Data-Driven Computational Methods: Parameter and Operator Estimations

This short book is a survey of computational methods for the stochastic modeling of dynamical systems. From the introduction: “we survey numerical methods that leverage observational data to estimate parameters in a dynamical model when the parametric model is available and to approximate the model non parametrically when such a parametric model is not available”.

 

The parametric approach is reviewed, in chapters 2 and 3, with a short description of Markov-Chain Monte-Carlo methods, and the ensemble Kalman filters. Chapters 4 and 5 are transitional and offer a review of topics such as orthogonal polynomial basis in Hilbert spaces, the Galerkin method and Mercer’s theorem. These are then used in the final chapter on diffusion forecasting.

 

Although this book collects material from a graduate course on Uncertainty Quantification Methods, I am not completely sure it would be suitable as a standalone textbook for a course, or for self-study. Some exercises would be a valuable addition to the book, perhaps in the form of simulation problems. The MATLAB code used for the examples in the book can be downloaded from the publisher’s website; the scripts are short, well commented and can be understood without difficulty (even if you are not a MATLAB expert). However, I wonder if an open-source language, like Scilab, Octave or Python, would have been a better choice (not everyone has a MATLAB license). 

 

The prerequisites for reading this text are a graduate-level understanding of: stochastic processes, functional analysis, Riemann geometry and general knowledge of Bayesian inference methods. Three short appendices review some of the relevant material, but I think it is still necessary to have had courses on topics like stochastic calculus or functional calculus.

 

Overall, I think this book could be a good complement to a general text like, for example, Sullivan’s Introduction to Uncertainty Quantification.  But both books lack, in my opinion, a discussion of the algorithmic efficiency and the implementation issues that one often encounters in the industrial applications of data-driven parameter estimations. 

 

As a side note, it would be interesting to see how the methods presented in this book are related to the methods used in statistical learning theory.

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 Fabio Mainardi (fabio.mainardi@rd.nestle.com) is a mathematician working as a senior data scientist at Nestlé Research. His mathematical interests are number theory, functional analysis, discrete mathematics and probability.