Variance-Weighted Averaging

Given multiple observations with different variances $\sigma^{2}_i$, we want to fuse the observations. This is the case, for example, when multiple measurements of the same quantity are made by different sensors at the same time, or when the same sensor measures a quantity assumed to be constant while the observation noise varies over time. The goal is to obtain a weighted average of the individual observations of the form

$$\bar{x} = \sum_i w_i x_i$$ (2.63)

By definition, the variance of the variable $\bar{x}$ is

$$\sigma^2_{\bar{x}} = \sum_i w^{2}_i \sigma^2_{i}$$ (2.64)

The optimal solution (the maximum-likelihood estimator) is obtained by minimizing this quantity subject to the additional constraint $\sum_i w_i = 1$.

The weight that minimizes this quantity is

$$w_i = \frac{ \frac{1}{\sigma^{2}_i} } { \sum_j \frac{1}{\sigma^{2}_j} }$$ (2.65)

In this way, the variance of the average is lower than the variance of the individual measurement instruments and is equal to

$$\sigma^{2}_{\bar{x}} = \frac{1}{\sum 1/\sigma^{2}_i}$$ (2.66)

A direct consequence is that $n$ readings of the same sensor and the same quantity can be combined (assuming observation noise with constant variance), even when acquired at different times. The final variance is reduced to

$$\sigma^{2}_{\bar{x}} = \frac{ \sigma^{2}_0 } {n}$$ (2.67)

This result can be constructed iteratively through the recurrence:

$$\bar{x}_{i+1} = (1 - k) \bar{x}_i + k x_{i+1} \quad k = \frac{\sigma^2_{\bar{x} } } { \sigma^2_{\bar{x} } + \sigma^2_{i+1} }$$ (2.68)

with $k$ the blending factor. Written in this form, the estimate of the quantity has the same form as the one-dimensional Kalman filter (compare this result with that of Section 3.2.1): in the absence of process noise, the gain $k$ tends to zero.