Factorial | EPFL Graph Search

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. The factorial of n also equals the product of n with the next smaller factorial: \begin{align} n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \ &= n\times(n-1)!\ \end{align} For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many a

Un modèle factoriel dynamique pour séries temporelles

Andrei Zenide

This work deals with factorial models for multiple time series. Its core content puts it at the interface between statistics and finance. After a brief description of the historical link between the two sciences, it reviews the literature on factorial models that are close to the model introduced in this work called The dynamical factor analysis model for time series. This model makes the hypothesis that the observed time series are influenced by a common factor, difficult to define and impossible to measure. No a priori structure is put on the factor, at each point time the value of the factor is considered as a new parameter that has to be estimated. As a consequence of this fact, the number of parameters is large and it is not possible to provide the usual asimptotic properties of the estimations by letting the number of periods tend to infi- nity. Asymptotic results in our context have been obtained by increasing the number of time series. The model makes the hypothesis that there is a linear dependence between the time series and the factor with coefficients, that are not constant over time, but rather follow a smooth random walk. This mean that the linear structure of the model is evolving slowly from a period to another. All the information which is not contained in the factor and the coefficients is considered as white noise. Using the normal distribution makes the estimation more easy and opens a toolbox of statistical methods that have been developed for this kind of data. The model is a part of the family of state space models and the Kalman filter is an essential ingredient of our estimator. The effort was concentrated on the elaboration of the structure of the model. Its complexity was constrained by the difficulty of estimation. The final shape of the model does not allow an analytical solution of the optimization problem introduced by the maximum likelihood estimation of the parameters. Numerical solutions have been found and compared with the parameters for simulated data. Some others models have been developed as simpler versions of the dynamical factor model for time series. The case where the factor can be observed has been studied and a new method for the estimation has been provided and compared with the existing methods. A second study considers a latent factor model without noise. Two methods for the estimation of the factor have been provided. The last chapter contains a detailed description of the main statistical tools used during this work. The links with the previous chapters are highlighted followed by comments.

EPFL2004

Un modèle factoriel dynamique pour séries temporelles

Andrei Zenide

EPFL2004