# A Two-Step Algorithm for Handling Block-Wise Missing Data in Multi-Omics
## 2 Methods
### Block-Wise Missingness and Profiles
\[\mathbf{y}_{m}=\sum_{m\in pf}^{n_{m}}\sum_{i=1}^{S}\alpha_{m}^{i}\mathbf{X}_{m}^{i}\mathbf{ \beta}^{i}+\mathbf{\varepsilon}, \tag{2}\]  
where given the profile \(m\), then \(\mathbf{X}_{m}^{i}\), \(i=1,\ldots,S\), represents the \(n_{m}\times p_{i}\) submatrix of the \(i\)-th source, \(n_{m}\) is the number of samples that contain \(m\) in their profiles, \(\alpha_{m}^{i}\) is the weight related to the matrix \(\mathbf{X}_{m}^{i}\), and \(\mathbf{y}_{m}\) denotes the \(n_{m}\) dimensional vector restricted to that samples. It will be useful to define \(\mathbf{X}_{m}=(\mathbf{X}_{m}^{1},\ldots,\mathbf{X}_{m}^{S})\) and \(\mathbf{\alpha}_{m}=(\alpha_{m}^{1},\ldots,\alpha_{m}^{S})\) to refer to all the model coefficients for the profile \(m\). Continuing with the previous example, by taking samples in profile \(m=6\) and those that have complete data from all sources in that profile, that is, sources 1 and 2, we will obtain the regression matrix \(\mathbf{X}_{6}=(\mathbf{X}_{6}^{1},\mathbf{X}_{6}^{2},\mathbf{X}_{6}^{3})\). According to Table 2, the matrix will contain the samples of profiles 6 and 7; and, according to Table 1, the samples of profile 6, unlike those of profile 7, have the missing block of source 3. We arrange the samples in matrix \(\mathbf{X}_{m}\) such that the initial rows correspond to the smallest profile, followed by rows from increasingly larger profiles, until \(\mathbf{X}_{m}\) is complete. This arrangement reveals a box structure in \(\mathbf{X}_{m}\). For example, \(\mathbf{X}_{6}\) takes the following form:  
\[\mathbf{X}_{6}=(\mathbf{X}_{6}^{1},\mathbf{X}_{6}^{2},\mathbf{X}_{6}^{3})= \begin{pmatrix}X_{ \text{Profile 6, Source 1}}&X_{\text{Profile 6, Source 2}}&\mathbf{0}\\ \hline X_{\text{Profile 7, Source 1}}&X_{\text{Profile 7, Source 2}}&X_{\text{Profile 7, Source 3}}\end{pmatrix} . \tag{3}\]  
We should point out here that since we will consider the model in (2), each weight \(\alpha_{6}^{i}\) will multiply the submatrix \(\mathbf{X}_{6}^{i}\). For example, although \(\mathbf{X}_{6}^{3}\) contains missing values, it will not be considered because \(\alpha_{6}^{3}=0\). We will have  
\[\mathbf{\alpha}_{6}=(\alpha_{6}^{1},\alpha_{6}^{2},\alpha_{6}^{3})=(\alpha_{6}^{1 },\alpha_{6}^{2},0).\]  
In general, if the source \(i\) in \(\mathbf{X}_{m}^{i}\) is not part of the profile \(m\) we assign \(\alpha_{m}^{i}=0\) and so it is not acknowledged for that profile. Naturally, the vector \(\mathbf{y}_{m}\) also forms a box-like structure when arranged by rows in the same manner as \(\mathbf{X}_{m}\). In this example,  
\[\mathbf{y}_{6}= \begin{pmatrix}\mathbf{y}_{\text{Profile 6}}\\ \hline\mathbf{y}_{\text{Profile 7}}\end{pmatrix} .\]  
Thus, in this specific case, the objective is to learn three model coefficients, \(\mathbf{\beta}^{1}\), \(\mathbf{\beta}^{2}\), and \(\mathbf{\beta}^{3}\) independently for each data source, as well as the vectors \(\mathbf{\alpha}_{m}=(\alpha_{m}^{1},\alpha_{m}^{2},\alpha_{m}^{3})\), with \(m\in pf\), which combine them. Observe that, for the \(i\)-th data source, \(\mathbf{\beta}^{i}\) stays the same across profiles, while the component \(\alpha_{m}^{i}\) of the vector \(\mathbf{\alpha}^{i}\) may vary across each different profile \(m\). Furthermore, to simplify notation, the components \(\alpha_{m}^{i}\) related to missing sources are set to zero, instead of specifying vectors \(\mathbf{\alpha}^{i}\) of different dimensions (see Table 3). \begin{table}
\begin{tabular}{l l l l l} \hline
**Complete Data** & **Profile** & \(\mathbf{\alpha}^{1}\) & \(\mathbf{\alpha}^{2}\) & \(\mathbf{\alpha}^{3}\) \\ \hline source 1 & 4 & \(\alpha_{4}^{1}\) & 0 & 0 \\ source 2 & 2 & 0 & \(\alpha_{2}^{2}\) & 0 \\ source 3 & 1 & 0 & 0 & \(\alpha_{1}^{3}\) \\ sources 1, 2 & 6 & \(\alpha_{6}^{1}\) & \(\alpha_{6}^{2}\) & 0 \\ sources 1,