Describing long-term trends in precipitation using generalized additive models

With the current concern over climate change, descriptions of how rainfall patterns are changing over time can be useful. Observations of daily rainfall data over the last few decades provide information on these trends. Generalized linear models are typically used to model patterns in the occurrence and intensity of rainfall. These models describe rainfall patterns for an average year but are more limited when describing long-term trends, particularly when these are potentially non-linear. Generalized additive models (GAMS) provide a framework for modelling non-linear relationships by fitting smooth functions to the data. This paper describes how GAMS can extend the flexibility of models to describe seasonal patterns and long-term trends in the occurrence and intensity of daily rainfall using data from Mauritius from 1962 to 2001. Smoothed estimates from the models provide useful graphical descriptions of changing rainfall patterns over the last 40 years at this location. GAMS are particularly helpful when exploring non-linear relationships in the data. Care is needed to ensure the choice of smooth functions is appropriate for the data and modelling objectives. (c) 2008 Elsevier B.V. All rights reserved.

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller-Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated.

Diagnosis of variability and trends in a global precipitation dataset using a physically motivated statistical model

A physically motivated statistical model is used to diagnose variability and trends in wintertime ( October - March) Global Precipitation Climatology Project (GPCP) pentad (5-day mean) precipitation. Quasi-geostrophic theory suggests that extratropical precipitation amounts should depend multiplicatively on the pressure gradient, saturation specific humidity, and the meridional temperature gradient. This physical insight has been used to guide the development of a suitable statistical model for precipitation using a mixture of generalized linear models: a logistic model for the binary occurrence of precipitation and a Gamma distribution model for the wet day precipitation amount. The statistical model allows for the investigation of the role of each factor in determining variations and long-term trends. Saturation specific humidity q(s) has a generally negative effect on global precipitation occurrence and with the tropical wet pentad precipitation amount, but has a positive relationship with the pentad precipitation amount at mid- and high latitudes. The North Atlantic Oscillation, a proxy for the meridional temperature gradient, is also found to have a statistically significant positive effect on precipitation over much of the Atlantic region. Residual time trends in wet pentad precipitation are extremely sensitive to the choice of the wet pentad threshold because of increasing trends in low-amplitude precipitation pentads; too low a choice of threshold can lead to a spurious decreasing trend in wet pentad precipitation amounts. However, for not too small thresholds, it is found that the meridional temperature gradient is an important factor for explaining part of the long-term trend in Atlantic precipitation.

A family study of co-morbidity between generalized social phobia and generalized anxiety disorder in a non-clinic sample

Background: High rates of co-morbidity between Generalized Social Phobia (GSP) and Generalized Anxiety Disorder (GAD) have been documented. The reason for this is unclear. Family studies are one means of clarifying the nature of co-morbidity between two disorders. Methods: Six models of co-morbidity between GSP and GAD were investigated in a family aggregation study of 403 first-degree relatives of non-clinical probands: 37 with GSP, 22 with GAD, 15 with co-morbid GSP/GAD, and 41 controls with no history of GSP or GAD. Psychiatric data were collected for probands and relatives. Mixed methods (direct and family history interviews) were utilised. Results: Primary contrasts (against controls) found an increased rate of pure GSP in the relatives of both GSP probands and co-morbid GSP/GAD probands, and found relatives of co-morbid GSP/GAD probands to have an increased rate of both pure GAD and comorbid GSP/GAD. Secondary contrasts found (i) increased GSP in the relatives of GSP only probands compared to the relatives of GAD only probands; and (ii) increased GAD in the relatives of co-morbid GSP/GAD probands compared to the relatives of GSP only probands. Limitations: The study did not directly interview all relatives, although the reliability of family history data was assessed. The study was based on an all-female proband sample. The implications of both these limitations are discussed. Conclusions: The results were most consistent with a co-morbidity model indicating independent familial transmission of GSP and GAD. This has clinical implications for the treatment of patients with both disorders. (C) 2006 Elsevier B.V. All fights reserved.

