## 3 2D \(\ell^{1}\) Major Component Detection and Analysis
### \(\ell^{1}\) MCDA Step 2: Calculation of the Main Directions  
After the data have been centered in Step 1, we need to calculate the main directions in which the distribution extends. Standard PCA prescribes that the main directions of the distribution are orthogonal to each other and that the measure of the extension in each of the orthogonal directions is determined by the covariance matrix. Such structure is consistent with a standard assumption of Gaussian or near-Gaussian distribution of the data. This structure is a leading factor in keeping the computational cost of standard PCA (calculated by singular value decomposition) low. At the same time, its rigidity is a cause of the limited applicability of standard PCA. The central point of a symmetric univariate heavy-tailed distribution is its 50% quantile, the median of the distribution. The spread of a univariate heavy-tailed distribution around its central point is represented by the distance to other quantiles, for example, the 25% and 75% quantiles. The further away the 25% and 75% quantiles are from the central point, the more spread out (flatter, with heavier tails) the distribution is. One can calculate the 25% and 75% quantiles by calculating the median of the data between the 0% quantile and 50% quantile and the median of the data between the 50% quantile and 100% quantile, respectively. This procedure for calculating the 25% and 75% quantiles is used here because it can be generalized to higher dimensions. For symmetric distributions, one need, of course, calculate only the 25% or the 75% quantile, not both. The univariate situation provides the guideline for how we approach determining the properties of a multivariate heavy-tailed distribution. How heavy-tailed a multivariate distribution is in a given direction is estimated by the \"median radius\" of the data points in and near that direction. Assume for now that, as previously stated, the distribution is symmetric around the origin, that is, \"spokes\" are in precisely opposite directions. Since the distribution is symmetric, we map, without loss of generality, every original data point \(\mathbf{x}_{m}=(x_{m},y_{m})\) for which \(x_{m}<0\) or for which \(x_{m}=0\) and \(y_{m}>0\) to an origin-symmetric data point \((-x_{m},-y_{m})\). To avoid proliferation of notation, we denote each such data point \((-x_{m},-y_{m})\) also by \(\mathbf{x}_{m}\). For each data point \(\mathbf{x}_{m}\) (whether an original data point or a data point obtained by mapping to the origin-symmetric point), define the \(\ell^{1}\) direction \(\theta_{m}\) (analogous to an angle in \(\ell^{2}\) polar coordinates) to be the \(y\)-coordinate of the corresponding point on the \(\ell^{1}\) \"unit circle\" (unit diamond), that is,  
\[\theta_{m}=\frac{y_{m}}{|x_{m}|+|y_{m}|} \tag{8}\]  
Here, all of the coordinates \(x_{m}\) are nonnegative and the \(\theta_{m}\) are in the interval \([-1,1)\). In what follows, we will need periodic extension of the \(\theta_{m}\)'s. Let \(k=sM+m\) for some integer \(s\) and for \(m\), \(0\leq m\leq M-1\). The \(\theta_{k}\) for \(k\) outside the range \(0\leq m\leq M-1\) are defined in a natural manner as  
\[\theta_{k}=\theta_{m}+2s \tag{9}\]  
(This situation is a direct analogue of the fact that the standard \(\ell^{2}\) angle \(\alpha\) of a point can be represented by \(\alpha+2\pi s\) for any integer \(s\).) The original data points as well as the data points obtained in this manner can be represented in \"polar\" form as \((\theta_{k},r_{k})\) where \(r_{k}=|x_{k}|+|y_{k}|\) is the \(\ell^{1}\) radius of the data point. In what follows, we assume that the data points \((\theta_{m},r_{m})\) have been ordered so that the values of \(\theta_{m}\) increase monotonically with \(m\). To avoid proliferation of notation, we use the same notation \((\theta_{m},r_{m})\) for the sorted data that was used for the unsorted data.