MOTIVATION: A number of available program packages determine the significant enrichments and/or depletions of GO categories among a class of genes of interest. Whereas a correct formulation of the problem leads to a single exact null distribution, these GO tools use a large variety of statistical tests whose denominations often do not clarify the underlying P-value computations. SUMMARY: We review the different formulations of the problem and the tests they lead to: the binomial, chi2, equality of two probabilities, Fisher's exact and hypergeometric tests. We clarify the relationships existing between these tests, in particular the equivalence between the hypergeometric test and Fisher's exact test. We recall that the other tests are valid only for large samples, the test of equality of two probabilities and the chi2-test being equivalent. We discuss the appropriateness of one- and two-sided P-values, as well as some discreteness and conservatism issues. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Our aim is to stress the importance of Jacobian matrix conditioning for model validation. We also comment on Monari and Dreyfus (2002), where, following Rivals and Personnaz (2000), it is proposed to discard neural candidates that are likely to overfit and/or for which quantities of interest such as confidence intervals cannot be computed accurately. In Rivals and Personnaz (2000), we argued that such models are to be discarded on the basis of the condition number of their Jacobian matrix. But Monari and Dreyfus (2002) suggest making the decision on the basis of the computed values of the leverages, the diagonal elements of the projection matrix on the range of the Jacobian, or "hat" matrix: they propose to discard a model if computed leverages are outside some theoretical bounds, pretending that it is the symptom of the Jacobian rank deficiency. We question this proposition because, theoretically, the hat matrix is defined whatever the rank of the Jacobian and because, in practice, the computed leverages of very ill-conditioned networks may respect their theoretical bounds while confidence intervals cannot be estimated accurately enough, two facts that have escaped Monari and Dreyfus's attention. Wealso recall the most accurate way to estimate the leverages and the properties of these estimations. Finally, we make an additional comment concerning the performance estimation in Monari and Dreyfus (2002).

We study how statistical tools which are commonly used independently can advantageously be exploited together in order to improve neural network estimation and selection in nonlinear static modeling. The tools we consider are the analysis of the numerical conditioning of the neural network candidates, statistical hypothesis tests, and cross validation. We present and analyze each of these tools in order to justify at what stage of a construction and selection procedure they can be most useful. On the basis of this analysis, we then propose a novel and systematic construction and selection procedure for neural modeling. We finally illustrate its efficiency through large-scale simulations experiments and real-world modeling problems.

In nonlinear regression theory, the sandwich estimator of the covariance matrix of the model parameters is known as a consistent estimator, even when the parameterized model does not contain the regression. However, in the latter case, we emphasize the fact that the consistency of the sandwich holds only if the inputs of the training set are the values of independent identically distributed random variables. Thus, in the frequent practical modeling situation involving a training set whose inputs are deliberately chosen and imposed by the designer, we question the opportunity to use the sandwich estimator rather than the simple estimator based on the inverse squared Jacobian.

We propose a design procedure of neural internal model control systems for stable processes with delay. We show that the design of such nonadaptive indirect control systems necessitates only the training of the inverse of the model deprived from its delay, and that the presence of the delay thus does not increase the order of the inverse. The controller is then obtained by cascading this inverse with a rallying model which imposes the regulation dynamic behavior and ensures the robustness of the stability. A change in the desired regulation dynamic behavior, or an improvement of the stability, can be obtained by simply tuning the rallying model, without retraining the whole model reference controller. The robustness properties of internal model control systems being obtained when the inverse is perfect, we detail the precautions which must be taken for the training of the inverse so that it is accurate in the whole space visited during operation with the process. In the same spirit, we make an emphasis on neural models affine in the control input, whose perfect inverse is derived without training. The control of simulated processes illustrates the proposed design procedure and the properties of the neural internal model control system for processes without and with delay.

In response to Zhu and Rower (1996), a recent communication (Goutte, 1997) established that leave-one-out cross validation is not subject to the "no-free-lunch" criticism. Despite this optimistic conclusion, we show here that cross validation has very poor performances for the selection of linear models as compared to classic statistical tests. We conclude that the statistical tests are preferable to cross validation for linear as well as for nonlinear model selection.

Multi-class classification problems can be efficiently solved by partitioning the original problem into sub-problems involving only two classes: for each pair of classes, a (potentially small) neural network is trained using only the data of these two classes. We show how to combine the outputs of the two-class neural networks in order to obtain posterior probabilities for the class decisions. The resulting probabilistic pairwise classifier is part of a handwriting recognition system which is currently applied to check reading. We present results on real world data bases and show that, from a practical point of view, these results compare favorably to other neural network approaches.

The paper first summarizes a general approach to the training of recurrent neural networks by gradient-based algorithms, which leads to the introduction of four families of training algorithms. Because of the variety of possibilities thus available to the "neural network designer," the choice of the appropriate algorithm to solve a given problem becomes critical. We show that, in the case of process modeling, this choice depends on how noise interferes with the process to be modeled; this is evidenced by three examples of modeling of dynamical processes, where the detrimental effect of inappropriate training algorithms on the prediction error made by the network is clearly demonstrated.

We present a model of the specialist olfactory system of selected moth species and the cockroach. The model is built in a semirandom fashion, constrained by biological (physiological and anatomical) data. We propose a classification of the response patterns of individual neurons, based on the temporal aspects of the observed responses. Among the observations made in our simulations a number relate to data about olfactory information processing reported in the literature; others may serve as predictions and as guidelines for further investigations. We discuss the effect of the stochastic parameters of the model on the observed model behavior and on the ability of the model to extract features of the input stimulation. We conclude that a formal network, built with random connectivity, can suffice to reproduce and to explain many aspects of olfactory information processing at the first level of the specialist olfactory system of insects.

The paper proposes a general framework that encompasses the training of neural networks and the adaptation of filters. We show that neural networks can be considered as general nonlinear filters that can be trained adaptively, that is, that can undergo continual training with a possibly infinite number of time-ordered examples. We introduce the canonical form of a neural network. This canonical form permits a unified presentation of network architectures and of gradient-based training algorithms for both feedforward networks (transversal filters) and feedback networks (recursive filters). We show that several algorithms used classically in linear adaptive filtering, and some algorithms suggested by other authors for training neural networks, are special cases in a general classification of training algorithms for feedback networks.

The definition of the requirements for the design of a neural network associative memory, with on-chip training, in standard digital CMOS technology is addressed. Various learning rules that can be integrated in silicon and the associative memory properties of the resulting networks are investigated. The relationships between the architecture of the circuit and the learning rule are studied in order to minimize the extra circuitry required for the implementation of training. A 64-neuron associative memory with on-chip training has been manufactured, and its future extensions are outlined. Beyond the application to the specific circuit described, the general methodology for determining the accuracy requirements can be applied to other circuits and to other autoassociative memory architectures.

It is shown that neural network classifiers with single-layer training can be applied efficiently to complex real-world classification problems such as the recognition of handwritten digits. The STEPNET procedure, which decomposes the problem into simpler subproblems which can be solved by linear separators, is introduced. Provided appropriate data representations and learning rules are used, performance comparable to that obtained by more complex networks can be achieved. Results from two different databases are presented: an European database comprising 8700 isolated digits and a zip code database from the US Postal Service comprising 9000 segmented digits. A hardware implementation of the classifier is briefly described.