Layer finding

The commands to find layers are

findm
findma
findmb
findmab

and they fit layers of the form

$$\begin{eqnarray*} & &\rho_{ik}\kappa_{jk}[\ \mu_k\ ] \\ & &\rho_{ik}\kappa_{jk}... ...\ ]\\ & &\rho_{ik}\kappa_{jk}[\ \mu_k+\alpha_{ik}+\beta_{jk}\ ] \end{eqnarray*}$$

respectively, where $\rho_{ik}, \kappa_{jk}\in\{0,1\}$.

The commands above determine the number of rows or columns as well as which ones to include. The user can also influence the number of rows or columns in the layer. The commands

findmab r c
findmab 0 c
findmab r 0

are for finding layers with exactly $r$ rows and $c$ columns. Setting $r=0$ tells the algorithm to choose the number of rows, setting $c=0$ tells the algorithm to choose the number of columns, and setting them both to $0$ is the same as not providing $r$ and $c$. The commmands findm, findma, and findmb also have optional arguments $r$ and $c$. Either none or both of these arguments must be provided, as the program cannot tell whether a single argument should refer to rows or to columns.

User specified $r$ and $c$ values can conflict with user specified options for unisign or releases as described above. In such cases it is possible for the algorithm to find a smaller layer than the desired size.

When a layer has been found, the program prints out a short description of it, giving the number of rows, number of columns, and an anova table for the layer. If the user is satisfied with the candidate layer, the user can give the accept command and the layer will be inserted in the model. Then the current values of $Z_{ij}$ will be replaced by $Z_{ij}-\rho_{ik}\kappa_{jk}\theta_{ijk}$. Once the first layer has been accepted it is no longer possible to apply location or scale adjustments to the data.

The command shuffle repeats the most recent layer search on a shuffled version of the data. For $i=1,\ldots,n$ let $\pi_i$ be a random permutation of the values $1,\ldots,p$ and for $j=1,\ldots,p$ let $\tau_j$ be a random permutation of the values $1,\ldots,n$. A row shuffle turns $Z_{ij}$ into

$$\widetilde Z_{ij} = Z_{i\pi_i(j)}.$$

Then a column shuffle turns $\widetilde Z_{ij}$ into

$$\widetilde{\widetilde Z}_{ij} = \widetilde Z_{\tau_j(i)j}.$$

The command shuffle k repeats the most recent layer search on shuffled data $k$ times.

The command backfit b causes the program to go through $b$ rounds of ``backfitting''. In a round of backfitting, every accepted layer is revisited in order from first to most recent. When a layer is revisited its parameters $\mu_k$, $\alpha_{ik}$, and $\beta_{jk}$ are recomputed and the residual $Z_{ij}$ is correspondingly adjusted. Backfitting does not change the values of $\rho_{ik}$ or $\kappa_{jk}$. Backfitting can introduce violations of the unisign or row release criteria described above.