Maciej Krawczak

An approach to dimensionality reduction in time series

Many methods of dimensionality reduction of data series (time series) have been introduced over the past decades. Some of them rely on a symbolic representation of the original data, however in this case the obtained dimensionality reduction is not substantial. In this paper, we introduce a new approach referred to as Symbolic Essential Attributes Approximation (SEAA) to reduce the dimensionality of multidimensional time series. In such a way we form a new nominal representation of the original data series. The approach is based on the concept of data series envelopes and essential attributes generated by a multilayer neural network. The real-valued attributes are discretized, and in this way symbolic data series representation is formed. The SEAA generates a vector of nominal values of new attributes which form the compressed representation of original data series. The nominal attributes are synthetic, and while not being directly interpretable, they still retain important features of the original data series. A validation of usefulness of the proposed dimensionality reduction is carried out for classification and clustering tasks. The experiments have shown that even for a significant reduction of dimensionality, the new representation retains information about the data series sufficient for classification and clustering of the time series.

Application of the InterCriteria Decision Making Method to Universities Ranking

In this paper we present an application of the InterCriteria Decision Making (ICDM) approach to real data extracted from the Polish University Ranking System [13] in the years 2012–2014. The aim is to analyze the correlations between the indicators used by the Ranking System.

On Perturbation Measure of Sets : Properties

In this paper we describe a new measure of remoteness between sets described by nominal values. The introduced measures of perturbation of one set by another are considered instead of commonly used distance between two sets. The operations of the set theory are operated and the considered measures describe changes of the perturbed second set by adding the first one or vice versa. The values of the measure of sets’ perturbation are range between 0 and 1, and in general, are not symmetric – it means that the perturbation of one set by another is not the same as the perturbation of the second set by the first one.

On Perturbation Measure of Clusters: Application

In this paper we developed a new methodology for grouping objects described by nominal attributes. We introduced a measure of perturbation of one cluster by another cluster in order to create a junction of clusters. The developed method is hierarchical and agglomerative and can be characterized both by high speed of computation as well as surprising good accuracy of clustering. keywords cluster analysis, nominal attributes, sets theory.

A novel modeling methodology: generalized nets

The generalized net methodology was developed as a counterpart of Petri nets. The methodology allows to model different kinds of discrete dynamic systems. The basics of the theory of generalized nets is introduced and next the algorithm of generalized nets is described. Algebraic aspects of generalized nets as well as operator aspects of generalized nets are described. At the end, one possible application of generalized nets, namely for neural networks is shown. Here a neural network without any aggregation is considered.

A clustering algorithm based on distinguishability for nominal attributes

In this paper we developed a new methodology for grouping objects described by nominal attributes. We introduced a definition of conditions domination within each pair of cluster, and next the measure of ω-distinguishability of clusters for creating a junction of clusters. The developed method is hierarchical and agglomerative one and can be characterized both by high speed of computation as well as extremely good accuracy of clustering.

Nominal time series representation for the clustering problem

In this paper we considered time series dimension reduction for clustering problem. The techniques of reduction of dimension of time series is based on the concept of envelopes, aggregation of the envelopes and extracting essential attributes. Essential attributes were nominalized. The reduced representation of time series is characterized by nominal attributes. For such representation of time series we applied a definition of conditions domination within each pair of clusters. We proposed a hierarchical agglomerative approach to clustering nominal data. There is considered a case of data series clustering problem as an illustrative example.

On bilateral matching between fuzzy sets

In the paper, we describe the new measure of matching fuzzy sets. The introduced measure of perturbation of one fuzzy set by another fuzzy set is considered instead of commonly used distance between two fuzzy sets. The operations known in the fuzzy set theory are used and the perturbation of one fuzzy set by another fuzzy set is understood as a measure describing changes of the first fuzzy set after adding the second one. Obviously, the opposite case can also be considered wherein the second fuzzy set is perturbed by the first one. In general, the new measure is asymmetric and can provide more information compare to a distance between fuzzy sets. The values of such measures of fuzzy sets perturbation are in range between 0 and 1. In this paper several mathematical properties of the measure of fuzzy sets perturbation are studied, and the measure of sets perturbation is compared to other selected measures.

On Bilateral Matching between Multisets

In the paper we defined a new measure of remoteness between multisets. The development of the measure is based on the definition of sets perturbation originally developed by the authors. The sets perturbation definition is here extended to multisets perturbation, it means perturbation of one multiset by another multiset and/or vice-versa. In general these two measures are different, it means asymmetrical, and therefore can be called the bilateral measure of matching between two multisets. Therefore the measure cannot be considered as a distance between multisets.

Heuristic Dynamic Programming for Neural Networks Learning Part 1: Learning as a Control Problem

The learning process of multilayer neural networks can be considered as a multistage optimal control process. We introduce a gain parameter into the models of neurons. Setting the parameter to a small value makes the neuron model “almost linear”, and the learning process problem can be solved using computational tools specified for linear-quadratic systems optimization. In Part 1 the continuation methodology is applied for changing the gain parameter in order to reach 1.0. In Part 2, by considering the gain parameter as an additional control variable, the optimal value of the parameter can be found. The methodology we propose to call the heuristic dynamic programming.