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Courses matching "At least four doors, numerous goats, a car, a frog"

Applied Probability III

Many processes in the real world involve some random variation superimposed on a deterministic structure. For example, the experiment of flipping a coin is best studied by treating the outcome as a random one. Mathematical probability has its origins in games of chance with dice and cards, originating in the fifteenth and sixteenth centuries. This course aims to provide a basic toolkit for modelling and analyzing discrete-time problems in which there is a significant probabilistic component. We will consider Markov chain examples in the course including population branching processes (with application to genetics), random walks (with application to games), and more general discrete time examples using Martingales. Topics covered are: basic probability and measure theory, discrete time Markov chains, hitting probabilities and hitting time theorems, population branching processes, inhomogeneous random walks on the line, solidarity properties and communicating classes, necessary and sufficient conditions for transience and positive recurrence, global balance, partial balance, reversibility, Martingales, stopping times and stopping theorems with a link to Brownian motion.

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Probability and Statistics

Probability theory is the branch of mathematics that deals with modelling uncertainty. It is important because of its direct application in areas such as genetics, finance and telecommunications. It also forms the fundamental basis for many other areas in the mathematical sciences including statistics, modern optimisation methods and risk modelling. This course provides an introduction to probability theory, random variables and Markov processes. Topics covered are: probability axioms, conditional probability; Bayes' theorem; discrete random variables, moments, bounding probabilities, probability generating functions, standard discrete distributions; continuous random variables, uniform, normal, Cauchy, exponential, gamma and chi-square distributions, transformations, the Poisson process; bivariate distributions, marginal and conditional distributions, independence, covariance and correlation, linear combinations of two random variables, bivariate normal distribution; sequences of independent random variables, the weak law of large numbers, the central limit theorem; definition and properties of a Markov chain and probability transition matrices; methods for solving equilibrium equations, absorbing Markov chains.

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Events matching "At least four doors, numerous goats, a car, a frog"

Queues with Advance Reservations
15:10 Fri 21 Sep, 2007 :: G04 Napier Building University of Adelaide :: Prof. Peter Taylor :: Department of Mathematics and Statistics, University of Melbourne

Queues where, on "arrival", customers make a reservation for service at some time in the future are endemic. However there is surprisingly little about them in the literature. Simulations illustrate some interesting implications of the facility to make such reservations. For example introducing independent and identically distributed reservation periods into an Erlang loss system can either increase or decrease the blocking probability from that given by Erlang's formula, despite the fact that the process of 'reserved arrivals' is still Poisson. In this talk we shall discuss a number of ways of looking at such queues. In particular, we shall obtain various transient and stationary distributions associated with the "bookings diary" for the infinite server system. However, this does not immediately answer the question of how to calculate the above-mentioned blocking probabilities. We shall conclude with a few suggestions as to how this calculation might be carried out.
Another proof of Gaboriau-Popa
13:10 Fri 3 Jul, 2009 :: School Board Room :: Prof Greg Hjorth :: University of Melbourne

Gaboriau and Popa showed that a non-abelian free group on finitely many generators has continuum many measure preserving, free, ergodic, actions on standard Borel probability spaces. The original proof used the notion of property (T). I will sketch how this can be replaced by an elementary, and apparently new, dynamical property.
The two envelope problem
12:10 Wed 11 Aug, 2010 :: Napier 210 :: A/Prof Gary Glonek :: University of Adelaide

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The two envelope problem is a long standing paradox in probability theory. Although its formulation has elements in common with the celebrated Monty Hall problem, the underlying paradox is apparently far more subtle. In this talk, the problem will be explained and various aspects of the paradox will be discussed. Connections to Bayesian inference and other areas of statistics will be explored.
A spatial-temporal point process model for fine resolution multisite rainfall data from Roma, Italy
14:10 Thu 19 Aug, 2010 :: Napier G04 :: A/Prof Paul Cowpertwait :: Auckland University of Technology

A point process rainfall model is further developed that has storm origins occurring in space-time according to a Poisson process. Each storm origin has a random radius so that storms occur as circular regions in two-dimensional space, where the storm radii are taken to be independent exponential random variables. Storm origins are of random type z, where z follows a continuous probability distribution. Cell origins occur in a further spatial Poisson process and have arrival times that follow a Neyman-Scott point process. Cell origins have random radii so that cells form discs in two-dimensional space. Statistical properties up to third order are derived and used to fit the model to 10 min series taken from 23 sites across the Roma region, Italy. Distributional properties of the observed annual maxima are compared to equivalent values sampled from series that are simulated using the fitted model. The results indicate that the model will be of use in urban drainage projects for the Roma region.
At least four doors, numerous goats, a car, a frog, four lily pads and some probability
11:10 Wed 13 Oct, 2010 :: Napier 210 :: Dr Joshua Ross :: University of Adelaide

