Estimating conditional probability densities for periodic variables

Most of the common techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we introduce three novel techniques for tackling such problems, and investigate their performance using synthetic data. We then apply these techniques to the problem of extracting the distribution of wind vector directions from radar scatterometer data gathered by a remote-sensing satellite.

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densities by generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Modelling conditional probability distributions for periodic variables

Most of the common techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we introduce two novel techniques for tackling such problems, and investigate their performance using synthetic data. We then apply these techniques to the problem of extracting the distribution of wind vector directions from radar scatterometer data gathered by a remote-sensing satellite.

Modelling conditional probability distributions for periodic variables

Most of the common techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we apply two novel techniques to the problem of extracting the distribution of wind vector directions from radar catterometer data gathered by a remote-sensing satellite.

Modelling conditional probability distributions for periodic variables

Most conventional techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we introduce three related techniques for tackling such problems, and investigate their performance using synthetic data. We then apply these techniques to the problem of extracting the distribution of wind vector directions from radar scatterometer data gathered by a remote-sensing satellite.

Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Stochastic modelling of financial time series with memory and multifractal scaling

Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.

Large margin vector quantization

In this paper we describe the Large Margin Vector Quantization algorithm (LMVQ), which uses gradient ascent to maximise the margin of a radial basis function classifier. We present a derivation of the algorithm, which proceeds from an estimate of the class-conditional probability densities. We show that the key behaviour of Kohonen's well-known LVQ2 and LVQ3 algorithms emerge as natural consequences of our formulation. We compare the performance of LMVQ with that of Kohonen's LVQ algorithms on an artificial classification problem and several well known benchmark classification tasks. We find that the classifiers produced by LMVQ attain a level of accuracy that compares well with those obtained via LVQ1, LVQ2 and LVQ3, with reduced storage complexity. We indicate future directions of enquiry based on the large margin approach to Learning Vector Quantization.

Approximate Bayesian Computation for astronomical model analysis : a case study in galaxy demographics and morphological transformation at high redshift

Approximate Bayesian Computation’ (ABC) represents a powerful methodology for the analysis of complex stochastic systems for which the likelihood of the observed data under an arbitrary set of input parameters may be entirely intractable – the latter condition rendering useless the standard machinery of tractable likelihood-based, Bayesian statistical inference [e.g. conventional Markov chain Monte Carlo (MCMC) simulation]. In this paper, we demonstrate the potential of ABC for astronomical model analysis by application to a case study in the morphological transformation of high-redshift galaxies. To this end, we develop, first, a stochastic model for the competing processes of merging and secular evolution in the early Universe, and secondly, through an ABC-based comparison against the observed demographics of massive (Mgal > 1011 M⊙) galaxies (at 1.5 < z < 3) in the Cosmic Assembly Near-IR Deep Extragalatic Legacy Survey (CANDELS)/Extended Groth Strip (EGS) data set we derive posterior probability densities for the key parameters of this model. The ‘Sequential Monte Carlo’ implementation of ABC exhibited herein, featuring both a self-generating target sequence and self-refining MCMC kernel, is amongst the most efficient of contemporary approaches to this important statistical algorithm. We highlight as well through our chosen case study the value of careful summary statistic selection, and demonstrate two modern strategies for assessment and optimization in this regard. Ultimately, our ABC analysis of the high-redshift morphological mix returns tight constraints on the evolving merger rate in the early Universe and favours major merging (with disc survival or rapid reformation) over secular evolution as the mechanism most responsible for building up the first generation of bulges in early-type discs.

Measurement of the top quark mass with dilepton events selected using neuroevolution at CDF

We report a measurement of the top quark mass $M_t$ in the dilepton decay channel $t\bar{t}\to b\ell'^{+}\nu'_\ell\bar{b}\ell^{-}\bar{\nu}_{\ell}$. Events are selected with a neural network which has been directly optimized for statistical precision in top quark mass using neuroevolution, a technique modeled on biological evolution. The top quark mass is extracted from per-event probability densities that are formed by the convolution of leading order matrix elements and detector resolution functions. The joint probability is the product of the probability densities from 344 candidate events in 2.0 fb$^{-1}$ of $p\bar{p}$ collisions collected with the CDF II detector, yielding a measurement of $M_t= 171.2\pm 2.7(\textrm{stat.})\pm 2.9(\textrm{syst.})\mathrm{GeV}/c^2$.

Top quark mass measurement in the tt̅ all hadronic channel using a matrix element technique in pp̅ collisions at √s=1.96 TeV

We present a measurement of the top quark mass in the all-hadronic channel (\tt $\to$ \bb$q_{1}\bar{q_{2}}q_{3}\bar{q_{4}}$) using 943 pb$^{-1}$ of \ppbar collisions at $\sqrt {s} = 1.96$ TeV collected at the CDF II detector at Fermilab (CDF). We apply the standard model production and decay matrix-element (ME) to $\ttbar$ candidate events. We calculate per-event probability densities according to the ME calculation and construct template models of signal and background. The scale of the jet energy is calibrated using additional templates formed with the invariant mass of pairs of jets. These templates form an overall likelihood function that depends on the top quark mass and on the jet energy scale (JES). We estimate both by maximizing this function. Given 72 observed events, we measure a top quark mass of 171.1 $\pm$ 3.7 (stat.+JES) $\pm$ 2.1 (syst.) GeV/$c^{2}$. The combined uncertainty on the top quark mass is 4.3 GeV/$c^{2}$.

Measurement of the Top-Quark Mass with Dilepton Events Selected Using Neuroevolution at CDF

We report a measurement of the top quark mass $M_t$ in the dilepton decay channel $t\bar{t}\to b\ell'^{+}\nu'_\ell\bar{b}\ell^{-}\bar{\nu}_{\ell}$. Events are selected with a neural network which has been directly optimized for statistical precision in top quark mass using neuroevolution, a technique modeled on biological evolution. The top quark mass is extracted from per-event probability densities that are formed by the convolution of leading order matrix elements and detector resolution functions. The joint probability is the product of the probability densities from 344 candidate events in 2.0 fb$^{-1}$ of $p\bar{p}$ collisions collected with the CDF II detector, yielding a measurement of $M_t= 171.2\pm 2.7(\textrm{stat.})\pm 2.9(\textrm{syst.})\mathrm{GeV}/c^2$.

Bayesian approach to signal modelling

In this paper, an introduction to Bayesian methods in signal processing will be given. The paper starts by considering the important issues of model selection and parameter estimation and derives analytic expressions for the model probabilities of two simple models. The idea of marginal estimation of certain model parameter is then introduced and expressions are derived for the marginal probability densities for frequencies in white Gaussian noise and a Bayesian approach to general changepoint analysis is given. Numerical integration methods are introduced based on Markov chain Monte Carlo techniques and the Gibbs sampler in particular.

Solvation effects on equilibria: Triazoles and N-methyl piperidinol

We have performed calculations of the solvation effects on a number of equilibrium constants in water using a recently proposed hybrid quantum classical scheme in which the liquid environment is modelled using classical solvent molecules and the solute electronic structure is computed using modern quantum chemical methods. The liquid phase space is sampled from a fully classical simulation. We find that solvation effects on both triazole tautomeric equilibrium constants and piperidinol conformational equilibrium constants can be interpreted in terms of subtle differences in the local environment which can be seen in probability densities and radial distribution functions. Lower level calculations were performed for comparison and we conclude that the solvation thermodynamics can be predicted from a good classical model of solvent and solute molecules, but the implicit models that we tried are less successful.