Power Series Representations of Hypergeometric Type and Non-Holonomic Functions in Computer Algebra. Updated pdf file here.
I worked under the supervision of Prof. Dr. Wolfram Koepf. I give below brief details of my major results.
1. An algorithm, say mfoldHyper, that computes bases of linear combinations of all m-fold (m being a positive integer) hypergeometric term solutions of holonomic recurrence equations. This algorithm is new, and it has the advantage to linearize the computation of Laurent-Puiseux series: every linear combination of hypergeometric type series solution of a holonomic differential equation, even for many different values of m, is detected.
2. An algorithm that computes normal form representations for an interesting class of non-holonomic functions. A significant consequence of this algorithm is its ability to prove difficult identities.
3. A variant of van Hoeij's algorithm for computing hypergeometric term solutions of holonomic recurrence equations. Following the main steps of van Hoeij's algorithm, my variant computes hypergeometric term solutions of holonomic recurrence equations as fast as the original algorithm while using other methods. This algorithm is an integral part of mfoldHyper mentioned in 1.
Related papers can be found under the section Papers on the website.
Automatic Computation of Laurent-Puiseux series of Hypergeometric Type.
This is my first contact with the subject of my Ph.D. thesis. Notice that in this essay the hypergeometric type series is reduced to those whose recurrence equations with only two terms are immediately found.
Classification non-Supervisée et Suivi des Processus de Dynamique Forestière. (French)
The main purpose here is to model the birth, growth, and mortality of species of trees in the forest. The EM algorithm is used for clustering. Concerning the regression, the growth and the mortality are modeled using the binomial distribution, and the birth is modeled using the Poisson distribution.
Simulation of data: Sim.r, SimPois.r
Estimation of parameters with the EM algorithm: EM_MortaliteCroissance.r, EM_Recruitement.r
Extension, August 2017 (pdf): I wrote a proposal that gives an approach to extend this work by considering data collected from satellites and estimate the stock of carbon.