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High Dimensional (high + dimensional)
Selected AbstractsResolving the biodiversity paradoxECOLOGY LETTERS, Issue 8 2007James S. Clark Abstract The paradox of biodiversity involves three elements, (i) mathematical models predict that species must differ in specific ways in order to coexist as stable ecological communities, (ii) such differences are difficult to identify, yet (iii) there is widespread evidence of stability in natural communities. Debate has centred on two views. The first explanation involves tradeoffs along a small number of axes, including ,colonization-competition', resource competition (light, water, nitrogen for plants, including the ,successional niche'), and life history (e.g. high-light growth vs. low-light survival and few large vs. many small seeds). The second view is neutrality, which assumes that species differences do not contribute to dynamics. Clark et al. (2004) presented a third explanation, that coexistence is inherently high dimensional, but still depends on species differences. We demonstrate that neither traditional low-dimensional tradeoffs nor neutrality can resolve the biodiversity paradox, in part by showing that they do not properly interpret stochasticity in statistical and in theoretical models. Unless sample sizes are small, traditional data modelling assures that species will appear different in a few dimensions, but those differences will rarely predict coexistence when parameter estimates are plugged into theoretical models. Contrary to standard interpretations, neutral models do not imply functional equivalence, but rather subsume species differences in stochastic terms. New hierarchical modelling techniques for inference reveal high-dimensional differences among species that can be quantified with random individual and temporal effects (RITES), i.e. process-level variation that results from many causes. We show that this variation is large, and that it stands in for species differences along unobserved dimensions that do contribute to diversity. High dimensional coexistence contrasts with the classical notions of tradeoffs along a few axes, which are often not found in data, and with ,neutral models', which mask, rather than eliminate, tradeoffs in stochastic terms. This mechanism can explain coexistence of species that would not occur with simple, low-dimensional tradeoff scenarios. [source] Accelerating iterative solution methods using reduced-order models as solution predictorsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2006R. Markovinovi Abstract We propose the use of reduced-order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large-scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two-phase flow through heterogeneous porous media. In particular we considered implicit-pressure explicit-saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time-consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time-varying parameters or source terms. Copyright © 2006 John Wiley & Sons, Ltd. [source] Empirical orthogonal functions and related techniques in atmospheric science: A reviewINTERNATIONAL JOURNAL OF CLIMATOLOGY, Issue 9 2007A. Hannachi Abstract Climate and weather constitute a typical example where high dimensional and complex phenomena meet. The atmospheric system is the result of highly complex interactions between many degrees of freedom or modes. In order to gain insight in understanding the dynamical/physical behaviour involved it is useful to attempt to understand their interactions in terms of a much smaller number of prominent modes of variability. This has led to the development by atmospheric researchers of methods that give a space display and a time display of large space-time atmospheric data. Empirical orthogonal functions (EOFs) were first used in meteorology in the late 1940s. The method, which decomposes a space-time field into spatial patterns and associated time indices, contributed much in advancing our knowledge of the atmosphere. However, since the atmosphere contains all sorts of features, e.g. stationary and propagating, EOFs are unable to provide a full picture. For example, EOFs tend, in general, to be difficult to interpret because of their geometric properties, such as their global feature, and their orthogonality in space and time. To obtain more localised features, modifications, e.g. rotated EOFs (REOFs), have been introduced. At the same time, because these methods cannot deal with propagating features, since they only use spatial correlation of the field, it was necessary to use both spatial and time information in order to identify such features. Extended and complex EOFs were introduced to serve that purpose. Because of the importance of EOFs and closely related methods in atmospheric science, and because the existing reviews of the subject are slightly out of date, there seems to be a need to update our knowledge by including new developments that could not be presented in previous reviews. This review proposes to achieve precisely this goal. The basic theory of the main types of EOFs is reviewed, and a wide range of applications using various data sets are also provided. Copyright © 2007 Royal Meteorological Society [source] Reduction and identification methods for Markovian control systems, with application to thin film depositionINTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 2 2004Martha A. Gallivan Abstract Dynamic models of nanometer-scale phenomena often require an explicit consideration of interactions among a large number of atoms or molecules. The corresponding mathematical representation may thus be high dimensional, nonlinear, and stochastic, incompatible with tools in nonlinear control theory that are designed for low-dimensional deterministic equations. We consider here a general class of probabilistic systems that are linear in the state, but whose input enters as a function multiplying the state vector. Model reduction is accomplished by grouping probabilities that evolve together, and truncating states that are unlikely to be accessed. An error bound for this reduction is also derived. A system identification approach that exploits the inherent linearity is then developed, which generates all coefficients in either a full or reduced model. These concepts are then extended to extremely high-dimensional systems, in which kinetic Monte Carlo (KMC) simulations provide the input,output data. This work was motivated by our interest in thin film deposition. We demonstrate the approaches developed in the paper on a KMC simulation of surface evolution during film growth, and use the reduced model to compute optimal temperature profiles that minimize surface roughness. Copyright © 2004 John Wiley & Sons, Ltd. [source] A Landmark Analysis-Based Approach to Age and Sex Classification of the Skull of the Mediterranean Monk Seal (Monachus monachus) (Hermann, 1779)ANATOMIA, HISTOLOGIA, EMBRYOLOGIA, Issue 5 2009C. Brombin Summary This work aimed at applying geometric morphometric analysis techniques to the skull of the Mediterranean monk seal (Monachus monachus, Hermann, 1779). Inferential analyses were performed using a non-parameteric permutation framework based on a series of skulls of different age classes belonging to individuals of both sexes. Our goal was to establish whether a statistical approach based on osteometric measurements and surface analysis of photographs of the left lateral plane of the skull may lead to a different and scientifically sound method of age and sex classification in this critically endangered marine mammal. Our data indicate that non-parametric combination methodology enables the researcher to give local assessment using a combination with domains. Developing geometric morphometric techniques in a non-parametric permutation framework could be useful in solving high dimensional and small sample size problems as well as classification problems, including zoological classification of specimens within a specific population. The Mediterranean monk seal is believed to be the world's rarest pinniped and one of the most endangered mammals of the world, with fewer than 600 individuals currently surviving. The use of shape analysis would allow new insights into the biological characteristics of the monk seal by simply extracting potentially new information on age and size from museal specimens. [source] | |||||||||||||