Splines de nós livres para ajuste de curvas em problemas na área da saúde
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2021-02-25
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Universidade Estadual Paulista (Unesp)
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Encontrar modelos que expliquem a relação entre variáveis é uma das principais aplicações da estatística. Os modelos mais simples são os chamados modelos de regressão linear que são, muitas vezes, incapazes de aproximar apropriadamente a relação entre as variáveis, a qual nem sempre é linear. Para essas situações, podem ser usados os modelos de regressão não-linear, os quais demandam conhecimentos específicos do processo envolvido e ferramentas mais complexas para ajuste que os modelos lineares. Outra opção é aproximar funções não-lineares por expansões de base. Expansões de base por splines aproximam funções não-lineares por polinômios segmentados de ordem q contínuos até a sua derivada de ordem q-2. O maior desafio nesses métodos é encontrar o posicionamento correto dos nós (onde dividir os intervalos para melhor aproximar a função desejada). Métodos chamados de splines de nós livres consideram a posição dos nós, e muitas vezes o número, como parâmetros livres que são, então, definidos pelos dados. No contexto Bayesiano, existem metodologias fundamentadas no algoritmo MARS (Multivariate Adaptive Regression Splines), que selecionam os nós adaptativamente com o método de Monte Carlo via cadeias de Markov com saltos reversíveis (RJMCMC). O presente trabalho é focado no estudo de uma dessas metodologias, Bayesian Adaptive Splines Surfaces (BASS), no contexto de ajuste de curvas. A metodologia foi aplicada em dois problemas da área da saúde. Um com objetivo de validar um protocolo de crioablação vertebral analisando as curvas de temperatura na sonda de crioablação e em termopares a distâncias crescentes das sondas, e o outro com o objetivo de determinar o período ótimo, após ingestão da uréia 13C, para a realização do teste respiratório para detecção da presença de Helicobacter pylori analisando as curvas de DOB (Delta Over Baseline). Os resultados do método BASS foram comparados aos resultados de método de splines de nós livres que utiliza algoritmo genético para encontrar a posição dos nós. Apesar do método levar a pistas interessantes a respeito dos problemas da área da saúde, na avaliação dos modelos foram observados graves problemas de heterocedasticidade.
Finding models that explain the relationship among variables is one of the main applications of statistics. The simplest models are the so-called linear regression models which, many times, are unable to approximate properly the relationship among variables that is not always linear. For these situations, non-linear regression models can be used, which demand specific knowledge of the process involved and more complex tools for fitting than linear models. Another option is to approximate these non-linear functions through basis expansions. Spline basis expansion approximate non-linear functions by segmented polynomials of q order that are continuous up to its q-2 order derivative. The greatest challenge in these methods is to find the right placement of knots (where to divide the intervals to better approximate the desired function). Free-knot splines methodologies set the positions of the knots, and sometimes the number of knots, as free parameters that are, then, defined by the data. In the Bayesian context, there exist methodologies inspired by the MARS algorithm, that use reversible jump Markov chain Monte Carlo methods (RJMCMC) to adaptively select the knots' locations and number. The following work focus on the study of one of those methodologies, Bayesian Adaptive Splines Surfaces (BASS), in the context of curve fitting. The methodology was applied in two health problems. In one the goal is to validate a vertebral cryoablation protocol analyzing temperature curves in the cryoablation catheter and at the thermocouples placed at growing distances from the cryoablation catheter and in the other problem the goal is finding the optimal time, after ingestion of 13C urea, to take the breath test to Helicobacter pylori detection by analyzing DOB (Delta Over Baseline) curves. The BASS results were compared to a free-knot splines approach that uses a genetic algorithm to find the optimal positions of knots. Despite presenting interesting clues about the health problems, several heteroscedasticity problems were observed in model evaluation.
Finding models that explain the relationship among variables is one of the main applications of statistics. The simplest models are the so-called linear regression models which, many times, are unable to approximate properly the relationship among variables that is not always linear. For these situations, non-linear regression models can be used, which demand specific knowledge of the process involved and more complex tools for fitting than linear models. Another option is to approximate these non-linear functions through basis expansions. Spline basis expansion approximate non-linear functions by segmented polynomials of q order that are continuous up to its q-2 order derivative. The greatest challenge in these methods is to find the right placement of knots (where to divide the intervals to better approximate the desired function). Free-knot splines methodologies set the positions of the knots, and sometimes the number of knots, as free parameters that are, then, defined by the data. In the Bayesian context, there exist methodologies inspired by the MARS algorithm, that use reversible jump Markov chain Monte Carlo methods (RJMCMC) to adaptively select the knots' locations and number. The following work focus on the study of one of those methodologies, Bayesian Adaptive Splines Surfaces (BASS), in the context of curve fitting. The methodology was applied in two health problems. In one the goal is to validate a vertebral cryoablation protocol analyzing temperature curves in the cryoablation catheter and at the thermocouples placed at growing distances from the cryoablation catheter and in the other problem the goal is finding the optimal time, after ingestion of 13C urea, to take the breath test to Helicobacter pylori detection by analyzing DOB (Delta Over Baseline) curves. The BASS results were compared to a free-knot splines approach that uses a genetic algorithm to find the optimal positions of knots. Despite presenting interesting clues about the health problems, several heteroscedasticity problems were observed in model evaluation.
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Palavras-chave
RJMCMC, BASS, Splines, Regressão linear, Inferência Bayesiana, Linear regression, Bayesian inference