The effect of round-off error on long memory processes

We study how the round-off (or discretization) error changes the statistical properties of a Gaussian long memory process. We show that the autocovariance and the spectral density of the discretized process are asymptotically rescaled by a factor smaller than one, and we compute exactly this scaling factor. Consequently, we find that the discretized process is also long memory with the same Hurst exponent as the original process. We consider the properties of two estimators of the Hurst exponent, namely the local Whittle (LW) estimator and the Detrended Fluctuation Analysis (DFA). By using analytical considerations and numerical simulations we show that, in presence of round-off error, both estimators are severely negatively biased in finite samples. Under regularity conditions we prove that the LW estimator applied to discretized processes is consistent and asymptotically normal. Moreover, we compute the asymptotic properties of the DFA for a generic (i.e. non Gaussian) long memory process and we apply the result to discretized processes.Comment: 44 pages, 4 figures, 4 table

BioPhysConnectoR: Connecting Sequence Information and Biophysical Models

<p>Abstract</p> <p>Background</p> <p>One of the most challenging aspects of biomolecular systems is the understanding of the coevolution in and among the molecule(s).</p> <p>A complete, theoretical picture of the selective advantage, and thus a functional annotation, of (co-)mutations is still lacking. Using sequence-based and information theoretical inspired methods we can identify coevolving residues in proteins without understanding the underlying biophysical properties giving rise to such coevolutionary dynamics. Detailed (atomistic) simulations are prohibitively expensive. At the same time reduced molecular models are an efficient way to determine the reduced dynamics around the native state. The combination of sequence based approaches with such reduced models is therefore a promising approach to annotate evolutionary sequence changes.</p> <p>Results</p> <p>With the <monospace>R</monospace> package <monospace>BioPhysConnectoR</monospace> we provide a framework to connect the information theoretical domain of biomolecular sequences to biophysical properties of the encoded molecules - derived from reduced molecular models. To this end we have integrated several fragmented ideas into one single package ready to be used in connection with additional statistical routines in <monospace>R</monospace>. Additionally, the package leverages the power of modern multi-core architectures to reduce turn-around times in evolutionary and biomolecular design studies. Our package is a first step to achieve the above mentioned annotation of coevolution by reduced dynamics around the native state of proteins.</p> <p>Conclusions</p> <p><monospace>BioPhysConnectoR</monospace> is implemented as an <monospace>R</monospace> package and distributed under GPL 2 license. It allows for efficient and perfectly parallelized functional annotation of coevolution found at the sequence level.</p

Trendit poistavan fluktuaatioanalyysin edistyneet menetelmät ja niiden sovellukset laskennallisessa kardiologiassa

