Sequence Of Probability Distributions Research Articles

Solution of a model for pricing options with hedging strategy through Nonlinear Filters

A methodology is presented to estimate the solution states for a non-linear price problem, a model for pricing options with a hedging strategy in the F$\ddot{o}$llmer-Schweizer sense is defined. The problem is to determine the price of a contingent claim, that is a contract, that pays of an amount at time $t$ in a incomplete market, that is not possible to replicate a payoff by a controlled portfolio of the basic securities. Two algorithms are presented to estimate the solution of the presented problem, the nested sequential Monte Carlo (NSMC) and space-time particle filter (STPF) are defined from sequences of probability distributions. The methodology is validated to use real data from option Asian, the states in real-time are estimated, that is proposed on the basis of the a price model. The efficiency of the forecasts of the model is compared, reproducing accuracy in the estimates. Finally, one goodness-of-fit measure to validate the performance of the model are used, obtaining insignificant estimation error.

Property Suffix Array with Applications in Indexing Weighted Sequences

The suffix array is one of the most prevalent data structures for string indexing; it stores the lexicographically sorted list of suffixes of a given string. Its practical advantage compared to the suffix tree is space efficiency. In Property Indexing , we are given a string x of length n and a property Π, i.e., a set of Π- valid intervals over x so that a pattern p occurs in x if and only if x has an occurrence of p that lies entirely within an interval of Π. A suffix-tree-like index over the valid prefixes of suffixes of x can be built in time and space O ( n ). We show here how to directly build a suffix-array-like index, the Property Suffix Array (PSA), in time and space O ( n ). We mainly draw our motivation from weighted (probabilistic) sequences: sequences of probability distributions over a given alphabet. Given a probability threshold 1/ z , we say that a string p of length m matches a weighted sequence x of length n at starting position i if the product of probabilities of the letters of p at positions i , …, i + m − 1 in x is at least 1/ z . Our algorithm for building the PSA can be directly applied to build an O ( nz )-sized suffix-array-like index over x in time and space O ( nz ). Finally, we present extensive experimental results showing that our new indexing data structure is well suited for real-world applications.

Probability distribution as a path and its action integral

To describe the convergence in law of a sequence of probability distributions, “the principle of least action” is introduced nonparametrically into statistics. A probability measure should be treated as a path (in some sense) to apply calculus of variations, and it is shown that saddlepoints, which appear in the method of saddlepoint approximations, play a crucial role. An action integral, i.e., a functional of the saddlepoint, is defined as a definite integral of entropy. As a saddlepoint equation naturally appears in the Gâteaux derivative of that integral, a unique saddlepoint may be found as an optimal path for this variations problem. Consequently, by virtue of the unique correspondence between probability measures and saddlepoints, the convergence in law is clearly described by a decreasing sequence of action integrals. Thereby, a new criterion for evaluating the convergence is introduced into statistics and a novel interpretation of saddlepoints is provided.

On Adaptive Heuristics that Converge to Correlated Equilibrium

I study the path properties of adaptive heuristics that mimic the natural dynamics of play in a game and converge to the set of correlated equilibria. Despite their apparent differences, I show that these heuristics have an abstract representation as a sequence of probability distributions that satisfy a number of common properties. These properties arise due to the topological structure of the set of correlated equilibria. The characterizations that I obtain have useful applications in the study of the convergence of the heuristics.

On-line weighted pattern matching

A weighted sequence is a sequence of probability distributions over an alphabet of size σ. Weighted sequences arise naturally in many applications. We study the problem of weighted pattern matching in which we are given a string pattern P of length m, a weight threshold 1z, and a weighted text X arriving on-line. We say that P occurs in X at position i if the product of probabilities of the letters of P at positions i−m+1,…,i in X is at least 1z. We first discuss how to apply a known general scheme that transforms off-line pattern matching algorithms to on-line algorithms to obtain an on-line algorithm that requires O((σ+log⁡z)log⁡m) or O(σlog2⁡m) time per arriving position; with the space requirement however being O(mmin⁡(σ,z)). Our main result is a new algorithm that processes each arriving position of X in O(z+σ) time using O(m+z) extra space.

An efficient robust optimization method with random and interval uncertainties

This paper develops a new mixed uncertainty robust optimization (MURO) method with both random and interval uncertainties. Existing strategies in literature always treat the system performance as a sequence of probability distribution with the interval factors varying within their domains. Moreover, the robust design objective and constraints are modeled in the form of combination of interval mean and interval deviations of performances, which cannot offer a quantitative robustness measurement of a design. The new MURO method is based on the sensitivity region concept and a hybrid robustness index is developed to represent the possibility that the uncertain vector locates within the worst-case sensitivity region (WCSR). This proposed index can offer a more quantitative and intuitive way to evaluate the robustness of a design. With the hybrid indices, the traditional robust optimization problem can be converted to an ordinary optimization with the robustness index constraints. Two numerical examples and two engineering examples with different combinations of interval and random factors are illustrated to demonstrate the applicability and efficiency of the proposed algorithm. The comparison results show that the new method can reduce the conservatism of previous method significantly with fewer computational efforts.

