Milena Stankovic

A methodology for training set instance selection using mutual information in time series prediction

Training set instance selection is an important preprocessing step in many machine learning problems, including time series prediction, and has to be considered in practice in order to increase the quality of the predictions and possibly reduce training time. Recently, the usage of mutual information (MI) has been proposed in regression tasks, mostly for feature selection and for identifying the real data from data sets that contain noise and outliers. This paper proposes a new methodology for training set instance selection for long-term time series prediction. The proposed methodology combines a recursive prediction strategy and advanced instance selection criterion—the nearest neighbor based MI estimator. An application of the concept of MI is presented for the selection of training instances based on MI computation between initial training set instances and the current forecasting instance, for every prediction step. The novelty of the approach lies in the fact that it fits the initial training subset with the current forecasting instance, and consequently reduces the uncertainty of the prediction. In this way, by selecting instances which share a large amount of MI with the current forecasting instance in every prediction step, error propagation and accumulation can be reduced, both of which are well known shortcomings of the recursive prediction strategy, thus leading to better forecasting quality. Another element which sets this approach apart from others is that it is not proposed as an outlier detector, but for the instance selection of data which do not necessarily have to contain noise and outliers. The results obtained from the data sets from NN5 competition in time series prediction indicate that the proposed method increases the quality of long-term time series prediction, as well as reduces the amount of instances needed for building the model.

Calculation of Reed-Muller-Fourier coefficients of multiple-valued functions through multiple-place decision diagrams

We extend the method for the calculation of Walsh transform of binary switching functions through the binary decision diagrams to the calculation of Reed-Muller-Fourier transform of p-valued through multiple-place decision diagrams functions through multiple-place decision diagrams. The calculation of Reed-Muller coefficients of binary switching functions is involved as a special case for p=2.<<ETX>>

Contribution to the Study of Multiple-Valued Bent Functions

This paper is meant to be a contribution to the study of multiple-valued bent functions. Special care has been given to the introduction of a sound formalism to allow giving formal proof of some properties and to characterize bent functions. A class of bent functions, called strict bent is introduced and its characterization is given. It is shown that there are 486 two place ternary strict bent functions.

Representation of Multiple-Valued Bent Functions Using Vilenkin-Chrestenson Decision Diagrams

Bent functions are a class of discrete functions which exhibit the highest degree of nonlinearity. As such bent functions form an essential part of cryptographic systems. Original concept of bent functions defined in GF(2) can be extended to multiple-valued case. Multiple-valued bent functions are defined in therms of properties of their Vilenkin-Chrestenson spectra. Decision diagrams are a method of compact representation of discrete functions. Special types of decision diagrams have been introduced for various types of discrete functions. In this paper we demonstrate how Vilenkin-Chrestenson decision diagrams can be used for efficient representation of multiple-valued bent functions.

Design of Haar wavelet transforms and Haar spectral transform decision diagrams for multiple-valued functions

In spectral interpretation, decision diagrams (DDs) are defined in terms of some spectral transforms. For a given DD, the related transform is determined by an analysis of expansion rules used in the nodes and the related labels of edges. The converse task, design of a DD in terms of a given spectral transform often requires decomposition of basic functions in spectral transform to determine the corresponding expansion rules and labels at the edges. We point out that this problem relates to the assignment of nodes in Pseudo-Kronecker DDs(PKDDs). Due to that, we generalize the definition of Haar spectral transform DDs (HSTDDs) to multiple-valued (MV) functions. Conversely, from such defined HSTDDs, we derive various Haar transforms for MV functions.

The Pascal Triangle (1654), the Reed-Muller-Fourier Transform (1992), and the Discrete Pascal Transform (2005)

This paper makes a theoretical comparative analysis of the Reed-Muller-Fourier Transform, Pascal matrices based on the Pascal triangle, and the Discrete Pascal Transform. The Reed-Muller-Fourier Transform was not originated by a Pascal matrix, however it happens to show a strong family resemblance with it, sharing several basic properties. Its area of application is the multiple-valued switching theory, mainly to obtain polynomial expressions from the value vector of multiple-valued functions. The Discrete Pascal Transform was introduced over a decade later, based on an ad hoc modification of a Pascal matrix, for applications on picture processing. It is however shown that a Discrete Pascal Transform of size p, taken modulo p equals the special Reed-Muller-Fourier Transform for the same p and n = 1. The Sierpinski fractal is close related to the Pascal matrix. Data structures based on the Sierpinski triangle have been successfully used to solve special problems in switching theory. Some of them will be addressed in the paper.

Hyper-bent Multiple-Valued Functions

Hyper-bent functions constitute a subset of bent functions and are harder to approximate than bent functions, making them particularly attractive for cryptographic applications. In the multiple-valued world, up to now, characterization and generation of hyper-bent functions represent an interesting challenging mathematical problem. We show that multiple-valued hyper-bent functions constitute a reduced subset of the multiple-valued bent functions and give a simple characterization lemma. Finally we introduce a new concept, that of strict hyper-bent functions, and study some of the properties of these functions. The only mathematical requirements of the paper are college algebra and a basic knowledge of Galois fields.

Fibonacci decision diagrams and spectral Fibonacci decision diagrams

The authors define the Fibonacci decision diagrams (FibDDs) permitting representation of functions defined in a number of points different from N=2/sup n/ by decision diagrams consisting of nodes with two outgoing edges. We show the relationships between the FibDDs and the contracted Fibonacci codes. Then, we define the Spectral Fibonacci DDs (FibSTDDs) in terms of the generalized Fibonacci transforms. This broad family of transforms provides a corresponding family of FibSTDDs. These DDs allow compact representations of functions with simple Fibonacci spectra. Such representations may be useful in various tasks of signal processing, including image processing and systems design, where the generalized Fibonacci transforms have been efficiently used.

Calculation of the paired Haar transform through shared binary decision diagrams

Abstract A concept of paired Haar transform (PHT) for representation and efficient optimization of systems of incompletely Boolean functions has recently been introduced. In this article, a method to calculate PHT for incompletely specified switching functions through shared binary decision diagrams (SBDDs) is presented. The algorithm converts switching functions in the form of SBDDs into their paired Haar spectra and can operate on functions with many variables.

Foundations for Applications of Gibbs Derivatives in Logic Design and VLSI

New technologies and increased requirements for performances of digital systems require new mathematical theories and tools as a basis for future VLSI CAD systems. New or alternative mathematical approaches and concepts must be suitable to solve some concrete problems in VLSI and efficient algorithms for their efficient application should be provided. This paper is an attempt in this direction and relates with the recently renewed interest in arithmetic expressions for switching functions, instead representations in Boolean structures, and spectral techniques and differential operators in switching theory and applications. Logic derivatives are efficiently used in solving different tasks in logic design, as for example, fault detection, functional decomposition, detection of symmetries and co-symmetries of logic functions, etc. Their application is based on the property that by differential operators, we can measure the rate of change of a logic function. However, by logic derivatives, we can hardly distinguish the direction of the change of the function, since they are defined in finite algebraic structures. Gibbs derivatives are a class of differential operators on groups, which applied to logic functions, permit to overcome this disadvantage of logic derivatives. Therefore, they may be useful in logic design in the same areas where the logic derivatives have been already using. For such applications, it is important to provide fast algorithms for calculation of Gibbs derivatives on finite groups efficiently in terms of space and time. In this paper, we discuss the methods for efficient calculation of Gibbs derivatives. These methods should represent a basis for further applications of these and related operators in VLSI CAD systems.