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 VC dimension, empirical VC entropy, and margin-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.

Fast rates for estimation error and oracle inequalities for model section

We consider complexity penalization methods for model selection. These methods aim to choose a model to optimally trade off estimation and approximation errors by minimizing the sum of an empirical risk term and a complexity penalty. It is well known that if we use a bound on the maximal deviation between empirical and true risks as a complexity penalty, then the risk of our choice is no more than the approximation error plus twice the complexity penalty. There are many cases, however, where complexity penalties like this give loose upper bounds on the estimation error. In particular, if we choose a function from a suitably simple convex function class with a strictly convex loss function, then the estimation error (the difference between the risk of the empirical risk minimizer and the minimal risk in the class) approaches zero at a faster rate than the maximal deviation between empirical and true risks. In this paper, we address the question of whether it is possible to design a complexity penalized model selection method for these situations. We show that, provided the sequence of models is ordered by inclusion, in these cases we can use tight upper bounds on estimation error as a complexity penalty. Surprisingly, this is the case even in situations when the difference between the empirical risk and true risk (and indeed the error of any estimate of the approximation error) decreases much more slowly than the complexity penalty. We give an oracle inequality showing that the resulting model selection method chooses a function with risk no more than the approximation error plus a constant times the complexity penalty.

Krylov subspace approximations for the exponential Euler method: Error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error.

Auditory spatial acuity approximates the resolving power of space-specific neurons

The relationship between neuronal acuity and behavioral performance was assessed in the barn owl (Tyto alba), a nocturnal raptor renowned for its ability to localize sounds and for the topographic representation of auditory space found in the midbrain. We measured discrimination of sound-source separation using a newly developed procedure involving the habituation and recovery of the pupillary dilation response. The smallest discriminable change of source location was found to be about two times finer in azimuth than in elevation. Recordings from neurons in its midbrain space map revealed that their spatial tuning, like the spatial discrimination behavior, was also better in azimuth than in elevation by a factor of about two. Because the PDR behavioral assay is mediated by the same circuitry whether discrimination is assessed in azimuth or in elevation, this difference in vertical and horizontal acuity is likely to reflect a true difference in sensory resolution, without additional confounding effects of differences in motor performance in the two dimensions. Our results, therefore, are consistent with the hypothesis that the acuity of the midbrain space map determines auditory spatial discrimination.

The Bayesian conditional independence model for measurement error: applications in ecology

The measurement error model is a well established statistical method for regression problems in medical sciences, although rarely used in ecological studies. While the situations in which it is appropriate may be less common in ecology, there are instances in which there may be benefits in its use for prediction and estimation of parameters of interest. We have chosen to explore this topic using a conditional independence model in a Bayesian framework using a Gibbs sampler, as this gives a great deal of flexibility, allowing us to analyse a number of different models without losing generality. Using simulations and two examples, we show how the conditional independence model can be used in ecology, and when it is appropriate.

Spatial analysis of biomineralization associated gene expression from the mantle organ of the pearl oyster Pinctada maxima

Background: Biomineralization is a process encompassing all mineral containing tissues produced within an organism. One of the most dynamic examples of this process is the formation of the mollusk shell, comprising a variety of crystal phases and microstructures. The organic component incorporated within the shell is said to dictate this architecture. However general understanding of how this process is achieved remains ambiguous. The mantle is a conserved organ involved in shell formation throughout molluscs. Specifically the mantle is thought to be responsible for secreting the protein component of the shell. This study employs molecular approaches to determine the spatial expression of genes within the mantle tissue to further the elucidation of the shell biomineralization. Results: A microarray platform was custom generated (PmaxArray 1.0) from the pearl oyster Pinctada maxima. PmaxArray 1.0 consists of 4992 expressed sequence tags (ESTs) originating from mantle tissue. This microarray was used to analyze the spatial expression of ESTs throughout the mantle organ. The mantle was dissected into five discrete regions and analyzed for differential gene expression with PmaxArray 1.0. Over 2000 ESTs were determined to be differentially expressed among the tissue sections, identifying five major expression regions. In situ hybridization validated and further localized the expression for a subset of these ESTs. Comparative sequence similarity analysis of these ESTs revealed a number of the transcripts were novel while others showed significant sequence similarities to previously characterized shell related genes.

Stochastic simulation for spatial modelling of dynamic process in a living cell

One of the fundamental motivations underlying computational cell biology is to gain insight into the complicated dynamical processes taking place, for example, on the plasma membrane or in the cytosol of a cell. These processes are often so complicated that purely temporal mathematical models cannot adequately capture the complex chemical kinetics and transport processes of, for example, proteins or vesicles. On the other hand, spatial models such as Monte Carlo approaches can have very large computational overheads. This chapter gives an overview of the state of the art in the development of stochastic simulation techniques for the spatial modelling of dynamic processes in a living cell.

A review of locative media, mobile and embodied spatial interaction

Highlights ► Provides a review of the history and development of locative media. ► Outlines different human-computer interaction techniques applied in locative media. ► Discusses how locative media applications have changed interaction affordances in and of physical spaces. ► Discusses practices of people in urban settings that evolved through these new affordances. ► Provides an overview on methods to investigate and elaborate design principles for future locative media.