Adaptive model selection using empirical complexities

Given $n$ independent replicates of a jointly distributed pair $(X,Y)\in {\cal R}^d \times {\cal R}$, we wish to select from a fixed sequence of model classes ${\cal F}_1, {\cal F}_2, \ldots$ a deterministic prediction rule $f: {\cal R}^d \to {\cal R}$ whose risk is small. We investigate the possibility of empirically assessingthe {\em complexity} of each model class, that is, the actual difficulty of the estimation problem within each class. The estimated complexities are in turn used to define an adaptive model selection procedure, which is based on complexity penalized empirical risk.The available data are divided into two parts. The first is used to form an empirical cover of each model class, and the second is used to select a candidate rule from each cover based on empirical risk. The covering radii are determined empirically to optimize a tight upper bound on the estimation error. An estimate is chosen from the list of candidates in order to minimize the sum of class complexity and empirical risk. A distinguishing feature of the approach is that the complexity of each model class is assessed empirically, based on the size of its empirical cover.Finite sample performance bounds are established for the estimates, and these bounds are applied to several non-parametric estimation problems. The estimates are shown to achieve a favorable tradeoff between approximation and estimation error, and to perform as well as if the distribution-dependent complexities of the model classes were known beforehand. In addition, it is shown that the estimate can be consistent,and even possess near optimal rates of convergence, when each model class has an infinite VC or pseudo dimension.For regression estimation with squared loss we modify our estimate to achieve a faster rate of convergence.

Variance reduction methods for simulation of densities on Wiener space

We develop a general error analysis framework for the Monte Carlo simulationof densities for functionals in Wiener space. We also study variancereduction methods with the help of Malliavin derivatives. For this, wegive some general heuristic principles which are applied to diffusionprocesses. A comparison with kernel density estimates is made.

Model selection and error estimation

We study model selection strategies based on penalized empirical loss minimization. We point out a tight relationship between error estimation and data-based complexity penalization: any good error estimate may be converted into a data-based penalty function and the performance of the estimate is governed by the quality of the error estimate. We consider several penalty functions, involving error estimates on independent test data, empirical {\sc vc} dimension, empirical {\sc vc} entropy, andmargin-based quantities. We also consider the maximal difference between the error on the first half of the training data and the second half, and the expected maximal discrepancy, a closely related capacity estimate that can be calculated by Monte Carlo integration. Maximal discrepancy penalty functions are appealing for pattern classification problems, since their computation is equivalent to empirical risk minimization over the training data with some labels flipped.

Random measurement error and regression dilution bias.

Summary points: - The bias introduced by random measurement error will be different depending on whether the error is in an exposure variable (risk factor) or outcome variable (disease) - Random measurement error in an exposure variable will bias the estimates of regression slope coefficients towards the null - Random measurement error in an outcome variable will instead increase the standard error of the estimates and widen the corresponding confidence intervals, making results less likely to be statistically significant - Increasing sample size will help minimise the impact of measurement error in an outcome variable but will only make estimates more precisely wrong when the error is in an exposure variable

Incremental neural networks for function approximation

A new strategy for incremental building of multilayer feedforward neural networks is proposed in the context of approximation of functions from R-p to R-q using noisy data. A stopping criterion based on the properties of the noise is also proposed. Experimental results for both artificial and real data are performed and two alternatives of the proposed construction strategy are compared.

A fully adaptive numerical approximation for a two-dimensional epidemic model with nonlinear cross-diffusion

An epidemic model is formulated by a reactionâeuro"diffusion system where the spatial pattern formation is driven by cross-diffusion. The reaction terms describe the local dynamics of susceptible and infected species, whereas the diffusion terms account for the spatial distribution dynamics. For both self-diffusion and cross-diffusion, nonlinear constitutive assumptions are suggested. To simulate the pattern formation two finite volume formulations are proposed, which employ a conservative and a non-conservative discretization, respectively. An efficient simulation is obtained by a fully adaptive multiresolution strategy. Numerical examples illustrate the impact of the cross-diffusion on the pattern formation.

Spatio-temporal Brain Dynamics Mediating Post-error Behavioral Adjustments.

