Publication: Models and inference for population dynamics
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2012-06
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2012-09-25
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Abstract
En el presente trabajo abordamos el problema de la modelización e inferencia de la dinámica de las poblaciones de bacterias. Dado que las mediciones del crecimiento de bacterias en platillos de Petri, pueden fácilmente replicarse bajo las mismas condiciones experimentales, el estudio se centra en los casos donde los datos presentan una estructura jerárquica. El crecimiento de bacterias está muy influido por las condiciones ambientales, por ejemplo niveles de sal, temperatura o acidez y la relación de estos factores con el crecimiento es muy compleja. Por ello, en experimentos bajo distintas condiciones, es fundamental buscar modelos flexibles para relacionar el crecimiento con tales condiciones. En esta tesis, presentamos como objetivo desarrollar modelos predictivos capaces de combinar toda la información disponible, como por ejemplo la repetición de los experimentos, con el fin de lograr predicciones más precisas. Por otra parte, se propone también desarrollar un modelo mas general para el crecimiento aplicable a una gran variedad de microorganismos y bajo un gran número de combinaciones de las condiciones ambientales y ecológicas. Con estos objetivos en mente, proponemos el uso de modelos jerárquicos cuando se observan multiples curvas de crecimiento. De esta manera, la estimación de una única curva es mejorada a través de la información que brindan el resto de las curvas de crecimiento observadas. Adicionalmente, proponemos también el uso de técnicas no paramétricas para modelizar los procesos de crecimiento, sin necesidad de asumir que las poblaciones se comportan según cierta función paramétrica. En particular, utilizamos redes neuronales ya que tienen una gran capacidad de describir el comportamiento de modelos complejos y no lineales. Los procesos de crecimiento pueden presentar ciertas fluctuaciones estocásticas que no se deben a errores de medición. Los modelos que simplemente adicionan un error a una función determinística no son capaces de capturar la variabilidad total de estos procesos. En consecuencia, hemos desarrollado un modelo estocástico que presenta dos características deseables: las trayectorias de crecimiento son no-decrecientes y la función de medias del proceso es proporcional a la función de Gompertz de crecimiento. Finalmente, en este trabajo también se aborda el problema de la estimación de los modelos, para lo cual hemos preferido utilizar inferencia bayesiana ya que, entre otra cosas, brinda un enfoque unificado al tratar con diversos tipos de modelos, como por ejemplo, jerárquicos y redes neuronales. Por otra parte, la inferencia Bayesiana nos permite diferenciar entre distintas fuentes de incertidumbre a través del uso de distribuciones a priori jerárquicas. Asi mismo, permite la incorporación de información previa, ampliamante disponible en ciencias como la microbiología
In this dissertation we study the problem of modeling and inference for the dynamics of bacterial populations. Bacterial growth data taken from Petri-dish experiments is easily replicated. Moreover, external factors such as temperature, salinity or acidity of the environment are known to influence bacterial growth and therefore, experiments are often undertaken under a variety of conditions. This implies that often, bacterial growth data present a multilevel structure. The first issue that we wish to to address in this thesis is how to analyze data from multiple experiments in this context. The aim of our study is to develop a predictive model able to combine all available information, such as replicated experiments, in order to get more accurate predictions. Additionally, we wish to develop a more general model for microbial growth for a variety of organism types and under a larger number of combinations of environmental and ecological variables. To accomplish this challenges, we propose the use of hierarchical models when multiple growth curve data are observed. In this way, it is possible to improve the estimation of a single growth curve by incorporating information from the other bacterial growth curves. Additionally, we propose the use of non-parametric techniques to model the growth process, where it is not assumed that the population fits any parameterized model. In particular, we shall introduce models based on neural networks which can be used to fit very complex relationships. A growth process may display some stochastic fluctuations which are not due to measurement errors. Models which simply add an error to a deterministic function cannot necessarily capture the total variability of the growth process. Therefore, it