27 research outputs found
The valuation of power futures based on optimal dispatch
The pricing of contingent claims in the wholesale power market is a controversial topic. Important challenges come from the non-storability of electricity and the number of parameters that impact the market. We propose an equilibrium model based on the fundamentals of power generation. In a perfect competitive market, spot electricity prices are determined by the marginal cost of producing the last unit of power. Electricity can be viewed as a derivative of demand, fuels prices and carbon emission price. We extend the Pirrong-Jermakayan model such as to incorporate the main factors driving the marginal cost and the non-linearities of electricity prices with respect to fuels prices. As in the Pirrong-Jermakayan framework, any contingent claims on power must satisfy a high dimensional PDE that embeds a market price of risk, as load is not a traded asset. Analyzing the specificity of the marginal cost in power market, we simplify the problem for evaluating power futures so that it becomes computationally tractable. We test our model on the German EEX for "German Month Futures" with maturity of June and September 2008.power contingent claims, PDE valuation of financial derivatives, unit commitment, market price of risk, EEX
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Poincaré and the Three Body Problem
The purpose of the thesis is to present an account of Henri Poincare's famous memoir on the three body problem, the final version of which was published in Acta Mathematica in 1890 as the prize-winning entry in King Oscar II's 60th birthday competition. The memoir is reknowned both for its role in providing the foundations for Poincare's celebrated three volume Méthodes Nouvelles de la Mécanique Céleste, and for containing the first mathematical description of chaotic behaviour in a dynamical system.
A historical context is provided both through consideration of the problem itself and through a discussion of Poincaré's earlier work which relates to the mathematics developed in the memoir. The organisation of the Oscar competition, which was undertaken by Gösta Mittag-Leffler, is also described. This not only provides an insight into the late 19th century European mathematical community but also reveals that after the prize had been awarded Poincare found an important error in his work and substantially revised the memoir prior to its publication in Acta. The discovery of a printed version of the original memoir personally annotated by Poincaré has allowed for a detailed comparative study of the mathematics contained in both versions of the memoir. The error is explained and it is shown that it was only as a result of its correction that Poincaré discovered the chaotic behaviour now associated with the memoir.
The contemporary reception of the memoir is discussed and Poincaré's subsequent work in celestial mechanics and related topics is examined. Through the consideration of sources up to 1920 the influence and impact of the memoir on the progress of the three body problem and on dynamics in general is assessed
Improved estimation of Fourier coefficients for ill-posed inverse problems
In this dissertation we present and solve an ill-posed inverse problem which involves reproducing a function f(x) or its Fourier coefficients from the observed values of the function. The observations of the f(x) are made at n equidistant points on the unit interval with p observations being made at each point. The observations are effected by a random error with a known distribution.
First of all we present a very simple estimator for the Fourier coefficients of f(x). Then we present an iteration algorithm for improving the estimator for the Fourier coefficients. We show that the improved estimator we use is a simplified and improved version of the Maximum Likelihood Estimator.
Second, we introduce the mean squared error (MSE) for the estimators, which is the main measure of estimator performance. We show that a singly iterated estimator has a smaller MSE then a non-iterated estimator and a multiply iterated estimator has a smaller MSE then a singly iterated estimator. We also prove that the errors in estimating the Fourier Coefficients by the singly and multiply improved methods are normally distributed.
Third, we prove a theorem showing that as the sample size goes to infinity, the MSE of our estimator asymptotically approaches the theoretical minimum. That shows that our results are theoretically the best possible results.
Fourth, we perform simulations which numerically approximate MSE for a given set of f, error distributions, as well as the number of observation points. We approximate the MSE for the non-iterated error coefficient approximation as well as the singly iterated and multiply iterated ones. We show that indeed the MSE decreases with each iteration. We also plot an error histogram in each case showing that the errors are normally distributed.
