Priors Stabilizers and Basis Functions: From Regularization to Radial, Tensor and Additive Splines

We had previously shown that regularization principles lead to approximation schemes, as Radial Basis Functions, which are equivalent to networks with one layer of hidden units, called Regularization Networks. In this paper we show that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models, Breiman's hinge functions and some forms of Projection Pursuit Regression. In the probabilistic interpretation of regularization, the different classes of basis functions correspond to different classes of prior probabilities on the approximating function spaces, and therefore to different types of smoothness assumptions. In the final part of the paper, we also show a relation between activation functions of the Gaussian and sigmoidal type.

Notes on PCA, Regularization, Sparsity and Support Vector Machines

We derive a new representation for a function as a linear combination of local correlation kernels at optimal sparse locations and discuss its relation to PCA, regularization, sparsity principles and Support Vector Machines. We first review previous results for the approximation of a function from discrete data (Girosi, 1998) in the context of Vapnik"s feature space and dual representation (Vapnik, 1995). We apply them to show 1) that a standard regularization functional with a stabilizer defined in terms of the correlation function induces a regression function in the span of the feature space of classical Principal Components and 2) that there exist a dual representations of the regression function in terms of a regularization network with a kernel equal to a generalized correlation function. We then describe the main observation of the paper: the dual representation in terms of the correlation function can be sparsified using the Support Vector Machines (Vapnik, 1982) technique and this operation is equivalent to sparsify a large dictionary of basis functions adapted to the task, using a variation of Basis Pursuit De-Noising (Chen, Donoho and Saunders, 1995; see also related work by Donahue and Geiger, 1994; Olshausen and Field, 1995; Lewicki and Sejnowski, 1998). In addition to extending the close relations between regularization, Support Vector Machines and sparsity, our work also illuminates and formalizes the LFA concept of Penev and Atick (1996). We discuss the relation between our results, which are about regression, and the different problem of pattern classification.

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densities by generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Central notations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform. In this way very elaborated aspects of mathematical statistics can be understood easily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating, combination of likelihood and robust M-estimation functions are simple additions/ perturbations in A2(Pprior). Weighting observations corresponds to a weighted addition of the corresponding evidence. Likelihood based statistics for general exponential families turns out to have a particularly easy interpretation in terms of A2(P). Regular exponential families form finite dimensional linear subspaces of A2(P) and they correspond to finite dimensional subspaces formed by their posterior in the dual information space A2(Pprior). The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P. The discussion of A2(P) valued random variables, such as estimation functions or likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Comparing methods for dimensionality reduction when data are density functions

Functional Data Analysis (FDA) deals with samples where a whole function is observed for each individual. A particular case of FDA is when the observed functions are density functions, that are also an example of infinite dimensional compositional data. In this work we compare several methods for dimensionality reduction for this particular type of data: functional principal components analysis (PCA) with or without a previous data transformation and multidimensional scaling (MDS) for diferent inter-densities distances, one of them taking into account the compositional nature of density functions. The difeerent methods are applied to both artificial and real data (households income distributions)

Special Functions

Exam questions and solutions in LaTex

Special Functions pdf

Exam questions and solutions in PDF

Vector functions and curves

Exercises and solutions about vector functions and curves.

Uso del test de cuantificación sensitiva para el estudio de la neuropatía de fibra pequeña

Introducción La mutación genética Val30Met de la proteína transtiretina (TTR) es causante de la polineuropatía amiloidótica familiar, comprometiendo en fases iniciales las fibras nerviosas pequeñas (mielinizadas Aδ y amielínicas tipo C), involucradas en funciones autonómicas, nocicepción, percepción térmica y sudoración. Los métodos neurofisiológicos convencionales, no logran detectar dichas anormalidades, retardando el inicio de tratamientos específicos para la enfermedad. Metodología El objetivo principal fue evaluar el test de cuantificación sensitiva (QST) como método de detección temprana de anormalidades de fibra pequeña, en individuos Val30Met, seguidos en el Hospital Universitario Santa María, Lisboa. Se clasificaron los pacientes en 3 grupos, según sintomatología y examen neurológico. Se analizaron los umbrales para percepción de frío, dolor con el calor y vibración en los grupos, en correlación con controles sanos. Resultados 18 registros de controles sanos y 33 de individuos con la mutación, divididos en asintomáticos (24,2%), sintomáticos con examen neurológico normal (42,4%) y sintomáticos con examen neurológico anormal (33,3%). No se encontraron diferencias entre los pacientes asintomáticos y los controles. Los umbrales para frío (p=0,042) y en el dolor intermedio con el calor (HP 5) (p=0,007) se encuentran elevados en individuos Val30Met sintomáticos con examen normal. En los pacientes sintomáticos con alteraciones al examen, también se presentaron alteraciones en el intervalo entre el inicio y el dolor intermedio con el calor (HP 5-0,5) (p=0,009). Discusión Los umbrales de frío y de percepción de dolor con el calor, permiten detectar anormalidades en personas con la mutación TTR Val30Met, sintomáticos, incluyendo aquellos sin cambios objetivos al examen neurológico.