A complete representation of uncertainties in layer-counted paleoclimatic archives

Abstract. Accurate time series representation of paleoclimatic proxy records is challenging because such records involve dating errors in addition to proxy measurement errors. Rigorous attention is rarely given to age uncertainties in paleoclimatic research, although the latter can severely bias the results of proxy record analysis. Here, we introduce a Bayesian approach to represent layer-counted proxy records – such as ice cores, sediments, corals, or tree rings – as sequences of probability distributions on absolute, error-free time axes. The method accounts for both proxy measurement errors and uncertainties arising from layer-counting-based dating of the records. An application to oxygen isotope ratios from the North Greenland Ice Core Project (NGRIP) record reveals that the counting errors, although seemingly small, lead to substantial uncertainties in the final representation of the oxygen isotope ratios. In particular, for the older parts of the NGRIP record, our results show that the total uncertainty originating from dating errors has been seriously underestimated. Our method is next applied to deriving the overall uncertainties of the Suigetsu radiocarbon comparison curve, which was recently obtained from varved sediment cores at Lake Suigetsu, Japan. This curve provides the only terrestrial radiocarbon comparison for the time interval 12.5–52.8 kyr BP. The uncertainties derived here can be readily employed to obtain complete error estimates for arbitrary radiometrically dated proxy records of this recent part of the last glacial interval.

Local limits of Markov branching trees and their volume growth

We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed. We prove that under some natural assumption on this sequence of probabilities, when their sizes go to infinity, the trees converge in distribution to an infinite tree which also satisfies the Markov branching property. Furthermore, when this infinite tree has a single path from the root to infinity, we give conditions to ensure its convergence in distribution under appropriate rescaling of its distance and counting measure to a self-similar fragmentation tree with immigration. In particular, this allows us to determine how, in this infinite tree, the volume of the ball of radius R centred at the root asymptotically grows with R. Our unified approach will allow us to develop various new applications, in particular to different models of growing trees and cut-trees, and to recover known results. An illustrative example lies in the study of Galton-Watson trees: the distribution of a critical Galton-Watson tree conditioned on its size converges to that of Kesten's tree when the size grows to infinity. If furthermore, the offspring distribution has finite variance, under adequate rescaling, Kesten's tree converges to Aldous' self-similar CRT and the total size of the R first generations asymptotically behaves like R².

A stochastic search algorithm for voltage and reactive power control with switching costs and ZIP load model

This paper considers voltage and reactive power control models that take into account the operational costs of switching devices and the correlation between loads and voltage profiles. In addition to maintaining acceptable voltage at all points along the distribution feeder, the aim is to determine the proper settings of shunt capacitors and transformer load tap changers in order to minimize the energy loss or the energy consumption over a selected planning horizon. A random search algorithm is proposed for optimizing these models. The algorithm searches the optimal control schedule by randomly sampling from a sequence of probability distributions over the space of all possible settings of tap changers and shunt capacitors. The algorithm is shown to be globally convergent and its effectiveness is illustrated by comparing its performance with that of three other commonly used procedures on a 69-bus distribution system and a 24-bus system with distributed generation.

Transdimensional sequential Monte Carlo using variational Bayes — SMCVB

A new transdimensional Sequential Monte Carlo (SMC) algorithm called SMCVB is proposed. In an SMC approach, a weighted sample of particles is generated from a sequence of probability distributions which ‘converge’ to the target distribution of interest, in this case a Bayesian posterior distribution. The approach is based on the use of variational Bayes to propose new particles at each iteration of the SMCVB algorithm in order to target the posterior more efficiently. The variational-Bayes-generated proposals are not limited to a fixed dimension. This means that the weighted particle sets that arise can have varying dimensions thereby allowing us the option to also estimate an appropriate dimension for the model. This novel algorithm is outlined within the context of finite mixture model estimation. This provides a less computationally demanding alternative to using reversible jump Markov chain Monte Carlo kernels within an SMC approach. We illustrate these ideas in a simulated data analysis and in applications.

RMCMC: A system for updating Bayesian models

A system to update estimates from a sequence of probability distributions is presented. The aim of the system is to quickly produce estimates with a user-specified bound on the Monte Carlo error. The estimates are based upon weighted samples stored in a database. The stored samples are maintained such that the accuracy of the estimates and quality of the samples are satisfactory. This maintenance involves varying the number of samples in the database and updating their weights. New samples are generated, when required, by a Markov chain Monte Carlo algorithm. The system is demonstrated using a football league model that is used to predict the end of season table. The correctness of the estimates and their accuracy are shown in a simulation using a linear Gaussian model.

Computability of Probability Distributions and Characteristic Functions

As a part of our works on effective properties of probability distributions, we deal with the corresponding characteristic functions. A sequence of probability distributions is computable if and only if the corresponding sequence of characteristic functions is computable. As for the onvergence problem, the effectivized Glivenko's theorem holds. Effectivizations of Bochner's theorem and de Moivre-Laplace central limit theorem are also proved.

