Koopman (1997) Exact Initial Kalman Filtering and Smoothing for Nonstationary Time Series Models

Summary

Koopman derives an exact, analytically-clean solution to the diffuse-initialization problem of the Kalman filter — how to start the filter for state-space models whose initial state vector is partly non-stationary (regression effects, stochastic trends, seasonals, ARIMA components) and therefore has an improper (diffuse) prior. Rather than the usual numerical hack of setting the initial variance to a huge constant $\kappa$, he writes each filtering quantity as an exact function of $\kappa$, expands the inverse innovation variance in powers of $1/\kappa$, and lets $\kappa\to\infty$, yielding exact recursions with a trivial initialization. The exact smoother, the diffuse likelihood and score, and missing-observation handling all follow, with proofs based only on elementary matrix algebra.

Key Claims

• The diffuse-init problem. Nonstationary state-space models require an initial state $\alpha_1 = a + A\eta + B\delta$ with a diffuse component $\delta\sim N(0,\kappa I)$, $\kappa\to\infty$, so $P_1 = P_* + \kappa P_\infty$ ($P_*=AA'$, $P_\infty=BB'$). The ordinary Kalman filter cannot be run because $P_\infty\ne 0$ has no finite representation.
• Why the standard fixes are unsatisfactory. Replacing $\kappa$ by a large number (e.g. $10^7$; Harvey-Phillips, Burridge-Wallis) is inexact and prone to rounding error; Ansley-Kohn's (1985, 1990) transformation is exact and general but "difficult to implement," needs nonstandard software, and has a long proof; de Jong's (1991) diffuse Kalman filter (DKF) augments the recursions by $m$ columns and inverts an $m\times m$ block, is computationally inefficient, and has complicated smoothing corrections.
• Exact solution by $\kappa$-expansion. Writing $F_t = F_{*,t} + \kappa F_{\infty,t}$ and $M_t = M_{*,t} + \kappa M_{\infty,t}$, the innovation-variance inverse expands as $F_t^{-1} = F_{*,t}^{-1}... = F^{(1)} + \tfrac1\kappa F^{(2)} - \dots + O(\kappa^{-2})$; taking $\kappa\to\infty$ gives exact update equations for $a_{t+1}$, the finite part $P_{*,t+1}$, and the diffuse part $P_{\infty,t+1}$, initialized trivially at $P_{*,1}, P_{\infty,1}$.
• Automatic collapse. The rank of $P_{\infty,t}$ falls as diffuse information is used up ($r(P_{\infty,t+1}) \le r(P_{\infty,t}) - r(F_{\infty,t})$); once it hits zero the recursion reverts automatically to the ordinary Kalman filter — no manual switch, no fixed break point $d$.
• Exact smoothing, likelihood, score, missing data. An exact initial smoothing algorithm is given (Section 3); the diffuse likelihood and score vector are computed (Section 4, using Koopman-Shephard 1992); missing observations are handled by simply deleting the relevant rows of $Z_t, G_t$ — including in the initial period, which the AK transformation could not do cleanly.
• Cheap. For the univariate nonsingular case the extra cost over the ordinary update is only $\sim p^2 + 2p$ flops per step; the method is easy to implement in standard software.

Concepts Introduced or Extended

• Kalman Filter
• State-Space Representation

Entities Mentioned

• Siem Jan Koopman

Quotes

"In this article I pursue an analytical approach where I express the Kalman filter quantities in terms of κ explicitly and then let κ → ∞ to obtain the exact solution."

"This numerical solution is not exact and may generate inaccuracies due to numerical rounding errors."

My Take

This is a small, permanent piece of state-space machinery: everyone who fits an unobserved-components or ARIMA model in state-space form has to initialize a diffuse prior, and Koopman's $\kappa$-expansion is the version that actually got implemented (it underlies SsfPack and the Durbin-Koopman textbook treatment) precisely because it is exact yet needs only a trivial initialization and collapses back to the ordinary filter on its own. The intellectual move — don't approximate $\kappa$ numerically, carry it symbolically and take the limit — is the kind of thing that looks obvious only in retrospect, and the payoff is both accuracy (no $10^7$ rounding pathologies) and simplicity (elementary proofs, standard-software-friendly, natural missing-data handling) relative to the Ansley-Kohn transformation and de Jong's DKF it supersedes. For the wiki it is the rigorous answer to the "$P_{1|0}\to\infty$" hand-wave in the Kalman filter page, and it connects the state-space toolkit to the diffuse-likelihood estimation that nonstationary models require.