Implements the Kalman Filter, Kalman Smoother, and EM algorithm.
This class implements the Kalman Filter, Kalman Smoother, and EM Algorithm for a Linear Gaussian model specified by,
The Kalman Filter is an algorithm designed to estimate
. As all state transitions and observations are linear with Gaussian distributed noise, these distributions can be represented exactly as Gaussian distributions with mean filtered_state_means[t] and covariances filtered_state_covariances[t].Similarly, the Kalman Smoother is an algorithm designed to estimate
.The EM algorithm aims to find for
If we define
, then the EM algorithm works by iteratively finding,then by maximizing,
| Parameters: | transition_matrices : [n_timesteps-1, n_dim_state, n_dim_state] or [n_dim_state,n_dim_state] array-like
observation_matrices : [n_timesteps, n_dim_obs, n_dim_state] or [n_dim_obs, n_dim_state] array-like
transition_covariance : [n_dim_state, n_dim_state] array-like
observation_covariance : [n_dim_obs, n_dim_obs] array-like
transition_offsets : [n_timesteps-1, n_dim_state] or [n_dim_state] array-like
observation_offsets : [n_timesteps, n_dim_obs] or [n_dim_obs] array-like
initial_state_mean : [n_dim_state] array-like
initial_state_covariance : [n_dim_state, n_dim_state] array-like
random_state : optional, numpy random state
em_vars : optional, subset of [‘transition_matrices’, ‘observation_matrices’, ‘transition_offsets’, ‘observation_offsets’, ‘transition_covariance’, ‘observation_covariance’, ‘initial_state_mean’, ‘initial_state_covariance’] or ‘all’
n_dim_state: optional, integer
n_dim_obs: optional, integer
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Methods
| em(X[, y, n_iter, em_vars]) | Apply the EM algorithm |
| filter(X) | Apply the Kalman Filter |
| filter_update(filtered_state_mean, ...[, ...]) | Update a Kalman Filter state estimate |
| loglikelihood(X) | Calculate the log likelihood of all observations |
| sample(n_timesteps[, initial_state, ...]) | Sample a state sequence timesteps in length. |
| smooth(X) | Apply the Kalman Smoother |
Apply the EM algorithm
Apply the EM algorithm to estimate all parameters specified by em_vars. Note that all variables estimated are assumed to be constant for all time. See _em() for details.
| Parameters: | X : [n_timesteps, n_dim_obs] array-like
n_iter : int, optional
em_vars : iterable of strings or ‘all’
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Apply the Kalman Filter
Apply the Kalman Filter to estimate the hidden state at time
for given observations up to and including time t. Observations are assumed to correspond to times . The output of this method corresponding to time can be used in KalmanFilter.filter_update() for online updating.| Parameters: | X : [n_timesteps, n_dim_obs] array-like
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| Returns: | filtered_state_means : [n_timesteps, n_dim_state]
filtered_state_covariances : [n_timesteps, n_dim_state, n_dim_state] array
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Update a Kalman Filter state estimate
Perform a one-step update to estimate the state at time
give an observation at time and the previous estimate for time given observations from times . This method is useful if one wants to track an object with streaming observations.| Parameters: | filtered_state_mean : [n_dim_state] array
filtered_state_covariance : [n_dim_state, n_dim_state] array
observation : [n_dim_obs] array or None
transition_matrix : optional, [n_dim_state, n_dim_state] array
transition_offset : optional, [n_dim_state] array
transition_covariance : optional, [n_dim_state, n_dim_state] array
observation_matrix : optional, [n_dim_obs, n_dim_state] array
observation_offset : optional, [n_dim_obs] array
observation_covariance : optional, [n_dim_obs, n_dim_obs] array
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| Returns: | next_filtered_state_mean : [n_dim_state] array
next_filtered_state_covariance : [n_dim_state, n_dim_state] array
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Calculate the log likelihood of all observations
| Parameters: | X : [n_timesteps, n_dim_obs] array
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| Returns: | likelihood : float
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Sample a state sequence
timesteps in length.| Parameters: | n_timesteps : int
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| Returns: | states : [n_timesteps, n_dim_state] array
observations : [n_timesteps, n_dim_obs] array
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Apply the Kalman Smoother
Apply the Kalman Smoother to estimate the hidden state at time
for given all observations. See _smooth() for more complex output| Parameters: | X : [n_timesteps, n_dim_obs] array-like
