STABILITY ANALYSIS WITH APPLICATIONS OF A TWO-DIMENSIONAL DYNAMICAL SYSTEM ARISING FROM A STOCHASTIC MODEL FOR AN ASSET MARKET

We analyze the stability properties of equilibrium solutions and periodicity of orbits in a two-dimensional dynamical system whose orbits mimic the evolution of the price of an asset and the excess demand for that asset. The construction of the system is grounded upon a heterogeneous interacting agent model for a single risky asset market. An advantage of this construction procedure is that the resulting dynamical system becomes a macroscopic market model which mirrors the market quantities and qualities that would typically be taken into account solely at the microscopic level of modeling. The system`s parameters correspond to: (a) the proportion of speculators in a market; (b) the traders` speculative trend; (c) the degree of heterogeneity of idiosyncratic evaluations of the market agents with respect to the asset`s fundamental value; and (d) the strength of the feedback of the population excess demand on the asset price update increment. This correspondence allows us to employ our results in order to infer plausible causes for the emergence of price and demand fluctuations in a real asset market. The employment of dynamical systems for studying evolution of stochastic models of socio-economic phenomena is quite usual in the area of heterogeneous interacting agent models. However, in the vast majority of the cases present in the literature, these dynamical systems are one-dimensional. Our work is among the few in the area that construct and study analytically a two-dimensional dynamical system and apply it for explanation of socio-economic phenomena.

An Information Theory framework for two-stage binary image operator design

The design of translation invariant and locally defined binary image operators over large windows is made difficult by decreased statistical precision and increased training time. We present a complete framework for the application of stacked design, a recently proposed technique to create two-stage operators that circumvents that difficulty. We propose a novel algorithm, based on Information Theory, to find groups of pixels that should be used together to predict the Output Value. We employ this algorithm to automate the process of creating a set of first-level operators that are later combined in a global operator. We also propose a principled way to guide this combination, by using feature selection and model comparison. Experimental results Show that the proposed framework leads to better results than single stage design. (C) 2009 Elsevier B.V. All rights reserved.

Performance of balanced two-stage empirical predictors of realized cluster latent values from finite populations: A simulation study

Predictors of random effects are usually based on the popular mixed effects (ME) model developed under the assumption that the sample is obtained from a conceptual infinite population; such predictors are employed even when the actual population is finite. Two alternatives that incorporate the finite nature of the population are obtained from the superpopulation model proposed by Scott and Smith (1969. Estimation in multi-stage surveys. J. Amer. Statist. Assoc. 64, 830-840) or from the finite population mixed model recently proposed by Stanek and Singer (2004. Predicting random effects from finite population clustered samples with response error. J. Amer. Statist. Assoc. 99, 1119-1130). Predictors derived under the latter model with the additional assumptions that all variance components are known and that within-cluster variances are equal have smaller mean squared error (MSE) than the competitors based on either the ME or Scott and Smith`s models. As population variances are rarely known, we propose method of moment estimators to obtain empirical predictors and conduct a simulation study to evaluate their performance. The results suggest that the finite population mixed model empirical predictor is more stable than its competitors since, in terms of MSE, it is either the best or the second best and when second best, its performance lies within acceptable limits. When both cluster and unit intra-class correlation coefficients are very high (e.g., 0.95 or more), the performance of the empirical predictors derived under the three models is similar. (c) 2007 Elsevier B.V. All rights reserved.

On the cutting stock problem under stochastic demand

This paper addresses the one-dimensional cutting stock problem when demand is a random variable. The problem is formulated as a two-stage stochastic nonlinear program with recourse. The first stage decision variables are the number of objects to be cut according to a cutting pattern. The second stage decision variables are the number of holding or backordering items due to the decisions made in the first stage. The problem`s objective is to minimize the total expected cost incurred in both stages, due to waste and holding or backordering penalties. A Simplex-based method with column generation is proposed for solving a linear relaxation of the resulting optimization problem. The proposed method is evaluated by using two well-known measures of uncertainty effects in stochastic programming: the value of stochastic solution-VSS-and the expected value of perfect information-EVPI. The optimal two-stage solution is shown to be more effective than the alternative wait-and-see and expected value approaches, even under small variations in the parameters of the problem.

CRITICAL BEHAVIOR AND THRESHOLD OF COEXISTENCE OF A PREDATOR-PREY STOCHASTIC MODEL IN A 2D LATTICE

We investigate the critical behavior of a stochastic lattice model describing a predator-prey system. By means of Monte Carlo procedure we simulate the model defined on a regular square lattice and determine the threshold of species coexistence, that is, the critical phase boundaries related to the transition between an active state, where both species coexist and an absorbing state where one of the species is extinct. A finite size scaling analysis is employed to determine the order parameter, order parameter fluctuations, correlation length and the critical exponents. Our numerical results for the critical exponents agree with those of the directed percolation universality class. We also check the validity of the hyperscaling relation and present the data collapse curves.

