Modelling of non-stationary processes using radial basis function networks

This paper reports preliminary progress on a principled approach to modelling nonstationary phenomena using neural networks. We are concerned with both parameter and model order complexity estimation. The basic methodology assumes a Bayesian foundation. However to allow the construction of pragmatic models, successive approximations have to be made to permit computational tractibility. The lowest order corresponds to the (Extended) Kalman filter approach to parameter estimation which has already been applied to neural networks. We illustrate some of the deficiencies of the existing approaches and discuss our preliminary generalisations, by considering the application to nonstationary time series.

Functional Transfer Theorems for Maxima of Stationary Processes

2000 Mathematics Subject Classification: 60G70, 60F12, 60G10.

Evidence of the heterogeneous market hypothesis using wavelet multi -resolution analysis

In finance literature many economic theories and models have been proposed to explain and estimate the relationship between risk and return. Assuming risk averseness and rational behavior on part of the investor, the models are developed which are supposed to help in forming efficient portfolios that either maximize (minimize) the expected rate of return (risk) for a given level of risk (rates of return). One of the most used models to form these efficient portfolios is the Sharpe's Capital Asset Pricing Model (CAPM). In the development of this model it is assumed that the investors have homogeneous expectations about the future probability distribution of the rates of return. That is, every investor assumes the same values of the parameters of the probability distribution. Likewise financial volatility homogeneity is commonly assumed, where volatility is taken as investment risk which is usually measured by the variance of the rates of return. Typically the square root of the variance is used to define financial volatility, furthermore it is also often assumed that the data generating process is made of independent and identically distributed random variables. This again implies that financial volatility is measured from homogeneous time series with stationary parameters. In this dissertation, we investigate the assumptions of homogeneity of market agents and provide evidence for the case of heterogeneity in market participants' information, objectives, and expectations about the parameters of the probability distribution of prices as given by the differences in the empirical distributions corresponding to different time scales, which in this study are associated with different classes of investors, as well as demonstrate that statistical properties of the underlying data generating processes including the volatility in the rates of return are quite heterogeneous. In other words, we provide empirical evidence against the traditional views about homogeneity using non-parametric wavelet analysis on trading data, The results show heterogeneity of financial volatility at different time scales, and time-scale is one of the most important aspects in which trading behavior differs. In fact we conclude that heterogeneity as posited by the Heterogeneous Markets Hypothesis is the norm and not the exception.

Geochemistry Of Acid Mine Drainage From A Coal Mining Area And Processes Controlling Metal Attenuation In Stream Waters, Southern Brazil.

Acid drainage influence on the water and sediment quality was investigated in a coal mining area (southern Brazil). Mine drainage showed pH between 3.2 and 4.6 and elevated concentrations of sulfate, As and metals, of which, Fe, Mn and Zn exceeded the limits for the emission of effluents stated in the Brazilian legislation. Arsenic also exceeded the limit, but only slightly. Groundwater monitoring wells from active mines and tailings piles showed pH interval and chemical concentrations similar to those of mine drainage. However, the river and ground water samples of municipal public water supplies revealed a pH range from 7.2 to 7.5 and low chemical concentrations, although Cd concentration slightly exceeded the limit adopted by Brazilian legislation for groundwater. In general, surface waters showed large pH range (6 to 10.8), and changes caused by acid drainage in the chemical composition of these waters were not very significant. Locally, acid drainage seemed to have dissolved carbonate rocks present in the local stratigraphic sequence, attenuating the dispersion of metals and As. Stream sediments presented anomalies of these elements, which were strongly dependent on the proximity of tailings piles and abandoned mines. We found that precipitation processes in sediments and the dilution of dissolved phases were responsible for the attenuation of the concentrations of the metals and As in the acid drainage and river water mixing zone. In general, a larger influence of mining activities on the chemical composition of the surface waters and sediments was observed when enrichment factors in relation to regional background levels were used.

Hitting and returning to rare events for all alpha-mixing processes

We prove that for any a-mixing stationary process the hitting time of any n-string A(n) converges, when suitably normalized, to an exponential law. We identify the normalization constant lambda(A(n)). A similar statement holds also for the return time. To establish this result we prove two other results of independent interest. First, we show a relation between the rescaled hitting time and the rescaled return time, generalizing a theorem of Haydn, Lacroix and Vaienti. Second, we show that for positive entropy systems, the probability of observing any n-string in n consecutive observations goes to zero as n goes to infinity. (c) 2010 Elsevier B.V. All rights reserved.

