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.
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.

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.

Methods

bind(endog) Bind data to the statespace representation
initialize(initialization[, …])
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.

Attributes

design (array) Design matrix\(Z~(k\_endog \times k\_states \times nobs)\)
dtype (dtype) Datatype of currently active representation matrices
endog (array) The observation vector, alias for obs.
obs (array) Observation vector\(y~(k\_endog \times nobs)\)
obs_cov (array) Observation covariance matrix\(H~(k\_endog \times k\_endog \times nobs)\)
obs_intercept (array) Observation intercept\(d~(k\_endog \times nobs)\)
prefix (str) BLAS prefix of currently active representation matrices
selection (array) Selection matrix\(R~(k\_states \times k\_posdef \times nobs)\)
state_cov (array) State covariance matrix\(Q~(k\_posdef \times k\_posdef \times nobs)\)
state_intercept (array) State intercept\(c~(k\_states \times nobs)\)
time_invariant (bool) Whether or not currently active representation matrices are time-invariant
transition (array) Transition matrix\(T~(k\_states \times k\_states \times nobs)\)