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Title: Dynamic model for estimation of radon concentration in buildings Author: Barbora Vaňková Department: Department of probability and mathematical statistics Supervisor: Ing. Marek Brabec, Ph.D. Supervisor's e-mail address: mbrabec@cs.cas.cz Abstract: In the present work there is described the method for estimation of functi- onal data from discrete values and basic methods of functional data analysis. 1

During the last decade, many baseline estimation methods have been proposed, but many of these approaches are either only useful for specific kinds of analytical signals or require the adjustment of many parameters. This complicates the selection of an appropriate approach for each kind of chemical signal and the optimization of multiple parameters itself is not an easy task. In this work, an asymmetric least squares (ALS) approach is used with truncated and augmented Fourier basis functions to provide a universal basis space for baseline approximation for diverse analytical signals. The proposed method does not require extensive parameter adjustment or prior baseline information. The basis set used to model the baselines includes a Fourier series truncated to low frequency sines and cosines (consistent with the number of channels) which is then augmented with lower frequencies. The number of basis functions employed depends mainly on the frequency characteristics of the baseline, which is the only parameter adjustment required for baseline estimation. The weighting factor for the asymmetric least squares in this case is dependent mainly on the level of the noise. The adjustment of these two parameters can be easily performed by visual inspection of results. To estimate and eliminate the baseline from the analytical signals, a novel algorithm, called Truncated Fourier Asymmetric Least Squares (TFALS) was successfully developed and optimized. It does not require baseline representative signals or extensive parameter adjustments. The method is described only with parameters optimization using simulated signals. The results with simulated and experimental data sets having different baseline artefacts show that TFALS is a versatile, effective and easy-to-use baseline removal method.

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number m generates a complete or incomplete Fourier basis for a Cantor-type measure with scale g.