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In the process of determining, amongst other things, the optimal strategy for playing a game show, and explaining the apparent persistence of a population that can be shown to die out with certainty, we will encounter a car, numerous goats, at least four doors, a frog, four lily pads, and some applied probability.
Change detection in rainfall time series for Perth, Western Australia
12:10 Mon 16 May, 2011 :: 5.57 Ingkarni Wardli :: Farah Mohd Isa :: University of Adelaide

There have been numerous reports that the rainfall in south Western Australia, particularly around Perth has observed a step change decrease, which is typically attributed to climate change. Four statistical tests are used to assess the empirical evidence for this claim on time series from five meteorological stations, all of which exceed 50 years. The tests used in this study are: the CUSUM; Bayesian Change Point analysis; consecutive t-test and the Hotelling’s T²-statistic. Results from multivariate Hotelling’s T² analysis are compared with those from the three univariate analyses. The issue of multiple comparisons is discussed. A summary of the empirical evidence for the claimed step change in Perth area is given.
Inference and optimal design for percolation and general random graph models (Part I)
09:30 Wed 8 Jun, 2011 :: 7.15 Ingkarni Wardli :: Dr Andrei Bejan :: The University of Cambridge

The problem of optimal arrangement of nodes of a random weighted graph is discussed in this workshop. The nodes of graphs under study are fixed, but their edges are random and established according to the so called edge-probability function. This function is assumed to depend on the weights attributed to the pairs of graph nodes (or distances between them) and a statistical parameter. It is the purpose of experimentation to make inference on the statistical parameter and thus to extract as much information about it as possible. We also distinguish between two different experimentation scenarios: progressive and instructive designs.

We adopt a utility-based Bayesian framework to tackle the optimal design problem for random graphs of this kind. Simulation based optimisation methods, mainly Monte Carlo and Markov Chain Monte Carlo, are used to obtain the solution. We study optimal design problem for the inference based on partial observations of random graphs by employing data augmentation technique. We prove that the infinitely growing or diminishing node configurations asymptotically represent the worst node arrangements. We also obtain the exact solution to the optimal design problem for proximity (geometric) graphs and numerical solution for graphs with threshold edge-probability functions.

We consider inference and optimal design problems for finite clusters from bond percolation on the integer lattice $\mathbb{Z}^d$ and derive a range of both numerical and analytical results for these graphs. We introduce inner-outer plots by deleting some of the lattice nodes and show that the ëmostly populatedí designs are not necessarily optimal in the case of incomplete observations under both progressive and instructive design scenarios. Some of the obtained results may generalise to other lattices.

Inference and optimal design for percolation and general random graph models (Part II)
10:50 Wed 8 Jun, 2011 :: 7.15 Ingkarni Wardli :: Dr Andrei Bejan :: The University of Cambridge

The problem of optimal arrangement of nodes of a random weighted graph is discussed in this workshop. The nodes of graphs under study are fixed, but their edges are random and established according to the so called edge-probability function. This function is assumed to depend on the weights attributed to the pairs of graph nodes (or distances between them) and a statistical parameter. It is the purpose of experimentation to make inference on the statistical parameter and thus to extract as much information about it as possible. We also distinguish between two different experimentation scenarios: progressive and instructive designs.

We adopt a utility-based Bayesian framework to tackle the optimal design problem for random graphs of this kind. Simulation based optimisation methods, mainly Monte Carlo and Markov Chain Monte Carlo, are used to obtain the solution. We study optimal design problem for the inference based on partial observations of random graphs by employing data augmentation technique. We prove that the infinitely growing or diminishing node configurations asymptotically represent the worst node arrangements. We also obtain the exact solution to the optimal design problem for proximity (geometric) graphs and numerical solution for graphs with threshold edge-probability functions.

We consider inference and optimal design problems for finite clusters from bond percolation on the integer lattice $\mathbb{Z}^d$ and derive a range of both numerical and analytical results for these graphs. We introduce inner-outer plots by deleting some of the lattice nodes and show that the ëmostly populatedí designs are not necessarily optimal in the case of incomplete observations under both progressive and instructive design scenarios. Some of the obtained results may generalise to other lattices.

