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Conservative Quantization of Covariance Matrices with Applications to Decentralized Information Fusion^{ †}

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

**Covariance Quantization.**We propose two approaches to the conservative quantization of covariance matrices. The first scheme uses diagonal dominance [34]. As an alternative, we study a modified Cholesky decomposition and compare it to the first approach.**Fusion of Estimates.**We apply the quantization schemes to both an optimal fusion algorithm and covariance intersection in order to demonstrate that reliable estimates are attained.

## 2. Notation

## 3. Considered Problem

## 4. Conservative Quantization of Covariance Matrices

#### 4.1. Covariance Quantization Based on Diagonal Dominance

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 4.2. Covariance Quantization Based on Modified Cholesky Decomposition

**Theorem**

**5.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

## 5. Applications to Information Fusion

#### 5.1. Optimal Fusion and Covariance Intersection

#### 5.2. Unbiased Conservative Quantization of Estimates

**Corollary**

**3.**

**Theorem**

**8.**

**Proof.**

#### 5.3. Quantized Optimal Fusion and Covariance Intersection

- Quantize the estimates ${\underline{\mathrm{x}}}_{a}$ and ${\underline{\mathrm{x}}}_{b}$ so that the quantization results remain unbiased. Account for the potential increase in uncertainty due to the quantization process. Both goals are achieved by employing the unbiased, conservative estimate quantizer introduced in the previous subsection.
- Quantize the error covariance matrices of the quantized estimates conservatively. This is done using either the quantizer from Section 4.1 or the one from Section 4.2.
- Apply the Bar-Shalom–Campo formulas or covariance intersection to the quantized estimates and quantized error covariance matrices. Since the quantized estimates are unbiased and the quantized error covariance matrices are conservative the fusion result will also be unbiased and conservative.

## 6. Results and Discussion

#### 6.1. Evaluation of the Covariance Quantizers

#### 6.2. Evaluation of Quantized Optimal Fusion and Quantized Covariance Intersection

#### 6.3. Evaluation of Quantized Covariance Intersection in 2D Tracking Scenario

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**): Confidence ellipsoids of a covariance matrix (dark blue), its naively quantized version (light blue), and its conservatively quantized version (orange). (

**Right**): Density of a Gaussian random variable (dark blue), histogram (light blue), and mean (orange) of its quantized version.

**Figure 2.**Confidence ellipsoids of ${\mathbf{C}}_{aa}$ and ${\mathbf{C}}_{bb}$ (blue), $\mathbb{C}({\underline{\mathrm{x}}}_{f}-\underline{\mathrm{x}})$ (orange, solid), and ${\mathbf{C}}_{ff}$ (orange, dashed) when ${\mathbf{C}}_{ab}\ne \mathbf{0}$. The result obtained using optimal fusion with the erroneous assumption ${\mathbf{C}}_{ab}=0$ is shown on the left. The result achieved using CI is shown on the right.

**Figure 3.**Average Frobenius norm of the quantization error matrix ${\Delta}_{dd}$ of the diagonal dominance-based quantization approach (

**top**), and the relative improvement achieved by the modified Cholesky-based quantization approach (

**bottom**) for varying dimensions n and bits per codeword b.

**Figure 4.**Relative increase of actual MSE (solid)/averaged trace (dashed) of DD-OPT with respect to OPT for varying dimensions n and bits per codeword b. Top row with quantized estimate vector and quantized error covariance, bottom row only with quantized error covariance.

**Figure 5.**Relative increase of actual MSE (solid)/averaged trace (dashed) of MC-OPT with respect to DD-OPT for varying dimensions n and bits per codeword b. Top row with quantized estimate vector and quantized error covariance, bottom row only with quantized error covariance.

**Figure 6.**Relative increase of actual MSE (solid)/averaged trace (dashed) of DD-CI with respect to CI for varying dimensions n and bits per codeword b. Top row with quantized estimate vector and quantized error covariance, bottom row only with quantized error covariance.

**Figure 7.**Relative increase of actual MSE (solid)/averaged trace (dashed) of MC-CI with respect to DD-CI for varying dimensions n and bits per codeword b. Top row with quantized estimate vector and quantized error covariance, bottom row only with quantized error covariance.

**Figure 8.**The estimate MSE of sensor node b plotted over time step k for varying bits per codeword using DD-CI. ‘CI’ indicates the use of a 64 bit floating point representation for each scalar.

**Figure 9.**The estimate MSE of sensor node b plotted over time step k for varying bits per codeword using MC-CI. ‘CI’ indicates the use of a 64 bit floating point representation for each scalar.

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**MDPI and ACS Style**

Funk, C.; Noack, B.; Hanebeck, U.D. Conservative Quantization of Covariance Matrices with Applications to Decentralized Information Fusion. *Sensors* **2021**, *21*, 3059.
https://doi.org/10.3390/s21093059

**AMA Style**

Funk C, Noack B, Hanebeck UD. Conservative Quantization of Covariance Matrices with Applications to Decentralized Information Fusion. *Sensors*. 2021; 21(9):3059.
https://doi.org/10.3390/s21093059

**Chicago/Turabian Style**

Funk, Christopher, Benjamin Noack, and Uwe D. Hanebeck. 2021. "Conservative Quantization of Covariance Matrices with Applications to Decentralized Information Fusion" *Sensors* 21, no. 9: 3059.
https://doi.org/10.3390/s21093059