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

## TL;DR

This paper compares the second order statistics of arithmetic and geometric averages in data fusion for target tracking, analyzing their variance and mean square error in different fusion contexts, including complex Gaussian mixtures.

## Contribution

It provides a detailed analysis and comparison of second order statistics for AA and GA in v-fusion and f-fusion, including complex Gaussian mixture scenarios.

## Key findings

- GA generally has lower variance than AA in v-fusion.
- Differences between AA and GA are more pronounced in f-fusion with Gaussian mixtures.
- The analysis includes exact derivations and illustrative examples.

## 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.