kalmanUnivariate | R Documentation |

Univariate treatment (sequential processing) of the multivariate Kalman filter and smoother equations for fast implementation. Refer to Koopman and Durbin (2000).

```
kalmanUnivariate(X, a0_0, P0_0, A, Lambda, Sig_e, Sig_u)
```

`X` |
n x p, numeric matrix of (stationary) time series |

`a0_0` |
k x 1, initial state mean vector |

`P0_0` |
k x k, initial state covariance matrix |

`A` |
k x k, state transition matrix |

`Lambda` |
p x k, measurement matrix |

`Sig_e` |
p x p, measurement equation residuals covariance matrix (diagonal) |

`Sig_u` |
k x k, state equation residuals covariance matrix |

For full details of the univariate filtering approach, please refer to Mosley et al. (2023). Note that `n`

is the number of observations, `p`

is the number of time series, and `k`

is the number of states.

logl log-likelihood of the innovations from the Kalman filter

at_t `k \times n`

, filtered state mean vectors

Pt_t `k \times k \times n`

, filtered state covariance matrices

at_n `k \times n`

, smoothed state mean vectors

Pt_n `k \times k \times n`

, smoothed state covariance matrices

Pt_tlag_n `k \times k \times n`

, smoothed state covariance with lag

Koopman, S. J., & Durbin, J. (2000). Fast filtering and smoothing for multivariate state space models. *Journal of Time Series Analysis, 21*(3), 281-296.

Mosley, L., Chan, TS., & Gibberd, A. (2023). sparseDFM: An R Package to Estimate Dynamic Factor Models with Sparse Loadings.

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