Technology and Business Skills

$Result = {Technology} ^ {Business Skills}$

Technology	Business Skills	Result
0	*	0
1	0	1
1	1	1
1	2	1
2	0	1
2	1	2
2	2	4
2	3	8

Estimation - Kalman Filter II

From the point of view of the measurement, we can make a prediction of the measurement ${\textbf{z}_1}'$ from ${\textbf{x}_1}'$, and we can get $({\textbf{z}_1}', \textbf{H}_1{\textbf{P}_1}'\textbf{H}_1^T)$.

The measurement is $(\textbf{z}_1, \textbf{R}_1)$, the estimate is

$\textbf{z}_1^* = (\textbf{I}-\textbf{G}_1){\textbf{z}_1}'+\textbf{G}_1\textbf{z}_1$

a good estimate comes with

$\textbf{G}_1=\textbf{H}_1{\textbf{P}_1}'\textbf{H}_1^T(\textbf{H}_1{\textbf{P}_1}'\textbf{H}_1^T+\textbf{R}_1)^{-1} = \textbf{H}_1\textbf{K}_1$

$\textbf{z}_1^* = (\textbf{I}-\textbf{H}_1\textbf{K}_1)\textbf{H}_1{\textbf{x}_1}'+\textbf{H}_1\textbf{K}_1\textbf{z}_1$

$ = \textbf{H}_1(\textbf{I}-\textbf{K}_1\textbf{H}_1){\textbf{x}_1}'+\textbf{H}_1\textbf{K}_1\textbf{z}_1$

$\textbf{H}_1\textbf{x}_1^* = \textbf{H}_1(\textbf{I}-\textbf{K}_1\textbf{H}_1){\textbf{x}_1}'+\textbf{H}_1\textbf{K}_1\textbf{z}_1$

so $\textbf{K}_1$ can give a good estimate for $\textbf{z}_1^*$, it seems also imply that

 $(\textbf{I}-\textbf{K}_1\textbf{H}_1){\textbf{x}_1}'+\textbf{K}_1\textbf{z}_1$ can give a good estimate for $\textbf{x}_1^*$.

Estimation - Probability Distribution

Here is the chart of the probability distribution of a 1D example on the (prediction, measurement, estimate) for the normal distribution case.

As we can see, both the measurement and the prediction are unbiased, but have different probability distributions.  By adjusting the weight, we can find a better unbiased estimation.