#
Particle Filtering: A Priori Estimation of Observational Errors of a State-Space Model with Linear Observation Equation^{ †}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Particle Filter Algorithm

Algorithm 1 SIR Particle Filter |

Require:N, q, ${N}_{eff}$, T |

Initialize $\{{x}_{0}^{i},{w}_{0}^{i}\}$ |

for $t=1,2,...,T$ do |

1. Importance Sampling |

Sample ${\tilde{x}}_{t}^{i}\sim q\left({x}_{t}\right|{x}_{0:t-1}^{i},{y}_{t})$ |

Set ${\tilde{x}}_{0:t}^{i}=({x}_{1:t-1}^{i},{\tilde{x}}_{t}^{i}),$ |

Calculate importance weights |

${\tilde{w}}_{t}^{i}\propto {w}_{t-1}^{i}\frac{p\left({y}_{t}\right|{\tilde{x}}_{t}^{i})p\left({\tilde{x}}_{t}^{i}\right|{x}_{t-1}^{i})}{q\left({\tilde{x}}_{t}^{i}\right|{x}_{t-1}^{i},{y}_{t})}.$ |

end for |

for $i=1,2,\dots ,N$ do |

Normalize weights ${w}_{t}^{i}=\frac{{\tilde{w}}_{t}^{i}}{{\sum}_{i=1}^{N}{\tilde{w}}_{t}^{i}}$ |

2. Resampling |

if ${\widehat{N}}_{eff}\left(t\right)\ge {N}_{T}$ then |

for $i=1,2,...,N$ do |

${x}_{0:t}^{i}={\tilde{x}}_{0:t}^{i}$ |

end for |

else |

for $i=1,2,\dots ,N$do |

Sample with replacement index $j\left(i\right)$ according to the discrete weight distribution |

$P(j\left(i\right)=d)={w}_{t}^{d}$, $d=1,\dots ,N$ |

Set ${x}_{0:t}^{i}={\tilde{x}}_{0:t}^{j\left(i\right)}$ and ${w}_{t}^{i}=\frac{1}{N}$ |

end for |

end if |

end for |

## 3. The Missing Data Case—Estimation of Weights

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**2.**

**Proof.**

Algorithm 2 SIR Particle Filter for missing data with observational error estimation |

Require:N, q, ${N}_{eff}$, T |

Initialize $\{{x}_{0}^{i},{w}_{0}^{i}\}$ |

for$t=1,2,...,T$ do |

1. Importance Sampling |

Sample ${\tilde{x}}_{t}^{i}\sim q\left({x}_{t}\right|{x}_{0:t-1}^{i},{W}_{t})$ |

Set ${\tilde{x}}_{0:t}^{i}=({x}_{1:t-1}^{i},{\tilde{x}}_{t}^{i}),$ |

Produce observational error estimations ${\widehat{u}}_{t}^{i}$ for the missing components ${Z}_{t}$ and calculate importance weights |

${\tilde{w}}_{t}^{i}\propto {w}_{t-1}^{i}\frac{p\left({y}_{t}\right|{\tilde{x}}_{t}^{i})p\left({\tilde{x}}_{t}^{i}\right|{x}_{t-1}^{i})}{q\left({\tilde{x}}_{t}^{i}\right|{x}_{t-1}^{i},{y}_{t})}.$ |

end for |

for $i=1,2,\dots ,N$ do |

Normalize weights ${w}_{t}^{i}=\frac{{\tilde{w}}_{t}^{i}}{{\sum}_{i=1}^{N}{\tilde{w}}_{t}^{i}}$ |

2. Resampling |

if ${\widehat{N}}_{eff}\left(t\right)\ge {N}_{T}$ then |

for $i=1,2,\dots ,N$ do |

${x}_{0:t}^{i}={\tilde{x}}_{0:t}^{i}$ |

end for |

else |

for $i=1,2,\dots ,N$ do |

Sample with replacement index $j\left(i\right)$ according to the discrete weight distribution $P(j\left(i\right)=d)={w}_{t}^{d},d=1,\dots ,N$ |

Set ${x}_{0:t}^{i}={\tilde{x}}_{0:t}^{j\left(i\right)}$ and ${w}_{t}^{i}=\frac{1}{N}$ |

end for |

end if |

end for |

#### 3.1. Connection to Markov Systems and Contribution to the Study over Impoverishment

## 4. Simulating Example

#### 4.1. Contribution to the Missing Data Case

#### 4.2. Contribution to Impoverishment Prediction

**Remark**

**3.**

**Remark**

**4.**

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PF | Particle Filter |

MC | Monte Carlo |

MIPF | Multiple Imputation Particle Filter |

MAR | Missing At Random |

MS | Markov System |

iid | independent and identically distributed |

SIR | Sampling Importance Resampling |

HMSs | Homogeneous Markov Systems |

NHMSs | Non-homogeneous Markov Systems |

RMSE | Root Mean Square Error |

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**Figure 1.**Time-series of the hidden values, the observations, and the corresponding point estimations of the proposed method and MIPF imputations for the first component ${x}_{1,t}$ of the state process.