Mnesic performance and executive functions in patients with multiple sclerosis (MS)

The Episodic Memory (EM) and the Executive Functions (EF) are cognitive areas that are affected in patients with diagnosis of Multiple Sclerosis (MS). Nowadays there exists scarce works destined to explore the infl uence of the EF on measures of mnesic performance in MS. For this reason, we analyze the effect of the EF on the performance in a set of memory measures. We worked with a clinical group (n=36) and with a control group (n=36) compared by age and educational level. The results show that the clinical group obtained significantly low average values in all the mnesic indexes (with exception of recognition) and in all the executive measures. All the executive indexes showed significant associations with some of the indexes of mnesic performance. These findings suggest that the problems in the episodic memory in EM patients could be analyzed as the manifestation of a global disorder that could be similar to the one that involves the EF. 

Rehabilitation of executive functions: Implications and strategies

Executive Functions (EF) concern a range of abilitiesincluding problem-solving, planning, initiation, selfmonitoring,conscious attention, cope with new situationsand the ability to modify plans if necessary. It’s ahigh cognitive function that is crucial for a person to getengaged and maintain daily activities whilst keeping agood quality of life. Problems in the EF were formerlyknown as Dysexecutive Syndrome (DS). There are manymodels concerning DS, although the literature on thesubject still remains unclear. Several works appoint theeffects brought by elderly life, as well as abuse of drugsand some psychopathologies. These factors are knownto increase the distress of the frontal circuits and thatcould be associated to executive deficits. The effects ofDS would compromise individuals in day-to-day routine,academic, social and labor fields. There is a growingbody of studies trying to determine the causes, implications,associations and the best way to take care of theseeffects. This work intends to review DS, focusing on themost important fields related to this area, such as psychopathologyassociations, cognitive reserve, assessmentand cognitive rehabilitation programs.

Executive functions in school-aged children: age and socioeconomic status effects

Lately, the study of prefrontal executive functions in grade scholars has noticeably increased. The aim of this study is to investigate the influence of age and socioeconomic status (sEs) on executive tasks performance and to analyze those socioeconomic variables that predict a better execution. A sample of 254 children aged between 7 and 12 years from the city of santa Fe, Argentina and belonging to different socioeconomic status were tested. A bat- tery of executive functions sensitive to prefrontal function was used to obtain the results. These in- dicate a significant influence of age and SES on executive functions. The cognitive patterns follow a different path according to the development and sEs effect. Besides, it is revealed a pattern of low cognitive functioning in low-sEs children in all executive functions. Finally, from the variables included in this study, it was found that only the educational level of the mother and the housing conditions are associated to the children’s executive function. The results are discussed in terms of the influence of the cerebral maturation and the envi- ronmental variables in the executive functioning.

Type 2 diabetes; elderly; cognitive deficit; cognitive flexibility; executive functions Resumen

Through meditation, people become aware of what happens in the body and mind, accepting the present experiences as they are and getting a better understanding of the true nature of things. Meditation practices and its inclusion as an intervention technique, have generated great interest in identifying the brain mechanisms through which these practices operate. Different studies suggest that the practice of meditation is associated with the use of different neural networks as well as changes in brain structure and function, represented in higher concentration of gray matter structures at the hippocampus, the right anterior insula, orbital frontal cortex (OFC) and greater involvement of the anterior cingulate cortex (ACC). These and other unrelated studies, shows the multiple implications of the regular practice of mindfulness in the structures and functions of the brain and its relation to certain observable and subjective states in people who practice it. Such evidence enabling the inclusion of mindfulness in psychological therapy where multiple applications have been developed to prove its effectiveness in treating affective and emotional problems, crisis management, social skills, verbal creativity, addiction and craving management, family and caregivers stress of dementia patients and others. However, neuropsychological rehabilitation has no formal proposals for intervention from these findings. The aim of this paper is to propose use of Mindfulness in neuropsychological rehabilitation process, taking the positions and theory of A.R. Luria.

Neuropsychological profile in patients with temporal lobe epilepsy

The temporal lobe epilepsy (TLE) is the most common type of refractory epilepsy in adults. There is a wide consensus regarding the commitment of memory in temporal lobe epilepsy with hippocampal sclerosis. However, the consensus is not as widespread with respect to the other functions such as attention, executive functions, language and intellectual performance. For this study we analyzed retrospectively a group of 76 patients with refractory epilepsy, 48 patients with temporal lobe epilepsy (23 with right lateralization and 25 with left lateralization) and 28 patients with extratemporal epilepsy. We applied a battery of neuropsychological tests used in the Epilepsy Surgery Program at Hospital de Egas Moniz, Lisbon, Portugal. Our results show that the battery of neuropsychological tests is internally consistent in the evaluation of patients with TLE. We have also found that patients with TLE have standard generalized deficits witch could be indicative of areas of engagement besides the hippocampus. One interesting finding was the fact that interference verbal memory (long term memory) remains adequate, suggesting that this function is not compromised in TLE. In addition to the general pattern of cognitive deficits, we can see the impact of the disease at the socio-demographic level, and we can also establish a relationship with neurobiological findings previously described in the literature

Distribución hiperbólica generalizada: una aplicación en la selección de portafolios y en la cuantificación de medidas de riesgo de mercado

En este trabajo se implementa una metodología para incluir momentos de orden superior en la selección de portafolios, haciendo uso de la Distribución Hiperbólica Generalizada, para posteriormente hacer un análisis comparativo frente al modelo de Markowitz.