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We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, therefore the whole series, converge to zero much more rapidly compared to the Taylor expansion of the same order. Furthermore, using numerical analysis using sixth-order polynomial expansion, we demonstrate that the Legendre polynomial approximation yields an error at least an order of magnitude smaller than the analogous Taylor series approximation. This strongly suggests that Legendre expansions, instead of Taylor expansions, should be used when global accuracy is important., Comment: 6 pages, 1 figure

Wavelet coefficients are estimated recursively at progressively coarser scales recursively. As a result, the estimation is prone to multiplicative propagation of truncation errors due to quantization and round-off at each stage. Yet, the influence of this propagation on wavelet filter output has not been explored systematically. Through numerical error analysis of a simple, generic sub-band coding scheme with a half-band low pass finite impulse-response filter for down sampling, we show that truncation error in estimated wavelet filter coefficients can quickly reach unacceptable levels, and may render the results unreliable especially at coarser scales., Comment: 9 pages, 4 figures

Many networks in natural and human-made systems exhibit scale-free properties and are small worlds. Now we show that people's understanding of complex systems in their cognitive maps also follow a scale-free topology (P_k = k^-lambda, lambda [1.24,3.03]; r^2 <= 0.95). People focus on a few attributes, as indicated by a fat tail in the probability distribution of total degree. These few attributes are related with many other variables in the system. Many more attributes have very few connections. The scale-free properties in the cognitive maps of people arise despite the fact that their average distances are not different (Wilcoxon sign-rank test, W=78, p=0.75) than random networks of the same size and connection density. The scale-free property manifests itself in the higher hierarchical structure compared to random networks (Wilcoxon sign-rank test, W=12, p=0.03). People use relatively short explanations to describe systems. These findings may help us to better understand people's perceptions, especially when it comes to decision-making, conflict resolution, politics and management., Comment: 8 pages, 2 figures

Ecological systems are governed by complex interactions which are mainly nonlinear. In order to capture this complexity and nonlinearity, statistical models recently gained popularity. However, although these models are commonly applied in ecology, there are no studies to date aiming to assess the applicability and performance. We provide an overview for nature of the wide range of the data sets and predictive variables, from both aquatic and terrestrial ecosystems with different scales of time-dependent dynamics, and the applicability and robustness of predictive modeling methods on such data sets by comparing different statistical modeling approaches. The methods considered k-NN, LDA, QDA, generalized linear models (GLM) feedforward multilayer backpropagation networks and pseudo-supervised network ARTMAP. For ecosystems involving time-dependent dynamics and periodicities whose frequency are possibly less than the time scale of the data considered, GLM and connectionist neural network models appear to be most suitable and robust, provided that a predictive variable reflecting these time-dependent dynamics included in the model either implicitly or explicitly. For spatial data, which does not include any time-dependence comparable to the time scale covered by the data, on the other hand, neighborhood based methods such as k-NN and ARTMAP proved to be more robust than other methods considered in this study. In addition, for predictive modeling purposes, first a suitable, computationally inexpensive method should be applied to the problem at hand a good predictive performance of which would render the computational cost and efforts associated with complex variants unnecessary., Comment: 34 pages, 2 figures. Submitted to Ecological Informatics

We review the use of artificial neural networks, particularly the feedforward multilayer perceptron with back-propagation for training (MLP), in ecological modelling. Overtraining on data or giving vague references to how it was avoided is the major problem. Various methods can be used to determine when to stop training in artificial neural networks: 1) early stopping based on cross-validation, 2) stopping after a analyst defined error is reached or after the error levels off, 3) use of a test data set. We do not recommend the third method as the test data set is then not independent of model development. Many studies used the testing data to optimize the model and training. Although this method may give the best model for that set of data it does not give generalizability or improve understanding of the study system. The importance of an independent data set cannot be overemphasized as we found dramatic differences in model accuracy assessed with prediction accuracy on the training data set, as estimated with bootstrapping, and from use of an independent data set. The comparison of the artificial neural network with a general linear model (GLM) as a standard procedure is recommended because a GLM may perform as well or better than the MLP. MLP models should not be treated as black box models but instead techniques such as sensitivity analyses, input variable relevances, neural interpretation diagrams, randomization tests, and partial derivatives should be used to make the model more transparent, and further our ecological understanding which is an important goal of the modelling process. Based on our experience we discuss how to build a MLP model and how to optimize the parameters and architecture., Comment: 22 pages, 2 figures. Presented in ISEI3 (2002). Ecological Modelling in press

Separate artificial neural network (ANN) models were developed from data in two geographical regions and years apart for a marsh-nesting bird, the red-winged blackbird Agelaius phoeniceus. Each model was independently tested on the spatially and temporally distinct data from the other region to determine how generalizable it was. The first model was developed to predict occurrence of nests in two wetlands on Lake Erie, Ohio in 1995 and 1996. The second model was developed to predict breeding success in two marshes in Connecticut, USA in 1969 and 1970. Independent variables were vegetation durability, stem density, stem/nest height, distance to open water, distance to edge, and water depth. With input variable relevances, sensitivity analyses and neural interpretation diagrams we were able to understand how the different models predicted nest occurrence and breeding success and compare their differences and similarities. Both models also predicted increasing nest occurrence/breeding success with increasing water depth under the nest and increasing distance to edge. However, relationships for prediction differed in the models. Generalizability of the models was poor except when the marshes had similar values of important variables in the model. ANN models performed better than generalized linear models (GLM) on marshes with similar structures. Generalizability of the models did not differ in nest occurrence and breeding success data. Extensive testing also showed that the GLMs were not necessarily more generalizable than ANNs, suggesting that ANN models make good definitions of a study system but are too specific to generalize well to other ecologically complex systems unless input variable distributions are very similar., Comment: 42 pages, 3 figures. Presented in ISEI3 conference (2002). Ecological Modeling in press

Although ARTMAP and ART-based models were introduced in early 70's they were not used in characterizing and classifying ecological observations. ART-based models have been extensively used for classification models based on satellite imagery. This report, to our knowledge, is the first application of ART-based methods and specifically ARTMAP for predicting habitat selection and spatial distribution of species. We compare the performance of ARTMAP to assess the breeding success of three bird species (Lanius senator, Hippolais pallida, and Calandrella brachydactyla) based on multi-spectral satellite imagery and environmental variables. ARTMAP is superior both in terms of performance (percent correctly classified - pcc = 1.00) and generalizability (pcc >0.96) to those of feedforward multilayer backpropogation (>0.87, >0.65), linear and quadratic discriminant analysis (>0.48, >0.46) and k-nearest neighbor (>0.82, >0.66) methods. Compared to other methods, ARTMAP is able to incorporate new observations with far less computational effort and can easily add data to already trained models., Comment: 14 pages, 1 figure. Submitted to Ecology as a Statistical Report