Event attenuation by weighting

Imagine an event that is attenuated, but not removed, by filters $\st S$ and $\st N$.  

$$\begin{array} {c} \sv \epsilon_1 = \st S \sv x \\ \sv \epsilon_2 = \st N \sv x,\end{array}$$ (94)

where $\sv x$ contains the event, $\st S$ is the signal filter, $\st N$ is the noise filter, $\sv \epsilon_1$ is the response of the event to filter $\st S$,and $\sv \epsilon_2$ is the response of the event to filter $\st N$. 

$$\begin{array} {c} \varepsilon_1 = \Sigma \sv \epsilon_1^2 \\ \varepsilon_2 = \Sigma \sv \epsilon_2^2\end{array}$$ (95)

is a measure of the power of the filtered events. When getting a least-squares solution for equations such as () and (), the events included in $\sv x$ will be distributed between signal and noise as $1/\varepsilon_1$ for the signal and $1/\varepsilon_2$ for the noise.

This distribution may be changed by modifying the system of equations. Consider, for example, this system:  

$$\sv 0 \approx \left( \begin{array} {c} \lambda \st S \\ \s... ...\left( \begin{array} {c} \sv 0 \\ \st N\sv d\end{array}\right).$$ (96)

If $\lambda$ is less than 1, $1/\varepsilon_1$ will increase, allowing relatively more of the event into the signal. If $\lambda$ is greater than 1, $1/\varepsilon_2$ will increase, allowing relatively more of the event into the noise. If $\lambda$ is very large, only events that are almost perfectly removed by $\st S$ will be allowed into the signal.

Even for events that are perfectly predicted and removed by filters $\st S$ and $\st N$,the distribution of events may be controlled by the weighting in equation (), which is the prediction equation with the initial estimate of the signal as the data. In this case, the $\lambda$ controls the final distribution of events in the null space of $\st S$ and $\st N$.Once again, if $\lambda$ is less than 1, $1/\varepsilon_1$ will increase, forcing relatively more of the event into the signal. If $\lambda$ is greater than 1, $1/\varepsilon_2$ will increase, forcing relatively more of the event into the noise.

Once again, the weighting in system () may be thought of in terms of using $\st S\sv s$ and $\st N\sv n$ as levelers. If $\st S\sv s$ is weighted higher than $\st N\sv n$, the least-squares solutions of $\sv s$ and $\sv n$ will be modified since the values of $\varepsilon_1$ and $\varepsilon_2$ are modified. In the unlikely event that either $\st S\sv s$ or $\st N\sv n$ actually becomes zero, the weighting becomes unimportant, since one of the conditions is fit perfectly and no better solution could be found. In practical situations, both $\st S\sv s$ and $\st N\sv n$ will have some residual and can only be minimized.