Probability density estimation by diffusion
15:10 Fri 10 Jun, 2011 :: 7.15 Ingkarni Wardli :: Prof Dirk Kroese :: University of Queensland

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One of the beautiful aspects of Mathematics is that seemingly disparate areas can often have deep connections. This talk is about the fundamental connection between probability density estimation, diffusion processes, and partial differential equations. Specifically, we show how to obtain efficient probability density estimators by solving partial differential equations related to diffusion processes. This new perspective leads, in combination with Fast Fourier techniques, to very fast and accurate algorithms for density estimation. Moreover, the diffusion formulation unifies most of the existing adaptive smoothing algorithms and provides a natural solution to the boundary bias of classical kernel density estimators. This talk covers topics in Statistics, Probability, Applied Mathematics, and Numerical Mathematics, with a surprise appearance of the theta function. This is joint work with Zdravko Botev and Joe Grotowski.
Configuration spaces in topology and geometry
15:10 Fri 9 Sep, 2011 :: 7.15 Ingkarni Wardli :: Dr Craig Westerland :: University of Melbourne

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Configuration spaces of points in R^n give a family of interesting geometric objects. They and their variants have numerous applications in geometry, topology, representation theory, and number theory. In this talk, we will review several of these manifestations (for instance, as moduli spaces, function spaces, and the like), and use them to address certain conjectures in number theory regarding distributions of number fields.
Mixing, dynamics, and probability
15:10 Fri 2 Mar, 2012 :: B.21 Ingkarni Wardli :: A/Prof Gary Froyland :: University of New South Wales

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Many interesting natural phenomena are hard to predict. When modelled as a dynamical system, this unpredictability is often the result of rapid separation of nearby trajectories. Viewing the dynamics as acting on a probability measure, the mixing property states that two measurements (or random variables), evaluated at increasingly separated times, become independent in the time-separation limit. Thus, the later measurement becomes increasingly difficult to predict, given the outcome of the earlier measurement. If this approach to independence occurs exponentially quickly in time, one can profitably use linear operator tools to analyse the dynamics. I will give an overview of these techniques and show how they can be applied to answer mathematical questions, describe observed behaviour in fluid mixing, and analyse models of the ocean and atmosphere.
Forecasting electricity demand distributions using a semiparametric additive model
15:10 Fri 16 Mar, 2012 :: B.21 Ingkarni Wardli :: Prof Rob Hyndman :: Monash University

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Electricity demand forecasting plays an important role in short-term load allocation and long-term planning for future generation facilities and transmission augmentation. Planners must adopt a probabilistic view of potential peak demand levels, therefore density forecasts (providing estimates of the full probability distributions of the possible future values of the demand) are more helpful than point forecasts, and are necessary for utilities to evaluate and hedge the financial risk accrued by demand variability and forecasting uncertainty. Electricity demand in a given season is subject to a range of uncertainties, including underlying population growth, changing technology, economic conditions, prevailing weather conditions (and the timing of those conditions), as well as the general randomness inherent in individual usage. It is also subject to some known calendar effects due to the time of day, day of week, time of year, and public holidays. I will describe a comprehensive forecasting solution designed to take all the available information into account, and to provide forecast distributions from a few hours ahead to a few decades ahead. We use semi-parametric additive models to estimate the relationships between demand and the covariates, including temperatures, calendar effects and some demographic and economic variables. Then we forecast the demand distributions using a mixture of temperature simulation, assumed future economic scenarios, and residual bootstrapping. The temperature simulation is implemented through a new seasonal bootstrapping method with variable blocks. The model is being used by the state energy market operators and some electricity supply companies to forecast the probability distribution of electricity demand in various regions of Australia. It also underpinned the Victorian Vision 2030 energy strategy.
Financial risk measures - the theory and applications of backward stochastic difference/differential equations with respect to the single jump process
12:10 Mon 26 Mar, 2012 :: 5.57 Ingkarni Wardli :: Mr Bin Shen :: University of Adelaide

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This is my PhD thesis submitted one month ago. Chapter 1 introduces the backgrounds of the research fields. Then each chapter is a published or an accepted paper. Chapter 2, to appear in Methodology and Computing in Applied Probability, establishes the theory of Backward Stochastic Difference Equations with respect to the single jump process in discrete time. Chapter 3, published in Stochastic Analysis and Applications, establishes the theory of Backward Stochastic Differential Equations with respect to the single jump process in continuous time. Chapter 2 and 3 consist of Part I Theory. Chapter 4, published in Expert Systems With Applications, gives some examples about how to measure financial risks by the theory established in Chapter 2. Chapter 5, accepted by Journal of Applied Probability, considers the question of an optimal transaction between two investors to minimize their risks. It's the applications of the theory established in Chapter 3. Chapter 4 and 5 consist of Part II Applications.
Correcting Errors in RSA Private Keys
12:10 Mon 23 Apr, 2012 :: 5.57 Ingkarni Wardli :: Mr Wilko Henecka :: University of Adelaide

