Synthetic biology is an emerging field that integrates and applies engineering design methods to biological systems. Its aim is to make biology an "engineerable" science. Over the years, biologists and engineers alike have abstracted biological systems into functional models that behave similarly to electric circuits, thus the creation of the subfield of genetic circuits. Mathematical models have been devised to simulate the behavior of genetic circuits in silico. Most models can be classified into deterministic and stochastic models. The work in this dissertation is for stochastic models. Although ordinary differential equation (ODE) models are generally amenable to simu- late genetic circuits, they wrongly assume that a system's chemical species vary continuously and deterministically, thus making erroneous predictions when applied to highly stochastic systems. Stochastic methods have been created to take into account the variability, un- predictability, and discrete nature of molecular populations. The most popular stochastic method is the stochastic simulation algorithm (SSA). These methods provide a single path of the overall pool of possible system's behavior. A common practice is to take several inde- pendent SSA simulations and take the average of the aggregate. This approach can perform iv well in low noise systems. However, it produces incorrect results when applied to networks that can take multiple modes or that are highly stochastic. Incremental SSA or iSSA is a set of algorithms that have been created to obtain ag- gregate information from multiple SSA runs. The marginal probability density evolution (MPDE) algorithm is a subset of iSSA which seeks to reveal the most likely "qualitative" behavior of a genetic circuit by providing a marginal probability function or statistical enve- lope for every species in the system, under the appropriate conditions. MPDE assumes that species are statistically independent given the rest of the system. This assumption is satisfied by some systems. However, most of the interesting biological systems, both synthetic and in nature, have correlated species forming conservation laws. Species correlation imposes con- straints in the system that are broken by MPDE. This work seeks to devise a mathematical method and algorithm to correct conservation constraints errors in MPDE. Furthermore, it aims to identify these constraints a priori and efficiently deliver a trustworthy result faithful to the true behavior of the system.

La presente investigación tiene como objetivo central identificar cuáles son los principales determinantes que influyen en la brecha de género en la inclusión financiera del Perú durante 2016. En tal sentido, se utiliza la Encuesta Nacional de Demanda de Servicios Financieros y Nivel de Cultura Financiera (ENIF, 2016), en la cual se encuestó a 6,303 individuos seleccionados al azar, formando una muestra representativa de todo el Perú, y se realiza la estimación de modelos de elección discreta (logit y probit). Además, se calculan los impactos marginales de las variables socioeconómicas sobre la posesión de cuentas de ahorro y tarjetas de crédito tanto para varones como mujeres. Los resultados muestran que el nivel educativo es la variable que genera un mayor aumento en la probabilidad de acceder al sistema financiero aunque no de manera muy diferenciada entre géneros; mientras que la posesión de activos, relación de parentesco, residencia y estado civil generan impactos menores en el género femenino. / The main objective of this research is to identify the main determinants that influence the gender gap in the financial inclusion of Peru during 2016. In this sense, the National Survey of Demand for Financial Services and Level of Financial Culture (ENIF, 2016) is used, in which 6,303 randomly selected individuals were surveyed, forming a representative sample of Peru. The estimation of discrete choice models (logit and probit) is made. In addition, the marginal impacts of socioeconomic variables on the possession of savings accounts and credit cards for both men and women are calculated. The results show that the educational level is the variable that generates a greater increase in the probability of accessing the financial system although not in a very differentiated way between genders; while the possession of assets, kinship relationship, residence and marital status generate minor impacts on the female gender. / Trabajo de investigación

We consider a longitudinal system in which transitions between the states are governed by a discrete time finite state space stochastic process X. Our aim, using aggregated sample survey data of the form typically collected by official statistical agencies, is to undertake model based inference for the underlying process X. We will develop inferential techniques for continuing sample surveys of two distinct types. First, longitudinal surveys in which the same individuals are sampled in each cycle of the survey. Second, cross-sectional surveys which sample the same population in successive cycles but with no attempt to track particular individuals from one cycle to the next. Some of the basic results have appeared in Davis et al (2001) and Davis et al (2002).¶ Longitudinal surveys provide data in the form of transition frequencies between the states of X. In Chapter Two we develop a method for modelling and estimating the one-step transition probabilities in the case where X is a non-homogeneous Markov chain and transition frequencies are observed at unit time intervals. However, due to their expense, longitudinal surveys are typically conducted at widely, and sometimes irregularly, spaced time points. That is, the observable frequencies pertain to multi-step transitions. Continuing to assume the Markov property for X, in Chapter Three, we show that these multi-step transition frequencies can be stochastically interpolated to provide accurate estimates of the one-step transition probabilities of the underlying process. These estimates for a unit time increment can be used to calculate estimates of expected future occupation time, conditional on an individual’s state at initial point of observation, in the different states of X.¶ For reasons of cost, most statistical collections run by official agencies are cross-sectional sample surveys. The data observed from an on-going survey of this type are marginal frequencies in the states of X at a sequence of time points. In Chapter Four we develop a model based technique for estimating the marginal probabilities of X using data of this form. Note that, in contrast to the longitudinal case, the Markov assumption does not simplify inference based on marginal frequencies. The marginal probability estimates enable estimation of future occupation times (in each of the states of X) for an individual of unspecified initial state. However, in the applications of the technique that we discuss (see Sections 4.4 and 4.5) the estimated occupation times will be conditional on both gender and initial age of individuals.¶ The longitudinal data envisaged in Chapter Two is that obtained from the surveillance of the same sample in each cycle of an on-going survey. In practice, to preserve data quality it is necessary to control respondent burden using sample rotation. This is usually achieved using a mechanism known as rotation group sampling. In Chapter Five we consider the particular form of rotation group sampling used by the Australian Bureau of Statistics in their Monthly Labour Force Survey (from which official estimates of labour force participation rates are produced). We show that our approach to estimating the one-step transition probabilities of X from transition frequencies observed at incremental time intervals, developed in Chapter Two, can be modified to deal with data collected under this sample rotation scheme. Furthermore, we show that valid inference is possible even when the Markov property does not hold for the underlying process.