\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\cov}{\operatorname{Cov}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}} \renewcommand{\one}{\unicode{x1D7D9}}\]

A random-sampling variant/generalisation of the Kalman-Bucy filter. That also describes particle filters, but the randomisation is different than those. We can do both types of randomisation. This sent has a few tweaks that make it more tenable in tricky situations with high dimensional state spaces or nonlinearities in inconvenient places. A popular data assimilation method for spatiotemporal models.

## Tutorial introductions

Katzfuss, Stroud, and Wikle (2016); Roth et al. (2017); Fearnhead and KΓΌnsch (2018), are all pretty good. Schillings and Stuart (2017) has been recommended by Haber, Lucka, and Ruthotto (2018) as the canonical modern version. Wikle and Berliner (2007) presents it in a broader data assimilation context, although it is too curt to be helpful for me. Mandel (2009) is slightly longer. The inventor of the method explains it in Geir Evensen (2003), but I could make neither head nor tail of that, since it uses too much oceanography terminology. Roth et al. (2017) is probably the best for my background. Let us copy their notation.

We start from the discrete-time state-space models; the basic one is the linear system \[ \begin{aligned} x_{k+1} &=F x_{k}+G v_{k}, \\ y_{k} &=H x_{k}+e_{k}, \end{aligned} \] with state \(x\in\mathbb{R}^n\) and the measurement \(y\in\mathbb{R}^m\). The initial state \(x_{0}\), the process noise \(v_{k}\), and the measurement noise \(e_{k}\) are mutually independent such that \[\begin{aligned} \Ex x_{0}&=\hat{x}_{0}\\ \Ex v_{k}&=0\\ \Ex e_{k}&=0\\ \cov x_{0} &=P_{0}\\ \cov v_{k} & =Q\\ \cov e_{k}&=R \end{aligned}\] and all are Gaussian.

The Kalman filter propagates state estimates \(\hat{x}_{k \mid k}\) and covariance matrices \(P_{k \mid k}\) for this model.
The KF *update* or *prediction* or *forecast* is given by the step
\[
\begin{aligned}
&\hat{x}_{k+1 \mid k}=F \hat{x}_{k \mid k} \\
&P_{k+1 \mid k}=F P_{k \mid k} F^{\top}+G Q G^{\top}
\end{aligned}
\]
We predict the observations forward using these state estimates via
\[
\begin{aligned}
\hat{y}_{k \mid k-1} &=H \hat{x}_{k \mid k-1}, \\
S_{k} &=H P_{k \mid k-1} H^{\top}+R .
\end{aligned}
\]
Given these and an actual observation, we update the state estimates using a *gain matrix*, \(K_{k}\)
\[
\begin{aligned}
\hat{x}_{k \mid k} &=\hat{x}_{k \mid k-1}+K_{k}\left(y_{k}-\hat{y}_{k \mid k-1}\right) \\
&=\left(I-K_{k} H\right) \hat{x}_{k \mid k-1}+K_{k} y_{k}, \\
P_{k \mid k} &=\left(I-K_{k} H\right) P_{k \mid k-1}\left(I-K_{k} H\right)^{\top}+K_{k} R K_{k}^{\top}.
\end{aligned}
\]
in what geoscience types refer to as an *analysis* update.
The variance-minimising gain is given
\[
K_{k}=P_{k \mid k-1} H^{\top} S_{k}^{-1}=M_{k} S_{k}^{-1},
\]
where \(M_{k}\) is the cross-covariance between the state and output predictions.

In the Ensemble Kalman filter, we approximate some of these quantities of interest using samples; this allows us to relax the assumption of Gaussianity and gets us computational savings in certain problems of interest. That does sound very similar to particle filters, and indeed there is a relation.

