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.
Readonly Instance Variables
 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 multiply 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 perform 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/KalmanandBayesianFiltersinPython
 read online:
 http://nbviewer.ipython.org/github/rlabbe/KalmanandBayesianFiltersinPython/blob/master/table_of_contents.ipynb
 PDF version (often lags the two sources above)
 https://github.com/rlabbe/KalmanandBayesianFiltersinPython/blob/master/Kalman_and_Bayesian_Filters_in_Python.pdf
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/KalmanandBayesianFiltersinPython
This is licensed under an MIT license. See the readme.MD file for more information.
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.
Examples
See my book Kalman and Bayesian Filters in Python https://github.com/rlabbe/KalmanandBayesianFiltersinPython
Attributes
x (numpy.array(dim_x, 1)) State estimate vector P (numpy.array(dim_x, dim_x)) Covariance matrix 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 You may read the following attributes. 
__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 : (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.

update_steadystate
(z)[source]¶ 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.
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.
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 : (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.

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 state (prior) using the Kalman filter state propagation equations.
Parameters: u : np.array
Optional control vector. If nonzero, 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.

predict_steadystate
(u=0, B=None)[source]¶ 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 nonzero, 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.

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 : listlike
list of measurements at each time step self.dt Missing measurements must be represented by ‘None’.
Fs : listlike, 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 : listlike, 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 : listlike, 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 : listlike, 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 : listlike, 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 : listlike, 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.
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
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 RauchTungStriebal 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 : listlike 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 : listlike 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
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
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)

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
Returns: (x, P) : tuple
State vector and covariance array of the prediction.

get_update
(z=None)[source]¶ Computes the new estimate based on measurement ``z`. Does not alter 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.

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
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.

log_likelihood
(z)[source]¶ log likelihood of the measurement z. This should only be called after a call to update(). Calling after predict() will yield an incorrect result.

likelihood
(z, allow_zero=False)[source]¶ likelihood of measurement z.
Computed from the loglikelihood. the loglikelihood 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 you want to allow zero, set allow_zero to True.
But really, this is a bad measure because of the scaling that is involved  try to use loglikelihood in your equations!

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.
 208212. (2006)


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 : (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

filterpy.kalman.
predict
(x, P, F=1, Q=0, u=0, B=1, alpha=1.0)[source]¶ 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 nonzero, 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

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 : listlike
list of measurements at each time step. Missing measurements must be represented by ‘None’.
Fs : listlike
list of values to use for the state transition matrix matrix.
Qs : listlike
list of values to use for the process error covariance.
Hs : listlike
list of values to use for the measurement matrix.
Rs : listlike
list of values to use for the measurement error covariance.
Bs : listlike, 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 : listlike, 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.
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
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)