Generating multiple random exams using mathematica package and LATEX

[EN]In the framework of the European Higher Education Area, the assessment has been one of the most important aspects considered. In the Spanish Universities, one of the main differences with regard to the previous system is the incorporation of the continuous assessment to the evaluation process that is understood in several ways depending on the Universities, the courses and the lecturers. Focus on our context, a course of Mathematics of the first academic year in the Faculty of Business Administration at the University of Las Palmas de Gran Canaria (Spain), the continuous assessment has brought the preparation of a large amount of different tests to evaluate the students enrolled in it, therefore the incorporation of new tools and skills in order to make the teaching–learning process easier and more dynamic has become a need mainly in those degrees with a large number of students as the case we consider here. In this work we provide an efficient and effective way to elaborate random multiple-choice examina tion tests (although essay exams are also possible) by using Mathematica package and LATEXin order to make easier to the lectures the preparation of a large number of mid-term tests for a large number of students.

Estimating the number of motor units using random sums with independently thinned terms

The problem of estimating the numbers of motor units N in a muscle is embedded in a general stochastic model using the notion of thinning from point process theory. In the paper a new moment type estimator for the numbers of motor units in a muscle is denned, which is derived using random sums with independently thinned terms. Asymptotic normality of the estimator is shown and its practical value is demonstrated with bootstrap and approximative confidence intervals for a data set from a 31-year-old healthy right-handed, female volunteer. Moreover simulation results are presented and Monte-Carlo based quantiles, means, and variances are calculated for N in{300,600,1000}.

Posterior Simulation in the Generalized Linear Model with Semiparmetric Random Effects

Generalized linear mixed models with semiparametric random effects are useful in a wide variety of Bayesian applications. When the random effects arise from a mixture of Dirichlet process (MDP) model, normal base measures and Gibbs sampling procedures based on the Pólya urn scheme are often used to simulate posterior draws. These algorithms are applicable in the conjugate case when (for a normal base measure) the likelihood is normal. In the non-conjugate case, the algorithms proposed by MacEachern and Müller (1998) and Neal (2000) are often applied to generate posterior samples. Some common problems associated with simulation algorithms for non-conjugate MDP models include convergence and mixing difficulties. This paper proposes an algorithm based on the Pólya urn scheme that extends the Gibbs sampling algorithms to non-conjugate models with normal base measures and exponential family likelihoods. The algorithm proceeds by making Laplace approximations to the likelihood function, thereby reducing the procedure to that of conjugate normal MDP models. To ensure the validity of the stationary distribution in the non-conjugate case, the proposals are accepted or rejected by a Metropolis-Hastings step. In the special case where the data are normally distributed, the algorithm is identical to the Gibbs sampler.

Estimating and quantifying uncertainties on level sets using the Vorob'ev expectation and deviation with Gaussian process models

Several methods based on Kriging have recently been proposed for calculating a probability of failure involving costly-to-evaluate functions. A closely related problem is to estimate the set of inputs leading to a response exceeding a given threshold. Now, estimating such a level set—and not solely its volume—and quantifying uncertainties on it are not straightforward. Here we use notions from random set theory to obtain an estimate of the level set, together with a quantification of estimation uncertainty. We give explicit formulae in the Gaussian process set-up and provide a consistency result. We then illustrate how space-filling versus adaptive design strategies may sequentially reduce level set estimation uncertainty.

Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

Multi-objective optimization algorithms aim at finding Pareto-optimal solutions. Recovering Pareto fronts or Pareto sets from a limited number of function evaluations are challenging problems. A popular approach in the case of expensive-to-evaluate functions is to appeal to metamodels. Kriging has been shown efficient as a base for sequential multi-objective optimization, notably through infill sampling criteria balancing exploitation and exploration such as the Expected Hypervolume Improvement. Here we consider Kriging metamodels not only for selecting new points, but as a tool for estimating the whole Pareto front and quantifying how much uncertainty remains on it at any stage of Kriging-based multi-objective optimization algorithms. Our approach relies on the Gaussian process interpretation of Kriging, and bases upon conditional simulations. Using concepts from random set theory, we propose to adapt the Vorob’ev expectation and deviation to capture the variability of the set of non-dominated points. Numerical experiments illustrate the potential of the proposed workflow, and it is shown on examples how Gaussian process simulations and the estimated Vorob’ev deviation can be used to monitor the ability of Kriging-based multi-objective optimization algorithms to accurately learn the Pareto front.

