## 2. EK Method  
Consider a standardized data matrix \(\mathbf{Y}\left(n\times p\right)\) with elements \(y_{ij}\) (\(i=1,\ldots n\); \(j=1,\ldots p\)) and \(n\geq p\) (if \(n<p\) the matrix should be first transposed), for which we wished to determine the best lower rank approximation. For this, [PERSON]'s [14] cross-validation scheme can be used, which quantifies the idea of \"acceptable accuracy\" in terms of predicting the elements of \(\mathbf{Y}\). The scheme is based on the fact that element \(y_{ij}\) in the \(i\)-th row and in the \(j\)-th column of \(\mathbf{Y}\) can be written as a multiplicative model using SVD, that is:  
\[y_{ij}=\sum_{t=1}^{p}u_{it}\ s_{t}\ v_{tj} \tag{1}\]  
[PERSON] [8] used this representation as a basis for determining the dimensionality of a multivariate dataset: if the data structure is essentially m-dimensional, then the variation in the remaining (p-m) dimensions can be treated as random noise. For this reason, it can be assumed that the main characteristics of the data are found in the space of the first \(m\) components. This means that the data can be written according to an m-component model, such as:  
\[y_{ij}=\sum_{t=1}^{m}u_{it}\ s_{t}\ v_{tj}+\varepsilon_{ij} \tag{2}\]  
where \(\varepsilon_{ij}\) is a residual term and setting \(\varepsilon_{ij}\) equal to zero, the expression results in a predictor of \(y_{ij}\)  
\[\hat{y}_{ij}^{(m)}=\sum_{t=1}^{m}u_{it}\ s_{t}\ v_{tj} \tag{3}\]  
Since the calculations of \(u_{it}\), \(s_{t}\) and \(v_{tj}\) involve all values of \(\mathbf{Y}\), the predictor of \(y_{ij}\) uses the value of \(y_{ij}\) itself. This is an undesirable feature in cross-validation because it can be a source of bias. To avoid this bias, [PERSON] and [PERSON] [14] suggested the following scheme: suppose we are looking for the prediction of \(y_{ij}\) in \(\mathbf{Y}\), then, the \(i\)-th row from \(\mathbf{Y}\) is deleted and the SVD for the \(((n-1)\times p)\) resulting matrix \(\mathbf{Y}^{(-i)}\) is calculated as \(\mathbf{Y}^{(-i)}=\overline{\mathbf{U}\mathbf{D}\mathbf{V}}^{T}\), \(\overline{\mathbf{U}}=(\overline{u}_{sh})\), \(\overline{\mathbf{V}}=(\overline{v}_{sh})\), \(\overline{\mathbf{D}}=\left(\overline{d}_{1},\ldots,\overline{d}_{p}\right)\). The next step is to delete the \(j\)-th column from \(\mathbf{Y}\) and obtain the SVD for the \((n\times(p-1))\) matrix \(\mathbf{Y}_{(-j)}\) as \(\mathbf{Y}_{(-j)}=\overline{\mathbf{U}}\overline{\mathbf{D}}\overline{\mathbf{V}}^{T}\), \(\widetilde{\mathbf{U}}=(\overline{u}_{sh})\), \(\widetilde{\mathbf{V}}=(\overline{v}_{sh})\), \(\widetilde{\mathbf{D}}=\left(\widetilde{d}_{1}^{\prime},\ldots,\widetilde{d}_{p- 1}\right)\). The matrices \(\overline{\mathbf{U}}\), \(\overline{\mathbf{V}}\), \(\widetilde{\mathbf{U}}\), and \(\widetilde{\mathbf{V}}\) are orthonormal, while \(\widetilde{\mathbf{D}}\) and \(\overline{\mathbf{D}}\) are diagonal. By combining the two SVDs, \(\mathbf{Y}^{(-i)}\) and \(\mathbf{Y}_{(-j)}\), a predictor of \(\mathbf{Y}_{ij}\) based on the \(m\) component model is given by  
\[\hat{y}_{ij}^{(m)}=\sum_{h=1}^{m}\widetilde{u}_{ih}\left(\widetilde{d}_{h} \sqrt{p/(p-1)}\right)^{\frac{1}{2}}\overline{y}_{jh}\left(\overline{d}_{h} \sqrt{n/(n-1)}\right)^{\frac{1}{2}} \tag{4}\]  
This predictor is slightly modified relative to the one proposed by [PERSON] and [PERSON] [14] following the work of [PERSON] et al. [9] and [PERSON] et al. [3] because the inclusion of the constants \(\sqrt{p/(p-1)}\) and \(\sqrt{n/(n-1)}\) improves the quality of the predictions. On the other hand, in order to avoid computational problems, a parity check should be done in each prediction by matching the sign of \(\left(\widetilde{u}_{ih}\left(\widetilde{d}_{h}\sqrt{p/(p-1)}\right)^{\frac {1}{2}}\right)\left(\overline{\hat{u}}_{jh}\left(\overline{d}_{h}\sqrt{n/(n-1 )}\right)^{\frac{1}{2}}\right)\) in (4) to the sign of \(u_{ih}d_{h}v_{jh}\) obtained from the SVD of the \(\mathbf{Y}\) matrix for each \(m\).