AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generated

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Rauch–Tung–Striebel fixed-interval Gaussian smoother

Definition

Fix a finite-horizon linear-Gaussian model from Linear-Gaussian state-space models with independent noises and data $y_{1 : T}$ for which Exact linear-Gaussian Kalman filter is admissible. Its forward output contains filtered means and covariances $m_{k ∣ k}, P_{k ∣ k}$ and predicted means and covariances $m_{k + 1 ∣ k}, P_{k + 1 ∣ k}$ . Assume each forecast covariance used below is positive definite, meaning $u^{T} P_{k + 1 ∣ k} u > 0$ for every nonzero $u \in R^{d_{k + 1}}$ . Positive semidefinite means the same quadratic form is nonnegative for every $u$ .

The algorithm takes that forward output as input, initializes $m_{T ∣ T} : = m_{T ∣ T}^{f i l t e r}, P_{T ∣ T} : = P_{T ∣ T}^{f i l t e r}$ and for $k = T - 1, T - 2, \dots, 0$ performs the following steps:

Form the backward gain $C_{k} = P_{k ∣ k} F_{k + 1}^{T} P_{k + 1 ∣ k}^{- 1} .$ For the computation, factor $P_{k + 1 ∣ k} = L_{k} L_{k}^{T}$ with the unique positive-diagonal Cholesky factor from A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique, then compute the rows of $U_{k}$ in increasing order by $(U_{k})_{i, \cdot} = \frac{(F_{k + 1} P_{k ∣ k})_{i, \cdot} - \sum_{j = 1}^{i - 1} (L_{k})_{i j} (U_{k})_{j, \cdot}}{(L_{k})_{i i}}, i = 1, \dots, d_{k + 1},$ and compute the rows of $C_{k}^{T}$ in decreasing order by $(C_{k}^{T})_{i, \cdot} = \frac{(U_{k})_{i, \cdot} - \sum_{j = i + 1}^{d_{k + 1}} (L_{k})_{j i} (C_{k}^{T})_{j, \cdot}}{(L_{k})_{i i}}, i = d_{k + 1}, \dots, 1.$ These recurrences give $L_{k} U_{k} = F_{k + 1} P_{k ∣ k}$ and $L_{k}^{T} C_{k}^{T} = U_{k}$ , so $C_{k} = P_{k ∣ k} F_{k + 1}^{T} P_{k + 1 ∣ k}^{- 1}$ ; no inverse is formed in the procedure.
Update the smoothed mean and covariance: $m_{k ∣ T} = m_{k ∣ k} + C_{k} (m_{k + 1 ∣ T} - m_{k + 1 ∣ k})$ $P_{k ∣ T} = P_{k ∣ k} + C_{k} (P_{k + 1 ∣ T} - P_{k + 1 ∣ k}) C_{k}^{T}$

Return $(m_{k ∣ T}, P_{k ∣ T})_{k = 0}^{T}$ . For $T = 0$ the loop is empty and the output is the initial filtered pair. The algorithm is undefined if a forecast covariance to be solved is singular; it makes no pseudoinverse, complexity, rounding-error, or numerical-stability claim. The theorem on this page proves that the outputs are moments of the fixed-interval conditional laws under these hypotheses.

Depends on

Used by

Dependency tree · two levels

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Sources

Simo Särkkä, Bayesian Filtering and Smoothing, Chapter 8, §8.2 (standard reference, not scraped)
Kody Law, Andrew M. Stuart, and Konstantinos Zygalakis, Data Assimilation: A Mathematical Introduction, Chapter 3, §3.1 (standard reference, not scraped)