Instead of maintaining the \(n\)-dimensional estimate \(\hat{x}_{k \mid k}\) and the \(n \times n\) covariance \(P_{k \mid k}\) as such, we maintain an ensemble of \(N<n\) sampled state realizations
\[X_{k}:=\left[x_{k}^{(i)}\right]_{i=1}^{N}.\]
This notation is intended to imply that we are treating these realisations as an \(n \times N\) matrix \(X_{k \mid k}\) with columns \(x_{k}^{(i)}\).
We introduce the following notation for ensemble moments:
\[
\begin{aligned}
&\bar{x}_{k \mid k}=\frac{1}{N} X_{k \mid k} \one \\
&\bar{P}_{k \mid k}=\frac{1}{N-1} \widetilde{X}_{k \mid k} \widetilde{X}_{k \mid k}^{\top},
\end{aligned}
\]
where \(\one=[1, \ldots, 1]^{\top}\) is an \(N\)-dimensional vector and
\[
\widetilde{X}_{k \mid k}=X_{k \mid k}-\bar{x}_{k \mid k} \one^{\top}=X_{k \mid k}\left(I_{N}-\frac{1}{N} \one \one^{\top}\right)
\]
is an ensemble of *anomalies*/*deviations* from \(\bar{x}_{k \mid k}\), which I would call it the *centred version*.
We attempt to match the moments of the ensemble with those realised by a true Kalman filter, in the sense that
\[
\begin{aligned}
&\bar{x}_{k \mid k}:=\frac{1}{N} \sum_{i=1}^{N} x_{k}^{(i)} \approx \hat{x}_{k \mid k}, \\
&\bar{P}_{k \mid k}:=\frac{1}{N-1} \sum_{i=1}^{N}\left(x_{k}^{(i)}-\bar{x}_{k \mid k}\right)\left(x_{k}^{(i)}-\bar{x}_{k \mid k}\right)^{\top} \approx P_{k \mid k} .
\end{aligned}
\]
The forecast step computes \(X_{k+1 \mid k}\) such that its moments are close to \(\hat{x}_{k+1 \mid k}\) and \(P_{k+1 \mid k}\).
An ensemble of \(N\) independent process noise realizations \(V_{k}:=\left[v_{k}^{(i)}\right]_{i=1}^{N}\) with zero mean and covariance \(Q\), is used in
\[
X_{k+1 \mid k}=F X_{k \mid k}+G V_{k}.
\]

Next the \(X_{k \mid k-1}\) is adjusted to obtain the filtering ensemble \(X_{k \mid k}\) by applying an update to each ensemble member: With some gain matrix \(\bar{K}_{k}\) the KF update is applied to the ensemble by the update \[ X_{k \mid k}=\left(I-\bar{K}_{k} H\right) X_{k \mid k-1}+\bar{K}_{k} y_{k} \one^{\top} . \] This does not yet approximate the update of the full Kalman observation β there is no term \(\bar{K}_{k} R \bar{K}_{k}^{\top}\); We have a choice how to implement that.

### Stochastic EnKF update

In the stochastic method, we use artificial zero-mean measurement noise realizations \(E_{k}:=\left[e_{k}^{(i)}\right]_{i=1}^{N}\) with covariance \(R\). \[ X_{k \mid k}=\left(I-\bar{K}_{k} H\right) X_{k \mid k-1}+\bar{K}_{k} y_{k} \one^{\top}-\bar{K}_{k} E_{k} . \] The resulting \(X_{k \mid k}\) has the correct ensemble mean and covariance, \(\hat{x}_{k \mid k}\) and \(P_{k \mid k}\).

If we define a predicted output ensemble \[ Y_{k \mid k-1}=H X_{k \mid k-1}+E_{k} \] that evokes the classic Kalman update (and encapsulates information about) \(\hat{y}_{k \mid k-1}\) and \(S_{k}\), we can rewrite this update into one that resembles the Kalman update: \[ X_{k \mid k}=X_{k \mid k-1}+\bar{K}_{k}\left(y_{k} \one^{\top}-Y_{k \mid k-1}\right) . \]

Now, the gain matrix \(\bar{K}_{k}\) in the classic KF is computed from the covariance matrices of the predicted state and output. In the EnKF, the required \(M_{k}\) and \(S_{k}\) must be estimated from the prediction ensembles. The obvious way of doing that is to once again centre the ensemble, \[ \begin{aligned} &\widetilde{X}_{k \mid k-1}=X_{k \mid k-1}\left(I_{N}-\frac{1}{N} \one \one^{\top}\right) \\ &\widetilde{Y}_{k \mid k-1}=Y_{k \mid k-1}\left(I_{N}-\frac{1}{N} \one \one^{\top}\right) \end{aligned} \] and use the sample covariances \[ \begin{aligned} \bar{M}_{k} &=\frac{1}{N-1} \widetilde{X}_{k \mid k-1} \widetilde{Y}_{k \mid k-1}^{\top}, \\ \bar{S}_{k} &=\frac{1}{N-1} \widetilde{Y}_{k \mid k-1} \widetilde{Y}_{k \mid k-1}^{\top} . \end{aligned} \] The gain \(\bar{K}_{k}\) is then the solution to the system of linear equations, \[ \bar{K}_{k} \widetilde{Y}_{k \mid k-1} \widetilde{Y}_{k \mid k-1}^{\top}=\widetilde{X}_{k \mid k-1} \widetilde{Y}_{k \mid k-1}^{\top} \]

### Deterministic update

Resemblance to unscented/sigma-point filtering also apparent. TBD.

