Estimation - Data Fusion

From the previous example, we know if we want to merge two data, we also need the standard deviation or variance of the data in order to make a better estimation.  By following the tradition, we will use variance for the following derivations.

Say the data fusion process is defined as: $d_f = \mathcal{F}(d_1,d_2)$, $d_i = (x_i,\sigma_i^2)$

To have a good estimate means to have an unbiased $x_f$ with minimizing the $\sigma_f^2$ at the same time.

Say $x_f = \alpha_1x_1+\alpha_2x_2$, and $\alpha_1+\alpha_2=1$

Then $\sigma_f^2 = \alpha_1^2\sigma_1^2+\alpha_2^2\sigma_2^2+2\alpha_1\alpha_2\mathcal{C}(d_1,d_2)$, $\mathcal{C}(d_1,d_2)$ is the covariance of the data.

If both data are uncorrelated, $\sigma_f^2 = \alpha_1^2\sigma_1^2+\alpha_2^2\sigma_2^2$

With the uncorrelated case, let $\alpha_2=1-\alpha_1$, and

$ \frac{\partial}{\partial \alpha_1} \sigma_f^2 = 2\alpha_1\sigma_1^2+(2\alpha_1-2)\sigma_2^2=0$

then $\alpha_1 = \sigma_2^2/(\sigma_1^2+\sigma_2^2) $, $\alpha_2 = \sigma_1^2/(\sigma_1^2+\sigma_2^2) $