Pattern and process: competition causes regular spacing of individuals within plant populations

1 We used simulated and experimental plant populations to analyse mortality-driven pattern formation under size-dependent competition. Larger plants had an advantage under size-asymmetric but not under symmetric competition. Initial patterns were random or clumped. 2 The simulations were individual-based and spatially explicit. Size-dependent competition was modelled with different rules to partition overlapping zones of influence. 3 The experiment used genotypes of Arabidopsis thaliana with different morphological plasticity and hence size-dependent competition. Compared with wild types, transgenic individuals over-expressed phytochrome A and had decreased plasticity because of disabled phytochrome-mediated shade avoidance. Therefore, competition among transgenics was more asymmetric compared with wild-types. 4 Density-dependent mortality under symmetric competition did not substantially change the initial spatial pattern. Conversely, simulations under asymmetric competition and experimental patterns of transgenic over-expressors showed patterns of survivors that deviated substantially from random mortality independent of initial patterns. 5 Small-scale initial patterns of wild types were regular rather than random or clumped. We hypothesize that this small-scale regularity may be explained by early shade avoidance of seedlings in their cotyledon stage. 6 Our experimental results support predictions from an individual-based simulation model and support the conclusion that regular spatial patterns of surviving individuals should be interpreted as evidence for strong, asymmetric competitive interactions and subsequent density-dependent mortality.

Multifractional Poisson process, multistable subordinator and related limit theorems

We introduce a multistable subordinator, which generalizes the stable subordinator to the case of time-varying stability index. This enables us to define a multifractional Poisson process. We study properties of these processes and establish the convergence of a continuous-time random walk to the multifractional Poisson process.

Wireless Optical Communications for Intra-Spacecraft Networks Based on OCDMA with Random Optical Codes

In recent years, spacial agencies have shown a growing interest in optical wireless as an alternative to wired and radio-frequency communications. The use of these techniques for intra-spacecraft communications reduces the effect of take-off acceleration and vibrations on the systems by avoiding the need for rugged connectors and provides a significant mass reduction. Diffuse transmission also eases the design process as terminals can be placed almost anywhere without a tight planification to ensure the proper system behaviour. Previous studies have compared the performance of radio-frequency and infrared optical communications. In an intra-satellite environment optical techniques help reduce EMI related problems, and their main disadvantages - multipath dispersion and the need for line-of-sight - can be neglected due to the reduced cavity size. Channel studies demonstrate that the effect of the channel can be neglected in small environments if data bandwidth is lower than some hundreds of MHz.

Node sampling using random centrifugal walks

Sampling a network with a given probability distribution has been identified as a useful operation. In this paper we propose distributed algorithms for sampling networks, so that nodes are selected by a special node, called the source, with a given probability distribution. All these algorithms are based on a new class of random walks, that we call Random Centrifugal Walks (RCW). A RCW is a random walk that starts at the source and always moves away from it. Firstly, an algorithm to sample any connected network using RCW is proposed. The algorithm assumes that each node has a weight, so that the sampling process must select a node with a probability proportional to its weight. This algorithm requires a preprocessing phase before the sampling of nodes. In particular, a minimum diameter spanning tree (MDST) is created in the network, and then nodes weights are efficiently aggregated using the tree. The good news are that the preprocessing is done only once, regardless of the number of sources and the number of samples taken from the network. After that, every sample is done with a RCW whose length is bounded by the network diameter. Secondly, RCW algorithms that do not require preprocessing are proposed for grids and networks with regular concentric connectivity, for the case when the probability of selecting a node is a function of its distance to the source. The key features of the RCW algorithms (unlike previous Markovian approaches) are that (1) they do not need to warm-up (stabilize), (2) the sampling always finishes in a number of hops bounded by the network diameter, and (3) it selects a node with the exact probability distribution.

Random exams using Sweave

The adaptation of the Spanish University to the European Higher Education Area (EEES in Spanish) demands the integration of new tools and skills that would make the teaching- learning process easier. This adaptation involves a change in the evaluation methods, which goes from a system where the student was evaluated with a final exam, to a new system where we include a continuous evaluation in which the final exam may represent at most 50% in the vast majority of the Universities. Devising a new and fair continuous evaluation system is not an easy task to do. That would mean a student’s’ learning process follow-up by the teachers, and as a consequence an additional workload on existing staff resources. Traditionally, the continuous evaluation is associated with the daily work of the student and a collection of the different marks partly or entirely based on the work they do during the academic year. Now, small groups of students and an attendance control are important aspects to take into account in order to get an adequate assessment of the students. However, most of the university degrees have groups with more than 70 students, and the attendance control is a complicated task to perform, mostly because it consumes significant amounts of staff time. Another problem found is that the attendance control would encourage not-interested students to be present at class, which might cause some troubles to their classmates. After a two year experience in the development of a continuous assessment in Statistics subjects in Social Science degrees, we think that individual and periodical tasks are the best way to assess results. These tasks or examinations must be done in classroom during regular lessons, so we need an efficient system to put together different and personal questions in order to prevent students from cheating. In this paper we provide an efficient and effective way to elaborate random examination papers by using Sweave, a tool that generates data, graphics and statistical calculus from the software R and shows results in PDF documents created by Latex. In this way, we will be able to design an exam template which could be compiled in order to generate as many PDF documents as it is required, and at the same time, solutions are provided to easily correct them.