The additive measurement noise model we have used the \(e_{k}\) for should not affect the cross covariance \(M_k\). Thus it is reasonable to make the substitution \[ \widetilde{Y}_{k \mid k-1}\longrightarrow \widetilde{Z}_{k \mid k-1}=H \widetilde{X}_{k \mid k-1} \] to get a less noisy update \[ \begin{aligned} \bar{M}_{k} &=\frac{1}{N-1} \widetilde{X}_{k \mid k-1} \widetilde{Z}_{k \mid k-1}^{\top} \\ \bar{S}_{k} &=\frac{1}{N-1} \widetilde{Z}_{k \mid k-1} \widetilde{Z}_{k \mid k-1}^{\top}+R \end{aligned} \] The Kalman gain \(\bar{K}_{k}\) is then computed as in the KF. Or we can take it to be a matrix square-root \(R^{\frac{1}{2}}\) with \(R^{\frac{1}{2}} R^{\frac{\top}{2}}=R\) and then factorize \[ \bar{S}_{k}=\left[\begin{array}{cc}\frac{1}{\sqrt{N-1}} \widetilde{Z}_{k \mid k-1}\quad R^{\frac{1}{2}}\end{array}\right] \left[\begin{array}{c}\frac{1}{\sqrt{N-1}} \widetilde{Z}^{\top}_{k \mid k-1} \\ R^{\frac{\top}{2}}\end{array}\right]. \]

TBD: EAKF and ETKF (Tippett et al. 2003) which deterministically propagate an estimate \[ P_{k \mid k}^{\frac{1}{2}} P_{k \mid k}^{\frac{\top}{2}}=P_{k \mid k} \] which introduces less sampling noise. Roth et al. (2017) explain it as rewriting the measurement update to use a square root \(P_{k \mid k-1}^{\frac{1}{2}}\) and in particular the ensemble approximation \(\frac{1}{N-1} \widetilde{X}_{k \mid k-1}\) : \[ \begin{aligned} P_{k \mid k} &=\left(I-K_{k} H\right) P_{k \mid k-1} \\ &=P_{k \mid k-1}^{\frac{1}{2}}\left(I-P_{k \mid k-1}^{\frac{\top}{2}} H^{\top} S_{k}^{-1} H P_{k \mid k-1}^{\frac{1}{2}}\right) P_{k \mid k-1}^{\frac{\top}{2}} \\ & \approx \frac{1}{N-1} \widetilde{X}_{k \mid k-1}\left(I-\frac{1}{N-1} \widetilde{Z}_{k \mid k-1}^{\top} \bar{S}_{k}^{-1} \widetilde{Z}_{k \mid k-1}\right) \widetilde{X}_{k \mid k-1}^{\top}. \end{aligned} \] Factorising, \[ \left(I-\frac{1}{N-1} \widetilde{Z}_{k \mid k-1}^{\top} \bar{S}_{k}^{-1} \widetilde{Z}_{k \mid k-1}\right)=\Pi_{k}^{\frac{1}{2}} \Pi_{k}^{\frac{\top}{2}}, \] The \(\Pi_{k}^{\frac{1}{2}}\in\mathbb{R}^{N\times N}\) can be used to create a deviation ensemble \[ \tilde{X}_{k \mid k}=\tilde{X}_{k \mid k-1} \Pi_{k}^{\frac{1}{2}} \] that correctly encodes \(P_{k \mid k}\) without using random perturbations. The actual filtering is achieved by updating each sample according to \[ \bar{x}_{k \mid k}=\left(I-\bar{K}_{k} H\right) F_{x_{k-1 \mid k-1}}+\bar{K}_{k} y_{k}, \] where \(\bar{K}_{k}\) is computed from the deviation ensembles.

## As least-squares

TBD. Permits calculating the operations without forming covariance matrices.

## Going nonlinear

TBD

## Monte Carlo moves in the ensemble

The ensemble is rank deficient. Question: When can we sample other states from the ensemble to improve the rank by stationary posterior moves?

## Managing overconfidence

TBD

## Ensemble methods in smoothing

Katzfuss, Stroud, and Wikle (2016) claims there are two major approaches to smoothing: Stroud et al. (2010) -type reverse methods, and the EnKS (Geir Evensen and van Leeuwen 2000) which augments the states with lagged copies rather than doing a reverse pass.

Here are some other papers I saw N. K. Chada, Chen, and Sanz-Alonso (2021); Luo et al. (2015); White (2018); Zhang et al. (2018).

## System identification in

Can we use ensemble methods for online parameter estimation? Apparently. G. Evensen (2009); Malartic, Farchi, and Bocquet (2021); Moradkhani et al. (2005); Fearnhead and KΓΌnsch (2018).

## Theoretical basis for probabilists

Bishop and Del Moral (2020); Del Moral, Kurtzmann, and Tugaut (2017); Garbuno-Inigo et al. (2020); Kelly, Law, and Stuart (2014); Le Gland, Monbet, and Tran (2009); Taghvaei and Mehta (2019).

## Relation to particle filters

Intimate. See particle filters.

## Schillingβs filter

Claudia Schillingβs filter (Schillings and Stuart 2017) is an elegant version which looks somehow more general than the original but also simpler. Haber, Lucka, and Ruthotto (2018) use it to train neural nets (!) and show a rather beautiful connection to stochastic gradient descent in section 3.2.

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