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| Returns: | smoothed_state_means : [n_timesteps, n_dim_state]
smoothed_state_covariances : [n_timesteps, n_dim_state]
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Kalman Filter based on Cholesky decomposition
| Parameters: | transition_matrices : [n_timesteps-1, n_dim_state, n_dim_state] or [n_dim_state,n_dim_state] array-like
observation_matrices : [n_timesteps, n_dim_obs, n_dim_obs] or [n_dim_obs, n_dim_obs] array-like
transition_covariance : [n_dim_state, n_dim_state] array-like
observation_covariance : [n_dim_obs, n_dim_obs] array-like
transition_offsets : [n_timesteps-1, n_dim_state] or [n_dim_state] array-like
observation_offsets : [n_timesteps, n_dim_obs] or [n_dim_obs] array-like
initial_state_mean : [n_dim_state] array-like
initial_state_covariance : [n_dim_state, n_dim_state] array-like
random_state : optional, numpy random state
em_vars : optional, subset of [‘transition_matrices’, ‘observation_matrices’, ‘transition_offsets’, ‘observation_offsets’, ‘transition_covariance’, ‘observation_covariance’, ‘initial_state_mean’, ‘initial_state_covariance’] or ‘all’
n_dim_state: optional, integer
n_dim_obs: optional, integer
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Methods
| em | |
| filter | |
| filter_update | |
| loglikelihood | |
| sample | |
| smooth |
Kalman Filter based on UDU’ decomposition
| Parameters: | transition_matrices : [n_timesteps-1, n_dim_state, n_dim_state] or [n_dim_state,n_dim_state] array-like
observation_matrices : [n_timesteps, n_dim_obs, n_dim_obs] or [n_dim_obs, n_dim_obs] array-like
transition_covariance : [n_dim_state, n_dim_state] array-like
observation_covariance : [n_dim_obs, n_dim_obs] array-like
transition_offsets : [n_timesteps-1, n_dim_state] or [n_dim_state] array-like
observation_offsets : [n_timesteps, n_dim_obs] or [n_dim_obs] array-like
initial_state_mean : [n_dim_state] array-like
initial_state_covariance : [n_dim_state, n_dim_state] array-like
random_state : optional, numpy random state
em_vars : optional, subset of [‘transition_matrices’, ‘observation_matrices’, ‘transition_offsets’, ‘observation_offsets’, ‘transition_covariance’, ‘observation_covariance’, ‘initial_state_mean’, ‘initial_state_covariance’] or ‘all’
n_dim_state: optional, integer
n_dim_obs: optional, integer
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Methods
| em | |
| filter | |
| filter_update | |
| loglikelihood | |
| sample | |
| smooth |
Implements the General (aka Augmented) Unscented Kalman Filter governed by the following equations,
Notice that although the input noise to the state transition equation and the observation equation are both normally distributed, any non-linear transformation may be applied afterwards. This allows for greater generality, but at the expense of computational complexity. The complexity of UnscentedKalmanFilter.filter() is
where is the number of time steps, is the size of the state space, and is the size of the observation space.If your noise is simply additive, consider using the AdditiveUnscentedKalmanFilter
| Parameters: | transition_functions : function or [n_timesteps-1] array of functions
observation_functions : function or [n_timesteps] array of functions
transition_covariance : [n_dim_state, n_dim_state] array
observation_covariance : [n_dim_obs, n_dim_obs] array
initial_state_mean : [n_dim_state] array
initial_state_covariance : [n_dim_state, n_dim_state] array
n_dim_state: optional, integer
n_dim_obs: optional, integer
random_state : optional, int or RandomState
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Methods
| filter(Z) | Run Unscented Kalman Filter |
| filter_update(filtered_state_mean, ...[, ...]) | Update a Kalman Filter state estimate |
| sample(n_timesteps[, initial_state, ...]) | Sample from model defined by the Unscented Kalman Filter |
| smooth(Z) | Run Unscented Kalman Smoother |
Run Unscented Kalman Filter
| Parameters: | Z : [n_timesteps, n_dim_state] array
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| Returns: | filtered_state_means : [n_timesteps, n_dim_state] array
filtered_state_covariances : [n_timesteps, n_dim_state, n_dim_state] array
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Update a Kalman Filter state estimate
Perform a one-step update to estimate the state at time
give an observation at time and the previous estimate for time given observations from times . This method is useful if one wants to track an object with streaming observations.| Parameters: | filtered_state_mean : [n_dim_state] array
filtered_state_covariance : [n_dim_state, n_dim_state] array
observation : [n_dim_obs] array or None
transition_function : optional, function
transition_covariance : optional, [n_dim_state, n_dim_state] array
observation_function : optional, function