Fractals are ubiquitous in nature. A defining characteristic of fractality is self-similarity; the phenomenon looks similar when observed at multiple scales, which implies the existence of a power law scaling relation. Detrended fluctuation analysis (DFA) is a popular tool for studying these fractal scaling relations. Power laws become linear relationships in logarithmic scales, and conventionally these scaling exponents are determined by simple linear regression in approximately linear regions in doubly logarithmic plots. However, in practice the scaling is hardly ever exact, and its behavior may often vary at different scales. This thesis extends the fluctuation analysis by introducing robust tools for determining these scaling exponents. A method based on the Kalman smoother is utilized for extracting a whole spectrum of exponents as a function of the scale. The method is parameter-free and resistant to statistical noise, which distinguishes it from prior efforts for determining such local scale exponents. Additionally, an optimization scheme is presented to obtain data-adaptive segmentation of approximately linear regimes. Based on integer linear programming model, the procedure may readily be customized for various purposes. This versatility is demonstrated by applying the method to a group of data to find a common segmentation that is particularly well-suited for machine learning applications. First, the methods are are employed in exploring the details of the scaling by analyzing simulated data with known scaling properties. These findings provide insight into the interpretation of earlier results. Second, the methods are applied to the study of heart rate variability. The beating of the heart follows fractal-like patterns, and deviations in these complex variations may be indicative of cardiac diseases. In this context DFA is traditionally performed by extracting two scaling exponents, for short- and long-range correlations, respectively. This has been criticized as an oversimplification, which is corroborated by the results of this thesis. The heart rate exhibits richer fractal-like variability, which becomes apparent in the full scaling spectra. The additional information provided by these methods facilitate improved classification of cardiac conditions.Fraktaaleja esiintyy kaikkialla luonnossa. Fraktaalisuuden ominaispiirre on itsesimilaarisuus, eli ilmiö näyttää samankaltaiselta, kun sitä tarkastellaan useassa eri skaalassa. Tämä johtaa siihen, että ilmiön skaalautuvuus noudattaa potenssilakia. Tällaisia fraktaalisia skaalausrelaatioita voidaan tutkia trendit poistavan fluktuaatioanalyysin avulla (Detrended Fluctuation Analysis, DFA). Logaritminen skaala muuntaa potenssilait lineaarisiksi riippuvuuksiksi, ja tavallisesti skaalauseksponentit määritetään logaritmisista kuvaajista lineaarisen regression avulla. Kuitenkaan skaalautuvuus ei lähes koskaan ole täydellistä, ja se voi myös muuttua eri skaaloilla. Tämä työ laajentaa fluktuaatioanalyysia esittelemällä paranneltuja menetelmiä näiden skaalauseksponenttien määrittämiseen. Kokonainen spektri skaalauseksponentteja skaalan funktiona määritetään hyödyntämällä Kalman-suodinta. Tämän menetelmän etuja verrattuna aikaisempiin tapoihin määrittää paikallisia skaalauseksponentteja ovat sen parametrivapaa esitys ja vakaus myös kohinaisissa tapauksissa. Lisäksi esitetään lineaariseen kokonaislukuoptimointiin perustuva menetelmä, jonka avulla skaalautuvuudessa voidaan erottaa alueita, jotka noudattavat likimäärin potenssilakia. Tämän mallin muokkaaminen eri tarpeisiin on myös suoraviivaista. Mallia sovelletaan yhteisen segmentaation etsimiseksi datajoukolle, mikä on tarpeen erityisesti koneoppimisen menetelmiä varten. Esitettyjen menetelmien avulla tutkitaan ensin simuloituja prosesseja, joiden teoreettinen skaalautuminen tunnetaan. Menetelmien mahdollistama yksityiskohtainen analyysi selittää aikaisempia havaintoja DFA:n käyttäytymisessä. Menetelmiä sovelletaan myös sykevälivaihtelun fraktaalianalyysiin. Terveen sydämen sykeväleissä on fraktaalisia piirteitä, joita eri sairaudet muokkaavat ja hävittävät. Sykevälivaihtelun fraktaalisuutta on perinteisesti kuvattu lyhyen- ja pitkän kantaman skaalauseksponenteilla. Tätä kahden eksponentin mallia on kritisoitu riittämättömäksi, ja tämän työn tulokset vahvistavat tätä näkökulmaa. Skaalauseksponenttispektri paljastaa, että sykevälivaihtelun fraktaalisuus on kahden eksponentin mallia monimuotoisempaa. Esitetyillä menetelmillä saatava lisäinformaatio mahdollistaa aikaisempaa tarkemman sydänsairauksien luokittelun

Dynamics of Macrosystems; Proceedings of a Workshop, September 3-7, 1984

There is an increasing awareness of the important and persuasive role that instability and random, chaotic motion play in the dynamics of macrosystems. Further research in the field should aim at providing useful tools, and therefore the motivation should come from important questions arising in specific macrosystems. Such systems include biochemical networks, genetic mechanisms, biological communities, neutral networks, cognitive processes and economic structures. This list may seem heterogeneous, but there are similarities between evolution in the different fields. It is not surprising that mathematical methods devised in one field can also be used to describe the dynamics of another. IIASA is attempting to make progress in this direction. With this aim in view this workshop was held at Laxenburg over the period 3-7 September 1984. These Proceedings cover a broad canvas, ranging from specific biological and economic problems to general aspects of dynamical systems and evolutionary theory

