Plaid Model

The plaid model was introduced by Lazzeroni & Owen (2000). To fix ideas, we suppose that $X$ is a matrix of gene expression data. The entry $X_{ij}$ describes the level of expression of gene $i$ in sample $j$. The Plaid model approximates this data by a sum

$$\sum_{k=1}^K \rho_{ik}\kappa_{jk}\theta_{ijk}$$ (1)

where $\rho_{ik}, \kappa_{jk}\in\{0,1\}$ are membership indicator variables. The row $i$ is said to be in layer $k$ if and only if $\rho_{ik}=1$. Similarly column $j$ is in layer $k$ if and only if $\kappa_{jk}=1$. The values $\theta_{ijk}$ represent Analysis of Variance (anova) models fit to the rows and columns in the layer. There are four types of anova model:

$$\begin{eqnarray*} \theta_{ijk} & = & \mu_k \\ \theta_{ijk} & = & \mu_k + \alpha... ...beta_{jk} \\ \theta_{ijk} & = & \mu_k + \alpha_{ik}+\beta_{jk}. \end{eqnarray*}$$

If the $\alpha_{ik}$ term is present then it must satisfy $\sum_i\rho_{ik}\alpha_{ik}=0$. A $\beta_{jk}$ term must satisfy $\sum_j\kappa_{jk}\beta_{jk}=0$.

The algorithm adds layers to the model one at a time. It searches for layer $K$ in the residual

$$Z_{ij} = X_{ij} - \sum_{k=1}^{K-1} \rho_{ik}\kappa_{jk}\theta_{ijk}$$ (2)

left over from the first $K-1$ layers.