A Mixture model with random-effects components for clustering correlated gene-expression profiles

Motivation: The clustering of gene profiles across some experimental conditions of interest contributes significantly to the elucidation of unknown gene function, the validation of gene discoveries and the interpretation of biological processes. However, this clustering problem is not straightforward as the profiles of the genes are not all independently distributed and the expression levels may have been obtained from an experimental design involving replicated arrays. Ignoring the dependence between the gene profiles and the structure of the replicated data can result in important sources of variability in the experiments being overlooked in the analysis, with the consequent possibility of misleading inferences being made. We propose a random-effects model that provides a unified approach to the clustering of genes with correlated expression levels measured in a wide variety of experimental situations. Our model is an extension of the normal mixture model to account for the correlations between the gene profiles and to enable covariate information to be incorporated into the clustering process. Hence the model is applicable to longitudinal studies with or without replication, for example, time-course experiments by using time as a covariate, and to cross-sectional experiments by using categorical covariates to represent the different experimental classes. Results: We show that our random-effects model can be fitted by maximum likelihood via the EM algorithm for which the E(expectation) and M(maximization) steps can be implemented in closed form. Hence our model can be fitted deterministically without the need for time-consuming Monte Carlo approximations. The effectiveness of our model-based procedure for the clustering of correlated gene profiles is demonstrated on three real datasets, representing typical microarray experimental designs, covering time-course, repeated-measurement and cross-sectional data. In these examples, relevant clusters of the genes are obtained, which are supported by existing gene-function annotation. A synthetic dataset is considered too.

Emulation of random output simulators

Computer models, or simulators, are widely used in a range of scientific fields to aid understanding of the processes involved and make predictions. Such simulators are often computationally demanding and are thus not amenable to statistical analysis. Emulators provide a statistical approximation, or surrogate, for the simulators accounting for the additional approximation uncertainty. This thesis develops a novel sequential screening method to reduce the set of simulator variables considered during emulation. This screening method is shown to require fewer simulator evaluations than existing approaches. Utilising the lower dimensional active variable set simplifies subsequent emulation analysis. For random output, or stochastic, simulators the output dispersion, and thus variance, is typically a function of the inputs. This work extends the emulator framework to account for such heteroscedasticity by constructing two new heteroscedastic Gaussian process representations and proposes an experimental design technique to optimally learn the model parameters. The design criterion is an extension of Fisher information to heteroscedastic variance models. Replicated observations are efficiently handled in both the design and model inference stages. Through a series of simulation experiments on both synthetic and real world simulators, the emulators inferred on optimal designs with replicated observations are shown to outperform equivalent models inferred on space-filling replicate-free designs in terms of both model parameter uncertainty and predictive variance.

Molecular phase space transport in water:non-stationary random walk model

Molecular transport in phase space is crucial for chemical reactions because it defines how pre-reactive molecular configurations are found during the time evolution of the system. Using Molecular Dynamics (MD) simulated atomistic trajectories we test the assumption of the normal diffusion in the phase space for bulk water at ambient conditions by checking the equivalence of the transport to the random walk model. Contrary to common expectations we have found that some statistical features of the transport in the phase space differ from those of the normal diffusion models. This implies a non-random character of the path search process by the reacting complexes in water solutions. Our further numerical experiments show that a significant long period of non-stationarity in the transition probabilities of the segments of molecular trajectories can account for the observed non-uniform filling of the phase space. Surprisingly, the characteristic periods in the model non-stationarity constitute hundreds of nanoseconds, that is much longer time scales compared to typical lifetime of known liquid water molecular structures (several picoseconds).

Solvable model for distribution networks on random graphs

We propose a simple model that captures the salient properties of distribution networks, and study the possible occurrence of blackouts, i.e., sudden failings of large portions of such networks. The model is defined on a random graph of finite connectivity. The nodes of the graph represent hubs of the network, while the edges of the graph represent the links of the distribution network. Both, the nodes and the edges carry dynamical two state variables representing the functioning or dysfunctional state of the node or link in question. We describe a dynamical process in which the breakdown of a link or node is triggered when the level of maintenance it receives falls below a given threshold. This form of dynamics can lead to situations of catastrophic breakdown, if levels of maintenance are themselves dependent on the functioning of the net, once maintenance levels locally fall below a critical threshold due to fluctuations. We formulate conditions under which such systems can be analyzed in terms of thermodynamic equilibrium techniques, and under these conditions derive a phase diagram characterizing the collective behavior of the system, given its model parameters. The phase diagram is confirmed qualitatively and quantitatively by simulations on explicit realizations of the graph, thus confirming the validity of our approach. © 2007 The American Physical Society.