Gibbs random graphs on point processes

Consider a discrete locally finite subset Gamma of R(d) and the cornplete graph (Gamma, E), with vertices Gamma and edges E. We consider Gibbs measures on the set of sub-graphs with vertices Gamma and edges E` subset of E. The Gibbs interaction acts between open edges having a vertex in common. We study percolation properties of the Gibbs distribution of the graph ensemble. The main results concern percolation properties of the open edges in two cases: (a) when Gamma is sampled from a homogeneous Poisson process; and (b) for a fixed Gamma with sufficiently sparse points. (c) 2010 American Institute of Physics. [doi:10.1063/1.3514605]

Singular Perturbation for the Discounted Continuous Control of Piecewise Deterministic Markov Processes

This paper deals with the expected discounted continuous control of piecewise deterministic Markov processes (PDMP`s) using a singular perturbation approach for dealing with rapidly oscillating parameters. The state space of the PDMP is written as the product of a finite set and a subset of the Euclidean space a""e (n) . The discrete part of the state, called the regime, characterizes the mode of operation of the physical system under consideration, and is supposed to have a fast (associated to a small parameter epsilon > 0) and a slow behavior. By using a similar approach as developed in Yin and Zhang (Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Applications of Mathematics, vol. 37, Springer, New York, 1998, Chaps. 1 and 3) the idea in this paper is to reduce the number of regimes by considering an averaged model in which the regimes within the same class are aggregated through the quasi-stationary distribution so that the different states in this class are replaced by a single one. The main goal is to show that the value function of the control problem for the system driven by the perturbed Markov chain converges to the value function of this limit control problem as epsilon goes to zero. This convergence is obtained by, roughly speaking, showing that the infimum and supremum limits of the value functions satisfy two optimality inequalities as epsilon goes to zero. This enables us to show the result by invoking a uniqueness argument, without needing any kind of Lipschitz continuity condition.

Further results on the relationship between mu-invariant measures and quasi-stationary distributions for absorbing continuous-time Markov chains

This note considers continuous-time Markov chains whose state space consists of an irreducible class, C, and an absorbing state which is accessible from C. The purpose is to provide results on mu-invariant and mu-subinvariant measures where absorption occurs with probability less than one. In particular, the well-known premise that the mu-invariant measure, m, for the transition rates be finite is replaced by the more natural premise that m be finite with respect to the absorption probabilities. The relationship between mu-invariant measures and quasi-stationary distributions is discussed. (C) 2000 Elsevier Science Ltd. All rights reserved.

New methods for determining quasi-stationary distributions for Markov chains

We shall be concerned with the problem of determining quasi-stationary distributions for Markovian models directly from their transition rates Q. We shall present simple conditions for a mu-invariant measure m for Q to be mu-invariant for the transition function, so that if m is finite, it can be normalized to produce a quasi-stationary distribution. (C) 2000 Elsevier Science Ltd. All rights reserved.

Limiting conditional distributions for birth-death processes

In a recent paper [16], one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

A new wavelet-based method for the solution of the population balance equation

A new wavelet-based method for solving population balance equations with simultaneous nucleation, growth and agglomeration is proposed, which uses wavelets to express the functions. The technique is very general, powerful and overcomes the crucial problems of numerical diffusion and stability that often characterize previous techniques in this area. It is also applicable to an arbitrary grid to control resolution and computational efficiency. The proposed technique has been tested for pure agglomeration, simultaneous nucleation and growth, and simultaneous growth and agglomeration. In all cases, the predicted and analytical particle size distributions are in excellent agreement. The presence of moving sharp fronts can be addressed without the prior investigation of the characteristics of the processes. (C) 2001 Published by Elsevier Science Ltd.

A new wavelet-based adaptive method for solving population balance equations

A new wavelet-based adaptive framework for solving population balance equations (PBEs) is proposed in this work. The technique is general, powerful and efficient without the need for prior assumptions about the characteristics of the processes. Because there are steeply varying number densities across a size range, a new strategy is developed to select the optimal order of resolution and the collocation points based on an interpolating wavelet transform (IWT). The proposed technique has been tested for size-independent agglomeration, agglomeration with a linear summation kernel and agglomeration with a nonlinear kernel. In all cases, the predicted and analytical particle size distributions (PSDs) are in excellent agreement. Further work on the solution of the general population balance equations with nucleation, growth and agglomeration and the solution of steady-state population balance equations will be presented in this framework. (C) 2002 Elsevier Science B.V. All rights reserved.

Optimal design and operation of compact simulated moving bed processes for enantioseparations