is also important to consider fully stochastic models. Another objective of this thesis is to provide a new, stochastic growth curve model of this type. In general, in the literature on growth curve modeling, most work has been carried out using weighted least squares techniques and other classical approaches. However, the Bayesian approach brings a unified approach to the handling of complex models, such as hierarchical models and neural networks and allows us to differentiate, through the use of hierarchical prior distributions, between various sources of variability, which is an important issue in predictive microbiology. Furthermore, the Bayesian approach permits the incorporation of prior information which is abundant in experimental sciences. One of the main difficulties with the Bayesian approach for practical purposes is that often, complex algorithms have to be devised for the implementation of these techniques, which is a disadvantage to non specialists. Therefore, a further objective of this thesis is to show that Bayesian inference can be implemented for many of the models proposed using a relatively simple algorithm based on a generally available free software package which can be used without the need to fine tune special samplers. In summary, this thesis aims to provide a statistical framework for the analysis of bacterial growth processes. Modeling and prediction play a key role in the field of microbiology as a valuable tool for making recommendations on food safety and human health and hence, improvements in the methods available are of interest. The rest of the thesis is structured as follows. In Chapter 1, we present a brief description of the main population growth models, focusing in the advantages and disadvantages of each one. Then we show that, given a single sample of growth curve data from one of these models, it is straightforward to implement both classical and Bayesian inference for these models. We concentrate on the Bayesian approach which is growing in interest because of its capability to incorporate information from a variety of widely available sources such as laboratory experiments, field measurements and expert judgements and for the possibility to distinguish formally between different sources of uncertainty. In particular, we show that the free software package WinBUGS can be used to implement Bayesian inference for simple bacterial growth models. In Chapter 2, we consider the case when various replications of Petri dish experiments under identical conditions are observed. In such cases, we would expect the individual growth curves to be similar and this suggests the use of hierarchical models to capture the relationship between the diffeerent growth curves. As in Chapter 1, we illustrate that the hierarchical model we use, based on the well known Gompertz curve, can be fitted using WinBUGS. In Chapter 3, we then consider the case of Petri dish experiments under different environmental conditions. The relationship between the growth curve parameters and the environmental factors is complex, and here we consider the use of neural networks to model this relationship. Two basic models are considered. Firstly, we introduce a neural network based secondary model which is based on a Gompertz curve where the parameters of the growth curve are modeled as a function of the environmental factors. Secondly, we consider the direct modeling of the growth curve using neural networks. As previously, inference is carried out using a Bayesian approach implemented via WinBUGS. These first three chapters demonstrate that WinBUGS can be a powerful and fleexible tool able to handle very complex models. We show that in practice, it is relatively straightforward to implement complex models in WinBUGS which allows microbiological researchers to conduct Bayesian inference in a simple way, without the necessity to design complex MCMC algorithms and instead to concentrate on the model building aspects of the problem. In the first three chapters, we concentrate on models in discrete time which have the restriction that, for example they may be difiult to implement if data are observed at irregular time intervals. In contrast, in Chapter 4 we develop a new, continuous time, stochastic growth curve model. We show by means of simulations that our proposed model has the potential to capture the the variability observed in replications of the same experiment under identical conditions. Also, we illustrate that by modifying the parameter values, different shaped growth curves can be generated. Finally, we introduce two approaches to Bayesian inference for our model. Firstly, in a simple case of the model, we introduce a Gibbs sampling algorithm and secondly, for the full model, we consider the use of an approximate Bayesian computing algorithm.
