Functional ANOVA approaches for detecting changes in air pollution during the COVID-19 pandemic

This research was funded by project PID2020-113961GB-I00 of the Spanish Ministry of Science and Innovation (also supported by the FEDER program), project FQM-307 of the Government of Andalusia (Spain) and the PhD grant (FPU18/01779) awarded to Christian Acal. The authors also thank the support of the University of Granada, Spain, under project for young researchers PPJIB2020-01.Faced with novel coronavirus outbreak, the most hard-hit countries adopted a lockdown strategy to contrast the spread of virus. Many studies have already documented that the COVID-19 control actions have resulted in improved air quality locally and around the world. Following these lines of research, we focus on air quality changes in the urban territory of Chieti-Pescara (Central Italy), identified as an area of criticality in terms of air pollution. Concentrations of NO2, PM10, PM2.5 and benzene are used to evaluate air pollution changes in this Region. Data were measured by several monitoring stations over two specific periods: from 1st February to 10 th March 2020 (before lockdown period) and from 11st March 2020 to 18 th April 2020 (during lockdown period). The impact of lockdown on air quality is assessed through functional data analysis. Our work makes an important contribution to the analysis of variance for functional data (FANOVA). Specifically, a novel approach based on multivariate functional principal component analysis is introduced to tackle the multivariate FANOVA problem for independent measures, which is reduced to test multivariate homogeneity on the vectors of the most explicative principal components scores. Results of the present study suggest that the level of each pollutant changed during the confinement. Additionally, the differences in the mean functions of all pollutants according to the location and type of monitoring stations (background vs traffic), are ascribable to the PM10 and benzene concentrations for pre-lockdown and during-lockdown tenure, respectively. FANOVA has proven to be beneficial to monitoring the evolution of air quality in both periods of time. This can help environmental protection agencies in drawing a more holistic picture of air quality status in the area of interest.Spanish Ministry of Science and Innovation - FEDER program PID2020-113961GB-I00Government of Andalusia (Spain) FQM-307University of Granada, Spain PPJIB2020-01- FPU18/0177

Basis expansion approaches for functional analysis of variance with repeated measures

The methodological contribution in this paper is motivated by biomechanical studies where data characterizing human movement are waveform curves representing joint measures such as flexion angles, velocity, acceleration, and so on. In many cases the aim consists of detecting differences in gait patterns when several independent samples of subjects walk or run under different conditions (repeated measures). Classic kinematic studies often analyse discrete summaries of the sample curves discarding important information and providing biased results. As the sample data are obviously curves, a Functional Data Analysis approach is proposed to solve the problem of testing the equality of the mean curves of a functional variable observed on several independent groups under different treatments or time periods. A novel approach for Functional Analysis of Variance (FANOVA) for repeated measures that takes into account the complete curves is introduced. By assuming a basis expansion for each sample curve, two-way FANOVA problem is reduced to Multivariate ANOVA for the multivariate response of basis coefficients. Then, two different approaches for MANOVA with repeated measures are considered. Besides, an extensive simulation study is developed to check their performance. Finally, two applications with gait data are developed

New Modeling Approaches Based on Varimax Rotation of Functional Principal Components

Functional Principal Component Analysis (FPCA) is an important dimension reduction technique to interpret themainmodes of functional data variation in terms of a small set of uncorrelated variables. The principal components can not always be simply interpreted and rotation is one of the main solutions to improve the interpretation. In this paper, two new functional Varimax rotation approaches are introduced. They are based on the equivalence between FPCA of basis expansion of the sample curves and Principal Component Analysis (PCA) of a transformation of thematrix of basis coefficients. The first approach consists of a rotation of the eigenvectors that preserves the orthogonality between the eigenfunctions but the rotated principal component scores are not uncorrelated. The second approach is based on rotation of the loadings of the standardized principal component scores that provides uncorrelated rotated scores but non-orthogonal eigenfunctions. A simulation study and an application with data from the curves of infections by COVID-19 pandemic in Spain are developed to study the performance of these methods by comparing the results with other existing approaches.Spanish Ministry of Science, Innovation and Universities (FEDER program) MTM2017-88708-PGovernment of Andalusia (Spain) FQM-307 FPU18/0177