Predicting random effects with an expanded finite population mixed model

Prediction of random effects is an important problem with expanding applications. In the simplest context, the problem corresponds to prediction of the latent value (the mean) of a realized cluster selected via two-stage sampling. Recently, Stanek and Singer [Predicting random effects from finite population clustered samples with response error. J. Amer. Statist. Assoc. 99, 119-130] developed best linear unbiased predictors (BLUP) under a finite population mixed model that outperform BLUPs from mixed models and superpopulation models. Their setup, however, does not allow for unequally sized clusters. To overcome this drawback, we consider an expanded finite population mixed model based on a larger set of random variables that span a higher dimensional space than those typically applied to such problems. We show that BLUPs for linear combinations of the realized cluster means derived under such a model have considerably smaller mean squared error (MSE) than those obtained from mixed models, superpopulation models, and finite population mixed models. We motivate our general approach by an example developed for two-stage cluster sampling and show that it faithfully captures the stochastic aspects of sampling in the problem. We also consider simulation studies to illustrate the increased accuracy of the BLUP obtained under the expanded finite population mixed model. (C) 2007 Elsevier B.V. All rights reserved.

Stochastic simulations of the Madden-Julian oscillation activity

The Madden-Julian oscillation (MJO) is the most prominent form of tropical intraseasonal variability. This study investigated the following questions. Do inter-annual-to-decadal variations in tropical sea surface temperature (SST) lead to substantial changes in MJO activity? Was there a change in the MJO in the 1970s? Can this change be associated to SST anomalies? What was the level of MJO activity in the pre-reanalysis era? These questions were investigated with a stochastic model of the MJO. Reanalysis data (1948-2008) were used to develop a nine-state first order Markov model capable to simulate the non-stationarity of the MJO. The model is driven by observed SST anomalies and a large ensemble of simulations was performed to infer the activity of the MJO in the instrumental period (1880-2008). The model is capable to reproduce the activity of the MJO during the reanalysis period. The simulations indicate that the MJO exhibited a regime of near normal activity in 1948-1972 (3.4 events year(-1)) and two regimes of high activity in 1973-1989 (3.9 events) and 1990-2008 (4.6 events). Stochastic simulations indicate decadal shifts with near normal levels in 1880-1895 (3.4 events), low activity in 1896 1917 (2.6 events) and a return to near normal levels during 1918-1947 (3.3 events). The results also point out to significant decadal changes in probabilities of very active years (5 or more MJO events): 0.214 (1880-1895), 0.076 (1896-1917), 0.197 (1918-1947) and 0.193 (1948-1972). After a change in behavior in the 1970s, this probability has increased to 0.329 (1973-1989) and 0.510 (1990-2008). The observational and stochastic simulations presented here call attention to the need to further understand the variability of the MJO on a wide range of time scales.

Robust tori in a double-waved Hamiltonian model

A Hamiltonian system perturbed by two waves with particular wave numbers can present robust tori, which are barriers created by the vanishing of the perturbed Hamiltonian at some defined positions. When robust tori exist, any trajectory in phase space passing close to them is blocked by emergent invariant curves that prevent the chaotic transport. Our results indicate that the considered particular solution for the two waves Hamiltonian model shows plenty of robust tori blocking radial transport. (C) 2010 Elsevier B.V. All rights reserved.

Sampling, WLS, and Mixed Models

Mixed models may be defined with or without reference to sampling, and can be used to predict realized random effects, as when estimating the latent values of study subjects measured with response error. When the model is specified without reference to sampling, a simple mixed model includes two random variables, one stemming from an exchangeable distribution of latent values of study subjects and the other, from the study subjects` response error distributions. Positive probabilities are assigned to both potentially realizable responses and artificial responses that are not potentially realizable, resulting in artificial latent values. In contrast, finite population mixed models represent the two-stage process of sampling subjects and measuring their responses, where positive probabilities are only assigned to potentially realizable responses. A comparison of the estimators over the same potentially realizable responses indicates that the optimal linear mixed model estimator (the usual best linear unbiased predictor, BLUP) is often (but not always) more accurate than the comparable finite population mixed model estimator (the FPMM BLUP). We examine a simple example and provide the basis for a broader discussion of the role of conditioning, sampling, and model assumptions in developing inference.

Comparing diagnostic tests with missing data

When missing data occur in studies designed to compare the accuracy of diagnostic tests, a common, though naive, practice is to base the comparison of sensitivity, specificity, as well as of positive and negative predictive values on some subset of the data that fits into methods implemented in standard statistical packages. Such methods are usually valid only under the strong missing completely at random (MCAR) assumption and may generate biased and less precise estimates. We review some models that use the dependence structure of the completely observed cases to incorporate the information of the partially categorized observations into the analysis and show how they may be fitted via a two-stage hybrid process involving maximum likelihood in the first stage and weighted least squares in the second. We indicate how computational subroutines written in R may be used to fit the proposed models and illustrate the different analysis strategies with observational data collected to compare the accuracy of three distinct non-invasive diagnostic methods for endometriosis. The results indicate that even when the MCAR assumption is plausible, the naive partial analyses should be avoided.