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Independent perturbed-observation reference members follow the Kalman filter

Statement

Assume AC and the finite-horizon linear-Gaussian model of Linear-Gaussian state-space models with independent noises, with $R_{k} ≻ 0$ for $1 \leq k \leq T$ . Fix the observation values $y_{1 : T}$ . For each $N$ , initialize $U_{N, i}^{a, 0}$ iid with the model's $N (m_{0}, P_{0})$ prior. At each time use independent per-member noises $q_{N, i}^{k} \sim N (0, Q_{k})$ and $η_{N, i}^{k} \sim N (0, R_{k})$ , independent across $i, k$ and from the preceding ensemble, and form $U_{N, i}^{f, k} = F_{k} U_{N, i}^{a, k - 1} + q_{N, i}^{k}, U_{N, i}^{a, k} = U_{N, i}^{f, k} + K_{k} (y_{k} + η_{N, i}^{k} - H_{k} U_{N, i}^{f, k}),$ where $K_{k} = P_{k ∣ k - 1} H_{k}^{T} (H_{k} P_{k ∣ k - 1} H_{k}^{T} + R_{k})^{- 1}$ is the deterministic gain from Exact linear-Gaussian Kalman filter. Then, for every $k \leq T$ , the analysis members $U_{N, 1}^{a, k}, \dots, U_{N, N}^{a, k}$ are iid with law $N (m_{k ∣ k}, P_{k ∣ k})$ , the exact Kalman filtering law at the fixed data $y_{1 : k}$ .

Facts & Assumptions

Given: AC, the finite-horizon linear-Gaussian model, $R_{k} ≻ 0$ , and the independent per-member initialization and perturbations in the Statement.

[A1]

AC is the axiom required by the multivariate-normal and Kalman suppliers. The Axiom of Choice

[F1]

The state and observation equations are affine with mutually independent Gaussian initial, process, and observation noises; their covariance matrices may be positive semidefinite. Linear-Gaussian state-space models with independent noises

[F2]

A finite independent stack of multivariate normal vectors is jointly multivariate normal, including for singular covariance blocks. A finite stack of independent Gaussian vectors is jointly Gaussian

[F3]

The Kalman algorithm predicts $m^{f} = F_{k} m$ , $P^{f} = F_{k} P F_{k}^{T} + Q_{k}$ and updates by $m^{a} = m^{f} + K_{k} (y_{k} - H_{k} m^{f})$ , $P^{a} = P^{f} - K_{k} S_{k} K_{k}^{T}$ . Exact linear-Gaussian Kalman filter

[F4]

Under the stated model and positive-definite innovation matrices, these recursions are exact Gaussian conditional laws at every observation history. The finite-horizon linear-Gaussian Kalman filter is exact

[F5]

A multivariate normal law permits singular positive-semidefinite covariance and is closed under the defining affine projections. Multivariate normal law, including singular covariance

Proof

Given: The hypotheses in the Statement.

1.1A1F1F5givenbase

At $k = 0$ , the analysis members are iid with law $N (m_{0}, P_{0})$ by their initialization, so the claim holds at the base time.

2.1A1F1F2F5givenstep 1.1IH

Assume the induction hypothesis: the members at time $k - 1$ are iid with law $N (m_{k - 1 ∣ k - 1}, P_{k - 1 ∣ k - 1})$ . Each forecast is an affine image of the independent Gaussian pair $(U_{N, i}^{a, k - 1}, q_{N, i}^{k})$ . By [F2] the pair is jointly Gaussian; every scalar projection of its affine image is a scalar Gaussian projection, so [F5] makes the forecasts iid Gaussian with mean $m^{f} = F_{k} m_{k - 1 ∣ k - 1}$ and covariance $P^{f} = F_{k} P_{k - 1 ∣ k - 1} F_{k}^{T} + Q_{k}$ .

3.1F1F2F3F5step 2.1algebra

Since $P^{f} ⪰ 0$ and $R_{k} ≻ 0$ , $S = H_{k} P^{f} H_{k}^{T} + R_{k} ≻ 0$ , so $K = P^{f} H_{k}^{T} S^{- 1}$ is defined. The independent Gaussian pair $(U_{N, i}^{f, k}, y_{k} + η_{N, i}^{k})$ and [F2] show that $U_{N, i}^{a, k} = A U_{N, i}^{f, k} + K (y_{k} + η_{N, i}^{k})$ , with $A = I - K H_{k}$ , is Gaussian. Its mean is $m^{f} + K (y_{k} - H_{k} m^{f})$ and its covariance is $A P^{f} A^{T} + K R_{k} K^{T} = P^{f} - K S K^{T},$ because $K S K^{T} = P^{f} H_{k}^{T} K^{T} = K H_{k} P^{f}$ . These are precisely the Kalman update parameters in [F3].

4.1A1F1F4step 1.1step 2.1step 3.1

For distinct member indices the pairs of forecast and perturbed observation are independent by the input independence and the induction hypothesis; applying the same deterministic affine update separately to each pair preserves independence. Therefore the updated members are iid with the Gaussian law from step 3.1. By [A1] and [F4] that law is the exact filtering law at the fixed data.

5.1step 1.1step 4.1discharge-induction: step 1.1step 4.1∎

The initialization in step 1.1 is the base case, and step 4.1 proves the claim at time $k$ from the induction hypothesis at time $k - 1$ . Induction therefore gives the asserted iid exact filtering law for every $k \leq T$ .

Depends on

Used by

The perturbed-observation EnKF converges to the Kalman filter at fixed horizon Theorem

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Sources

Jan Mandel, Loren Cobb, and Jonathan D. Beezley, On the Convergence of the Ensemble Kalman Filter, §3 (standard reference, not scraped)
K. J. H. Law, A. M. Stuart, and K. C. Zygalakis, Data Assimilation: A Mathematical Introduction, Chapter 4 §4.2.3 (standard reference, not scraped)