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Let pk=(N,e) be an RSA public key with corresponding secret key sk=(d,p,q,...). Assume that we obtain partial error-free information of sk, e.g., assume that we obtain half of the most significant bits of p. Then there are well-known algorithms to recover the full secret key. As opposed to these algorithms that allow for correcting erasures of the key sk, we present for the first time a heuristic probabilistic algorithm that is capable of correcting errors in sk provided that e is small. That is, on input of a full but error-prone secret key sk' we reconstruct the original sk by correcting the faults. More precisely, consider an error rate of d in [0,1), where we flip each bit in sk with probability d resulting in an erroneous key sk'. Our Las-Vegas type algorithm allows to recover sk from sk' in expected time polynomial in logN with success probability close to 1, provided that d is strictly less than 0.237. We also obtain a polynomial time Las-Vegas factorization algorithm for recovering the factorization (p,q) from an erroneous version with error rate d strictly less than 0.084.
Change detection in rainfall times series for Perth, Western Australia
12:10 Mon 14 May, 2012 :: 5.57 Ingkarni Wardli :: Ms Farah Mohd Isa :: University of Adelaide

Media...
There have been numerous reports that the rainfall in south Western Australia, particularly around Perth has observed a step change decrease, which is typically attributed to climate change. Four statistical tests are used to assess the empirical evidence for this claim on time series from five meteorological stations, all of which exceed 50 years. The tests used in this study are: the CUSUM; Bayesian Change Point analysis; consecutive t-test and the Hotelling's T^2-statistic. Results from multivariate Hotelling's T^2 analysis are compared with those from the three univariate analyses. The issue of multiple comparisons is discussed. A summary of the empirical evidence for the claimed step change in Perth area is given.
The change of probability measure for jump processes
12:10 Mon 28 May, 2012 :: 5.57 Ingkarni Wardli :: Mr Ahmed Hamada :: University of Adelaide

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In financial derivatives pricing theory, it is very common to change the probability measure from historical measure "real world" into a Risk-Neutral measure as a development of the non arbitrage condition. Girsanov theorem is the most known example of this technique and is used when prices randomness is modelled by Brownian motions. Other genuine candidates for modelling market randomness that have proved efficiency in recent literature are jump process, so how can a change of measure be performed for such processes? This talk will address this question by introducing the non arbitrage condition, discussing Girsanov theorem for diffusion and jump processes and presenting a concrete example.
A brief introduction to Support Vector Machines
12:30 Mon 4 Jun, 2012 :: 5.57 Ingkarni Wardli :: Mr Tyman Stanford :: University of Adelaide

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Support Vector Machines (SVMs) are used in a variety of contexts for a range of purposes including regression, feature selection and classification. To convey the basic principles of SVMs, this presentation will focus on the application of SVMs to classification. Classification (or discrimination), in a statistical sense, is supervised model creation for the purpose of assigning future observations to a group or class. An example might be determining healthy or diseased labels to patients from p characteristics obtained from a blood sample. While SVMs are widely used, they are most successful when the data have one or more of the following properties: The data are not consistent with a standard probability distribution. The number of observations, n, used to create the model is less than the number of predictive features, p. (The so-called small-n, big-p problem.) The decision boundary between the classes is likely to be non-linear in the feature space. I will present a short overview of how SVMs are constructed, keeping in mind their purpose. As this presentation is part of a double post-grad seminar, I will keep it to a maximum of 15 minutes.
The fundamental theorems of invariant theory, classical and quantum
15:10 Fri 10 Aug, 2012 :: B.21 Ingkarni Wardli :: Prof Gus Lehrer :: The University of Sydney

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Let V = C^n, and let (-,-) be a non-degenerate bilinear form on V , which is either symmetric or anti-symmetric. Write G for the isometry group of (V , (-,-)); thus G = O_n (C) or Sp_n (C). The first fundamental theorem (FFT) provides a set of generators for End_G(V^{\otimes r} ) (r = 1, 2, . . . ), while the second fundamental theorem (SFT) gives all relations among the generators. In 1937, Brauer formulated the FFT in terms of his celebrated 'Brauer algebra' B_r (\pm n), but there has hitherto been no similar version of the SFT. One problem has been the generic non-semisimplicity of B_r (\pm n), which caused H Weyl to call it, in his work on invariants 'that enigmatic algebra'. I shall present a solution to this problem, which shows that there is a single idempotent in B_r (\pm n), which describes all the relations. The proof is through a new 'Brauer category', in which the fundamental theorems are easily formulated, and where a calculus of tangles may be used to prove these results. There are quantum analogues of the fundamental theorems which I shall also discuss. There are numerous applications in representation theory, geometry and topology. This is joint work with Ruibin Zhang.
Continuous random walk models for solute transport in porous media
15:10 Fri 17 Aug, 2012 :: B.21 Ingkarni Wardli :: Prof Pavel Bedrikovetski :: The University of Adelaide

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The classical diffusion (thermal conductivity) equation was derived from the Master random walk equation and is parabolic. The main assumption was a probabilistic distribution of the jump length while the jump time is constant. Distribution of the jump time along with the jump length adds the second time derivative into the averaged equations, but the equation becomes ... elliptic! Where from to take an extra initial condition? We discuss how to pose the well-posed flow problem, exact 1d solution and numerous engineering applications. This is joint work with A. Shapiro and H. Yuan.
Probability, what can it tell us about health?
13:10 Tue 9 Oct, 2012 :: 7.15 Ingkarni Wardli :: Prof Nigel Bean :: School of Mathematical Sciences

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Clinical trials are the way in which modern medical systems test whether individual treatments are worthwhile. Sophisticated statistics is used to try and make the conclusions from clinical trials as meaningful as possible. What can a very simple probability model then tell us about the worth of multiple treatments? What might the implications of this be for the whole health system?

