The dynamics of laser droplet generation

We propose an experimental setup allowing for the characterization of laser droplet generation in terms of the underlying dynamics, primarily showing that the latter is deterministically chaotic by means of nonlinear time series analysis methods. In particular, we use a laser pulse to melt the end of a properly fed vertically placed metal wire. Due to the interplay of surface tension, gravity force and light-metal interaction, undulating pendant droplets are formed at the molten end, which eventually completely detach from the wire as a consequence of their increasing mass. We capture the dynamics of this process by employing a high-speed infrared camera, thereby indirectly measuring the temperature of the wire end and the pendant droplets. The time series is subsequently generated as the mean value over the pixel intensity of every infrared snapshot. Finally, we employ methods of nonlinear time series analysis to reconstruct the phase space from the observed variable and test it against determinism and stationarity. After establishing that the observed laser droplet generation is a deterministic and dynamically stationary process, we calculate the spectra of Lyapunov exponents. We obtain a positive largest Lyapunov exponent and a negative divergence, i.e., sum of all the exponents, thus indicating that the observed dynamics is deterministically chaotic with an attractor as solution in the phase space. In addition to characterizing the dynamics of laser droplet generation, we outline industrial applications of the process and point out the significance of our findings for future attempts at mathematical modeling.Comment: 7 two-column pages, 8 figures; accepted for publication in Chaos [supplementary material available at http://www.matjazperc.com/chaos/laser.html

Generalizations of pairwise comparison models

V disertaciji se osredotočimo na modele parnih primerjav. Ti modeli so dobro uveljavljeni v statistiki in so zelo uporabni na številnih področjih, zlasti v športu, ki je bil motivacija za naše delo. Predstavimo več posplošitev dobro znanega Bradley-Terryjevega modela za parne primerjave. Najpomembnejša razširitev, ki jo predlagamo, je modeliranje časovno spremenljivih latentnih moči z Gaussovimi procesi. Kot alternativo uporabimo tudi baricentrično racionalno interpolacijo. Izid parnih primerjav pogosto ni odvisen samo od latentne moči, ampak tudi od različnih zunanjih spremenljivk. Iz tega razloga razširimo naše modele tako, da omogočajo vključitev dodatnih kovariatov. V športu so pogosto na voljo podatki o stavnicah. Znano je, da so stavne kvote vir natančnih verjetnosti izida, saj so tako stavnice kot javnost motivirani, da upoštevajo vse razpoložljive informacije. Naši modeli omogočajo vključitev teh informacij v obliki verjetnosti izida. V empiričnem delu primerjamo več modelov in inferenčnih metod na športnih podatkih. Pokažemo, da z uporabo Gaussovih procesov dosežemo boljše rezultate v primerjavi z baricentrično racionalno interpolacijo, ker so Gaussovi procesi bolj prilagodljivi pri modeliranju naglih sprememb v latentni moči. Pri uporabi baricentrične racionalne interpolacije se je izkazalo, da z bayesovskim pristopom dobimo boljše rezultate kot z oceno največjega verjetja. V primeru Gaussovih procesov pa z bayesovskim pristopom dobimo podobne rezultate kot pri uporabi Laplaceove aproksimacije, če hiper-parametre slednje določimo s križno validacijo in ne z robnim verjetjem. S primeri iz športa smo potrdili, da uporaba informacij o verjetnosti izida iz stavnih kvot dodatno izboljša napovedi modelov in tako dobimo najboljše rezultate. Z uporabo teh dodatnih informacij zmanjšamo negotovost cenilk parametrov. Implementirali smo tudi več algoritmov Monte Carlo z markovskimi verigami, posebej prilagojenih za Gaussove procese. Algoritem Metropolis-Hastings in različice Gibbsovega algoritma smo primerjali s Hamiltonskim Monte Carlo algoritmom. Slednji algoritem, v različici, ki je implementirana v Stanu, vrne boljše rezultate v smislu efektivne velikosti vzorca in časovne učinkovitosti.In this thesis we focus on pairwise comparison models. These models are well established in the field of statistics and are highly applicable in many areas, in particular in sports which motivated our work. We present several generalizations of the famous Bradley-Terry model for pairwise comparisons. The main extension we propose is to model the underlying time-varying latent strengths as Gaussian processes. Alternatively, we also apply barycentric rational interpolation and compare the two approaches. The outcome of pairwise comparisons are often not dependent solely on latent strengths but also on various external variables. For this reason we also extend our models to allow additional covariates. In sports there are often bookmaker odds data available. Bookmaker odds are known to be a source of accurate outcome probabilities, because bookmakers and the public are incentivized to use all available information. Our models allow for the inclusion of the information in the form of outcome probabilities. The empirical part of our research consists of comparing several models and inference methods on various sports data sets. We demonstrated that using Gaussian processes is advantageous compared to barycentric rational interpolation as they are more flexible in modelling sudden changes in the underlying latent strengths. When using barycentric rational interpolation, Bayesian inference gives better results than maximum likelihood estimation. In the case of Gaussian processes, Bayesian inference will give similar results as using Laplace approximation if hyper-parameters of the latter are determined by cross-validation rather than by marginal likelihood. With examples from sports we confirmed that using information from outcome probabilities from bookmaker odds further improves models\u27 predictions and gives the best results. Using this additional information also results in less uncertainty in parameter estimates. We also implemented several Markov chain Monte Carlo algorithms specifically adapted for inference for Gaussian processes. We compared the Metropolis-Hastings algorithm and variants of the Gibbs algorithm with the Hamiltonian Monte Carlo algorithm. The latter algorithm as implemented in Stan gives superior results with respect to effective sample size and time efficiency

Stability diagrams and chatter avoidance in horizontal band sawing

The paper presents recurrence plot based stability analysis of the horizontal band sawing process of structural steel profiles. The analysis is performed in the parameter space defined by the cutting speed, the distance between the blade supports, and the feed rate. The corresponding stability diagrams have been constructed using the recurrence plot characteristic, the determinism of the sound pressure emitted by the process, which quantifies the process predictability. The topology of the experimentally obtained stability diagrams revealed non-linear non-monotonic dynamic behaviour, which made two different chatter avoidance strategies possible by cutting speed variation

A Bayesian approach to time-varying latent strengths in pairwise comparisons.

The famous Bradley-Terry model for pairwise comparisons is widely used for ranking objects and is often applied to sports data. In this paper we extend the Bradley-Terry model by allowing time-varying latent strengths of compared objects. The time component is modelled with barycentric rational interpolation and Gaussian processes. We also allow for the inclusion of additional information in the form of outcome probabilities. Our models are evaluated and compared on toy data set and real sports data from ATP tennis matches and NBA games. We demonstrated that using Gaussian processes is advantageous compared to barycentric rational interpolation as they are more flexible to model discontinuities and are less sensitive to initial parameters settings. However, all investigated models proved to be robust to over-fitting and perform well with situations of volatile and of constant latent strengths. When using barycentric rational interpolation it has turned out that applying Bayesian approach gives better results than by using MLE. Performance of the models is further improved by incorporating the outcome probabilities