Andrew Zardecki

Rule-Based Forecasting

Fuzzy rule-based systems and related techniques, chiefly fuzzy basis functions expansions, are applied to time series forecasting and anomaly detection in temporal and spatial patterns. The usefulness of different techniques is compared using the simple parity classification problem as an example. Forecasting of a time series is analyzed, together with a brief discussion of chaotic and noisy patterns. As a by-product of the rule-based forecasting, an edge detection algorithm for digital images is obtained.

Unstructured surface mesh adaptation using the Laplace-Beltrami target metric approach

This paper develops a set of adaptive surface mesh equations by using a harmonic morphism, which is a special case of a harmonic map. These equations are applicable both to structured and unstructured surface meshes, provided that the underlying surface is given in a parametric form. By representing the target metric of the mesh as a sum of a coarse-grained component and a component quadratic in surface gradients, an improved surface mesh may be obtained. The weak form of the grid equations is solved using the finite element approximation, which reduces the grid equations to a nonlinear, algebraic set. Examples of structured and unstructured meshes are used to illustrate the applicability of the proposed approach.

Continued Fractions in Time Series Forecsting

Through their ergodic properties, continued fractions provide a fascinating example of a dynamical system whose properties we attempt to encode in a library of rules. For an irrational number, randomly selected from a unit interval, the probability distribution of partial quotients can be derived from the ergodic theorem, allowing one to distinguish a random time series from a time series whose elements are drawn from the probability distribution resulting from the ergodic hypothesis. Applications of ergodicity to modular transformations and chaotic cosmology are sketched. In addition to being an object of study, continued fractions are used as a tool to overcome the curse of dimensionality in rule-based forecasting. To this end, we encode the successive (possibly resealed) values of a time series, as the partial quotients of a continued fraction, resulting in a number from the unit interval. The accuracy of a ruled-based system utilizing this coding is investigated to some extent. Qualitative criteria for the applicability of the algorithm are formulated.

Noise contaminated transmittance

We compare the efficiency of a classifier based on probabilistic neural networks and the general least squares method. Both methods must accommodate noise due to uncertainty in the measured spectrum. The evaluation of both methods is based on a simulated transmittance spectrum, in which the received signal is supplemented by an additive admixture of noise. To obtain a realistic description of the noise mode, we generate several hundred laser pulses for each wavelength under consideration. These pulses have a predetermined correlation matrix for different wavelengths; furthermore, they are composed of three components accounting for the randomness of the observed spectrum. The first component is the correlated 1/f noise; the second component is due to uncorrelated 1/f noise; the third one is the uncorrelated white noise. The probabilistic neural network fails to retrieve the species concentration correctly for large noise levels; on the other hand, its predictions being confined to a fixed number of concentration bins, the network produces relatively small variances. To a large extent, the general least square method avoids the false alarms. It reproduces the average concentrations correctly; however, the concentration variances can be large.