Variance-Weighted Averaging

Given multiple observations with different variances $\sigma^{2}_i$, we want to fuse the observations. This occurs, for example, when multiple measurements of the same quantity are obtained from 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)

Thus, the variance of the average is lower than the variance of the individual measuring 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 from the same sensor measuring 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 sequence:

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

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