Decentralized Learning in Finite Markov Chains: Revisited

The convergence proof in the paper ldquoDecentralized learning in finite Markov chains,rdquo published in the IEEE Transactions on Automatic Control, vol. AC-31, no. 6, pp. 519-526, 1986, is incomplete. This note first provides a sufficient condition for the existence of a unique optimal policy for infinite-horizon average-cost Markov decision processes (MDPs), making the convergence result established by Wheeler and Narendra preserved with the condition. We then present a novel simulation-based decentralized algorithm, called ldquosampled joint-strategy fictitious play for MDPrdquo for average MDPs based on the recent study by Garcia of a decentralized approach to discrete optimization via fictitious play applied to games with identical payoffs. We establish a stronger almost-sure convergence result than Wheeler and Narendra's, showing that the sequence of probability distributions over the policy space for a given MDP generated by the algorithm converges to a unique optimal policy with probability one.

On adaptive resampling strategies for sequential Monte Carlo methods

Sequential Monte Carlo (SMC) methods are a class of techniques to sample approximately from any sequence of probability distributions using a combination of importance sampling and resampling steps. This paper is concerned with the convergence analysis of a class of SMC methods where the times at which resampling occurs are computed online using criteria such as the effective sample size. This is a popular approach amongst practitioners but there are very few convergence results available for these methods. By combining semigroup techniques with an original coupling argument, we obtain functional central limit theorems and uniform exponential concentration estimates for these algorithms.

A Generalization of Wigner’s Law

We present a generalization of Wigner’s semicircle law: we consider a sequence of probability distributions \((p_1,p_2,\dots)\), with mean value zero and take an N × N real symmetric matrix with entries independently chosen from p N and analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N → ∞ for certain p N the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the k th moment of p N (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when p N does not depend on N we obtain Wigner’s law: if all moments of a distribution are finite, the distribution of eigenvalues is a semicircle.

Sequential Monte Carlo Samplers

SummaryWe propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space, each distribution being known up to a normalizing constant. These probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential Monte Carlo methods. This methodology allows us to derive simple algorithms to make parallel Markov chain Monte Carlo algorithms interact to perform global optimization and sequential Bayesian estimation and to compute ratios of normalizing constants. We illustrate these algorithms for various integration tasks arising in the context of Bayesian inference.

Sequential Monte Carlo Samplers CUED Technical Report

In this paper, we propose a general methodology to sample sequentially from a sequence of probability distributions known up to a normalizing constant and defined on a common space. These probability distributions are approximated by a cloud of weighted random samples which are propagated over time using Sequential Monte Carlo methods. This methodology allows us not only to derive simple algorithms to make parallel Markov chain Monte Carlo runs interact in a principled way, but also to obtain new methods for global optimization and sequential Bayesian estimation. We demonstrate the performance of these algorithms through simulation for various integration and global optimization tasks arising in the context of Bayesian inference.

On the Markov Property of Quantised State Measuirement Sequences

This paper is concerned with the representation of stable autonomous discrete–time systems with quantised state measurements. It is assumed that the state space R n is partitioned into disjoint regions Q x which are enumerated. Only the number z( k) of the region Q x to which the state x( k) belongs can be measured. As a consequence, the initial state x(0) ∈ R n is unknown and assumed to be uniformly distributed over the set Q x( z(0)) associated with the measurement z(0) . The paper shows that the measurement sequence z(k) (k=0, 1, …) is, in general, not a Markov chain. Hence, the sequence of probability distributions cannot be represented exactly by a stochastic automaton whose state set equals the set of measurement symbols.

Method for generating a speech recognition model for a non-vocabulary utterance

The method disclosed herein facilitates the generation of a recognition model for a non-standard word uttered by a user in the context of a large vocabulary speech recognition system in which standard vocabulary models are represented by sequences of probability distributions for various acoustic symbols. Along with the probability distributions, a corresponding plurality of converse probability functions are precalculated which represent the likelihood that a particular probability distribution would correspond to a given input acoustic symbol. For a non-standard word uttered, a corresponding sequence of acoustic symbols is generated and, for each such symbol in the sequence, the most likely probability distribution is selected using the converse probability functions. For successive symbols in the utterance, a corresponding sequence of custom converse probability functions are generated, each of which is a composite of weighted contributions from the corresponding precalculated converse probability function and the converse probability functions corresponding to time-adjacent symbols in the utterance. The resulting sequence of custom converse probability functions identify a corresponding sequence of probability distributions which constitute a model of the word uttered, which model incorporates contextual information from the utterance.

Simulated Annealing and Adaptive Search in Global Optimization

Simulated annealing is a class of sequential search techniques for solving continuous global optimization problems. In this paper we attempt to help explain the success of simulated annealing for this class of problems by studying an idealized version of this algorithm, which we call adaptive search. The prototypical adaptive search algorithm generates a sequence of improving points drawn conditionally from samples from a corresponding sequence of probability distributions. Under the condition that the sequence of distributions stochastically dominate in objective function value the uniform distribution, we show that the expected number of improving points required to achieve the global optimum within a prespecified error grows at most linearly in the dimension of the problem for a large class of global optimization problems. Moreover, we derive a cooling schedule for simulated annealing, which follows in a natural way from the definition of the adaptive search algorithm.