Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Frequency domain analysis of stationary time series

This thesis provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalised autocovariance function of a Gaussian stationary random process. The generalised autocovariance function is the inverse Fourier transform of a power transformation of the spectral density, and encompasses the traditional and inverse autocovariance functions. Its nonparametric estimator is based on the inverse discrete Fourier transform of the same power transformation of the pooled periodogram. The general result is then applied to the class of Gaussian stationary ARMA processes and its implications are discussed. We illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule-Walker system of equations in the generalised autocovariance estimator. Selection of the pooling parameter, which characterizes the nonparametric estimator of the generalised autocovariance, controlling its resolution, is addressed by using a multiplicative periodogram bootstrap to estimate the finite-sample distribution of the estimator. A multivariate extension of recently introduced spectral models for univariate time series is considered, and an algorithm for the coefficients of a power transformation of matrix polynomials is derived, which allows to obtain the Wold coefficients from the matrix coefficients characterizing the generalised matrix cepstral models. This algorithm also allows the definition of the matrix variance profile, providing important quantities for vector time series analysis. A nonparametric estimator based on a transformation of the smoothed periodogram is proposed for estimation of the matrix variance profile.

A Nonlinear Panel Unit Root Test under Cross Section Dependence

We propose a nonlinear heterogeneous panel unit root test for testing the null hypothesis of unit-roots processes against the alternative that allows a proportion of units to be generated by globally stationary ESTAR processes and a remaining non-zero proportion to be generated by unit root processes. The proposed test is simple to implement and accommodates cross sectional dependence. We show that the distribution of the test statistic is free of nuisance parameters as (N, T) −! 1. Monte Carlo simulation shows that our test holds correct size and under the hypothesis that data are generated by globally stationary ESTAR processes has a better power than the recent test proposed in Pesaran [2007]. Various applications are provided.

Diagnostic accuracy of G-CSF, IL-8, and IL-1ra in critically ill children with suspected infection.

OBJECTIVE: To elucidate the diagnostic accuracy of granulocyte colony-stimulating factor (G-CSF), interleukin-8 (IL-8), and interleukin-1 receptor antagonist (IL-1ra) in identifying patients with sepsis among critically ill pediatric patients with suspected infection. DESIGN AND SETTING: Nested case-control study in a multidisciplinary neonatal and pediatric intensive care unit (PICU) PATIENTS: PICU patients during a 12-month period with suspected infection, and plasma available from the time of clinical suspicion (254 episodes, 190 patients). MEASUREMENTS AND RESULTS: Plasma levels of G-CSF, IL-8, and IL-1ra. Episodes classified on the basis of clinical and bacteriological findings into: culture-confirmed sepsis, probable sepsis, localized infection, viral infection, and no infection. Plasma levels were significantly higher in episodes of culture-confirmed sepsis than in episodes with ruled-out infection. The area under the receiver operating characteristic curve was higher for IL-8 and G-CSF than for IL-1ra. Combining IL-8 and G-CSF improved the diagnostic performance, particularly as to the detection of Gram-negative sepsis. Sensitivity was low (<50%) in detecting Staphylococcus epidermidis bacteremia or localized infections. CONCLUSIONS: In this heterogeneous population of critically ill children with suspected infection, a model combining plasma levels of IL-8 and G-CSF identified patients with sepsis. Negative results do not rule out S. epidermidis bacteremia or locally confined infectious processes. The model requires validation in an independent data-set.

A note on moments of dividends

We reconsider a formula for arbitrary moments of expected discounted dividend payments in a spectrally negative L,vy risk model that was obtained in Renaud and Zhou (2007, [4]) and in Kyprianou and Palmowski (2007, [3]) and extend the result to stationary Markov processes that are skip-free upwards.