Simulated moving bed (SMB) chromatography is attracting more and more attention since it is a powerful technique for complex separation tasks. Nowadays, more than 60% of preparative SMB units are installed in the pharmaceutical and in the food in- dustry [SDI, Preparative and Process Liquid Chromatography: The Future of Process Separations, International Strategic Directions, Los Angeles, USA, 2002. http://www. strategicdirections.com]. Chromatography is the method of choice in these ¯elds, be- cause often pharmaceuticals and ¯ne-chemicals have physico-chemical properties which di®er little from those of the by-products, and they may be thermally instable. In these cases, standard separation techniques as distillation and extraction are not applicable. The noteworthiness of preparative chromatography, particulary SMB process, as a sep- aration and puri¯cation process in the above mentioned industries has been increasing, due to its °exibility, energy e±ciency and higher product purity performance. Consequently, a new SMB paradigm is requested by the large number of potential small- scale applications of the SMB technology, which exploits the °exibility and versatility of the technology. In this new SMB paradigm, a number of possibilities for improving SMB performance through variation of parameters during a switching interval, are pushing the trend toward the use of units with smaller number of columns because less stationary phase is used and the setup is more economical. This is especially important for the phar- maceutical industry, where SMBs are seen as multipurpose units that can be applied to di®erent separations in all stages of the drug-development cycle. In order to reduce the experimental e®ort and accordingly the coast associated with the development of separation processes, simulation models are intensively used. One impor- tant aspect in this context refers to the determination of the adsorption isotherms in SMB chromatography, where separations are usually carried out under strongly nonlinear conditions in order to achieve higher productivities. The accurate determination of the competitive adsorption equilibrium of the enantiomeric species is thus of fundamental importance to allow computer-assisted optimization or process scale-up. Two major SMB operating problems are apparent at production scale: the assessment of product quality and the maintenance of long-term stable and controlled operation. Constraints regarding product purity, dictated by pharmaceutical and food regulatory organizations, have drastically increased the demand for product quality control. The strict imposed regulations are increasing the need for developing optically pure drugs.(...)

A nonlinear threshold model for the dependence of extremes of stationary sequences

One of the main implications of the efficient market hypothesis (EMH) is that expected future returns on financial assets are not predictable if investors are risk neutral. In this paper we argue that financial time series offer more information than that this hypothesis seems to supply. In particular we postulate that runs of very large returns can be predictable for small time periods. In order to prove this we propose a TAR(3,1)-GARCH(1,1) model that is able to describe two different types of extreme events: a first type generated by large uncertainty regimes where runs of extremes are not predictable and a second type where extremes come from isolated dread/joy events. This model is new in the literature in nonlinear processes. Its novelty resides on two features of the model that make it different from previous TAR methodologies. The regimes are motivated by the occurrence of extreme values and the threshold variable is defined by the shock affecting the process in the preceding period. In this way this model is able to uncover dependence and clustering of extremes in high as well as in low volatility periods. This model is tested with data from General Motors stocks prices corresponding to two crises that had a substantial impact in financial markets worldwide; the Black Monday of October 1987 and September 11th, 2001. By analyzing the periods around these crises we find evidence of statistical significance of our model and thereby of predictability of extremes for September 11th but not for Black Monday. These findings support the hypotheses of a big negative event producing runs of negative returns in the first case, and of the burst of a worldwide stock market bubble in the second example. JEL classification: C12; C15; C22; C51 Keywords and Phrases: asymmetries, crises, extreme values, hypothesis testing, leverage effect, nonlinearities, threshold models

Depleted arc volcanism in the Alboran Sea and shoshonitic volcanism in Morocco: geochemical and isotopic constraints on Neogene tectonic processes

Samples of volcanic rocks from Alboran Island, the Alboran Sea floor and from the Gourougou volcanic centre in northern Morocco have been analyzed for major and trace elements and Sr-Nd isotopes to test current theories on the tectonic geodynamic evolution of the Alboran Sea. The Alboran Island samples are low-K tholeiitic basaltic andesites whose depleted contents of HFS elements (similar to0.5xN-MORB), especially Nb (similar to0.2xN-MORB), show marked geochemical parallels with volcanics from immature intra-oceanic arcs and back-arc basins. Several of the submarine samples have similar compositions, one showing low-Ca boninite affinity. Nd-143/Nd-144 ratios fall in the same range as many island-arc and back-arc basin samples, whereas Sr-87/Sr-86 ratios (on leached samples) are somewhat more radiogenic. Our data point to active subduction taking place beneath the Alboran region in Miocene times, and imply the presence of an associated back-arc spreading centre. Our sea floor suite includes a few more evolved dacite and rhyolite samples with (Sr-87/Sr-86)(0) up to 0.717 that probably represent varying degrees of crustal melting. The shoshonite and high-K basaltic andesite lavas from Gourougou have comparable normalized incompatible-element enrichment diagrams and Ce/Y ratios to shoshonitic volcanics from oceanic island arcs, though they have less pronounced Nb deficits. They are much less LIL- and LREE-enriched than continental arc analogues and post-collisional shoshonites from Tibet. The magmas probably originated by melting in subcontinental lithospheric mantle that had experienced negligible subduction input. Sr-Nd isotope compositions point to significant crustal contamination which appears to account for the small Nb anomalies. The unmistakable supra-subduction zone (SSZ) signature shown by our Alboran basalts and basaltic andesite samples refutes geodynamic models that attribute all Neogene volcanism in the Alboran domain to decompression melting of upwelling asthenosphere arising from convective thinning of over-thickened lithosphere. Our data support recent models in which subsidence is caused by westward rollback of an eastward-dipping subduction zone beneath the westemmost Mediterranean. Moreover, severance of the lithosphere at the edges of the rolling-back slab provides opportunities for locally melting lithospheric mantle, providing a possible explanation for the shoshonitic volcanism seen in northern Morocco and more sporadically in SE Spain. (C) 2004 Elsevier B.V. All rights reserved.