NN2Poly: A Polynomial Representation for Deep Feed-Forward Artificial Neural Networks.

Interpretability of neural networks (NNs) and their underlying theoretical behavior remain an open field of study even after the great success of their practical applications, particularly with the emergence of deep learning. In this work, NN2Poly is proposed: a theoretical approach to obtain an explicit polynomial model that provides an accurate representation of an already trained fully connected feed-forward artificial NN a multilayer perceptron (MLP). This approach extends a previous idea proposed in the literature, which was limited to single hidden layer networks, to work with arbitrarily deep MLPs in both regression and classification tasks. NN2Poly uses a Taylor expansion on the activation function, at each layer, and then applies several combinatorial properties to calculate the coefficients of the desired polynomials. Discussion is presented on the main computational challenges of this method, and the way to overcome them by imposing certain constraints during the training phase. Finally, simulation experiments as well as applications to real tabular datasets are presented to demonstrate the effectiveness of the proposed method.

Using survey data to estimate the impact of the omicron variant on vaccine efficacy against COVID-19 infection.

Symptoms-based detection of SARS-CoV-2 infection is not a substitute for precise diagnostic tests but can provide insight into the likely level of infection in a given population. This study uses symptoms data collected in the Global COVID-19 Trends and Impact Surveys (UMD Global CTIS), and data on variants sequencing from GISAID. This work, conducted in January of 2022 during the emergence of the Omicron variant (subvariant BA.1), aims to improve the quality of infection detection from the available symptoms and to use the resulting estimates of infection levels to assess the changes in vaccine efficacy during a change of dominant variant; from the Delta dominant to the Omicron dominant period. Our approach produced a new symptoms-based classifier, Random Forest, that was compared to a ground-truth subset of cases with known diagnostic test status. This classifier was compared with other competing classifiers and shown to exhibit an increased performance with respect to the ground-truth data. Using the Random Forest classifier, and knowing the vaccination status of the subjects, we then proceeded to analyse the evolution of vaccine efficacy towards infection during different periods, geographies and dominant variants. In South Africa, where the first significant wave of Omicron occurred, a significant reduction of vaccine efficacy is observed from August-September 2021 to December 2021. For instance, the efficacy drops from 0.81 to 0.30 for those vaccinated with 2 doses (of Pfizer/BioNTech), and from 0.51 to 0.09 for those vaccinated with one dose (of Pfizer/BioNTech or Johnson & Johnson). We also extended the study to other countries in which Omicron has been detected, comparing the situation in October 2021 (before Omicron) with that of December 2021. While the reduction measured is smaller than in South Africa, we still found, for instance, an average drop in vaccine efficacy from 0.53 to 0.45 among those vaccinated with two doses. Moreover, we found a significant negative (Pearson) correlation of around - 0.6 between the measured prevalence of Omicron in several countries and the vaccine efficacy in those same countries. This prediction, in January of 2022, of the decreased vaccine efficacy towards Omicron is in line with the subsequent increase of Omicron infections in the first half of 2022.

Accurate Prediction of Children's ADHD Severity Using Family Burden Information: A Neural Lasso Approach.

The deep lasso algorithm (dlasso) is introduced as a neural version of the statistical linear lasso algorithm that holds benefits from both methodologies: feature selection and automatic optimization of the parameters (including the regularization parameter). This last property makes dlasso particularly attractive for feature selection on small samples. In the two first conducted experiments, it was observed that dlasso is capable of obtaining better performance than its non-neuronal version (traditional lasso), in terms of predictive error and correct variable selection. Once that dlasso performance has been assessed, it is used to determine whether it is possible to predict the severity of symptoms in children with ADHD from four scales that measure family burden, family functioning, parental satisfaction, and parental mental health. Results show that dlasso is able to predict parents' assessment of the severity of their children's inattention from only seven items from the previous scales. These items are related to parents' satisfaction and degree of parental burden.

Towards a mathematical framework to inform neural network modelling via polynomial regression.

Even when neural networks are widely used in a large number of applications, they are still considered as black boxes and present some difficulties for dimensioning or evaluating their prediction error. This has led to an increasing interest in the overlapping area between neural networks and more traditional statistical methods, which can help overcome those problems. In this article, a mathematical framework relating neural networks and polynomial regression is explored by building an explicit expression for the coefficients of a polynomial regression from the weights of a given neural network, using a Taylor expansion approach. This is achieved for single hidden layer neural networks in regression problems. The validity of the proposed method depends on different factors like the distribution of the synaptic potentials or the chosen activation function. The performance of this method is empirically tested via simulation of synthetic data generated from polynomials to train neural networks with different structures and hyperparameters, showing that almost identical predictions can be obtained when certain conditions are met. Lastly, when learning from polynomial generated data, the proposed method produces polynomials that approximate correctly the data locally.

Estimating the COVID-19 Prevalence in Spain With Indirect Reporting via Open Surveys.

During the initial phases of the COVID-19 pandemic, accurate tracking has proven unfeasible. Initial estimation methods pointed toward case numbers that were much higher than officially reported. In the CoronaSurveys project, we have been addressing this issue using open online surveys with indirect reporting. We compare our estimates with the results of a serology study for Spain, obtaining high correlations (R squared 0.89). In our view, these results strongly support the idea of using open surveys with indirect reporting as a method to broadly sense the progress of a pandemic.

Variable selection with P-splines in functional linear regression: Application in graft-versus-host disease.

This paper focuses on the problems of estimation and variable selection in the functional linear regression model (FLM) with functional response and scalar covariates. To this end, two different types of regularization (L1 and L2 ) are considered in this paper. On the one hand, a sample approach for functional LASSO in terms of basis representation of the sample values of the response variable is proposed. On the other hand, we propose a penalized version of the FLM by introducing a P-spline penalty in the least squares fitting criterion. But our aim is to propose P-splines as a powerful tool simultaneously for variable selection and functional parameters estimation. In that sense, the importance of smoothing the response variable before fitting the model is also studied. In summary, penalized (L1 and L2 ) and nonpenalized regression are combined with a presmoothing of the response variable sample curves, based on regression splines or P-splines, providing a total of six approaches to be compared in two simulation schemes. Finally, the most competitive approach is applied to a real data set based on the graft-versus-host disease, which is one of the most frequent complications (30% -50%) in allogeneic hematopoietic stem-cell transplantation.

Comparing quantile residual life functions by confidence bands.

In this article we present a nonparametric method for constructing confidence bands for the difference of two quantile residual life (qrl) functions. These bands provide evidence for two random variables ordering with respect to the qrl order. The comparison of qrl functions is of importance, specially in the treatment of cancer when there exists a possibility of benefiting from a new secondary therapy. A qrl function is the quantile of the remaining life of a surviving subject, as it varies with time. We show the applicability of this approach in Medicine and Ecology. A simulation study has been carried out to evaluate and illustrate the performance and the consistency of this new methodology.