Higher-order dynamics on complex networks

L’estudi de les xarxes complexes ha esdevingut un nou paradigma a l’hora d’entendre i modelar sistemes físics. Uns dels principals punts d’interès són les dinàmiques que hi podem modelar. Però com en tot model, la quantitat de informació que podem representar-hi està limitada per la seva complexitat. La motivació principal d’aquesta tesi és l’estudi de l’efecte que un increment de la complexitat estructural, relacional i temporal té sobre tres importants àrees d’estudi: l’evolució de la cooperació, la propagació de malalties, i l’estudi de la mobilitat humana. En aquest treball hem utilitzat dilemes socials per estudiar com evoluciona la cooperació dins d’una població. Incrementant l’ordre de complexitat estructural de les xarxes, permetent que els individus és puguin relacionar en diferents contextos socials, s’ha mostrat cabdal a l’hora d’explicar algunes característiques sobre l’aparició de comportaments altruistes. Utilitzant aquestes noves estructures, les xarxes multicapa, permetem als membres de la població cooperar en determinat contextos i de no fer-ho en d’altres i això, com analíticament demostrem, augmenta l’espectre d’escenaris allà on cooperació i defecció poden sobreviure. Seguidament, estudiem els models de propagació de malalties des de el punt de vista dels enllaços entre individus. Amb aquest augment de la complexitat relacional dels models epidèmics, aconseguim extreure informació que ens permet, entre altres coses, definir una mesura d’influència d’un enllaç a la propagació de l’epidèmia. Utilitzem aquest fet per a proposar una nova mesura de contenció, basada en l’eliminació dels enllaços més influents, que es mostra més eficient que altres mètodes previs. Finalment, proposem un mètode per a descriure la mobilitat que permet capturar patrons recurrents i heterogeneïtats en els temps que els individus estan en un lloc abans de desplaçar-se a un altre. Aquestes propietats són intrínseques a la mobilitat humana i el fet de poder-les capturar, tot i el cost d’augmentar l’ordre temporal, és crític, com demostrem, a l’hora de modelar com les epidèmies és difonen per mitja del moviment de les persones.El estudio de redes complejas se ha convertido en un nuevo paradigma para comprender y modelar sistemas físicos. Uno de los principales puntos de interés son las dinámicas que podemos modelar. Pero como en todo modelo, la cantidad de información que podemos representar está limitada por su complejidad. La motivación principal de esta tesis es estudiar el efecto que un incremento de la complejidad estructural, relacional y temporal tiene sobre tres importantes áreas de estudio: la evolución de la cooperación, la propagación de enfermedades, y el estudio de la movilidad humana. En este trabajo hemos utilizado dilemas sociales para estudiar cómo evoluciona la cooperación dentro de una población. Incrementando el orden de complejidad estructural de las redes, permitiendo que los individuos se puedan relacionar en diferentes contextos sociales, se ha demostrado capital para explicar algunas de las características sobre la aparición de comportamientos altruistas. Utilizando estas nuevas estructuras, las redes multicapa, permitimos a los miembros de la población cooperar en determinados contextos y no hacerlo en otros, con lo que, como demostramos analíticamente, aumenta el espectro de escenarios en los que la cooperación y la defección pueden sobrevivir. A continuación, estudiamos modelos de propagación de enfermedades desde el punto de vista de los enlaces entre individuos. Con este aumento de complejidad relacional de los modelos epidémicos, conseguimos extraer información que nos permite, entre otras cosas, definir una medida de contención, basada en la eliminación de los enlaces más influyentes, que se muestra más eficaz que otros métodos previos. Finalmente, proponemos un método para describir la movilidad que permite capturar patrones recurrentes y heterogeneidades en los tiempos que los individuos están en un lugar antes de desplazarse a otro. Estas propiedades son intrínsecas a la movilidad humana y el hecho de poder capturarlas, a pesar de incrementar el orden temporal, es crítico, como demostramos, para modelar cómo las epidemias se difunden por medio del movimiento de las personas.The study of complex networks has become a new paradigm to understand and model physical systems. One of the points of interest is the dynamics that we can model. However, as with any model, the amount of information that we can represent is limited by its complexity. The primary motivation of this thesis is the study of the effect that an increase in structural, relational and temporal complexity has on three critical areas of study: the evolution of cooperation, epidemic spreading and human mobility. In this work, we have used social dilemmas to study how cooperation within a population evolves. Increasing the order of structural complexity of the networks, allowing individuals to interact in different social contexts, has shown to be crucial to explain some features about the emergence of altruistic behaviors. Using these new structures, multilayer networks, we allow members of the population to cooperate in specific contexts and defect in others, and this, as we analytically demonstrate, increases the spectrum of scenarios where both strategies can survive. Next, we study the models of epidemic spreading from the point of view of the links between individuals. With this increase in the relational complexity of the epidemic models, we can extract information that allows us, among other things, to define a measure of the contribution of a link to the spreading. We use this metric to propose a new containment measure, based on the elimination of the most influential links, which is more effective than other previous methods. Finally, we propose a method to describe mobility that allows capturing recurrent and heterogeneous patterns in the times that individuals stay in a place before moving to another. These properties are intrinsic to human mobility, and the fact of being able to capture them, despite the cost of increasing the temporal order is critical, as we demonstrate, when it comes to modeling how epidemics spread through the movement of the people