This talk is based on research currently being conducted with a physician at a major Adelaide hospital. It requires no health knowledge and was not tested on animals. All you need is an enquiring and open mind.
Numerical Free Probability: Computing Eigenvalue Distributions of Algebraic Manipulations of Random Matrices
15:10 Fri 2 Nov, 2012 :: B.20 Ingkarni Wardli :: Dr Sheehan Olver :: The University of Sydney

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Suppose that the global eigenvalue distributions of two large random matrices A and B are known. It is a remarkable fact that, generically, the eigenvalue distribution of A + B and (if A and B are positive definite) A*B are uniquely determined from only the eigenvalue distributions of A and B; i.e., no information about eigenvectors are required. These operations on eigenvalue distributions are described by free probability theory. We construct a numerical toolbox that can efficiently and reliably calculate these operations with spectral accuracy, by exploiting the complex analytical framework that underlies free probability theory.
Asymptotic independence of (simple) two-dimensional Markov processes
15:10 Fri 1 Mar, 2013 :: B.18 Ingkarni Wardli :: Prof Guy Latouche :: Universite Libre de Bruxelles

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The one-dimensional birth-and death model is one of the basic processes in applied probability but difficulties appear as one moves to higher dimensions. In the positive recurrent case, the situation is singularly simplified if the stationary distribution has product-form. We investigate the conditions under which this property holds, and we show how to use the knowledge to find product-form approximations for otherwise unmanageable random walks. This is joint work with Masakiyo Miyazawa and Peter Taylor.
A stability theorem for elliptic Harnack inequalities
15:10 Fri 5 Apr, 2013 :: B.18 Ingkarni Wardli :: Prof Richard Bass :: University of Connecticut

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Harnack inequalities are an important tool in probability theory, analysis, and partial differential equations. The classical Harnack inequality is just the one you learned in your graduate complex analysis class, but there have been many extensions, to different spaces, such as manifolds, fractals, infinite graphs, and to various sorts of elliptic operators. A landmark result was that of Moser in 1961, where he proved the Harnack inequality for solutions to a class of partial differential equations. I will talk about the stability of Harnack inequalities. The main result says that if the Harnack inequality holds for an operator on a space, then the Harnack inequality will also hold for a large class of other operators on that same space. This provides a generalization of the result of Moser.
Markov decision processes and interval Markov chains: what is the connection?
12:10 Mon 3 Jun, 2013 :: B.19 Ingkarni Wardli :: Mingmei Teo :: University of Adelaide

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Markov decision processes are a way to model processes which involve some sort of decision making and interval Markov chains are a way to incorporate uncertainty in the transition probability matrix. How are these two concepts related? In this talk, I will give an overview of these concepts and discuss how they relate to each other.
The Hamiltonian Cycle Problem and Markov Decision Processes
15:10 Fri 2 Aug, 2013 :: B.18 Ingkarni Wardli :: Prof Jerzy Filar :: Flinders University

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We consider the famous Hamiltonian cycle problem (HCP) embedded in a Markov decision process (MDP). More specifically, we consider a moving object on a graph G where, at each vertex, a controller may select an arc emanating from that vertex according to a probabilistic decision rule. A stationary policy is simply a control where these decision rules are time invariant. Such a policy induces a Markov chain on the vertices of the graph. Therefore, HCP is equivalent to a search for a stationary policy that induces a 0-1 probability transition matrix whose non-zero entries trace out a Hamiltonian cycle in the graph. A consequence of this embedding is that we may consider the problem over a number of, alternative, convex - rather than discrete - domains. These include: (a) the space of stationary policies, (b) the more restricted but, very natural, space of doubly stochastic matrices induced by the graph, and (c) the associated spaces of so-called "occupational measures". This approach to the HCP has led to both theoretical and algorithmic approaches to the underlying HCP problem. In this presentation, we outline a selection of results generated by this line of research.
Shannon entropy as a diagnostic tool for PDEs in conservation form
15:10 Fri 16 Aug, 2013 :: B.18 Ingkarni Wardli :: Prof Philip Broadbridge :: La Trobe University