Optimal behavior relies on flexible adaptation to environmental requirements, notably based on the detection of errors. The impact of error detection on subsequent behavior typically manifests as a slowing down of RTs following errors. Precisely how errors impact the processing of subsequent stimuli and in turn shape behavior remains unresolved. To address these questions, we used an auditory spatial go/no-go task where continual feedback informed participants of whether they were too slow. We contrasted auditory-evoked potentials to left-lateralized go and right no-go stimuli as a function of performance on the preceding go stimuli, generating a 2 × 2 design with "preceding performance" (fast hit [FH], slow hit [SH]) and stimulus type (go, no-go) as within-subject factors. SH trials yielded SH trials on the following trials more often than did FHs, supporting our assumption that SHs engaged effects similar to errors. Electrophysiologically, auditory-evoked potentials modulated topographically as a function of preceding performance 80-110 msec poststimulus onset and then as a function of stimulus type at 110-140 msec, indicative of changes in the underlying brain networks. Source estimations revealed a stronger activity of prefrontal regions to stimuli after successful than error trials, followed by a stronger response of parietal areas to the no-go than go stimuli. We interpret these results in terms of a shift from a fast automatic to a slow controlled form of inhibitory control induced by the detection of errors, manifesting during low-level integration of task-relevant features of subsequent stimuli, which in turn influences response speed.

A new estimator to minimize the error due to breathing in the measurement of respiratory impedance

This paper presents a new respiratory impedance estimator to minimize the error due to breathing. Its practical reliability was evaluated in a simulation using realistic signals. These signals were generated by superposing pressure and flow records obtained in two conditions: 1) when applying forced oscillation to a resistance- inertance- elastance (RIE) mechanical model; 2) when healthy subjects breathed through the unexcited forced oscillation generator. Impedances computed (4-32 Hz) from the simulated signals with the new estimator resulted in a mean value which was scarcely biased by the added breathing (errors less than 1 percent in the mean R, I , and E ) and had a small variability (coefficients of variation of R, I, and E of 1.3, 3.5, and 9.6 percent, respectively). Our results suggest that the proposed estimator reduces the error in measurement of respiratory impedance without appreciable extracomputational cost.

Local and global error models to improve uncertainty quantification

In groundwater applications, Monte Carlo methods are employed to model the uncertainty on geological parameters. However, their brute-force application becomes computationally prohibitive for highly detailed geological descriptions, complex physical processes, and a large number of realizations. The Distance Kernel Method (DKM) overcomes this issue by clustering the realizations in a multidimensional space based on the flow responses obtained by means of an approximate (computationally cheaper) model; then, the uncertainty is estimated from the exact responses that are computed only for one representative realization per cluster (the medoid). Usually, DKM is employed to decrease the size of the sample of realizations that are considered to estimate the uncertainty. We propose to use the information from the approximate responses for uncertainty quantification. The subset of exact solutions provided by DKM is then employed to construct an error model and correct the potential bias of the approximate model. Two error models are devised that both employ the difference between approximate and exact medoid solutions, but differ in the way medoid errors are interpolated to correct the whole set of realizations. The Local Error Model rests upon the clustering defined by DKM and can be seen as a natural way to account for intra-cluster variability; the Global Error Model employs a linear interpolation of all medoid errors regardless of the cluster to which the single realization belongs. These error models are evaluated for an idealized pollution problem in which the uncertainty of the breakthrough curve needs to be estimated. For this numerical test case, we demonstrate that the error models improve the uncertainty quantification provided by the DKM algorithm and are effective in correcting the bias of the estimate computed solely from the MsFV results. The framework presented here is not specific to the methods considered and can be applied to other combinations of approximate models and techniques to select a subset of realizations

On the consequences of misspecifing assumptions concerning residuals distribution in a repeated measures and nonlinear mixed modelling context

In this paper we describe the results of a simulation study performed to elucidate the robustness of the Lindstrom and Bates (1990) approximation method under non-normality of the residuals, under different situations. Concerning the fixed effects, the observed coverage probabilities and the true bias and mean square error values, show that some aspects of this inferential approach are not completely reliable. When the true distribution of the residuals is asymmetrical, the true coverage is markedly lower than the nominal one. The best results are obtained for the skew normal distribution, and not for the normal distribution. On the other hand, the results are partially reversed concerning the random effects. Soybean genotypes data are used to illustrate the methods and to motivate the simulation scenarios

Surface collective oscillations of metal clusters and spheres: Random-phase-approximation sum-rules approach

Using the once and thrice energy-weighted moments of the random-phase-approximation strength function, we have derived compact expressions for the average energy of surface collective oscillations of clusters and spheres of metal atoms. The L=0 volume mode has also been studied. We have carried out quantal and semiclassical calculations for Na and Ag systems in the spherical-jellium approximation. We present a rather thorough discussion of surface diffuseness and quantal size effects on the resonance energies.