Second Order Statistics Analysis and Comparison between Arithmetic and Geometric Average Fusion

TL;DR

This paper compares the second-order statistics of arithmetic-average (AA) and geometric-average (GA) fusion across $v$-fusion (random variables) and $f$-fusion (PDFs/PHDs), including Gaussian and Poisson multitarget scenarios with false alarms and misdetections. It derives bias, variance, and MSE properties, revealing that AA preserves unbiasedness under unbiased inputs while GA can introduce bias, and that GA often offers tighter variance and MSE in $f$-fusion, especially for Gaussian fusions. For $v$-fusion, variance can be reduced below the smallest input variance under suitable correlation and weight choices, whereas in $f$-fusion GA generally dominates AA in variance bounds; MSE analyses corroborate these trends with specific conditions. In PHD averaging, a hybrid approach—GA for localization and AA for cardinality—balances peak sharpness and robustness, offering practical guidance for distributed multitarget tracking systems.

Abstract

Two fundamental approaches to information averaging are based on linear and logarithmic combination, yielding the arithmetic average (AA) and geometric average (GA) of the fusing initials, respectively. In the context of target tracking, the two most common formats of data to be fused are random variables and probability density functions, namely $v$-fusion and $f$-fusion, respectively. In this work, we analyze and compare the second order statistics (including variance and mean square error) of AA and GA in terms of both $v$-fusion and $f$-fusion. The case of weighted Gaussian mixtures representing multitarget densities in the presence of false alarms and misdetection (whose weight sums are not necessarily unit) is also considered, the result of which appears significantly different from that for a single target. In addition to exact derivation, exemplifying analysis and illustrations are provided.

Figures (8)

• Figure 1: Variances of the AA and GA of two correlated, approximate-Gaussian-distributed variables with mean $\mu_1=50$ and variance $\Sigma_1=100$, and with mean $\mu_2 =60$ and variance $\Sigma_2=200$, respectively, under three different correlation coefficient $\rho$s.
• Figure 2: Variances of the AA and GA of two correlated Poisson-distributed variables with rates $\lambda_1=10,\lambda_2=12$ (and so $\alpha = \frac{\lambda_2}{\lambda_1}=1.2$), under three different correlation coefficient $\rho$s.
• Figure 3: MSEs of the AA and GA of two independent, approximate-Gaussian-distributed variables with mean $\mu_1=50$ and variance $\Sigma_1=100$, and with mean $\mu_2 =60$ and variance $\Sigma_2=200$, respectively, in the case of three different real variables $\theta =45, 55, 65$, respectively.
• Figure 4: MSEs of the AA and GA of two correlated, approximate-Gaussian-distributed variables (with correlation coefficient $\rho=0.70736$) with mean $\mu_1=50$ and variance $\Sigma_1=100$, and with mean $\mu_2 =60$ and variance $\Sigma_2=200$, respectively, in the case of three different real variables $\theta =45, 55, 65$, respectively.
• Figure 5: Variances and MSEs of the AA and of the GA, of two Gaussian PDFs $f_{\hat{\theta}_1}(x) = \mathcal{N}(x;50, 100)$ and $f_{\hat{\theta}_2}(x) = \mathcal{N}(x; 60, 200)$ regarding different real variables $\theta \in [40, 80]$ and different fusing weights $w_1 \in [0, 1]$.