Nonlinear Stochastic Models for Water Level Dynamics in Closed Lakes

This paper presents the results of investigation of nonlinear mathematical models of the behavior of closed lakes using the example of the Caspian Sea. Forecasting the level of the Caspian Sea is crucial both for the economy of the region and for the region's environment. The Caspian Sea is a closed reservoir; it is well known that its level changes considerably due to a variety of factors including global climate change. A series of forecasts exists based on different methods and taking into account some of the following factors: the influence of the sun's activity; the atmospheric circulation; the changing shape of the world's ocean; geological phenomena; the river inflow; and the velocity of evaporation. All of these models were calculated based on the linearization of the relations considered. For the last two decades, the most popular model has been the linear stochastic equation of water balance. This model was used as the base of the well known project of reversing the flow of the northward-flowing rivers. But the real behavior of the Caspian Sea contradicted the forecasting done using this model. One of the reasons of the failure was ignorance of the relations mentioned above. We are inclined to think however that the main reason for failure was that the forecast used a linear equation. The goal of the present paper is to analyze and generalize, from the modern mathematical point of view, the forecasting methodology for the level of the Caspian Sea, including the nonlinear effects crucial influence on the dynamics of sea level. In particular, the mathematical problems concerning the nonlinear stochastic equations are considered

Analysis of Affective State as Covariate in Human Gait Identification

There is an increased interest in the need for a noninvasive and nonintrusive biometric identification and recognition system such as Automatic Gait Identification (AGI) due to the rise in crime rates in the US, physical assaults, and global terrorism in public places. AGI, a biometric system based on human gait, can recognize people from a distance and current literature shows that AGI has a 95.75% success rate in a closely controlled laboratory environment. Also, this success rate does not take into consideration the effect of covariate factors such as affective state (mood state); and literature shows that there is a lack of understanding of the effect of affective state on gait biometrics. The purpose of this study was to determine the percent success rate of AGI in an uncontrolled outdoor environment with affective state as the main variable. Affective state was measured using the Profile of Mood State (POMS) scales. Other covariate factors such as footwear or clothes were not considered in this study. The theoretical framework that grounded this study was Murray\u27s theory of total walking cycle. This study included the gait signature of 24 participants from a population of 62 individuals, sampled based on simple random sampling. This quantitative research used empirical methods and a Fourier Series Analysis. Results showed that AGI has a 75% percent success rate in an uncontrolled outdoor environment with affective state. This study contributes to social change by enhancing an understanding of the effect of affective state on gait biometrics for positive identification during and after a crime such as bank robbery when the use of facial identification from a surveillance camera is either not clear or not possible. This may also be used in other countries to detect suicide bombers from a distance

Computationally intensive Value at Risk calculations

Market risks are the prospect of financial losses- or gains- due to unexpected changes in market prices and rates. Evaluating the exposure to such risks is nowadays of primary concern to risk managers in financial and non-financial institutions alike. Until late 1980s market risks were estimated through gap and duration analysis (interest rates), portfolio theory (securities), sensitivity analysis (derivatives) or "what-if" scenarios. However, all these methods either could be applied only to very specific assets or relied on subjective reasoning. --

Pathways to randomness in the economy: Emergent nonlinearity and chaos in economics and finance

This paper: (1) Gives a general argument why research on nonlinear science in general and chaos in particular is important in economics and finance. (2) Puts forth two definitions of stochastic nonlinearity (IID-Linearity and MDS-Linearity) for nonlinear time series analysis and argues for their usefulness as organizing concepts not only for discussion of nonlinearity testinf but also for building a new class of structural asset pricing models. (3) Shows how to use ideas from interacting particle systems theory to build structural asset pricing models that turn IID-Linear or MDS-Linear earnings processes into non MDS-Linear equilibrium returns processes.

Understanding cognitive differences in processing competing visualizations of complex systems

