(x, P, u=u, B=B, F=F, Q=Q)
means_p[i, :] = x
covariances_p[i, :, :] = P
if saver is not None:
saver.save()
else:
for i, (z, F, Q, H, R, B, u) in enumerate(zip(zs, Fs, Qs, Hs, Rs, Bs, us)):
x, P = predict(x, P, u=u, B=B, F=F, Q=Q)
means_p[i, :] = x
covariances_p[i, :, :] = P
x, P = update(x, P, z, R=R, H=H)
means[i, :] = x
covariances[i, :, :] = P
if saver is not None:
saver.save()
return (means, covariances, means_p, covariances_p)
def rts_smoother(Xs, Ps, Fs, Qs):
"""
Runs the Rauch-Tung-Striebal Kalman smoother on a set of
means and covariances computed by a Kalman filter. The usual input
would come from the output of `KalmanFilter.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.
Fs : list-like collection of numpy.array
State transition matrix of the Kalman filter at each time step.
Qs : list-like collection of numpy.array, optional
Process noise of the Kalman filter at each time step.
Returns
-------
x : numpy.ndarray
smoothed means
P : numpy.ndarray
smoothed state covariances
K : numpy.ndarray
smoother gain at each step
pP : numpy.ndarray
predicted state covariances
Examples
--------
.. code-block:: Python
zs = [t + random.randn()*4 for t in range (40)]
(mu, cov, _, _) = kalman.batch_filter(zs)
(x, P, K, pP) = rts_smoother(mu, cov, kf.F, kf.Q)
"""
if len(Xs) != len(Ps):
raise ValueError('length of Xs and Ps must be the same')
n = Xs.shape[0]
dim_x = Xs.shape[1]
# smoother gain
K = zeros((n, dim_x, dim_x))
x, P, pP = Xs.copy(), Ps.copy(), Ps.copy()
for k in range(n-2, -1, -1):
pP[k] = dot(dot(Fs[k], P[k]), Fs[k].T) + Qs[k]
#pylint: disable=bad-whitespace
K[k] = dot(dot(P[k], Fs[k].T), linalg.inv(pP[k]))
x[k] += dot(K[k], x[k+1] - dot(Fs[k], x[k]))
P[k] += dot(dot(K[k], P[k+1] - pP[k]), K[k].T)
return (x, P, K, pP)
[docs]class Saver(object):
"""
Deprecated. Use filterpy.common.Saver instead.
Helper class to save the states of the KalmanFilter class.
Each time you call save() the current states are appended to lists.
Generally you would do this once per epoch - predict/update.
Once you are done filtering you can optionally call to_array()
to convert all of the lists to numpy arrays. You cannot safely call
save() after calling to_array().
Examples
--------
.. code-block:: Python
kf = KalmanFilter(...whatever)
# initialize kf here
saver = Saver(kf) # save data for kf filter
for z in zs:
kf.predict()
kf.update(z)
saver.save()
saver.to_array()
# plot the 0th element of the state
plt.plot(saver.xs[:, 0, 0])
"""
[docs] def __init__(self, kf, save_current=True):
""" Construct the save object, optionally saving the current
state of the filter"""
warnings.warn(
'Use filterpy.common.Saver instead of this, as it works for any filter clase',
DeprecationWarning)
self.xs = []
self.Ps = []
self.Ks = []
self.ys = []
self.xs_prior = []
self.Ps_prior = []
self.kf = kf
if save_current:
self.save()
[docs] def save(self):
""" save the current state of the Kalman filter"""
kf = self.kf
self.xs.append(np.copy(kf.x))
self.Ps.append(np.copy(kf.P))
self.Ks.append(np.copy(kf.K))
self.ys.append(np.copy(kf.y))
self.xs_prior.append(np.copy(kf.x_prior))
self.Ps_prior.append(np.copy(kf.P_prior))
[docs] def to_array(self):
""" convert all of the lists into np.array"""
self.xs = np.array(self.xs)
self.Ps = np.array(self.Ps)
self.Ks = np.array(self.Ks)
self.ys = np.array(self.ys)
self.xs_prior = np.array(self.xs_prior)
self.Ps_prior = np.array(self.Ps_prior)
```