Impact on the Virtual Learning Environment Due to COVID-19

As a result of the global pandemic caused by COVID-19, universities have carried out teaching in a digital way, accelerating the inclusion and use of technologies in methodological adaptation. The research aims to ascertain the perception that students at the Faculty of Education Sciences of the University of Granada have regarding the pedagogical model adopted in the virtual learning environment during confinement through the second semester of the 2019–2020 academic year. The information collection method was an online questionnaire, using simple random sampling with proportional affixing 0.5, 95% confidence level and maximum permissible error of 4.7%. The results demonstrate a generalised dissatisfaction of the students, being fundamental to carry out the transition of the educational processes and training of the teaching staff. The implementation of active methodologies increases due to the virtual condition, specifically the flipped classroom methodology, but students manifest generalised dissatisfaction regarding the adequate methodological development and the involvement of professors. There is an outstanding use of e-mail and the virtual learning platform (PRADO), although they consider that they do not have the appropriate knowledge about image editors, video, computer graphics, synchronous response systems and anti-plagiarism tools. The students surveyed express that the tutoring functions, tasks and beliefs of the teaching staff in e-learning are not satisfactory.Spanish Ministry of Science, Innovation and Universities FPU18/0177

Implementation of the flipped classroom and its longitudinal impact on improving academic performance

The authors thank the support of the Spanish Ministry of Science, Innovation and Universities the PhD grant (FPU18/01779) awarded to Christian Acal. We thank the involvement, innovation and collaboration from the professors part of the SEJ-622 Research Group, accredited and financed by the Junta de Andalucia, who are carrying out the proposals and development of these active methodologies at the University of Granada, for the purpose of providing experiences at all educational levels.The objective has been to know the impact of the flipped classroom methodology on the academic performance of students during their training process in relation to the traditional methodology over time, in order to establish baselines in the academic grades in both models. The research is of a quasi-experimental type of non-equivalent groups, with a longitudinal trend design in the data collection process. The entire available population has been selected, with 1.236 students participating, exploring the grades as an analytical resource, from the 2010/2011 to the 2019/2020 academic years. The results show statistically significant differences in the improvement of academic performance with the flipped classroom methodology. Furthermore, the results reinforce that the flipped teaching model effectively promotes students’ interest, their capacity for autonomous learning and personal and cooperative relationships.Spanish Government FPU18/01779Junta de Andaluci

Holistic Variability Analysis in Resistive Switching Memories Using a Two-Dimensional Variability Coefficient

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsami.2c22617We present a new methodology to quantify the variability of resistive switching memories. Instead of statistically analyzing few data points extracted from current versus voltage (I− V) plots, such as switching voltages or state resistances, we take into account the whole I−V curve measured in each RS cycle. This means going from a one-dimensional data set to a two-dimensional data set, in which every point of each I−V curve measured is included in the variability calculation. We introduce a new coefficient (named two-dimensional variability coefficient, 2DVC) that reveals additional variability information to which traditional one-dimensional analytical methods (such as the coefficient of variation) are blind. This novel approach provides a holistic variability metric for a better understanding of the functioning of resistive switching memoriesConsejería de Conocimiento, Investigación y Universidad, Junta de Andalucía (Spain)FEDER: B-TIC-624-UGR20, PID2020-113961GB-I00, A-FQM-66-UGR20, FQM-307IMAG María de Maeztu CEX2020-001105-M/AEI/10.13039/501100011033King Abdullah University of Science and Technolog

One Cut‐Point Phase‐Type Distributions in Reliability. An Application to Resistive Random Access Memories

A new probability distribution to study lifetime data in reliability is introduced in this paper. This one is a first approach to a non‐homogeneous phase‐type distribution. It is built by considering one cut‐point in the non‐negative semi‐line of a phase‐type distribution. The density function is defined and the main measures associated, such as the reliability function, hazard rate, cumulative hazard rate and the characteristic function, are also worked out. This new class of dis‐ tributions enables us to decrease the number of parameters in the estimate when inference is con‐ sidered. Additionally, the likelihood distribution is built to estimate the model parameters by maximum likelihood. Several applications considering Resistive Random Access Memories com‐ pare the adjustment when phase type distributions and one cut‐point phase‐type distributions are considered. The developed methodology has been computationally implemented in R‐cran.This paper is partially supported by the project FQM‐307 of the Government of Andalu‐ sia (Spain), by the project PID2020‐113961GB‐I00 of the Spanish Ministry of Science and Innovation (also supported by the European Regional Development Fund program, ERDF) and by the project PPJIB2020‐01 of the University of Granada. Additionally, the first and second authors acknowledge financial support by the IMAG–María de Maeztu grant CEX2020‐001105‐M/AEI/10.13039/501100011033. They also acknowledge the financial support of the Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía (Spain) and the FEDER programme for projects A.TIC.117.UGR18, IE2017‐5414, B.TIC.624.UGR20 and A‐FQM‐66‐UGR20