Bootstrapping Autoregressions with Conditional Heteroskedasticity of Unknown Form

Conditional heteroskedasticity is an important feature of many macroeconomic and financial time series. Standard residual-based bootstrap procedures for dynamic regression models treat the regression error as i.i.d. These procedures are invalid in the presence of conditional heteroskedasticity. We establish the asymptotic validity of three easy-to-implement alternative bootstrap proposals for stationary autoregressive processes with m.d.s. errors subject to possible conditional heteroskedasticity of unknown form. These proposals are the fixed-design wild bootstrap, the recursive-design wild bootstrap and the pairwise bootstrap. In a simulation study all three procedures tend to be more accurate in small samples than the conventional large-sample approximation based on robust standard errors. In contrast, standard residual-based bootstrap methods for models with i.i.d. errors may be very inaccurate if the i.i.d. assumption is violated. We conclude that in many empirical applications the proposed robust bootstrap procedures should routinely replace conventional bootstrap procedures for autoregressions based on the i.i.d. error assumption.

The feeling of fluent perception: A single experience from multiple asynchronous sources

Zeki and co-workers recently proposed that perception can best be described as locally distributed, asynchronous processes that each create a kind of microconsciousness, which condense into an experienced percept. The present article is aimed at extending this theory to metacognitive feelings. We present evidence that perceptual fluency-the subjective feeling of ease during perceptual processing-is based on speed of processing at different stages of the perceptual process. Specifically, detection of briefly presented stimuli was influenced by figure-ground contrast, but not by symmetry (Experiment 1) or the font (Experiment 2) of the stimuli. Conversely, discrimination of these stimuli was influenced by whether they were symmetric (Experiment 1) and by the font they were presented in (Experiment 2), but not by figure-ground contrast. Both tasks however were related with the subjective experience of fluency (Experiments 1 and 2). We conclude that subjective fluency is the conscious phenomenal correlate of different processing stages in visual perception.

Element concentrations and Osmium, Platinum and Rhenium isotope ratios of ODP Site 153-920 abyssal peridotites

Abyssal peridotites are normally thought to be residues of melting of the mid-ocean ridge basalt (MORB) source and are presumably a record of processes affecting the upper mantle. Samples from a single section of abyssal peridotite from the Kane Transform area in the Atlantic Ocean were examined for 190Pt-186Os and 187Re-187Os systematics. They have uniform 186Os/188Os ratios with a mean of 0.1198353 +/- 7, identical to the mean of 0.1198340 +/-12 for Os-Ir alloys and chromitites believed to be representative of the upper mantle. While the Pt/Os ratios of the upper mantle may be affected locally by magmatic processes, these data show that the Pt/Os ratio for the bulk upper mantle has not deviated by more than about +/- 30% from a chondritic Pt/Os ratio over 4.5 billion years. These observations are consistent with the addition of a chondritic late veneer after core separation as the primary control on the highly siderophile element budget of the terrestrial upper mantle. The 187Os/188Os of the samples range from 0.12267 to 0.12760 and correlate well with Pt and Pt/Os, but not Re/Os. These relationships may be explained by variable amounts of partial melting with changing D(Re), reflecting in part garnet in the residue, with a model-dependent melting age between about 600 and 1700 Ma. A model where the correlation between Pt/Os and 187Os/188Os results from multiple ancient melting events, in mantle peridotites that were later juxtaposed by convection, is also consistent with these data. This melting event or events are evidently unrelated to recent melting under mid-ocean ridges, because recent melting would have disturbed the relationship between Pt/Os and 187Os/188Os. Instead, this section of abyssal peridotite may be a block of refractory mantle that remained isolated from the convecting portions of the upper mantle for 600 Ma to >1 Ga. Alternatively, Pt and Os may have been sequestered during more recent melting and possibly melt/rock reaction processes, thereby preserving an ancient melting history. If representative of other abyssal peridotites, then the rocks from this suite with subchondritic 187Os/188Os are not simple residues of recent MORB source melting at ridges, but instead have a more complex history. This suite of variably depleted samples projects to an undepleted present-day Pt/Os of about 2.2 and 187Os/188Os of about 0.128-0.129, consistent with estimates for the primitive upper mantle.

