statsmodels.tsa.statespace.representation.Representation

class statsmodels.tsa.statespace.representation.Representation(k_endog, k_states, k_posdef=None, initial_variance=1000000.0, nobs=0, dtype=<class 'numpy.float64'>, design=None, obs_intercept=None, obs_cov=None, transition=None, state_intercept=None, selection=None, state_cov=None, statespace_classes=None, **kwargs)[source]

State space representation of a time series process

Parameters:

k_endog : array_like or integer

The observed time-series process \(y\) if array like or the number of variables in the process if an integer.

k_states : int

The dimension of the unobserved state process.

k_posdef : int, optional

The dimension of a guaranteed positive definite covariance matrix describing the shocks in the measurement equation. Must be less than or equal to k_states. Default is k_states.

initial_variance : float, optional

Initial variance used when approximate diffuse initialization is specified. Default is 1e6.

initialization : Initialization object or string, optional

Initialization method for the initial state. If a string, must be one of {‘diffuse’, ‘approximate_diffuse’, ‘stationary’, ‘known’}.

initial_state : array_like, optional

If initialization=’known’ is used, the mean of the initial state’s distribution.

initial_state_cov : array_like, optional

If initialization=’known’ is used, the covariance matrix of the initial state’s distribution.

nobs : integer, optional

If an endogenous vector is not given (i.e. k_endog is an integer), the number of observations can optionally be specified. If not specified, they will be set to zero until data is bound to the model.

dtype : np.dtype, optional

If an endogenous vector is not given (i.e. k_endog is an integer), the default datatype of the state space matrices can optionally be specified. Default is np.float64.

design : array_like, optional

The design matrix, \(Z\). Default is set to zeros.

obs_intercept : array_like, optional

The intercept for the observation equation, \(d\). Default is set to zeros.

obs_cov : array_like, optional

The covariance matrix for the observation equation \(H\). Default is set to zeros.

transition : array_like, optional

The transition matrix, \(T\). Default is set to zeros.

state_intercept : array_like, optional

The intercept for the transition equation, \(c\). Default is set to zeros.

selection : array_like, optional

The selection matrix, \(R\). Default is set to zeros.

state_cov : array_like, optional

The covariance matrix for the state equation \(Q\). Default is set to zeros.

**kwargs

Additional keyword arguments. Not used directly. It is present to improve compatibility with subclasses, so that they can use **kwargs to specify any default state space matrices (e.g. design) without having to clean out any other keyword arguments they might have been passed.

Notes

A general state space model is of the form

\[\begin{split}y_t & = Z_t \alpha_t + d_t + \varepsilon_t \\ \alpha_t & = T_t \alpha_{t-1} + c_t + R_t \eta_t \\\end{split}\]

where \(y_t\) refers to the observation vector at time \(t\), \(\alpha_t\) refers to the (unobserved) state vector at time \(t\), and where the irregular components are defined as

\[\begin{split}\varepsilon_t \sim N(0, H_t) \\ \eta_t \sim N(0, Q_t) \\\end{split}\]

The remaining variables (\(Z_t, d_t, H_t, T_t, c_t, R_t, Q_t\)) in the equations are matrices describing the process. Their variable names and dimensions are as follows

Z : design \((k\_endog \times k\_states \times nobs)\)

d : obs_intercept \((k\_endog \times nobs)\)

H : obs_cov \((k\_endog \times k\_endog \times nobs)\)

T : transition \((k\_states \times k\_states \times nobs)\)

c : state_intercept \((k\_states \times nobs)\)

R : selection \((k\_states \times k\_posdef \times nobs)\)

Q : state_cov \((k\_posdef \times k\_posdef \times nobs)\)

In the case that one of the matrices is time-invariant (so that, for example, \(Z_t = Z_{t+1} ~ \forall ~ t\)), its last dimension may be of size \(1\) rather than size nobs.

References

[*]Durbin, James, and Siem Jan Koopman. 2012. Time Series Analysis by State Space Methods: Second Edition. Oxford University Press.

Attributes

nobs (int) The number of observations.
k_endog (int) The dimension of the observation series.
k_states (int) The dimension of the unobserved state process.
k_posdef (int) The dimension of a guaranteed positive definite covariance matrix describing the shocks in the measurement equation.
shapes (dictionary of name:tuple) A dictionary recording the initial shapes of each of the representation matrices as tuples.
initialization (str) Kalman filter initialization method. Default is unset.
initial_variance (float) Initial variance for approximate diffuse initialization. Default is 1e6.

Methods

bind(endog) Bind data to the statespace representation
initialize(initialization[, …]) Create an Initialization object if necessary
initialize_approximate_diffuse([variance]) Initialize the statespace model with approximate diffuse values.
initialize_diffuse() Initialize the statespace model as stationary.
initialize_known(constant, stationary_cov) Initialize the statespace model with known distribution for initial state.
initialize_stationary() Initialize the statespace model as stationary.