Master's degree in Statistics and Operations Research

Stochastic Growth Modeling: Hubbert Curves, Bayesian Inference, and Applications.

Date:

June 22, 23, 25, and 26. AFTERNOON: 15 to 19h (Mo, Tu, Th) and 15 to 18h (Fr).

Instructor

Istoni da Luz Sant Ana

Lecturer, Faculty of Biology, University of Barcelona.
PhD in Stochastic Growth Models – University of Granada - Spain, with international experience in Brazil, Puerto Rico, and the USA. Expertise in applied statistics, data science, stochastic modeling, epidemiology., Bayesian inference, and  R. 
Over 15 years of experience teaching statistics and biostatistics at undergraduate and graduate levels, with a strong focus on data science applied to life sciences and health.
Proven research track record: leading and participating in multiple funded projects in life sciences, public health, and probabilistic modeling; publications in indexed international journals.
Significant experience supervising MSc and PhD theses, and guiding complex statistical analysis projects. Fluent in Portuguese, Spanish, Catalan and English. Good knowledge in  French.

Language

Castellano, English

Description

This summer school course introduces stochastic modeling of growth phenomena through diffusion processes, with a strong emphasis on Bayesian statistical inference. The course connects growth curves, such as the Hubbert curves, with stochastic differential equation models that explicitly account for uncertainty, data limitations, and intrinsic variability in real-world processes. Participants will learn how to perform parameter inference from discrete-time observations, simulate diffusion processes, and generate probabilistic predictions. A central component of the course is the study of First Passage Times, which provide a natural probabilistic framework for analyzing critical events in growth dynamics, such as peak production times or stabilization thresholds. The methodology is illustrated through both simulated data and real-world applications. These include oil production modeling, where the timing and uncertainty of the production peak are of primary interest, and CD4 cell count dynamics in HIV-positive patients, where diffusion models are used to assess when immune defenses stabilize. Throughout the course, theory is tightly integrated with practical implementation in R, enabling participants to fit models, conduct Bayesian inference, and interpret results in applied stochastic modeling contexts.

Course goals

The goal of this course is to guide students through a structured learning path, moving from theoretical foundations to applied stochastic modeling of growth phenomena. By the end of the course, students will be able to: 1. Understand growth curves and their associated stochastic diffusion processes, with particular attention to processes whose transition densities are known, allowing for analytical tractability and efficient simulation. 2. Formulate stochastic models appropriate to real-world growth phenomena, translating deterministic growth curves into diffusion processes that capture randomness and uncertainty. 3. Apply parameter estimation procedures for diffusion processes, focusing on discrete-time sampling schemes and simulation-based Bayesian methods commonly used in practice. 4. Perform statistical inference on the parameters of diffusion processes, including Bayesian inference and posterior analysis, and evaluate how different estimation strategies influence other properties of the process, such as dynamics, uncertainty, and stability. 5. Assess model adequacy through optimal data fitting, ensuring that the chosen model provides a realistic representation of the observed phenomenon while maintaining strong inferential foundations. 6. Evaluate the predictive performance of stochastic models, emphasizing uncertainty quantification and the conditions under which reliable predictions can be made. 7. Fit, validate, and interpret models using simulated and real datasets, integrating inference, prediction, and interpretation into a coherent modeling workflow. Overall, the course aims to provide students with the theoretical insight and computational skills needed to develop, estimate, and evaluate stochastic diffusion models associated with growth processes, ensuring both explanatory power and predictive capability.

Course contents

  • General Introduction
    Overview of growth phenomena across disciplines (economics, ecology, energy, epidemiology, and social systems). Motivation for growth modeling, historical development of growth curves, and the role of uncertainty and stochasticity in real-world data. Introduction to the course structure, objectives, and datasets to be used.

  • Growth Curves
    Review of classical deterministic growth models, including exponential, logistic, Gompertz, and related curves. Detailed discussion of the Hubbert curve and the Double Hubbert curve, with emphasis on their mathematical properties, interpretability of parameters, and typical applications. Comparison between different growth models and their limitations.

  • Diffusion Processes Associated with Growth Curves
    Transformation of deterministic growth curves into stochastic diffusion processes. Introduction to stochastic differential equations (SDEs) as a modeling framework for growth under uncertainty. Interpretation of drift and diffusion components, connection between diffusion processes and macroscopic growth dynamics, and illustrative examples.

  • Inference for Diffusion Processes
    Statistical inference for diffusion models, with a focus on Bayesian approaches. Likelihood-based Bayesian inference and simulation-based methods for parameter estimation, including discretization schemes and data augmentation. Prior specification, posterior inference, uncertainty quantification, and model diagnostics using simulated and real data.

  • First-Passage Times in Diffusion Processes
    Concept and theory of first-passage (hitting) times in diffusion processes. Analytical and numerical approaches for estimating first-passage time distributions. Interpretation in applied contexts such as peak timing, threshold crossing, and regime shifts in growth dynamics.

  • Model Fitting, Prediction, and Applications
    Practical implementation of growth and diffusion models in R. Simulation of trajectories, parameter estimation, forecasting, and predictive uncertainty. Model comparison and validation. Applications to real-world case studies, including interpretation of results and discussion of strengths, limitations, and possible extensions of the models.

Prerequisites

Participants are expected to have prior knowledge of the R programming language, including data manipulation, basic visualization, and the ability to write simple functions. A solid understanding of basic statistics and probability theory is required, covering concepts such as random variables, probability distributions, expectation, variance, and hypothesis testing. Familiarity with statistical inference is essential, and prior exposure to Bayesian statistical inference—such as likelihood functions, prior and posterior distributions, and basic Bayesian modeling—is strongly recommended. The course is intended for graduate and advanced undergraduate students with a quantitative background and assumes a working knowledge of mathematical and statistical reasoning.

Targeted at

This course is targeted at graduate students (Master’s and PhD) and advanced undergraduate students with a strong quantitative background who are interested in statistical modeling, stochastic processes, and Bayesian inference. It is particularly suitable for students in statistics, applied mathematics, data science, economics, engineering, environmental sciences, and related fields, as well as researchers seeking to deepen their understanding of growth curves, diffusion processes, and Bayesian modeling approaches through simulation and applied examples.

Teaching Methodology and Activities

Teaching will follow an interactive, in-person format that integrates theory and practice. Lectures will be complemented by hands-on R sessions, where students will implement Bayesian growth and diffusion models using simulated and real datasets. Emphasis will be placed on experimentation, model interpretation, and critical discussion. Complete R scripts for all exercises will be provided, allowing students to reproduce and extend the analyses beyond the classroom.

Evaluation

Evaluation will be based on daily hands-on activities carried out during class sessions. Students will work on guided and open-ended exercises that require the use of the R programming language to fit models to both simulated and real datasets. These activities will involve parameter inference, data simulation, forecasting, and interpretation of model outputs within a Bayesian framework. Assessment will emphasize active participation, correct implementation of statistical models, and the ability to critically interpret results rather than formal examinations. Students will be evaluated on the completion and quality of the daily exercises, including code functionality, clarity of reasoning, and discussion of results.

Software requirements

Participants are required to have the most recent version of R installed, along with RStudio as the integrated development environment. Throughout the course, several R packages will be used for modeling, simulation, inference, and visualization, including fptdApprox, nleqslv, ggplot2, and yuima.

All software and packages are open-source and available for free. Participants are expected to install and test the required software prior to the start of the course. Installation instructions and example scripts will be provided in advance to ensure that all participants can fully engage in the hands-on computational activities.