In this dissertation we study the problem of modeling and inference for the dynamics of bacterial populations. Bacterial growth data taken from Petri-dish experiments is easily replicated. Moreover, external factors such as temperature, salinity or acidity of the environment are known to influence bacterial growth and therefore, experiments are often undertaken under a variety of conditions. This implies that often, bacterial growth data present a multilevel structure. The first issue that we wish to to address in this thesis is how to analyze data from multiple experiments in this context. The aim of our study is to develop a predictive model able to combine all available information, such as replicated experiments, in order to get more accurate predictions. Additionally, we wish to develop a more general model for microbial growth for a variety of organism types and under a larger number of combinations of environmental and ecological variables. To accomplish this challenges, we propose the use of hierarchical models when multiple growth curve data are observed. In this way, it is possible to improve the estimation of a single growth curve by incorporating information from the other bacterial growth curves. Additionally, we propose the use of non-parametric techniques to model the growth process, where it is not assumed that the population fits any parameterized model. In particular, we shall introduce models based on neural networks which can be used to fit very complex relationships. A growth process may display some stochastic fluctuations which are not due to measurement errors. Models which simply add an error to a deterministic function cannot necessarily capture the total variability of the growth process. Therefore, it is also important to consider fully stochastic models. Another objective of this thesis is to provide a new, stochastic growth curve model of this type. In general, in the literature on growth curve modeling, most work has been carried out using weighted least squares techniques and other classical approaches. However, the Bayesian approach brings a unified approach to the handling of complex models, such as hierarchical models and neural networks and allows us to differentiate, through the use of hierarchical prior distributions, between various sources of variability, which is an important issue in predictive microbiology. Furthermore, the Bayesian approach permits the incorporation of prior information which is abundant in experimental sciences. One of the main difficulties with the Bayesian approach for practical purposes is that often, complex algorithms have to be devised for the implementation of these techniques, which is a disadvantage to non specialists. Therefore, a further objective of this thesis is to show that Bayesian inference can be implemented for many of the models proposed using a relatively simple algorithm based on a generally available free software package which can be used without the need to fine tune special samplers. In summary, this thesis aims to provide a statistical framework for the analysis of bacterial growth processes. Modeling and prediction play a key role in the field of microbiology as a valuable tool for making recommendations on food safety and human health and hence, improvements in the methods available are of interest. The rest of the thesis is structured as follows. In Chapter 1, we present a brief description of the main population growth models, focusing in the advantages and disadvantages of each one. Then we show that, given a single sample of growth curve data from one of these models, it is straightforward to implement both classical and Bayesian inference for these models. We concentrate on the Bayesian approach which is growing in interest because of its capability to incorporate information from a variety of widely available sources such as laboratory experiments, field measurements and expert judgements and for the possibility to distinguish formally between different sources of uncertainty. In particular, we show that the free software package WinBUGS can be used to implement Bayesian inference for simple bacterial growth models. In Chapter 2, we consider the case when various replications of Petri dish experiments under identical conditions are observed. In such cases, we would expect the individual growth curves to be similar and this suggests the use of hierarchical models to capture the relationship between the diffeerent growth curves. As in Chapter 1, we illustrate that the hierarchical model we use, based on the well known Gompertz curve, can be fitted using WinBUGS. In Chapter 3, we then consider the case of Petri dish experiments under different environmental conditions. The relationship between the growth curve parameters and the environmental factors is complex, and here we consider the use of neural networks to model this relationship. Two basic models are considered. Firstly, we introduce a neural network based secondary model which is based on a Gompertz curve where the parameters of the growth curve are modeled as a function of the environmental factors. Secondly, we consider the direct modeling of the growth curve using neural networks. As previously, inference is carried out using a Bayesian approach implemented via WinBUGS. These first three chapters demonstrate that WinBUGS can be a powerful and fleexible tool able to handle very complex models. We show that in practice, it is relatively straightforward to implement complex models in WinBUGS which allows microbiological researchers to conduct Bayesian inference in a simple way, without the necessity to design complex MCMC algorithms and instead to concentrate on the model building aspects of the problem. In the first three chapters, we concentrate on models in discrete time which have the restriction that, for example they may be difiult to implement if data are observed at irregular time intervals. In contrast, in Chapter 4 we develop a new, continuous time, stochastic growth curve model. We show by means of simulations that our proposed model has the potential to capture the the variability observed in replications of the same experiment under identical conditions. Also, we illustrate that by modifying the parameter values, different shaped growth curves can be generated. Finally, we introduce two approaches to Bayesian inference for our model. Firstly, in a simple case of the model, we introduce a Gibbs sampling algorithm and secondly, for the full model, we consider the use of an approximate Bayesian computing algorithm.
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Keywords
Inferencia estadística, Modelo matemático, Estadística bayesiana, Procesos estocásticos