observation_covariance : optional, [n_dim_obs, n_dim_obs] array
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| Returns: | next_filtered_state_mean : [n_dim_state] array
next_filtered_state_covariance : [n_dim_state, n_dim_state] array
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Sample from model defined by the Unscented Kalman Filter
| Parameters: | n_timesteps : int
initial_state : optional, [n_dim_state] array
random_state : optional, int or Random
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Run Unscented Kalman Smoother
| Parameters: | Z : [n_timesteps, n_dim_state] array
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| Returns: | smoothed_state_means : [n_timesteps, n_dim_state] array
smoothed_state_covariances : [n_timesteps, n_dim_state, n_dim_state] array
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Implements the Unscented Kalman Filter with additive noise. Observations are assumed to be generated from the following process,
While less general the general-noise Unscented Kalman Filter, the Additive version is more computationally efficient with complexity
where is the number of time steps and is the size of the state space.| Parameters: | transition_functions : function or [n_timesteps-1] array of functions
observation_functions : function or [n_timesteps] array of functions
transition_covariance : [n_dim_state, n_dim_state] array
observation_covariance : [n_dim_obs, n_dim_obs] array
initial_state_mean : [n_dim_state] array
initial_state_covariance : [n_dim_state, n_dim_state] array
n_dim_state: optional, integer
n_dim_obs: optional, integer
random_state : optional, int or RandomState
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Methods
| filter(Z) | Run Unscented Kalman Filter |
| filter_update(filtered_state_mean, ...[, ...]) | Update a Kalman Filter state estimate |
| sample(n_timesteps[, initial_state, ...]) | Sample from model defined by the Unscented Kalman Filter |
| smooth(Z) | Run Unscented Kalman Smoother |
Run Unscented Kalman Filter
| Parameters: | Z : [n_timesteps, n_dim_state] array
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| Returns: | filtered_state_means : [n_timesteps, n_dim_state] array
filtered_state_covariances : [n_timesteps, n_dim_state, n_dim_state] array
|
Update a Kalman Filter state estimate
Perform a one-step update to estimate the state at time
give an observation at time and the previous estimate for time given observations from times . This method is useful if one wants to track an object with streaming observations.| Parameters: | filtered_state_mean : [n_dim_state] array
filtered_state_covariance : [n_dim_state, n_dim_state] array
observation : [n_dim_obs] array or None
transition_function : optional, function
transition_covariance : optional, [n_dim_state, n_dim_state] array
observation_function : optional, function
observation_covariance : optional, [n_dim_obs, n_dim_obs] array
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| Returns: | next_filtered_state_mean : [n_dim_state] array
next_filtered_state_covariance : [n_dim_state, n_dim_state] array
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Sample from model defined by the Unscented Kalman Filter
| Parameters: | n_timesteps : int
initial_state : optional, [n_dim_state] array
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Run Unscented Kalman Smoother
| Parameters: | Z : [n_timesteps, n_dim_state] array
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| Returns: | smoothed_state_means : [n_timesteps, n_dim_state] array
smoothed_state_covariances : [n_timesteps, n_dim_state, n_dim_state] array
|
Implements the Unscented Kalman Filter with additive noise. Observations are assumed to be generated from the following process,
While less general the general-noise Unscented Kalman Filter, the Additive version is more computationally efficient with complexity
where is the number of time steps and is the size of the state space.| Parameters: | transition_functions : function or [n_timesteps-1] array of functions
observation_functions : function or [n_timesteps] array of functions
transition_covariance : [n_dim_state, n_dim_state] array
observation_covariance : [n_dim_obs, n_dim_obs] array
initial_state_mean : [n_dim_state] array
initial_state_covariance : [n_dim_state, n_dim_state] array
n_dim_state: optional, integer
n_dim_obs: optional, integer
random_state : optional, int or RandomState
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Methods
| filter(Z) | Run Unscented Kalman Filter |
| filter_update(filtered_state_mean, ...[, ...]) | Update a Kalman Filter state estimate |
| sample(n_timesteps[, initial_state, ...]) | Sample from model defined by the Unscented Kalman Filter |
| smooth(Z) | Run Unscented Kalman Smoother |