Node-link diagrams are used represent systems having different elements and relationships among the elements. Representing the systems using visualizations like node-link diagrams provides cognitive aid to individuals in understanding the system and effectively managing these systems. Using appropriate visual tools aids in task completion by reducing the cognitive load of individuals in understanding the problems and solving them. However, the visualizations that are currently developed lack any cognitive processing based evaluation. Most of the evaluations (if any) are based on the result of tasks performed using these visualizations. Therefore, the evaluations do not provide any perspective from the point of the cognitive processing required in working with the visualization. This research focuses on understanding the effect of different visualization types and complexities on problem understanding and performance using a visual problem solving task. Two informationally equivalent but visually different visualizations - geon diagrams based on structural object perception theory and UML diagrams based on object modeling - are investigated to understand the cognitive processes that underlie reasoning with different types of visualizations. Specifically, the two visualizations are used to represent interdependent critical infrastructures. Participants are asked to solve a problem using the different visualizations. The effectiveness of the task completion is measured in terms of the time taken to complete the task and the accuracy of the result of the task. The differences in the cognitive processing while using the different visualizations are measured in terms of the search path and the search-steps of the individual. The results from this research underscore the difference in the effectiveness of the different diagrams in solving the same problem. The time taken to complete the task is significantly lower in geon diagrams. The error rate is also significantly lower when using geon diagrams. The search path for UML diagrams is more node-dominant but for geon diagrams is a distribution of nodes, links and components (combinations of nodes and links). Evaluation dominates the search-steps in geon diagrams whereas locating steps dominate UML diagrams. The results also show that the differences in search path and search steps for different visualizations increase when the complexity of the diagrams increase. This study helps to establish the importance of cognitive level understanding of the use of diagrammatic representation of information for visual problem solving. The results also highlight that measures of effectiveness of any visualization should include measuring the cognitive process of individuals while they are doing the visual task apart from the measures of time and accuracy of the result of a visual task

Mediterranean-wide Green Vegetation Abundance for Land Degradation Assessment Derived from AVHRR NDVI and Surface Temperature 1989 to 2005

NOAA AVHRR data stemming from the MEDOKADS archive and ranging from 1989 to 2005 was processed and decomposed into their fractions of the vegetated, non-vegetated and the so called ¿cold¿ endmember. Decomposition occurred via Linear Unmixing within a triangle spanned up by NDVI (y-axis) and surface temperature (x-axis), separately for each of the 612 10-day composites. Endmembers were derived statistically using percentiles and the inverse relationship between NDVI and Ts. The cold endmember was fixed at -20 degrees Celsius, the vegetated endmember at NDVI = 0.7, the latter was then empirically corrected for illumination effects. Linear Unmixing occurred for the whole Mediterranean area, separately for a western and eastern window. Outcomes are the vegetation abundance, soil abundance and ¿cold¿ abundance, indicating the individual coverage of a pixel by each of these. The vegetation abundance was re-scaled to the so-called Grenn Vegetation Fraction (GVF), re-distributing the ¿cold¿ abundance on vegetation and soil abundance proportionally. Unmixing led to a higher stability of GVF data in comparison to NDVI data with regard to atmospheric effects. The data was post-processed for missing values and outliers and it was filtered. The GVF shows close parallelism on several test sites in comparison to a re-scaled NDVI within the endmember limits. The positive effect of the cold abundance, which is amongst other accounting for negative effects from poor atmospheric conditions and which was used to improve the GVF, could be clearly shown. Comparison with high and low resolution SPOT data shows a linear relationship and higher values for GVF. Squared GVF values were found to be closely correlated with independently derived high and low resolution vegetation cover (fCover), confirming this relationship known from literature. Coefficients of determination (R2), slope and offset of linear relations between squared GVF on one side and the two validation data sets on the other side were 0.69, 0.91, 0.07 and 0.58, 1.27, 0.06, respectively. In addition to the ¿per se¿ value of the derived abundances, validation results indicate that squared GVF may be used as approximation for vegetation cover.JRC.H.7-Land management and natural hazard