Identification of nonlinear systems using generalized kernel models

Nonlinear system identification is considered using a generalized kernel regression model. Unlike the standard kernel model, which employs a fixed common variance for all the kernel regressors, each kernel regressor in the generalized kernel model has an individually tuned diagonal covariance matrix that is determined by maximizing the correlation between the training data and the regressor using a repeated guided random search based on boosting optimization. An efficient construction algorithm based on orthogonal forward regression with leave-one-out (LOO) test statistic and local regularization (LR) is then used to select a parsimonious generalized kernel regression model from the resulting full regression matrix. The proposed modeling algorithm is fully automatic and the user is not required to specify any criterion to terminate the construction procedure. Experimental results involving two real data sets demonstrate the effectiveness of the proposed nonlinear system identification approach.

Thermodynamics of climate change: generalized sensitivities

Using a recent theoretical approach, we study how global warming impacts the thermodynamics of the climate system by performing experiments with a simplified yet Earth-like climate model. The intensity of the Lorenz energy cycle, the Carnot efficiency, the material entropy production, and the degree of irreversibility of the system change monotonically with the CO2 concentration. Moreover, these quantities feature an approximately linear behaviour with respect to the logarithm of the CO2 concentration in a relatively wide range. These generalized sensitivities suggest that the climate becomes less efficient, more irreversible, and features higher entropy production as it becomes warmer, with changes in the latent heat fluxes playing a predominant role. These results may be of help for explaining recent findings obtained with state of the art climate models regarding how increases in CO2 concentration impact the vertical stratification of the tropical and extratropical atmosphere and the position of the storm tracks.

A neural network enhanced generalized minimum variance self-tuning proportional, integral and derivative control algorithm for complex dynamic systems

A neural network enhanced proportional, integral and derivative (PID) controller is presented that combines the attributes of neural network learning with a generalized minimum-variance self-tuning control (STC) strategy. The neuro PID controller is structured with plant model identification and PID parameter tuning. The plants to be controlled are approximated by an equivalent model composed of a simple linear submodel to approximate plant dynamics around operating points, plus an error agent to accommodate the errors induced by linear submodel inaccuracy due to non-linearities and other complexities. A generalized recursive least-squares algorithm is used to identify the linear submodel, and a layered neural network is used to detect the error agent in which the weights are updated on the basis of the error between the plant output and the output from the linear submodel. The procedure for controller design is based on the equivalent model, and therefore the error agent is naturally functioned within the control law. In this way the controller can deal not only with a wide range of linear dynamic plants but also with those complex plants characterized by severe non-linearity, uncertainties and non-minimum phase behaviours. Two simulation studies are provided to demonstrate the effectiveness of the controller design procedure.

Generalized neurofuzzy network modeling algorithms using Bezier-Bernstein polynomial functions and additive decomposition

This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bezier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bezier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bezier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bezier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.

Generalized beta generated distributions

This article introduces generalized beta-generated (GBG) distributions. Sub-models include all classical beta-generated, Kumaraswamy-generated and exponentiated distributions. They are maximum entropy distributions under three intuitive conditions, which show that the classical beta generator skewness parameters only control tail entropy and an additional shape parameter is needed to add entropy to the centre of the parent distribution. This parameter controls skewness without necessarily differentiating tail weights. The GBG class also has tractable properties: we present various expansions for moments, generating function and quantiles. The model parameters are estimated by maximum likelihood and the usefulness of the new class is illustrated by means of some real data sets.

Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

Does increased hydrochemical model complexity decrease robustness?

The aim of this study was, within a sensitivity analysis framework, to determine if additional model complexity gives a better capability to model the hydrology and nitrogen dynamics of a small Mediterranean forested catchment or if the additional parameters cause over-fitting. Three nitrogen-models of varying hydrological complexity were considered. For each model, general sensitivity analysis (GSA) and Generalized Likelihood Uncertainty Estimation (GLUE) were applied, each based on 100,000 Monte Carlo simulations. The results highlighted the most complex structure as the most appropriate, providing the best representation of the non-linear patterns observed in the flow and streamwater nitrate concentrations between 1999 and 2002. Its 5% and 95% GLUE bounds, obtained considering a multi-objective approach, provide the narrowest band for streamwater nitrogen, which suggests increased model robustness, though all models exhibit periods of inconsistent good and poor fits between simulated outcomes and observed data. The results confirm the importance of the riparian zone in controlling the short-term (daily) streamwater nitrogen dynamics in this catchment but not the overall flux of nitrogen from the catchment. It was also shown that as the complexity of a hydrological model increases over-parameterisation occurs, but the converse is true for a water quality model where additional process representation leads to additional acceptable model simulations. Water quality data help constrain the hydrological representation in process-based models. Increased complexity was justifiable for modelling river-system hydrochemistry. Increased complexity was justifiable for modelling river-system hydrochemistry.