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After normalization, an evolving real non-negative function may be viewed as a probability density. From this we may derive the corresponding evolution law for Shannon entropy. Parabolic equations, hyperbolic equations and fourth-order diffusion equations evolve information in quite different ways. Entropy and irreversibility can be introduced in a self-consistent manner and at an elementary level by reference to some simple evolution equations such as the linear heat equation. It is easily seen that the 2nd law of thermodynamics is equivalent to loss of Shannon information when temperature obeys a general nonlinear 2nd order diffusion equation. With fourth order diffusion terms, new problems arise. We know from applications such as thin film flow and surface diffusion, that fourth order diffusion terms may generate ripples and they do not satisfy the Second Law. Despite this, we can identify the class of fourth order quasilinear diffusion equations that increase the Shannon entropy.
Is it possible to beat the lottery system?
12:10 Mon 24 Mar, 2014 :: B.19 Ingkarni Wardli :: Michael Lydeamore :: University of Adelaide

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Every week millions of people around the country buy tickets for a round of the lottery. Known as the "lotto", the chances of winning the big prize are less than 1 in 8 million, yet every week people will purchase a ticket. What if there was a smart way of betting which would increase your odds? A few weeks ago an article came across my desk with those very words: "Using this scheme you will win more". In this talk, we'll test those claims. Looking first at a basic counting argument, and then later moving the hard work over to a computer we'll find out if this betting scheme (and many others similar to it) will actually win you more or if just like playing in a casino, you'll still go bankrupt with probability 1.
A Hybrid Markov Model for Disease Dynamics
12:35 Mon 29 Sep, 2014 :: B.19 Ingkarni Wardli :: Nicolas Rebuli :: University of Adelaide

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Modelling the spread of infectious diseases is fundamental to protecting ourselves from potentially devastating epidemics. Among other factors, two key indicators for the severity of an epidemic are the size of the epidemic and the time until the last infectious individual is removed. To estimate the distribution of the size and duration of an epidemic (within a realistic population) an epidemiologist will typically use Monte Carlo simulations of an appropriate Markov process. However, the number of states in the simplest Markov epidemic model, the SIR model, is quadratic in the population size and so Monte Carlo simulations are computationally expensive. In this talk I will discuss two methods for approximating the SIR Markov process and I will demonstrate the approximation error by comparing probability distributions and estimates of the distributions of the final size and duration of an SIR epidemic.
Multiscale modelling of multicellular biological systems: mechanics, development and disease
03:10 Fri 6 Mar, 2015 :: Lower Napier LG24 :: Dr James Osborne :: University of Melbourne

When investigating the development and function of multicellular biological systems it is not enough to only consider the behaviour of individual cells in isolation. For example when studying tissue development, how individual cells interact, both mechanically and biochemically, influences the resulting tissues form and function. In this talk we present a multiscale modelling framework for simulating the development and function of multicellular biological systems (in particular tissues). Utilising the natural structural unit of the cell, the framework consists of three main scales: the tissue level (macro-scale); the cell level (meso-scale); and the sub-cellular level (micro-scale), with multiple interactions occurring between all scales. The cell level is central to the framework and cells are modelled as discrete interacting entities using one of a number of possible modelling paradigms, including lattice based models (cellular automata and cellular Potts) and off-lattice based models (cell centre and vertex based representations). The sub-cellular level concerns numerous metabolic and biochemical processes represented by interaction networks rendered stochastically or into ODEs. The outputs from such systems influence the behaviour of the cell level affecting properties such as adhesion and also influencing cell mitosis and apoptosis. At the tissue level we consider factors or restraints that influence the cells, for example the distribution of a nutrient or messenger molecule, which is represented by field equations, on a growing domain, with individual cells functioning as sinks and/or sources. The modular approach taken within the framework enables more realistic behaviour to be considered at each scale. This framework is implemented within the Open Source Chaste library (Cancer Heart and Soft Tissue Environment, (http://www.cs.ox.ac.uk/chaste/) and has been used to model biochemical and biomechanical interactions in various biological systems. In this talk we present the key ideas of the framework along with applications within the fields of development and disease.
Cricket and Maths
12:10 Mon 16 Mar, 2015 :: Napier LG29 :: Peter Ballard :: University of Adelaide

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Each game of international cricket has a scorecard. You don't need to know much maths to go through these scorecards and extract simple information, such as batting and bowling averages. However there is also the opportunity to use some more advanced maths. I will be using a bit of optimisation, probability and statistics to try to answer the questions: Which was the most dominant team ever? What scores are most likely? And are some players unlucky?
Groups acting on trees
12:10 Fri 10 Apr, 2015 :: Napier 144 :: Anitha Thillaisundaram :: Heinrich Heine University of Duesseldorf

From a geometric point of view, branch groups are groups acting spherically transitively on a spherically homogeneous rooted tree. The applications of branch groups reach out to analysis, geometry, combinatorics, and probability. The early construction of branch groups were the Grigorchuk group and the Gupta-Sidki p-groups. Among its many claims to fame, the Grigorchuk group was the first example of a group of intermediate growth (i.e. neither polynomial nor exponential). Here we consider a generalisation of the family of Grigorchuk-Gupta-Sidki groups, and we examine the restricted occurrence of their maximal subgroups.
The Mathematics of Crime
15:10 Fri 23 Oct, 2015 :: Ingkarni Wardli B21 :: Prof Andrea Bertozzi :: UCLA

