# 4. Singular Value Decomposition

    library(bage)

# Derivation of standardised principal components

## Data

Let \(\pmb{Y}\) be an \(A \times L\) matrix of values, where \(a=1,\cdots,A\) indexes age group and \(l=1,\cdots,L\) indexes some combination of classifying variables, such as country crossed with time. The values are real numbers, including negative numbers, such as log-transformed rates, or logit-transformed probabilities.

## Singular value decomposition

We perform a singular value decomposition on \(\pmb{Y}\), and retain only the first \(C < A\) components, to obtain \[\begin{equation} \pmb{Y} \approx \pmb{U} \pmb{D} \pmb{V}^\top \end{equation}\]

\(\pmb{U}\) is an \(A \times C\) matrix whose columns are left singular vectors. \(\pmb{D}\) is a \(C \times C\) diagonal matrix holding the singular values. \(\pmb{V}\) is a \(L \times C\) matrix whose columns are right singular vectors.

## Standardising

Let \(\pmb{m}_V\) be a vector, the \(c\)th element of which is the mean of the \(c\)th singular vector, \(\sum_{l=1}^L v_{lc} / L\). Similarly, let \(\pmb{s}_V\) be a vector, the \(c\)th element of which is the standard deviation of the \(c\)th singular vector, \(\sqrt{\sum_{l=1}^L (v_{lc} - m_c)^2 / (L-1)}\). Then define \[\begin{align} \pmb{M}_V & = \pmb{1} \pmb{m}_V^\top \\ \pmb{S}_V & = \text{diag}(\pmb{s}_V), \end{align}\] where \(\pmb{1}\) is an L-vector of ones. Let \(\tilde{\pmb{V}}\) be a standardized version of \(\pmb{V}\), \[\begin{equation} \tilde{\pmb{V}} = (\pmb{V} - \pmb{M}_V) \pmb{S}_V^{-1}. \end{equation}\]

We can now express \(\pmb{Y}\) as \[\begin{align} \pmb{Y} & \approx \pmb{U} \pmb{D} (\tilde{\pmb{V}} \pmb{S}_V + \pmb{M}_V)^\top \\ & = \pmb{U} \pmb{D} \pmb{S}_V \tilde{\pmb{V}}^\top + \pmb{U} \pmb{D}\pmb{M}_V ^\top \\ & = \pmb{A}\tilde{\pmb{V}}^\top + \pmb{B}. \end{align}\]

Furthermore, we can express matrix \(\pmb{B}\) as \[\begin{align} \pmb{B} & = \pmb{U} \pmb{D}\pmb{M}_V ^\top \\ & = \pmb{U} \pmb{D} \pmb{m}_V \pmb{1}^\top$ \\ & = \pmb{b} \pmb{1}^\top. \end{align}\]

## Result

Consider a randomly selected row \(\tilde{\pmb{v}}_l\) from \(\tilde{\pmb{V}}\). From the construction of \(\tilde{\pmb{V}}\), and the orthogonality of the columns of \(\pmb{V}\) \(\color{cyan}{\text{TODO-spell this out a bit more}}\), we obtain \[\begin{equation} \text{E}[\tilde{\pmb{v}}_l] = \pmb{0} \end{equation}\] and \[\begin{equation} \text{Var}[\tilde{\pmb{v}}_l] = \pmb{I}. \end{equation}\] This implies that if set \[\begin{equation} \pmb{y}' = \pmb{A} \pmb{z} + \pmb{b} \end{equation}\] where \[\begin{equation} \pmb{z} \sim \text{N}(\pmb{0}, \pmb{I}), \end{equation}\] then \(\pmb{y}'\) will look like a randomly-chosen column from \(\pmb{Y}\).

\(\color{cyan}{\text{TODO - illustrate with examples}}\)