## 3 2D \(\ell^{1}\) Major Component Detection and Analysis  
Reformulation of the PCA approach in the \(\ell^{1}\) norm involves more than adjusting individual steps in the standard PCA process that was outlined in Section 2. One must now accomplish the objective of determining the main directions and the magnitudes of the spread of the point cloud in those directions without the tools (singular values, inner products, orthogonal projection, averaging and second-order moments) of the standard \(\ell^{2}\)-based approach.  
The \(\ell^{1}\) Major Component Detection and Analysis (\(\ell^{1}\) MCDA) that we propose consists of the following two steps:  
1. Calculate the central point of the data and subtract it out of the data.
2. Calculate the main directions of the point cloud that results from Step 1 and the magnitudes of radial extension in those directions.  
Post-processing analogous to Step 6 of standard PCA will be part of a fully developed \(\ell^{1}\) MCDA in the future but will not be discussed in this paper.  
The point cloud under consideration is denoted by \(\{\mathbf{x}_{m}\}_{m=0}^{M-1}\). The distance between points \(\mathbf{x}\) and \(\mathbf{y}\) in the data space is denoted by \(d(\mathbf{x},\mathbf{y})\). Since the \(\ell^{1}\) norm requires fewer operations than the \(\ell^{2}\) norm (see Remark 1 below), we will use the \(\ell^{1}\) norm to define the distance function in the data space for the description of the algorithm in the present section and for the computational experiments discussed in Section 4. However, \(\ell^{1}\) MCDA works with any distance function in the data space that the user wishes to choose, for example, the \(\ell^{2}\) norm. The distance function does need not to satisfy the triangle inequality but does need to allow definition of angular coordinates. All \(\ell^{p}\) norms, \(1\leq p<\infty\), allow definition of angular coordinates. We do not require orthogonality of the main directions of the distribution but do allow orthogonality to be imposed based on outside information, for example, when we wish to identify major components of a point cloud that is known to be from a distribution with orthogonal main directions or when the data are geometric data with orthogonal main directions in a Euclidean space.  
_Remark 1_ Minimization of a \"linear\" \(\ell^{1}\) functional (sum of absolute values of linear components) is a linear programming problem that is generally more expensive to solve than minimization of a corresponding \(\ell^{2}\) functional (sum of squares of linear components), which is carried out by solving one linear system. Not surprisingly, therefore, the \(\ell^{1}\) MCDA that we will develop will be more expensive than standard \(\ell^{2}\)-based PCA. However, as a functional for measuring distance in the data space, the \(\ell^{1}\) norm is not an \(\ell^{1}\) minimization problem but rather simply a defined metric. As a metric in the data space, the \(\ell^{1}\) norm, which is a sum of absolute values, is computationally much cheaper than the \(\ell^{2}\) norm, which is a square root of a sum of products. Rotation in \(\ell^{1}\)-normed space consists of adding and/or subtracting quantities from the coordinates while rotation in \(\ell^{2}\)-normed space involves the more expensive operations of calculation of multiple sums of products. This situation suggests that using the \(\ell^{1}\) norm in the data space is meaningful. Even when the natural norm in the data space is the \(\ell^{2}\) norm, using the computationally cheaper \(\ell^{1}\) norm as an approximation of the \(\ell^{2}\) norm (rather than vice versa as has traditionally been the case) is a meaningful choice. The fact that \(\ell^{1}\)-based methods are more expensive than \(\ell^{2}\)-based methods in many higher-level situations does not change the fact that the \(\ell^{1}\) metric is much less expensive than the \(\ell^{2}\) metric at the lowest level of measuring distance in a data space.