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Law enforcement agencies across the US have discovered that partnering with a team of mathematicians and social scientists from UCLA can help them determine where crime is likely to occur. Dr. Bertozzi will talk about the fascinating story behind her participation on the UCLA team that developed a “predictive policing” computer program that zeros-in on areas that have the highest probability of crime. In addition, the use of mathematics in studying gang crimes and other criminal activities will also be discussed. Commercial use of the "predictive-policing" program allows communities to put police officers in the right place at the right time, stopping crime before it happens.
A Semi-Markovian Modeling of Limit Order Markets
13:00 Fri 11 Dec, 2015 :: Ingkarni Wardli 5.57 :: Anatoliy Swishchuk :: University of Calgary

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R. Cont and A. de Larrard (SIAM J. Financial Mathematics, 2013) introduced a tractable stochastic model for the dynamics of a limit order book, computing various quantities of interest such as the probability of a price increase or the diffusion limit of the price process. As suggested by empirical observations, we extend their framework to 1) arbitrary distributions for book events inter-arrival times (possibly non-exponential) and 2) both the nature of a new book event and its corresponding inter-arrival time depend on the nature of the previous book event. We do so by resorting to Markov renewal processes to model the dynamics of the bid and ask queues. We keep analytical tractability via explicit expressions for the Laplace transforms of various quantities of interest. Our approach is justified and illustrated by calibrating the model to the five stocks Amazon, Apple, Google, Intel and Microsoft on June 21st 2012. As in Cont and Larrard, the bid-ask spread remains constant equal to one tick, only the bid and ask queues are modelled (they are independent from each other and get reinitialized after a price change), and all orders have the same size. (This talk is based on our joint paper with Nelson Vadori (Morgan Stanley)).
On the Willmore energy
15:10 Fri 7 Oct, 2016 :: Napier G03 :: Dr Yann Bernard :: Monash University

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The Willmore energy of a surface captures its bending. Originally discovered 200 years ago by Sophie Germain in the context of elasticity theory, it has since then been rediscovered numerous times in several areas of science: general relativity, optics, string theory, conformal geometry, and cell biology. For example, our red blood cells assume a peculiar shape that minimises the Willmore energy. In this talk, I will present the thrilling history of the Willmore energy, its applications, and its main properties. The presentation will be accessible to all mathematicians as well as to advanced undergraduate students.
In space there is no-one to hear you scream
12:10 Tue 12 Sep, 2017 :: Inkgarni Wardli 5.57 :: A/Prof Gary Glonek :: School of Mathematical Sciences

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Modern data problems often involve data in very high dimensions. For example, gene expression profiles, used to develop cancer screening models, typically have at least 30,000 dimensions. When dealing with such data, it is natural to apply intuition from low dimensional cases. For example, in a sample of normal observations, a typical data point will be near the centre of the distribution with only a small number of points at the edges. In this talk, simple probability theory will be used to show that the geometry of data in high dimensional space is very different from what we can see in one and two-dimensional examples. We will show that the typical data point is at the edge of the distribution, a long way from its centre and even further from any other points.
The Markovian binary tree applied to demography and conservation biology
15:10 Fri 27 Oct, 2017 :: Ingkarni Wardli B17 :: Dr Sophie Hautphenne :: University of Melbourne

Markovian binary trees form a general and tractable class of continuous-time branching processes, which makes them well-suited for real-world applications. Thanks to their appealing probabilistic and computational features, these processes have proven to be an excellent modelling tool for applications in population biology. Typical performance measures of these models include the extinction probability of a population, the distribution of the population size at a given time, the total progeny size until extinction, and the asymptotic population composition. Besides giving an overview of the main performance measures and the techniques involved to compute them, we discuss recently developed statistical methods to estimate the model parameters, depending on the accuracy of the available data. We illustrate our results in human demography and in conservation biology.
Stochastic Modelling of Urban Structure
11:10 Mon 20 Nov, 2017 :: Engineering Nth N132 :: Mark Girolami :: Imperial College London, and The Alan Turing Institute