Sea surface fugacity of carbon dioxide measurements in the Pacific ocean

As part of the JGOFS field program, extensive CO2 partial-pressure measurements were made in the atmosphere and in the surface waters of the equatorial Pacific from 1992 to 1999. For the first time, we are able to determine how processes occurring in the western portion of the equatorial Pacific impact the sea-air fluxes of CO2 in the central and eastern regions. These 8 years of data are compared with the decade of the 1980s. Over this period, surface-water pCO2 data indicate significant seasonal and interannual variations. The largest decreases in fluxes were associated with the 1991-94 and 1997-98 El Niño events. The lower sea-air CO2 fluxes during these two El Niño periods were the result of the combined effects of interconnected large-scale and locally forced physical processes: (1) development of a low-salinity surface cap as part of the formation of the warm pool in the western and central equatorial Pacific, (2) deepening of the thermocline by propagating Kelvin waves in the eastern Pacific, and (3) the weakening of the winds in the eastern half of the basin. These processes serve to reduce pCO2 values in the central and eastern equatorial Pacific towards near-equilibrium values at the height of the warm phase of ENSO. In the western equatorial Pacific there is a small but significant increase in seawater pCO2 during strong El Niño events (i.e., 1982-83 and 1997-98) and little or no change during weak El Niño events (1991-94). The net effect of these interannual variations is a lower-than-normal CO2 flux to the atmosphere from the equatorial Pacific during El Niño. The annual average fluxes indicate that during strong El Niños the release to the atmosphere is 0.2-0.4 Pg C/yr compared to 0.8-1.0 Pg C/yr during non-El Niño years.

A distribution of tumor size at detection and its limiting form.

A distribution of tumor size at detection is derived within the framework of a mechanistic model of carcinogenesis with the object of estimating biologically meaningful parameters of tumor latency. Its limiting form appears to be a generalization of the distribution that arises in the length-biased sampling from stationary point processes. The model renders the associated estimation problems tractable. The usefulness of the proposed approach is illustrated with an application to clinical data on premenopausal breast cancer.

A Stochastic Approach for Investigation Ultrafast Phenomena in Semiconductors

2002 Mathematics Subject Classification: 65C05

Shared information in stationary states of stochastic processes

We present four estimators of the shared information (or interdepency) in ground states given that the coefficients appearing in the wave function are all real non-negative numbers and therefore can be interpreted as probabilities of configurations. Such ground states of Hermitian and non-Hermitian Hamiltonians can be given, for example, by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models. We consider in detail the last case, the system being a classical not a quantum one. Using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive, conformal invariant and nonconformal invariant massless phases. We show that like in the case of the quantum problem, the estimators verify the area law with logarithmic corrections when phase transitions take place.

A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic

For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martinez and Picco studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Phi on the space of probability distributions on {1, 2,.. }. In the case of a birth-death process, the components of Phi(nu) can be written down explicitly for any given distribution nu. Using this explicit representation, we will show that Phi preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefevre concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary.

Markovian Processes, Two-Sided Autoregressions and Finite-Sample Inference for Stationary and Nonstationary Autoregressive Processes.

In this paper, we develop finite-sample inference procedures for stationary and nonstationary autoregressive (AR) models. The method is based on special properties of Markov processes and a split-sample technique. The results on Markovian processes (intercalary independence and truncation) only require the existence of conditional densities. They are proved for possibly nonstationary and/or non-Gaussian multivariate Markov processes. In the context of a linear regression model with AR(1) errors, we show how these results can be used to simplify the distributional properties of the model by conditioning a subset of the data on the remaining observations. This transformation leads to a new model which has the form of a two-sided autoregression to which standard classical linear regression inference techniques can be applied. We show how to derive tests and confidence sets for the mean and/or autoregressive parameters of the model. We also develop a test on the order of an autoregression. We show that a combination of subsample-based inferences can improve the performance of the procedure. An application to U.S. domestic investment data illustrates the method.