`statespace_smoothing`#

Further smoothing results (DK §4.4.5-4.8): updating smoothed estimates, fixed-point and fixed-lag smoothing, other state smoothers, covariances of the smoothed state across time, and filtering and smoothing weights.

The recursions are written in terms of the conventional \(K_t\) and \(F_t^{-1}\); conventional_gain forms them in any period after the diffuse periods, whichever filter method was used. Unless stated, the routines are not defined with diffuse states (info is SS_ERR_UNSUPPORTED).

Other state smoothers#

All give the same \(\hat\alpha_t\) as state_smoother; they illustrate the algorithms of DK §4.6 and serve as checks.

subroutine fast_state_smoother(rep, fres, alphahat, info)
subroutine classical_state_smoother(rep, fres, alphahat, V, info)
subroutine two_filter_smoother(rep, fres, alphahat, V, info)
subroutine whittle_smoother(rep, fres, alphahat, info)

fast_state_smoother: DK §4.6.2: a backward pass for \(r_t\) only, then \(\hat\alpha_1 = a_1 + P_1 r_0\) (plus \(P_{\infty,1} r_0^{(1)}\) with diffuse states) and \(\hat\alpha_{t+1} = c_t + T_t \hat\alpha_t + R_t Q_t R_t' r_t\). Defined with diffuse states.
classical_state_smoother: The Rauch-Tung-Striebel form (DK §4.6.1): \(J_t = P_{t|t} T_t' P_{t+1}^{-1}\), \(\hat\alpha_t = a_{t|t} + J_t (\hat\alpha_{t+1} - a_{t+1})\), \(V_t = P_{t|t} + J_t (V_{t+1} - P_{t+1}) J_t'\), with \(P_{t+1}^{-1}\) applied as a pseudo-inverse.
two_filter_smoother: DK §4.6.4: a backward information filter for \(y_t, \dots, y_n\) combined with the forward filter.
whittle_smoother: DK §4.6.3: the backward recursion from the first-order conditions of the joint density of \(\alpha\) and Y, started from \(\hat\alpha_n = a_{n|n}\). It needs \(T_t\), \(R_t Q_t R_t'\) and \(H_t\) nonsingular and, as DK note, loses accuracy geometrically going back; usable for short series only.

Updating, fixed-point and fixed-lag smoothing#

subroutine update_smoothed(rep, fres, n0, alphahat, V, info)
  real(dp), intent(inout) :: alphahat(:, :), V(:, :, :)    ! (m, n), (m, m, n)
subroutine fixed_point_smoother(rep, fres, t, path_a, path_V, info)
  real(dp), intent(out) :: path_a(:, :), path_V(:, :, :)   ! (m, n-t+1), (m, m, n-t+1)
subroutine fixed_lag_smoother(rep, fres, j, alphahat, V, info)
  real(dp), intent(out) :: alphahat(:, :), V(:, :, :)      ! (m, n), (m, m, n)

update_smoothed: DK §4.4.5: columns 1..n0 hold estimates given \(y_1, \dots, y_{n_0}\) on entry and given all n observations on exit, by the recursions (DK eqs. 4.49-4.50) for each new observation.
fixed_point_smoother: DK §4.4.6: \(\hat\alpha_{t|k}\) and \(V_{t|k}\) for fixed t and k = t, …, n, in column k - t + 1.
fixed_lag_smoother: DK §4.4.6: \(\hat\alpha_{s|s+j}\) and \(V_{s|s+j}\) (DK eq. 4.51) for fixed j, in column s = 1, …, n - j; later columns are NaN.

Covariances across time#

subroutine smoothed_state_cov_between(rep, fres, sres, t, j, C, info)
  real(dp), intent(out) :: C(:, :)                 ! (m, m)
subroutine smoothed_state_autocov(rep, fres, sres, acov, info)
  real(dp), intent(out) :: acov(:, :, :)           ! (m, m, n-1)

smoothed_state_cov_between: \(Cov(\alpha_t, \alpha_j | Y_n)\) (DK §4.7); for j > t, \(P_t L_t' \cdots L_{j-1}' (I - N_{j-1} P_j)\). Periods t, …, j-1 must be past the diffuse periods.
smoothed_state_autocov: acov(:, :, t) is \(Cov(\alpha_{t+1}, \alpha_t | Y_n)\); NaN where the diffuse periods are involved.

Weights#

subroutine filtered_state_weights(rep, fres, Wa, Watt, info)
  real(dp), intent(out) :: Wa(:, :, :, :), Watt(:, :, :, :)   ! (m, p, n, n)
subroutine smoothed_state_weights(rep, W, C, A, info, j_list)
  real(dp), intent(out) :: W(:, :, :, :)     ! (m, p, n, n)
  real(dp), intent(out) :: C(:, :, :, :)     ! (m, m, n, n)
  real(dp), intent(out) :: A(:, :, :)        ! (m, m, n)
  integer, intent(in), optional :: j_list(:)

filtered_state_weights: DK §4.8.2 (Table 4.5): Wa(:, :, t, j) is the weight of \(y_j\) in \(a_t\), \(L_{t-1} \cdots L_{j+1} K_j\) for j < t; Watt the same for \(a_{t|t}\). Intercepts and the initial mean aside.
smoothed_state_weights: DK §4.8.3, extended to the intercepts and the prior mean: \(\hat\alpha_t = \sum_j W_{tj} (y_j - d_j) + \sum_j C_{tj} c_j + A_t a_1\). Defined in diffuse periods. Costs one smoother run per observation; j_list restricts the computation to some j.

Building blocks#

subroutine innovation_transition(rep, fres, t, L, info)
subroutine conventional_gain(rep, fres, t, K, Finv, info)

innovation_transition: \(L_t\) with \(a_{t+1} - \alpha_{t+1} = L_t (a_t - \alpha_t) + \dots\): \(T_t - K_t Z_t\) in conventional periods and \(T_t L_{t,p} \cdots L_{t,1}\), \(L_{t,i} = I - K_{t,i} Z^*_i\), in univariate ones (DK §4.3, §6.4).
conventional_gain: \(K_t = T_t P_t Z_t' F_t^{-1}\) and \(F_t^{-1}\) over the observed elements for a period after the diffuse periods, formed from the stored \(P_t\) and \(F_t\) where the filter did not store them.

statespace_smoothing#