# State and Parameter Estimation from Observed Signal Increments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Problem Formulation

**Example**

**1.**

## 3. Parameter Estimation from Noiseless Data

#### 3.1. Feedback Particle Filter

**Lemma**

**1**(Feedback particle filter)

**.**

**Remark**

**1.**

#### 3.2. Ensemble Kalman–Bucy Filter

## 4. State Estimation for Noisy Data

**Lemma**

**2**(Generalised Kushner–Stratonovich equation)

**.**

#### 4.1. Generalised Feedback Particle Filter Formulation

**Lemma**

**3**(Feedback particle filter with correlated innovation)

**.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

#### 4.2. Generalised Kalman–Bucy Filter

#### 4.3. Ensemble Kalman–Bucy Filter

## 5. Combined State and Parameter Estimation

#### 5.1. Feedback Particle Filter Formulation

#### 5.2. Ensemble Kalman–Bucy Filter

## 6. Numerical Results

#### 6.1. Parameter Estimation for the Ornstein–Uhlenbeck Process

#### 6.2. Averaging

#### 6.3. Homogenisation

#### 6.4. Nonparametric Drift and State Estimation

#### 6.5. Spde Parameter Estimation

#### 6.6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The Filtering Equations for Correlated Noise

**Lemma**

**A1.**

**Proof.**

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**Figure 1.**Results for the Ornstein–Uhlenbeck state and parameter estimation problem under different experimental settings: (

**a**) $Q=1/2$, $R=0.01$; (

**b**) $Q=1/2$, $R=0.0001$; (

**c**) $Q=1/2$, $R=0$ (pure parameter estimation); (

**d**) $Q=0.005$, $R=0.0001$. The ensemble size is set to $M=1000$ in all cases. Displayed are the ensemble mean ${\overline{a}}_{n}$ and the ensemble variance in ${\tilde{A}}_{n}$ and ${\tilde{X}}_{n}$. The variance of ${\tilde{X}}_{n}$ is zero when $R=0$ in case (

**b**).

**Figure 2.**Results for the averaged Ornstein–Uhlenbeck state and parameter estimation problem under different experimental settings: (

**a**) $Q=1/2$, $R=0.01$, $\u03f5=0.1$; (

**b**) $Q=1/2$, $R=0$, $\u03f5=0.1$ (pure parameter estimation); (

**c**) $Q=1/2$, $R=0.01$, $\u03f5=0.01$; (

**d**) $Q=1/2$, $R=0.01$, $\u03f5=0.01$ and subsampling by a factor of ten. The ensemble size is set to $M=1000$ in all cases. Displayed are the ensemble mean and the ensemble variance in ${\tilde{A}}_{n}$ and ${\tilde{X}}_{n}$. The variance of ${\tilde{X}}_{n}$ is zero when $R=0$ in case (

**b**).

**Figure 3.**Results for the averaged Ornstein-Uhlenbeck process, now with a smaller ensemble size M = 10. Otherwise, panels (

**a**–

**d**) correspond to the same experimental settings as in Figure 2.

**Figure 4.**Results for the homoginsation Ornstein–Uhlenbeck state and parameter estimation problem under different experimental settings: (

**a**) $Q=1/2$, $R=0.01$, $\u03f5=0.1$; (

**b**) $Q=1/2$, $R=0$, $\u03f5=0.1$ (pure parameter estimation); (

**c**) $Q=1/2$, $R=0.01$, $\u03f5=0.1$ and subsampling by a factor of fifty; (

**d**) $Q=1/2$, $R=0.01$, $\u03f5=0.1$ and subsampling by a factor of five hundred. The ensemble size is set to $M=10$ in all cases. Displayed are the ensemble mean and the ensemble variance in ${\tilde{A}}_{n}$ and ${\tilde{X}}_{n}$. The variance of ${\tilde{X}}_{n}$ is zero under (

**c**).

**Figure 5.**Results for the nonparametric drift and state estimation problem: (

**a**) reference drift function (thick line) and ensemble of drift functions drawn from the prior distribution; (

**b**) histogram of samples from the reference trajectory; (

**c**) reference drift function and its estimate (

**top**) and ensemble of drift functions (

**bottom**) at final time; (

**d**) ensemble of states and the true value at final time.

**Figure 6.**Results for SPDE parameter estimation: (

**a**) estimate of $\theta $ as a function of time as obtained by the ensemble Kalman–Bucy filter; (

**b**) evidence based on a Kalman–Bucy filter for state estimation applied to a sequence of parameter values $\theta \in \{0.2,0.3,\dots ,1.8\}$.

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**MDPI and ACS Style**

Nüsken, N.; Reich, S.; Rozdeba, P.J.
State and Parameter Estimation from Observed Signal Increments. *Entropy* **2019**, *21*, 505.
https://doi.org/10.3390/e21050505

**AMA Style**

Nüsken N, Reich S, Rozdeba PJ.
State and Parameter Estimation from Observed Signal Increments. *Entropy*. 2019; 21(5):505.
https://doi.org/10.3390/e21050505

**Chicago/Turabian Style**

Nüsken, Nikolas, Sebastian Reich, and Paul J. Rozdeba.
2019. "State and Parameter Estimation from Observed Signal Increments" *Entropy* 21, no. 5: 505.
https://doi.org/10.3390/e21050505