**Figure 2.**Time-series of the hidden values, the observations, and the corresponding point estimations of the proposed method and MIPF imputations for the second component ${x}_{2,t}$ of the state process.

**Figure 4.**Joint histogram of the particle sample ($i=1,\cdots ,N$) for the posterior estimation of both components of the whole hidden vector ${x}_{10}$. Red lines delimit the suggested grid cells of Table 3. The red diamond stands for the hidden state. The red star stands for the observation at this time point. The sides of the two squares are correspondingly one and two standard deviations ${\sigma}_{z}$ from the center of the diagram. The circles are inscribed in the corresponding squares.

**Table 1.**Comparison of the results over three methods: the basic PF algorithm, when all observations are available; the weight estimation method, which is proposed in this study; and MIPF for $n=5$ imputations. The methods are compared through the mean of RMSE and the time consumed over the 100 repeated implementations.

Method | Mean RMSE for (${\mathit{x}}_{1,\mathit{t}}$) | Mean RMSE for (${\mathit{x}}_{2,\mathit{t}}$) | Overall Mean Precision | Mean Time Elapsed (s) |
---|---|---|---|---|

Basic PF | 0.1610253 | 0.1566881 | 0.1588567 | 2.5570 |

Weight est. | 0.2065578 | 0.2102287 | 0.2083933 | 2.5491 |

MIPF | 0.2267527 | 0.2173670 | 0.2220598 | 4.9137 |

2nd Componet | $(-\mathit{\infty},0.89)$ | $[0.89,1.25)$ | $[1.25,+\mathit{\infty})$ | |
---|---|---|---|---|

1st Component | ||||

$(-\infty ,0.767)$ | 2 | 6 | 3 | |

$[0.767,1.13)$ | 5 | 77 | 1 | |

$[1.13,+\infty )$ | 1 | 4 | 1 |

**Table 3.**Frequency table of the particle distribution over the suggested grid at $t=10$ when the particles ${x}_{9}^{i},i=1,\cdots ,100$, move only according to the deterministic part of Equation (9).

2nd Componet | $(-\mathit{\infty},0.804)$ | $[0.804,1.16)$ | $[1.16,+\mathit{\infty})$ | |
---|---|---|---|---|

1st Component | ||||

$(-\infty ,0.813)$ | 0 | 0 | 0 | |

$[0.813,1.17)$ | 0 | 100 | 0 | |

$[1.17,+\infty )$ | 0 | 0 | 0 |

**Table 4.**Transition probability table for ${x}_{10}(-)$ to move with the addition of process noise ${v}_{10}$.

2nd Componet | $(-\mathit{\infty},0.804)$ | $[0.804,1.16)$ | $[1.16,+\mathit{\infty})$ | |
---|---|---|---|---|

1st Component | ||||

$(-\infty ,0.813)$ | 0.044 | 0.122 | 0.044 | |

$[0.813,1.17)$ | 0.122 | 0.335 | 0.122 | |

$[1.17,+\infty )$ | 0.044 | 0.122 | 0.044 |

**Table 5.**Frequency table for the expected numbers of the particles over the grid cells after the addition of process noise realizations at $t=10$.

2nd Componet | $(-\mathit{\infty},0.804)$ | $[0.804,1.16)$ | $[1.16,+\mathit{\infty})$ | |
---|---|---|---|---|

1st Component | ||||

$(-\infty ,0.813)$ | 4.4 | 12.2 | 4.4 | |

$[0.813,1.17)$ | 12.2 | 33.5 | 12.2 | |

$[1.17,+\infty )$ | 4.4 | 12.2 | 4.4 |

**Table 6.**Frequency table of the particle distribution over the grid around the mean of the sample in the end of $t=10$.

2nd Componet | $(-\mathit{\infty},1.01)$ | $[1.01,1.37)$ | $[1.37,+\mathit{\infty})$ | |
---|---|---|---|---|

1st Component | ||||

$(-\infty ,1.06)$ | 1 | 3 | 0 | |

$[1.06,1.42)$ | 6 | 90 | 0 | |

$[1.42,+\infty )$ | 0 | 0 | 0 |

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**MDPI and ACS Style**

Lykou, R.; Tsaklidis, G.
Particle Filtering: A Priori Estimation of Observational Errors of a State-Space Model with Linear Observation Equation. *Mathematics* **2021**, *9*, 1445.
https://doi.org/10.3390/math9121445

**AMA Style**

Lykou R, Tsaklidis G.
Particle Filtering: A Priori Estimation of Observational Errors of a State-Space Model with Linear Observation Equation. *Mathematics*. 2021; 9(12):1445.
https://doi.org/10.3390/math9121445

**Chicago/Turabian Style**

Lykou, Rodi, and George Tsaklidis.
2021. "Particle Filtering: A Priori Estimation of Observational Errors of a State-Space Model with Linear Observation Equation" *Mathematics* 9, no. 12: 1445.
https://doi.org/10.3390/math9121445