# Biclustering of Smart Building Electric Energy Consumption Data
## 2 Our Proposal
### Biclustering  
In this section, we present the model of bicluster. In particular, we focus on how to assess the quality of a bicluster, i.e., how to estimate the coherence of the elements contained in the bicluster. Let \(B=\{b_{1},\ldots,b_{N}\}\) be a set of buildings and \(D=\{d_{1},\ldots,d_{M}\}\) a set of days. The data can be viewed as an \(N\times M\) energy consumption matrix \(EM\). \(EM\) is a matrix of real numbers, where each \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Building** & **Year (Refurbishment)** & **Size (m\({}^{2}\))** & **Description** \\ \hline
4 & 1956 (2000) & 1948.16 & A two-story building with four large classrooms \\ \hline
8 & 1956 (1999) & 1948.16 & A two-story building with four large classrooms \\ \hline
11 & 1956 (2005) & 5231.01 & A four-story building with several classrooms and offices \\ \hline
20 & 1997 (-) & 6926.49 & A devoted biology research centre, which contains many biological laboratories and offices. \\ \hline
45 & 2010 (-) & 6992.18 & A newly-built two-story building with computer labs, classrooms and offices. This is the largest building. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Description of the selected buildings. The first column presents the year of construction and the year of the refurbishment if applicable. The second column presents the size in m\({}^{2}\). The third column presents a short description. entry \(e_{ij}\) corresponds to the electric energy consumption (expressed in kWh) of building \(i\) registered on day \(j\). Given such an \(N\times M\) matrix \(EM\) representing the data, a bicluster is defined as a \(I\times J\) sub-matrix \((I,J)\) of \(EM\) that exhibits some coherent tendency, where \(I\) and \(J\) are sets of rows and columns, respectively, and \(|I|\leq|N|\) and \(|J|\leq|M|\). We define the volume of a bicluster \((I,J)\) as the number of elements \(e_{ij}\) such that \(i\in I\) and \(j\in J\). In our case, rows stand for buildings, while columns stand for days. It follows that a bicluster represents the energy consumption of \(I\) buildings registered over \(J\) days. To determine whether a bicluster presents a coherent tendency, i.e., whether a group of buildings show a similar electric energy consumption behaviour during the same period of time, we used a measure called Transposed Virtual Error (\(VE^{t}\)) [30]. The lower is the value of \(VE^{t}\), the better the bicluster is considered. Before defining \(VE^{t}\), one important observation that can be extracted from an analysis of biclusters is that the range of values contained in a bicluster can vary substantially. Therefore, to make an appropriate comparison between each building and the general pattern, a previous standardisation process of the bicluster would enable the values of the bicluster to be scaled to a common range. This mechanism would also be responsible for softening the behaviour of a building, since the most important aspect is to characterise its tendency rather than its numerical values. Thus, we performed a standardisation of the biclusters \(\mathcal{B}\), whose elements \(\hat{b}_{ij}\) are obtained as follows:  
\[\hat{b}_{ij}=\frac{\hat{b}_{ij}-\
u_{r_{i}}}{\sigma_{r_{i}}},1\leq i\leq|I|,1 \leq j\leq|J| \tag{1}\]  
After having standardised the bicluster, we could proceed to compute \(VE^{t}\), which is a measure that allows simultaneously detecting different kinds of interesting patterns in a bicluster, which is an improvement over other measures, such as the Mean Square Residue [31]. The basic idea behind \(VE^{t}\) is to measure how rows follow the general tendency within the bicluster. The first step to compute \(VE^{t}\) of a bicluster \(\mathcal{B}\) is to obtain a new row from \(\mathcal{B}\), which represents a _virtual pattern_ (\(\rho=\{\rho_{1},\rho_{2},\ldots,\rho_{J}\}\)), where \(\rho_{j}\), \(1\leq j\leq J\), is defined as the mean of the \(j\)th column:  
\[\rho_{j}=\frac{1}{I}\sum_{i=1}^{I}e_{ij} \tag{2}\]  
The virtual pattern symbolizes the general behaviour of the set of buildings for the given bicluster.