A Complex Model via Phase-Type Distributions to Study Random Telegraph Noise in Resistive Memories

A new stochastic process was developed by considering the internal performance of macro-states in which the sojourn time in each one is phase-type distributed depending on time. The stationary distribution was calculated through matrix-algorithmic methods and multiple interesting measures were worked out. The number of visits distribution to a determine macro-state were analyzed from the respective differential equations and the Laplace transform. The mean number of visits to a macro-state between any two times was given. The results were implemented computationally and were successfully applied to study random telegraph noise (RTN) in resistive memories. RTN is an important concern in resistive random access memory (RRAM) operation. On one hand, it could limit some of the technological applications of these devices; on the other hand, RTN can be used for the physical characterization. Therefore, an in-depth statistical analysis to model the behavior of these devices is of essential importance.Spanish Ministry of Science, Innovation and Universities (FEDER program) MTM2017-88708-P TEC2017-84321-C4-3-RGovernment of Andalusia (Spain) FQM-307Andalusian Ministry of Economy, Knowledge, Companies and Universities A-TIC-117-UGR18 FPU18/0177

Phase-type distributions for studying variability in resistve memories

A new statistical approach has been developed to analyze Resistive Random Access Memory (RRAM) variability. The stochastic nature of the physical processes behind the operation of resistive memories makes variability one of the key issues to solve from the industrial viewpoint of these new devices. The statistical features of variability have been usually studied making use of Weibull distribution. However, this probability distribution does not work correctly for some resistive memories, in particular for those based on the Ni/HfO2/Si structure thar has been employed in this work. A completely new approach based on phase-type modelling is proposed in this paper to characterize the randomness of resistive memories operation. An in-depth comparision with experimental results shows that the fitted phase-type distribution works better than the Weibull distribution and also helps to understand the physics of the resistive memories.Spanish Ministry of Economy and Competitiveness (FEDER program) TEC2017-84321-C4-3-R MTM2017-88708-PIMB-CNM (CSIC) (Barcelona

