Theory for noise along with missing data

Data $\bold d$ space can be decomposed into known plus missing parts, $\bold d = \bold k + \bold m$.We partition an identity operator $\bold I$ on the data space into parts that separate the known from the missing data $\bold I = \bold K + \bold M$.Thus data space can be written as

$$\bold d \eq \bold K \bold d + \bold M \bold m$$ (17)

where $\bold K\bold d$ is zero-padded known data and all the components of $\bold m$ are freely adjustable.

Data space $\bold d$can also be decomposed into signal plus noise, $\bold d = \bold s + \bold n$.Thus  

$$\bold s + \bold n \eq \bold K \bold d + \bold M \bold m$$ (18)

Writing goals for signal and noise and then eliminating the noise by the constraint equation (18) gives

$$\begin{eqnarray} \bold 0&\approx&\bold N\bold n\eq \bold N(\bold K\bold d+\bold M\bold m-\bold s)\\ \bold 0&\approx&\bold S\bold s\eq \bold S \bold s\end{eqnarray}$$ (19) (20)

Putting this in matrix form we have the operator needed in computation

$$\left[ \begin{array} {c} \bold 0 \\ \bold 0 \end{array... ...{c} \bold N \bold K \bold d \\ \bold 0 \end{array} \right]$$ (21)

I have not had time to prepare an example.