# KalmanFilter¶

Implements a linear Kalman filter. For now the best documentation is my free book Kalman and Bayesian Filters in Python [1]

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. It’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.

These are the matrices (instance variables) which you must specify. All are of type numpy.array (do NOT use numpy.matrix) If dimensional analysis allows you to get away with a 1x1 matrix you may also use a scalar. All elements must have a type of float.

Instance Variables

You will have to assign reasonable values to all of these before running the filter. All must have dtype of float.

x : ndarray (dim_x, 1), default = [0,0,0...0]
filter state estimate
P : ndarray (dim_x, dim_x), default eye(dim_x)
covariance matrix
Q : ndarray (dim_x, dim_x), default eye(dim_x)
Process uncertainty/noise
R : ndarray (dim_z, dim_z), default eye(dim_x)
measurement uncertainty/noise
H : ndarray (dim_z, dim_x)
measurement function
F : ndarray (dim_x, dim_x)
state transistion matrix
B : ndarray (dim_x, dim_u), default 0
control transition matrix

Optional Instance Variables

alpha : float

Assign a value > 1.0 to turn this into a fading memory filter.

K : ndarray
Kalman gain that was used in the most recent update() call.
y : ndarray
Residual calculated in the most recent update() call. I.e., the different between the measurement and the current estimated state projected into measurement space (z - Hx)
S : ndarray
System uncertainty projected into measurement space. I.e., HPH’ + R. Probably not very useful, but it is here if you want it.
likelihood : float
Likelihood of last measurment update.
log_likelihood : float
Log likelihood of last measurment update.

Example

Here is a filter that tracks position and velocity using a sensor that only reads position.

First construct the object with the required dimensionality.

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:

f.x = np.array([[2.],    # position
[0.]])   # velocity


or just use a one dimensional array, which I prefer doing.

f.x = np.array([2., 0.])


Define the state transition matrix:

f.F = np.array([[1.,1.],
[0.,1.]])


Define the measurement function:

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 multipy by the uncertainty:

f.P *= 1000.


I could have written:

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

f.R = 5


I could have done this instead:

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:

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:

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 peform Kalman filtering. Use these if you are not a fan of objects.

Example

while True:
z, R = read_sensor()
x, P = predict(x, P, F, Q)
x, P = update(x, P, z, R, H)


References

 [1] Labbe, Roger. “Kalman and Bayesian Filters in Python”.
github repo:
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/table_of_contents.ipynb
PDF version (often lags the two sources above)
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Kalman_and_Bayesian_Filters_in_Python.pdf

Copyright 2015 Roger R Labbe Jr.

FilterPy library. http://github.com/rlabbe/filterpy

Supporting book at: https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

Kalman filter

class filterpy.kalman.KalmanFilter(dim_x, dim_z, dim_u=0)[source]

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.

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.

Attributes

 y measurement residual (innovation) K Kalman gain S system uncertainty in measurement space likelihood likelihood of measurement
 log_likelihood (scalar) Log likelihood of last measurement update.
__init__(dim_x, dim_z, dim_u=0)[source]

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.

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.
update(z, R=None, H=None)[source]

Add a new measurement (z) to the Kalman filter. If z is None, nothing is changed.

Parameters: z : np.array measurement for this update. z can be a scalar if dim_z is 1, otherwise it must be 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.
update_correlated(z, R=None, H=None)[source]

Add a new measurement (z) to the Kalman filter assuming that process noise and measurement noise are correlated as defined in the self.M matrix.

If z is None, nothing is changed.

Parameters: z : np.array measurement for this update. 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.
test_matrix_dimensions(z=None, H=None, R=None, F=None, Q=None)[source]

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.

predict(u=0, B=None, F=None, Q=None)[source]

Predict next position 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. B : np.array(dim_x, dim_z), or None Optional control transition matrix; a value of None in any position 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 in any position 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 in any position will cause the filter to use self.Q.
batch_filter(zs, Fs=None, Qs=None, Hs=None, Rs=None, Bs=None, us=None, update_first=False)[source]

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 : list-like, optional optional list of values to use for the state transition matrix matrix; a value of None in any position will cause the filter to use self.F for that time step. If Fs is None then self.F is used for all epochs. Qs : list-like, optional optional list of values to use for the process error covariance; a value of None in any position will cause the filter to use self.Q for that time step. If Qs is None then self.Q is used for all epochs. Hs : list-like, optional optional list of values to use for the measurement matrix; a value of None in any position will cause the filter to use self.H for that time step. If Hs is None then self.H is used for all epochs. Rs : list-like, optional optional list of values to use for the measurement error covariance; a value of None in any position will cause the filter to use self.R for that time step. If Rs is None then self.R is used for all epochs. Bs : list-like, optional 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. If Bs is None then self.B is used for all epochs. us : list-like, optional 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. 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

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)

rts_smoother(Xs, Ps, Fs=None, Qs=None)[source]

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 ‘x’ : numpy.ndarray smoothed means ‘P’ : numpy.ndarray smoothed state covariances ‘K’ : numpy.ndarray smoother gain at each step

Examples

zs = [t + random.randn()*4 for t in range (40)]

(mu, cov, _, _) = kalman.batch_filter(zs)
(x, P, K) = rts_smoother(mu, cov, kf.F, kf.Q)

get_prediction(u=0)[source]

Predicts the next state of the filter and returns it. Does not alter the state of the filter.

Parameters: u : np.array optional control input (x, P) : tuple State vector and covariance array of the prediction.
residual_of(z)[source]

returns the residual for the given measurement (z). Does not alter the state of the filter.

measurement_of_state(x)[source]

Helper function that converts a state into a measurement.

Parameters: x : np.array kalman state vector z : np.array measurement corresponding to the given state
likelihood

likelihood of measurement

alpha

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.
1. 208-212. (2006)
Q

Process uncertainty matrix

P

state covariance matrix

F

state transition matrix

B

control transition matrix

x

state vector.

K

Kalman gain

y

measurement residual (innovation)

S

system uncertainty in measurement space

filterpy.kalman.update(x, P, z, R, H=None, return_all=False)[source]

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 : numpy.array(dim_z, 1), or float measurement for this update. 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. 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
filterpy.kalman.predict(x, P, F=1, Q=0, u=0, B=1, alpha=1.0)[source]

Predict next position 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 Process noise matrix u : numpy.array, default 0. Control vector. If non-zero, it is multiplied by B to create the control input into the system. B : numpy.array, default 0. Optional control transition matrix. alpha : float, 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 x : numpy.array Prior state estimate vector P : numpy.array Prior covariance matrix
filterpy.kalman.batch_filter(x, P, zs, Fs, Qs, Hs, Rs, Bs=None, us=None, update_first=False)[source]

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; a value of None in any position will cause the filter to use self.F for that time step. Qs : list-like, list of values to use for the process error covariance; a value of None in any position will cause the filter to use self.Q for that time step. Hs : list-like, optional list of values to use for the measurement matrix; a value of None in any position will cause the filter to use self.H for that time step. Rs : list-like, optional list of values to use for the measurement error covariance; a value of None in any position will cause the filter to use self.R for that time step. 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. 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

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)