Avances en modelización estocástica y funcional de datos de alta dimensión

En muchos campos cient cos, es habitual encontrar magnitudes caracterizadas por la evoluci on de una variable aleatoria a lo largo de alg un continuo (proceso estoc astico). A pesar de que los datos experimentales medidos sobre estas variables son claramente funciones (curvas, super cies o im agenes), hist oricamente su tratamiento ha sido a trav es del an alisis multivariante o de series temporales, perdi endose informaci on importante. Por suerte, los grandes avances que ha experimentado el sector tecnol ogico en los ultimos a~nos, han facilitado el seguimiento y reconstrucci on de las funciones de forma r apida y sin esfuerzo, siendo posible trabajar con las funciones completas. En este escenario, es altamente probable tener datos de alta dimensi on, en los que el n umero de variables es mayor que el n umero de individuos muestreados. Este hecho hace que los m etodos estad sticos tradicionales no sean adecuados. Dependiendo del prop osito nal, en esta tesis se abordan estos datos desde dos perspectivas estad sticas diferentes y complementarias: el An alisis de Datos Funcional (FDA) y el An alisis de la Fiabilidad (RA) basado en las distribuciones de probabilidad Tipo Fase (PH). FDA surge ante la necesidad de construir m etodos que permitan modelizar datos funcionales, cuyas observaciones suelen ser curvas dependiendo del tiempo u otro argumento continuo. En las ultimas d ecadas, se viene realizando una intensa investigaci on en este campo, en el que se han generalizado la mayor a de las t ecnicas multivariantes, especialmente, m etodos de reducci on de la dimensi on, clasi caci on y regresi on. Destaca el An alisis de Componentes Principales (FPCA) porque reduce la dimensi on y explica la estructura de variabilidad en t erminos de un n umero peque~no de variables incorreladas. En el campo de la abilidad, uno de los objetivos es estudiar el comportamiento de sistemas complejos, cuyo funcionamiento est a condicionado por varios factores incontrolables. En este sentido, RA intenta identi car la distribuci on de probabilidad de los datos para arrojar luz sobre la variabilidad que hay detr as del funcionamiento de los sistemas. Una posibilidad es considerar los procesos Markovianos y las distribuciones PH. Esta clase de distribuciones es capaz de aproximar cualquier distribuci on no negativa tanto como se desee gracias a su versatilidad, y permite modelar problemas complejos con resultados bien estructurados. Las contribuciones metodol ogicas de esta tesis se desarrollan en base a problemas de gran inter es impulsados por datos relacionados con las Memorias Resistivas de Acceso Aleatorio (RRAMs) y la pandemia de COVID-19. Las RRAM despiertan un gran inter es porque son una de las principales fuentes de ingresos en la industria, mientras que para mitigar la propagaci on del virus, es crucial desarrollar modelos optimos que ayuden a tomar buenas decisiones. Un nuevo enfoque estad stico basado en las distribuciones PH es desarrollado para analizar la variabilidad de las RRAM, siendo esta uno de los aspectos clave a resolver. Tras un exhaustivo estudio experimental se muestra que las distribuciones PH funcionan mejor que cualquier otra distribuci on y adem as, ayudan a conocer mejor el comportamiento interno de las RRAM. Se construye un nuevo proceso estoc astico de macro-estados considerando el desempe~no interno de los mismos. El tiempo de permanencia en cada uno de estos macro-estado se distribuye mediante una PH. Se muestra como el comportamiento interno del proceso es Markoviano, pero tanto la homogeneidad como la Markovianidad desaparecen para el nuevo modelo de macro-estados. Tambi en se obtienen otras medidas asociadas al modelo. La nueva metodolog a permite modelar sistemas complejos de forma algor tmica, en particular, el ruido producido dentro de las RRAM. FPCA basado en la expansi on de Karhunen-Lo eve permite describir la evoluci on estoc astica de las RRAM. Sin embargo, es esencial identi car la distribuci on de las componentes principales (pc's) para modelizar todo el proceso. Para ello, se introduce una nueva clase de distribuciones, llamada distribuciones Tipo-fase Lineal (LPH). A partir de esta metodolog a se demuestra que, si las pc's siguen una distribuci on LPH, el proceso es caracterizado por una distribuci on LPH en cada punto. En relaci on a las pc's, a veces su interpretaci on no es inmediata y se necesita aplicar una rotaci on para facilitarla. En este sentido, se desarrollan dos nuevos enfoques de rotaci on Varimax funcional basado en la equivalencia entre el FPCA y PCA. El primer m etodo consiste en rotar los autovectores, mientras que el segundo rota las cargas de las puntuaciones de las pc's estandarizadas. Estas rotaciones son aplicadas para interpretar la variabilidad de las curvas de positivos por COVID-19 en las comunidades aut onomas espa~nolas. Adem as, se proponen dos nuevos enfoques param etricos y no param etricos para resolver el problema de la homogeneidad funcional, asumiendo la expansi on b asica de las curvas. Estos m etodos consisten en aplicar los test de homogeneidad multivariante sobre el vector de coe cientes b asicos y sobre el vector de las puntuaciones de las pc's. Esta metodolog a ayudar a a analizar qu e in uencia tienen el material y el grosor empleado en los procesos de fabricaci on sobre el funcionamiento de las RRAM. Para el caso de m as de una variable de respuesta funcional, se extiende la metodolog a anterior basada en el FPCA multivariante para probar la homogeneidad. En particular, se usa para comprobar si existen diferencias signi cativas entre los niveles de varios contaminantes seg un la localizaci on geogr a ca de las estaciones de monitoreo en la Regi on de Abruzzo, Italia. Adem as, se considera un enfoque de medidas repetidas para estudiar si el nivel de cada contaminante se redujo durante el con namiento establecido por el Gobierno Italiano durante la pandemia del COVID-19. Finalmente, se propone un modelo de regresi on m ultiple funci