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Urban systems are complex in nature and comprise of a large number of individuals that act according to utility, a measure of net benefit pertaining to preferences. The actions of individuals give rise to an emergent behaviour, creating the so-called urban structure that we observe. In this talk, I develop a stochastic model of urban structure to formally account for uncertainty arising from the complex behaviour. We further use this stochastic model to infer the components of a utility function from observed urban structure. This is a more powerful modelling framework in comparison to the ubiquitous discrete choice models that are of limited use for complex systems, in which the overall preferences of individuals are difficult to ascertain. We model urban structure as a realization of a Boltzmann distribution that is the invariant distribution of a related stochastic differential equation (SDE) that describes the dynamics of the urban system. Our specification of Boltzmann distribution assigns higher probability to stable configurations, in the sense that consumer surplus (demand) is balanced with running costs (supply), as characterized by a potential function. We specify a Bayesian hierarchical model to infer the components of a utility function from observed structure. Our model is doubly-intractable and poses significant computational challenges that we overcome using recent advances in Markov chain Monte Carlo (MCMC) methods. We demonstrate our methodology with case studies on the London retail system and airports in England.
Calculating optimal limits for transacting credit card customers
15:10 Fri 2 Mar, 2018 :: Horace Lamb 1022 :: Prof Peter Taylor :: University of Melbourne

Credit card users can roughly be divided into `transactors', who pay off their balance each month, and `revolvers', who maintain an outstanding balance, on which they pay substantial interest. In this talk, we focus on modelling the behaviour of an individual transactor customer. Our motivation is to calculate an optimal credit limit from the bank's point of view. This requires an expression for the expected outstanding balance at the end of a payment period. We establish a connection with the classical newsvendor model. Furthermore, we derive the Laplace transform of the outstanding balance, assuming that purchases are made according to a marked point process and that there is a simplified balance control policy which prevents all purchases in the rest of the payment period when the credit limit is exceeded. We then use the newsvendor model and our modified model to calculate bounds on the optimal credit limit for the more realistic balance control policy that accepts all purchases that do not exceed the limit. We illustrate our analysis using a compound Poisson process example and show that the optimal limit scales with the distribution of the purchasing process, while the probability of exceeding the optimal limit remains constant. Finally, we apply our model to some real credit card purchase data.

News matching "At least four doors, numerous goats, a car, a frog"

A/Prof Joshua Ross, 2017 Moran Medal recipient
Congratulations to Associate Professor Joshua Ross who has won the 2017 Moran Medal, awarded by the Australian Academy of Science. The Moran Medal recognises outstanding research by scientists up to 10 years post-PhD in applied probability, biometrics, mathematical genetics, psychometrics and statistics. Associate Professor Ross has made influential contributions to public health and conservation biology using mathematical modelling and statistics to help in decision making. Posted Fri 23 Dec 16.

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Publications matching "At least four doors, numerous goats, a car, a frog"

Publications
Hitting probabilities and hitting times for stochastic fluid flows the bounded model
Bean, Nigel; O'Reilly, Malgorzata; Taylor, P, Probability in the Engineering and Informational Sciences 23 (121–147) 2009
A total probability approach to flood frequency analysis in tidal river reaches
Need, Steven; Lambert, Martin; Metcalfe, Andrew, World Environmental and Water Resources Congress 2008 Ahupua'a, Honolulu 12/05/08
Algorithms for the Laplace-Stieltjes transforms of first return times for stochastic fluid flows
Bean, Nigel; O'Reilly, Malgorzata; Taylor, Peter, Methodology and Computing in Applied Probability 10 (381–408) 2008
Robust Optimal Portfolio Choice Under Markovian Regime-switching Model
Elliott, Robert; Siu, T, Methodology and Computing in Applied Probability 11 (145–157) 2008
Dynamic portfolio allocation, the dual theory of choice and probability distortion functions
Hamada, M; Sherris, M; Van Der Hoek, John, Astin Bulletin 31 (187–217) 2006
Methods of constrained and unconstrained approximation for mappings in probability spaces
Torokhti, Anatoli; Howlett, P; Pearce, Charles, chapter in Modern Applied Mathematics (Narosa Publishing House) 83–129, 2005
Oriented site percolation, phase transitions and probability bounds
Pearce, Charles; Fletcher, F, Journal of Inequalities in Pure and Applied Mathematics 6 (WWW 1–WWW 15) 2005
Identification of probability distributions within hidden state models of rainfall
Whiting, Julian; Lambert, Martin; Metcalfe, Andrew, 28th International Hydrology and Water Resources Symposium, Wollongong, NWS, Australia 10/11/03
The tree cut and merge algorithm for estimation of network reliability
Hui, Kin-Ping; Bean, Nigel; Kraetzl, Miro; Kroese, D, Probability in the Engineering and Informational Sciences 17 (23–45) 2003
What is a unit of capacity worth?
Chiera, Belinda; Taylor, Peter, Probability in the Engineering and Informational Sciences 16 (513–522) 2002
On the existence of a quasistationary measure for a Markov chain
Lasserre, J; Pearce, Charles, Annals of Probability 29 (437–446) 2001
Level-phase independence for GI/M/1-type markov chains
Latouche, Guy; Taylor, Peter, Journal of Applied Probability 37 (984–998) 2000

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