Layer preferences

This section describes strategies for adding the next layer to the model.

The algorithm can be tuned to prefer $\mu_k+\alpha_{ik}$ to have the same sign as $\mu_k$. To set this option give the command unisignrow. To remove this option give the command unisignrowoff. Similarly, commands unisigncol and unisigncoloff set and unset preferences for $\mu_k+\beta_{jk}$ to have the same sign as $\mu_k$. The unisign features are treated as preferences, and not as constraints by the algorithm. By default, the algorithm starts without any unisign preferences.

The command seekms causes the algorithm to prefer small intense layers to larger diffuse ones. These small layers may not reduce the sum of squared errors as much as in the default ( seekss) mode, but they can be more interpretable. The name seekms suggests that the sum of squared $\theta_{ijk}$ within such a layer is large compared to the number of parameters in the layer. seekms is implemented by changing the followup iterations. These are the iterations that take place after all $\rho_{ik}$ and $\kappa_{jk}$ for the new layer $k$ are either $0$ or $1$. seekms is implemented by following up the default search with a number of iterations that prefer small intense layers.

The algorithm can be tuned to release rows or columns that do not fit the present layer so well. The basic algorithm is a greedy search to reduce the sum of squared errors. As such rows or columns with very large entries can be selected to join the present layer. It may be better to release them and have them turn up in a subsequent layer where they fit better.

Setting rowrel .3 will cause the algorithm to release row $i$ from the candidate layer $k$ if

$$\sum_{j=1}^p \kappa_{jk} (Z_{ij}-\theta_{ijk})^2 > (1-.3)\sum_{j=1}^p \kappa_{jk} Z_{ij}^2.$$ (3)

Of course, the argument to rowrel need not be $0.3$. It can be interpreted as a minimum desired $R^2$ value for rows in the new layer. Row releases can be turned off by the command rowreloff. Commands colrel and colreloff apply analogously to columns. The algorithm starts in a default state with no row or column releases.

The rowrel and colrel filters are applied after the algorithm has identified a layer, so that every $\kappa_{jk}$ and $\rho_{ik}$ is $0$ or $1$. The releases are applied in parallel to all rows and or columns, and then the $\theta_{ijk}$ are updated. It is possible that all rows or that all columns get released producing a candidate layer with no data. A nonparallel algorithm that alternated between releasing only a few (possibly one) of the rows or columns most deserving to be released and then updated the layer might yield fewer empty layers, but could be very slow on large data sets.

Whatever choices are made for the unisign or release features, the $\mu_k$, $\alpha_{ik}$, and $\beta_{jk}$ values are always recomputed for the current $\rho_{ik}$ and $\kappa_{jk}$ values. This can cause the desired features to be violated.

The command strategy will print out the present state of all the layer seeking options.