on-sobre-funci on en t erminos de las pc's para la imputaci on de datos faltantes en una variable de respuesta funcional. Se asume que todos los predictores funcionales son completamente observados. Este m etodo permitir a la imputaci on de datos faltantes relacionados con el COVID-19. El contenido de esta tesis est a presentado como un compendio de siete publicaciones. Las versiones completas de los art culos est an incluidas en los Ap endices.In many scienti c elds, it is usual to nd magnitudes characterized by the evolution of a random variable over some continuum (stochastic process). Despite the experimental data measured on these variables are functions (curves, surfaces or images), historically their treatment has been through multivariate or time-series analysis, losing key information. Luckily, the great advances experimented by the technology sector in last years, have made easier the monitoring and reconstruction of the functions quickly and e ortless, being possible to work with the complete functions. In this scenario, there is a high probability of having high dimensional data, in which the number of variables is greater than the number of sampling individuals. This fact makes that traditional statistical methods could not be appropriate. Depending on the nal purpose, in this thesis these data are tackled from two di erent and complementary statistical perspectives: Functional Data Analysis (FDA) or Reliability Analysis (RA) based on Phase-type (PH) probability distributions. FDA arose facing the need of building robust tools to model and predict functional data, whose observations are normally curves depending on time or any other continuous argument. In the last two decades, FDA has been subject of intensive research in which most multivariate techniques have been generalized, specially dimension reduction, regression and classi cation methods. Functional Principal Component Analysis (FPCA) stands out because reduces the dimension and explains the variability structure in terms of a small number of uncorrelated variables. In the reliability eld, one of the main objectives is to study the behaviour of complex systems, whose operation is conditioned by several uncontrollable variables. In this sense, RA attempts to identify the probability distribution of the data to shed light about the variability behind the systems operation. A suitable solution is to contemplate the Markovian processes and the PH distributions. This class is known to be able to approximate any non-negative distribution as much as desired thanks to its versatility and to model complex problems with well-structured results. The methodological contributions of this thesis are elaborated in based to datadriven problems of great interest related to Resistive Random Access Memories (RRAMs) and COVID-19 pandemic. RRAMs awaken much expectation because are one important source of incomes in the industry, whereas for mitigating the spread of the virus, it is crucial developing suitable models to make correct decisions A new statistical approach based on PH distributions is developed to analyze the RRAM variability, which is one of the key issues to solve. A wide comparison with experimental data shows that the tted PH distributions works better than the classic probability distributions and helps to know the RRAM internal performance. A new stochastic process is built by considering the internal performance of macro-states in which the sojourn time is PH distributed. It is showed as the internal behaviour of the process is Markovian but both the homogeneity and Markovianity is lost for the new macro-state model. Other associated measures are also obtained. The new methodology allows the modeling of complex systems in an algorithmic way, in particular, the noise produced inside the RRAMs. FPCA based on Karhunen-Lo eve expansion enables to characterize the stochastic evolution of RRAMs. Nevertheless, it is essential to identify the distribution of the principal components (pc's) to describe the entire process. In this sense, a new class of distributions, Linear PH (LPH) distributions, are introduced. Speci cally, it was proved that if the principal components are LPH distributed then the process follows a LPH distribution at each point. In relation to pc's, sometimes their interpretation is not immediate and a rotation is needed to facilitate it. We develop two new functional Varimax rotation approaches based on the equivalence between FPCA and PCA. One method consists of rotating the eigenvectors, and the other one, rotates the loadings of the standardized pc's scores. They are applied to interpret the variability of the positive cases curves of COVID-19 in the Spanish autonomous communities. Additionally, two di erent parametric and non-parametric functional homogeneity testing approaches are proposed by assuming a basis expansion of sample curves. They consists of testing multivariate homogeneity on a vector of basis coe cients and on a vector of pc's scores, respectively. This fact will be useful to check the in uence of the material and thickness in the RRAM behaviour. For the case of more than one functional response variable, the previous methodology for testing homogeneity based on multivariate FPCA is extended. It is used to test if there are di erences between the levels of several pollutants in terms of the location of measuring stations in the Region Abruzzo, Italy. Also, an approach for repeated measures is considered to study if the level of each pollutant decreased during the lockdown established by the Italian Government for COVID-19 pandemic. Finally, a multiple function-on-function regression model in terms of pc's is proposed for the imputation of missing data for the functional response, by assuming that the multiple functional predictors are completely observed. This approach will enable to impute missing data related to COVID-19. The content of this thesis are presented as a compendium of seven publications. The full version of the papers is included in the Appendices.Tesis Univ. Granada