Tomislav Marošević

Multiple circle detection based on center-based clustering

Adaptation of the k-means algorithm is used by circle-centers.Incremental algorithm for searching for a globally optimal k-partition is proposed.A few known indexes for determining a number of clusters in partition are adopted.Hausdorff distance between two circles is adopted. The multiple circle detection problem has been considered in the paper on the basis of given data point set A ? R 2 . It is supposed that all data points from the set A come from k circles that should be reconstructed or detected. The problem has been solved by the application of center-based clustering of the set A , i.e. an optimal k-partition is searched for, whose clusters are determined by corresponding circle-centers. Thereby, the algebraic distance from a point to the circle is used. First, an adaptation of the well-known k-means algorithm is given in the paper. Also, the incremental algorithm for searching for an approximate globally optimal k-partition is proposed. The algorithm locates either a globally optimal k-partition or a locally optimal k-partition close to the global one. Since optimal partitions with 2, 3, ? clusters are determined successively in the algorithm, several well-known indexes for determining an appropriate number of clusters in a partition are adopted for this case. Thereby, the Hausdorff distance between two circles is used and adopted. The proposed method and algorithm are illustrated and tested on several numerical examples.

Modified indices of political power: a case study of a few parliaments

According to yes–no voting systems, players (e.g., parties in a parliament) have some influence on making some decisions. In formal voting situations, taking into account that a majority vote is needed for making a decision, the question of political power of parties can be considered. There are some well-known indices of political power e.g., the Shapley–Shubik index, the Banzhaf index, the Johnston index, the Deegan–Packel index. In order to take into account different political nature of the parties, as the main factor for forming a winning coalition i.e., a parliamentary majority, we give a modification of the power indices. For the purpose of comparison of these indices of political power from the empirical point of view, we consider the indices of power in some cases, i.e., in relation to a few parliaments.

An application of a few inequalities among sequences in electoral systems

Abstract We look at the concept of ‘favouring large states’ for divisor methods in proportional electoral systems, which is based on the comparison of the ratios of divisors. We show that it is possible in the ordered way to insert new divisor methods between any two divisor methods which have the property that one ‘favours large states’ over the other. It follows from a few sequences’ inequalities of the harmonic, geometric, arithmetic and quadratic means.

On Directional Bias of the L p -Norm

We look at the so-called directional bias of the L P -norm (1 ≤ p ≤ ∞) (see [1]). The L p -norm is a popular distance measure in the problems of estimating the distances among objects. Distance ‖x − y‖ p between any two points x and y depends on a particular set of orthogonal reference axes, except for the L 2 -norm (i.e. euclidean norm). The axes and the distance function should be chosen in accordance with the directional bias.