Controls on inorganic nitrogen leaching from Finnish catchments assessed using a sensitivity and uncertainty analysis of the INCA-N model

The semi-distributed, dynamic INCA-N model was used to simulate the behaviour of dissolved inorganic nitrogen (DIN) in two Finnish research catchments. Parameter sensitivity and model structural uncertainty were analysed using generalized sensitivity analysis. The Mustajoki catchment is a forested upstream catchment, while the Savijoki catchment represents intensively cultivated lowlands. In general, there were more influential parameters in Savijoki than Mustajoki. Model results were sensitive to N-transformation rates, vegetation dynamics, and soil and river hydrology. Values of the sensitive parameters were based on long-term measurements covering both warm and cold years. The highest measured DIN concentrations fell between minimum and maximum values estimated during the uncertainty analysis. The lowest measured concentrations fell outside these bounds, suggesting that some retention processes may be missing from the current model structure. The lowest concentrations occurred mainly during low flow periods; so effects on total loads were small.

Calibration of structure in a distributed forecasting model for a semiarid flash flood: dynamic surface storage and channel roughness

Flash floods pose a significant danger for life and property. Unfortunately, in arid and semiarid environment the runoff generation shows a complex non-linear behavior with a strong spatial and temporal non-uniformity. As a result, the predictions made by physically-based simulations in semiarid areas are subject to great uncertainty, and a failure in the predictive behavior of existing models is common. Thus better descriptions of physical processes at the watershed scale need to be incorporated into the hydrological model structures. For example, terrain relief has been systematically considered static in flood modelling at the watershed scale. Here, we show that the integrated effect of small distributed relief variations originated through concurrent hydrological processes within a storm event was significant on the watershed scale hydrograph. We model these observations by introducing dynamic formulations of two relief-related parameters at diverse scales: maximum depression storage, and roughness coefficient in channels. In the final (a posteriori) model structure these parameters are allowed to be both time-constant or time-varying. The case under study is a convective storm in a semiarid Mediterranean watershed with ephemeral channels and high agricultural pressures (the Rambla del Albujón watershed; 556 km 2 ), which showed a complex multi-peak response. First, to obtain quasi-sensible simulations in the (a priori) model with time-constant relief-related parameters, a spatially distributed parameterization was strictly required. Second, a generalized likelihood uncertainty estimation (GLUE) inference applied to the improved model structure, and conditioned to observed nested hydrographs, showed that accounting for dynamic relief-related parameters led to improved simulations. The discussion is finally broadened by considering the use of the calibrated model both to analyze the sensitivity of the watershed to storm motion and to attempt the flood forecasting of a stratiform event with highly different behavior.

Exact Search Directions for Optimization of Linear and Nonlinear Models Based on Generalized Orthonormal Functions

A novel technique for selecting the poles of orthonormal basis functions (OBF) in Volterra models of any order is presented. It is well-known that the usual large number of parameters required to describe the Volterra kernels can be significantly reduced by representing each kernel using an appropriate basis of orthonormal functions. Such a representation results in the so-called OBF Volterra model, which has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a nonlinear static mapping given by the Volterra polynomial series. Aiming at optimizing the poles that fully parameterize the orthonormal bases, the exact gradients of the outputs of the orthonormal filters with respect to their poles are computed analytically by using a back-propagation-through-time technique. The expressions relative to the Kautz basis and to generalized orthonormal bases of functions (GOBF) are addressed; the ones related to the Laguerre basis follow straightforwardly as a particular case. The main innovation here is that the dynamic nature of the OBF filters is fully considered in the gradient computations. These gradients provide exact search directions for optimizing the poles of a given orthonormal basis. Such search directions can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into account the error of estimation of the system output. The Levenberg-Marquardt algorithm is adopted here as the optimization procedure. Unlike previous related work, the proposed approach relies solely on input-output data measured from the system to be modeled, i.e., no information about the Volterra kernels is required. Examples are presented to illustrate the application of this approach to the modeling of dynamic systems, including a real magnetic levitation system with nonlinear oscillatory behavior.

A new family of generalized distributions

Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79-88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix `Kw`) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with an application to real data.