Finally, we look at some ways in which our problem can be expanded. Possible expansions include working on the problem in multiple dimensions, taking measurements of f at random points, or both of the above
Proceedings of the 1968 Summer Institute on Symbolic Mathematical Computation
Investigating symbolic mathematical computation using PL/1 FORMAC batch system and Scope FORMAC interactive syste
Learning Non-Parametric and High-Dimensional Distributions via Information-Theoretic Methods
Learning distributions that govern generation of data and estimation of related functionals are the foundations of many classical statistical problems. In the following dissertation we intend to investigate such topics when either the hypothesized model is non-parametric or the number of free parameters in the model grows along with the sample size. Especially, we study the above scenarios for the following class of problems with the goal of obtaining minimax rate-optimal methods for learning the target distributions when the sample size is finite. Our techniques are based on information-theoretic divergences and related mutual-information based methods. (i) Estimation in compound decision and empirical Bayes settings: To estimate the data-generating distribution, one often takes the following two-step approach. In the first step the statistician estimates the distribution of the parameters, either the empirical distribution or the postulated prior, and then in the second step plugs in the estimate to approximate the target of interest. In the literature, the estimation of empirical distribution is known as the compound decision problem and the estimation of prior is known as the problem of empirical Bayes. In our work we use the method of minimum-distance estimation for approximating these distributions. Considering certain discrete data setups, we show that the minimum-distance based method provides theoretically and practically sound choices for estimation. The computational and algorithmic aspects of the estimators are also analyzed. (ii) Prediction with Markov chains: Given observations from an unknown Markov chain, we study the problem of predicting the next entry in the trajectory. Existing analysis for such a dependent setup usually centers around concentration inequalities that uses various extraneous conditions on the mixing properties. This makes it difficult to achieve results independent of such restrictions. We introduce information-theoretic techniques to bypass such issues and obtain fundamental limits for the related minimax problems. We also analyze conditions on the mixing properties that produce a parametric rate of prediction errors
Robust network computation
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 91-98).In this thesis, we present various models of distributed computation and algorithms for these models. The underlying theme is to come up with fast algorithms that can tolerate faults in the underlying network. We begin with the classical message-passing model of computation, surveying many known results. We give a new, universally optimal, edge-biconnectivity algorithm for the classical model. We also give a near-optimal sub-linear algorithm for identifying bridges, when all nodes are activated simultaneously. After discussing some ways in which the classical model is unrealistic, we survey known techniques for adapting the classical model to the real world. We describe a new balancing model of computation. The intent is that algorithms in this model should be automatically fault-tolerant. Existing algorithms that can be expressed in this model are discussed, including ones for clustering, maximum flow, and synchronization. We discuss the use of agents in our model, and give new agent-based algorithms for census and biconnectivity. Inspired by the balancing model, we look at two problems in more depth.(cont.) First, we give matching upper and lower bounds on the time complexity of the census algorithm, and we show how the census algorithm can be used to name nodes uniquely in a faulty network. Second, we consider using discrete harmonic functions as a computational tool. These functions are a natural exemplar of the balancing model. We prove new results concerning the stability and convergence of discrete harmonic functions, and describe a method which we call Eulerization for speeding up convergence.by David Pritchard.M.Eng
Maximum likelihood sequence estimation from the lattice viewpoint.
by Mow Wai Ho.Thesis (M.Phil.)--Chinese University of Hong Kong, 1991.Bibliographies: leaves 98-104.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Channel Model and Other Basic Assumptions --- p.5Chapter 1.2 --- Complexity Measure --- p.8Chapter 1.3 --- Maximum Likelihood Sequence Estimator --- p.9Chapter 1.4 --- The Viterbi Algorithm ´ؤ An Implementation of MLSE --- p.11Chapter 1.5 --- Error Performance of the Viterbi Algorithm --- p.14Chapter 1.6 --- Suboptimal Viterbi-like Algorithms --- p.17Chapter 1.7 --- Trends of Digital Transmission and MLSE --- p.19Chapter 2 --- New Formulation of MLSE --- p.21Chapter 2.1 --- The Truncated Viterbi Algorithm --- p.21Chapter 2.2 --- Choice of Truncation Depth --- p.23Chapter 2.3 --- Decomposition of MLSE --- p.26Chapter 2.4 --- Lattice Interpretation of MLSE --- p.29Chapter 3 --- The Closest Vector Problem --- p.34Chapter 3.1 --- Basic Definitions and Facts About Lattices --- p.37Chapter 3.2 --- Lattice Basis Reduction --- p.40Chapter 3.2.1 --- Weakly Reduced Bases --- p.41Chapter 3.2.2 --- Derivation of the LLL-reduction Algorithm --- p.43Chapter 3.2.3 --- Improved Algorithm for LLL-reduced Bases --- p.52Chapter 3.3 --- Enumeration Algorithm --- p.57Chapter 3.3.1 --- Lattice and Isometric Mapping --- p.58Chapter 3.3.2 --- Enumerating Points in a Parallelepiped --- p.59Chapter 3.3.3 --- Enumerating Points in a Cube --- p.63Chapter 3.3.4 --- Enumerating Points in a Sphere --- p.64Chapter 3.3.5 --- Comparisons of Three Enumeration Algorithms --- p.66Chapter 3.3.6 --- Improved Enumeration Algorithm for the CVP and the SVP --- p.67Chapter 3.4 --- CVP Algorithm Using the Reduce-and-Enumerate Approach --- p.71Chapter 3.5 --- CVP Algorithm with Improved Average-Case Complexity --- p.72Chapter 3.5.1 --- CVP Algorithm for Norms Induced by Orthogonalization --- p.73Chapter 3.5.2 --- Improved CVP Algorithm using Norm Approximation --- p.76Chapter 4 --- MLSE Algorithm --- p.79Chapter 4.1 --- MLSE Algorithm for PAM Systems --- p.79Chapter 4.2 --- MLSE Algorithm for Unimodular Channel --- p.82Chapter 4.3 --- Reducing the Boundary Effect for PAM Systems --- p.83Chapter 4.4 --- Simulation Results and Performance Investigation for Example Channels --- p.86Chapter 4.5 --- MLSE Algorithm for Other Lattice-Type Modulation Systems --- p.91Chapter 4.6 --- Some Potential Applications --- p.92Chapter 4.7 --- Further Research Directions --- p.94Chapter 5 --- Conclusion --- p.96Bibliography --- p.10
Finite worldlength effects in fixed-point implementations of linear systems
Thesis (M.Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 173-194).by Vinay Mohta.M.Eng