Mathematical Aspects of Hopfield Models

Diese Dissertation behandelt zwei Modelle aus der statistischen Mechanik ungeordneter Systeme. Beide sind Varianten des Hopfield-Modells und gehören zur Klasse der Molekularfeldmodelle. Im ersten Teil behandeln wir den Fall mit p-Spin-Wechselwirkungen (p größer als 4 und gerade) und superextensiv vielen Mustern (deren Anzahl M wie die p-te Potenz der Systemgröße N wächst), wobei wir zwei verschiedene Energiefunktionen betrachten. Wir beweisen die Existenz einer kritischen Temperatur, bei welcher der sogenannte Replikaüberlapp von Null auf einen strikt positiven Wert springt. Wir geben obere und untere Schranken für ihren Wert an und zeigen, daß für die eine Wahl der Hamiltonfunktion beide gegen die kritische Temperatur (bis auf einen konstanten Faktor) des Random Energy Model konvergieren, falls p gegen Unendlich strebt. Diese kritische Temperatur fällt mit der kleinsten Temperatur zusammen, für welche die ausgeglühte freie Energie und der Erwartungswert der abgeschreckten freien Energie identisch sind. Der Zusammenhang zwischen diesen beiden Resultaten wird durch eine partielle Integrationsformel geliefert, welche mit Hilfe einer Störungsentwicklung der Boltzmannfaktoren bewiesen wird. Außerdem berechnen wir die Fluktuationen der freien Energie und erhalten, daß sie von der Ordnung Quadratwurzel von N sind. Weiterhin beweisen wir die Existenz einer kritischen Proportionalitätskonstanten für die Anzahl Muster, oberhalb welcher das Minimum der Hamiltonfunktion mit großer Wahrscheinlichkeit nicht in der Nähe eines der Muster angenommen wird. Dies bedeutet, daß, obwohl das Gibbsmaß sich bei kleinen Temperaturen auf einer kleinen Teilmenge des Zustandsraumes konzentriert, dies nicht in der Nähe der Muster geschieht. In einem zweiten Teil beweisen wir obere Schranken für die Norm von gewissen zufälligen Matrizen mit abhängigen Einträgen. Diese Abschätzungen werden im ersten Teil zur Berechnung der Fluktuationen der freien Energie benutzt. Sie werden mit der Chebyshev-Markov-Ungleichung, angewandt auf die Spur von hohen Potenzen der Matrizen, bewiesen. Das zentrale Resultat dazu ist eine Darstellung der Spur von diesen hohen Potenzen als Wege auf gewissen bipartiten Graphen. Dies transformiert das Berechnen des Erwartungswertes der Spur auf das kombinatorische Problem, die maximale Anzahl kreisförmiger Teilgraphen eines gegebenen Eulergraphen zu bestimmen. Die Resultate zeigen, dass die Abhängigkeit zwischen den Einträgen eine wichtige Rolle spielt und nicht vernachlässigt werden kann. Im letzten Teil schließlich betrachten wir ein Hopfield-Modell mit zwei Gauß'schen Mustern. Wir zeigen, da$szlig; überabzählbar viele extremale Gibbszustände existieren, welche durch den Einheitskreis indiziert werden. Diese Symmetrie wird zufällig gebrochen im Sinne, daß der Metazustand von einem Kontinuum von Paaren von extremalen Gibbsmaßen getragen wird, welche durch eine globale Spinsymmetrie verknüpft sind. Wir beweisen diese Resultate mit Hilfe der zufälligen Ratenfunktion des induzierten Maßes auf den Überlapparametern. Insbesondere zeigen wir, daß die Symmetriebrechung durch die Fluktuationen der Ratenfunktion auf den (entarteten) Minima ihrer Erwartung erzwungen wird. Diese Fluktuationen werden durch einen zufälligen Prozeß auf dem Einheitskreis beschrieben, dessen globale Minima die Menge (schlussendlich ein Paar) der extremalen Zustände auswählen.This thesis is concerned with two models from equilibrium statistical mechanics of disordered systems. Both of them are variants of the Hopfield model, and belong to the class of mean-field models. In the first part, we treat the case of p-spin interactions (with p larger than 4 and even) and super-extensively many patterns (their number M scaling as the (p-1)th power of the system size N). We consider two choices of the Hamiltonians. We find that there exists a critical temperature, at which the replica overlap changes from 0 to a strictly positive value. We give upper and lower bounds for its value, and show that for one choice of the Hamiltonian, both of them converge as p tends to infinity to the critical temperature (up to a constant factor) of the random energy model. This critical temperature coincides with the minimum temperature for which annealed free energy and mean of the quenched free energy are equal. The relation between the two results is furnished by an integration by parts formula that is proved by perturbative expansion of the Boltzmann factors. We also calculate the fluctuations of the free energy which are shown to be of the order of the square root of the system size N. Furthermore, we find that there exists a critical scaling constant for the number of patterns above which with large probability the minimum of the Hamiltonian is not realized in the vicinity of any of the patterns. This means that while there is a condensation for low temperatures, the Gibbs measure does not concentrate around the patterns. In a second part of the thesis, we prove upper bounds on the norm of certain random matrices with dependent entries. These estimates are used in Part I to prove the fluctuations of the free energy. They are proved by the Chebyshev-Markov inequality, applied to the trace of large powers of the matrices. The key ingredient is a representation of the trace of these large powers in terms of walks on an appropriate bipartite graph. This reduces the calculation of the expectation of the trace to the combinatorial problem of counting the maximum number of sub-circuits of a given circuit. The results show that the dependence between the entries cannot be neglected. Finally, in the last part, we consider a two pattern Hopfield model with Gaussian patterns. We show that there are uncountably many pure states indexed by the circle. This symmetry is randomly broken in the sense that the metastate is supported on a continuum of pairs of pure states that are related by a global (spin-flip) symmetry. We prove these results by studying the random rate function of the induced measure on the overlap parameters. In particular, the breaking of the symmetry is shown to be due to the fluctuations of this rate function at the (degenerate) minima of its expectation. These fluctuations are described by a random process on the circle whose global minima determine the chosen set (eventually a pair) of states