In this work, we propose two types of models for the analysis of regression and dependence of positive and continuous spatio-temporal data, and of continuous spatio-temporal data with possible asymmetry and/or heavy tails. For the first case, we propose two (possibly non stationary) random fields with Gamma and Weibull marginals. Both random fields are obtained transforming a rescaled sum of independent copies of squared Gaussian random fields. For the second case, we propose a random field with t marginal distribution. We then consider two possible generalizations allowing for possible asymmetry. In the first approach we obtain a skew-t random field mixing a skew Gaussian random field with an inverse square root Gamma random field. In the second approach we obtain a two piece t random field mixing a specific binary discrete random field with half-t random field.

We study the associated second order properties and in the stationary case, the geometrical properties. Since maximum likelihood estimation is computationally unfeasible, even for relatively small data-set, we propose the use of the pairwise likelihood. The effectiveness of our proposal for the gamma and weibull cases, is illustrated through a simulation study and a re-analysis of the Irish Wind speed data (Haslett and Raftery, 1989) without considering any prior transformation of the data as in previous statistical analysis. For the t and asymmetric t cases we present a simulated study in order to show the performance of our method.

Los modelos log-lineales son modelos estadísticos útiles para explicar la interrelación entre un conjunto de variables aleatorias discretas $X = (X_1, \ldots, X_p)$. El modelo se expresa a través del logaritmo de la función de distribución conjunta, $\log(P(X_1 = x_1,\ldots,X_p = x_p))$. Si una de estas variables, digamos $X_1: 0,1$, es considerada como variable respuesta y el resto como explicativas, podemos considerar a un modelo de regresión logística para modelar el logaritmo del momio $log(P (X_1 = 1|X_2, \ldots, X_p)/P(X_1 = 0|X_2,\ldots, X_p))$.

Estos dos modelos guardan una relación estrecha en el sentido que a través de cada uno de ellos puede expresarse la probabilidad condicional de $X_1$ dado el resto de las variables, $P (X_1 = 1|X_2 = x_2 ,\ldots, X_p = x_p )$. La probabilidad condicional $P (X_1 = 1|X_2 = x_2,\ldots,X_p = x_p)$ derivada del modelo log-lineal y la derivada de la regresión logística en general no son iguales.

En esta plática damos la condición bajo la cual esta probabilidad condicional es la misma bajo un modelos log-lineales jerárquico y bajo una regresión logística. Se presenta un ejemplo de aplicación que ilustra la utilidad de los modelos y la relación que guardan entre ellos. Adicionalmente presentamos los mismos ejemplos desde la perspectiva de los modelos gráficos dirigidos probabilísticos, también conocidos como redes Bayesianas. 

If X has a phase–type distribution and N is any positive discrete random variable, then we say that the distribution of X · N belongs to the class of NPH distributions. Such distributions preserve the tractability and generality of phase–type distributions (often allowing for explicit solutions to stochastic models and being dense in the class of distri- butions on the positive reals) but with a different tail behaviour which is basically dictated by the tail of N. We thereby gain a tool for specifying distributions with a “body” shaped by X and with a tail defined by N. After reviewing the construction and basic properties of distributions from the NPH class, we will consider the problem of their estimation. To this end we will employ the EM algorithm, using a similar method as for finite–dimensional phase–type distributions. We consider the the fitting of a NPH distribution to observed data, (left-,right and interval-) censored data, theoretical distributions, histograms, and a couple of examples. 

In this talk, I will demonstrate the use of dynamic models to fit (1) longitudinal count responses such as repeated number of yearly physician visits by an individual; (2) longitudinal binary responses such as repeated asthma status (yes or no) of an individual over several months, and (3) longitudinal multinomial responses such as repeated stress levels (low, medium, high) of an individual worker over a period of few years. These dynamic models are developed to accommodate the correlations of the repeated responses and then to find out the regression effects of certain primary covariates on the repeated responses. In some situations, it may be necessary to add a non-parametric function in certain secondary covariates to the regression function. The ex- tended models are referred to as the longitudinal semi-parametric models. Estimation theory for the model parameters and the non-parametric functions will be discussed. The models and the inference methodologies will also be illustrated by numerical examples.