## 3 2D \(\ell^{1}\) Major Component Detection and Analysis
### \(\ell^{1}\) MCDA Step 2: Calculation of the Main Directions
Due to statistical variability, we cannot find good approximations of the directions in which the distribution spreads farthest simply by identifying the locally largest values of \(r_{m}\) (as a function of \(m\)). One estimates the directions in which the multidimensional distribution spreads farthest and the extent to which it spreads in these directions by calculating the local maxima with respect to \(\theta\) of the median radius of the distribution. To find the directions in which the median radius is locally maximal, we have to approximate the data \((\theta_{m},r_{m})\) in a smooth manner and then find the directions \(\theta\) in which this approximation is locally maximal. A method for finding local maxima that relies on fitting the data with a global function would be computationally feasible in 2D. However, fitting a global surface to the data would be less computationally attractive in 3D and 4D (space + time) and not at all computationally useful in \(n\) dimensions for \(n>4\). For this reason, we adopt the following locally based algorithm for calculating values of a function \(r(\theta)\) that represents the median radius of the data points \((\theta_{m},r_{m})\) and for identifying local maxima of this function:  
1. Choose a point \(\theta_{m}\) from which to start. 2. Choose an integer \(q\) that represents the number of neighbors in each direction (index lower and higher than \(m\)) that will be included in a local domain \(D\). 3. On the local domain \(D=[\theta_{m-q},\theta_{m+q}]\), calculate the quadratic polynomial \(a_{0}+a_{1}\theta+a_{2}\theta^{2}\) that best fits the data in the \(\ell^{1}\) norm, that is, minimizes, \[\sum_{k=m-q}^{m+q}|a_{0}+a_{1}\theta_{k}+a_{2}\theta_{k}^{2}-r_{k}|\] (10) over all real numbers \(a_{0}\), \(a_{1}\) and \(a_{2}\). 4. Determine the location of the maximum of the quadratic polynomial on the local domain \(D\). If the location of the maximum is strictly inside \(D\), go to Step 5. If the location of the maximum is at \(\theta_{m-q}\) or \(\theta_{m+q}\), call this direction a new \(\theta_{m}\) and return to Step 3. 5. Refine the location and value of the maximum of the median radius in the following way. Calculate a quadratic polynomial on a local domain with a larger \(q^{*}\) around the node closest to the location of the approximate local maximum identified in Step 4. The maximum of this quadratic polynomial is the estimate of the maximum of the median radius. The above procedure is for calculating one local maximum. To calculate all local maxima for a distribution with several spokes, one needs to assume that the spokes are distinct from each other, that is, do not overlap in a way that two closely placed spokes appear nearly like one spoke. For example, one may have information that the spokes are locally separated by angular distances greater than or equal to a known lower bound. One then chooses starting points for multiple implementations of Step 1 that cover the interval \([-1,1)\) at distances that are slightly less than the lower bound. For applications for which local minima are important, one can calculate local minima analogously. _Remark 3_ The (unusual) situation in which all points lie on one or a few radial lines does not allow use of the local fitting method described above and can be taken care of by other procedures. In this case, the maxima of the median radius occur at the directions of the radial lines. From the clusters of points with identical \(\theta_{i}\), one identifies the directions of the lines and then calculates the one-dimensional median of the \(r_{i}\) in each cluster. Theoretical guidance for choosing the \(q\) and \(q^{*}\) of Steps 2 and 5 is not yet available but it is clear that \(q\) and \(q^{*}\) need to be chosen based on the structure of the distribution and the properties of the outliers in the data. There will certainly be a trade-off between accuracy and computational efficiency.