Large deviation principle for piecewise monotonic maps with density of periodic measures

We show that a piecewise monotonic map with positive topological entropy satisfies the level-2 large deviation principle with respect to the unique measure of maximal entropy under the conditions that the corresponding Markov diagram is irreducible and that the periodic measures of the map are dense in the set of ergodic measures. This result can apply to a broad class of piecewise monotonic maps, such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces.

Peak Estimation of Univariate Spectraby the Best L1 Piecewise Monotonic Approximation Method

We consider applications of the best L1 piecewise monotonic approximation method for the peak estimation of three sets of up to 2500 measurements of Raman, Infrared and Nuclear Magnetic Resonance (NMR)spectra. Peak estimation is an inherent problem of spectroscopy. The location of peaks and their intensities arethe signature of a sample of an organic or an inorganic compound. The diversity and the complexity of our measurements makes it a difficult test of the effectiveness of the method. We find that the method identifies efficientlypeaks and we compare to the results obtained by the analogous least squares calculations. These results havemany similarities and occasionally considerable differences due to both properties of the norms employed in theoptimization calculations and nature of the spectra. Our results may be helpful to subject analysts as part of theinformation on which decisions will be made for estimating peaks in sequences of spectra and to the developmentof new algorithms that are particularly suitable for peak estimation calculations.