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Finding Chaos in Noisy Systems
, 1991
"... In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is comp ..."
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Cited by 50 (1 self)
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In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behavior. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (Neural Networks and Thin Plate Splines) it is possible to consistently estimate the LE. The properties of these methods have been studied using simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (18201900). Based on a nonparametric analysis there is little evidence for lowdimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.
Behavior near zero of the distribution of GCV smoothing parameter estimates for splines
 Statistics and Probability Letters
, 1993
"... It has been noticed by several authors that there is a small but nonzero probability that the GCV estimate 2 of the smoothing parameter in spline and related smoothing problems will he extremely small, leading to gross undersmoothing. We obtain an upper bound to the probability that the GCV functio ..."
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Cited by 8 (6 self)
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It has been noticed by several authors that there is a small but nonzero probability that the GCV estimate 2 of the smoothing parameter in spline and related smoothing problems will he extremely small, leading to gross undersmoothing. We obtain an upper bound to the probability that the GCV function, whose minimizer provides,~, has a (possibly local) minimum at 0. This upper bound goes to 0 exponentially fast as the sample size gets large. For the mediumto smallsample case we study this probability both by Monte Carlo evaluation of a formula for the exact probability that the GCV function has a minimum at 0 as well as by replicated calculations of ~..
Bayesian fluorescence in situ hybridisation signal classification
, 2004
"... research has indicated the significance of accurate classification of fluorescence in situ hybridisation (FISH) signals for the detection of genetic abnormalities. Based on welldiscriminating features and a trainable neural network (NN) classifier, a previous system enabled highlyaccurate classifi ..."
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Cited by 5 (0 self)
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research has indicated the significance of accurate classification of fluorescence in situ hybridisation (FISH) signals for the detection of genetic abnormalities. Based on welldiscriminating features and a trainable neural network (NN) classifier, a previous system enabled highlyaccurate classification of valid signals and artefacts of two fluorophores. However, since this system employed several features that are considered independent, the naive Bayesian classifier (NBC) is suggested here as an alternative to the NN. The NBC independence assumption permits the decomposition of the highdimensional likelihood of the model for the data into a product of onedimensional probability densities. The naive independence assumption together with the Bayesian methodology allow the NBC to predict a posteriori probabilities of class membership using estimated classconditional densities in a close and simple form. Since the probability densities are the only parameters of the NBC, the misclassification rate of the model is determined exclusively by the quality of density estimation. Densities are evaluated by three methods: single Gaussian estimation (SGE; parametric method), Gaussian mixture model assuming spherical covariance matrices (GMM; semiparametric method) and kernel density estimation (KDE; nonparametric method). For lowdimensional densities, the GMM generally outperforms the KDE that tends to overfit the training set
A Full Bayesian Approach to Curve and Surface Reconstruction
, 1999
"... When interpolating incomplete data, one can choose a parametric model, or opt for a more general approach and use a nonparametric model which allows a very large class of interpolants. A popular nonparametric model for interpolating various types of data is based on regularization, which looks for ..."
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Cited by 4 (2 self)
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When interpolating incomplete data, one can choose a parametric model, or opt for a more general approach and use a nonparametric model which allows a very large class of interpolants. A popular nonparametric model for interpolating various types of data is based on regularization, which looks for an interpolant that is both close to the data and also "smooth" in some sense. Formally, this interpolant is obtained by minimizing an error functional which is the weighted sum of a "fidelity term" and a "smoothness term".