Global Sensitivity Analysis: The Primer [Anglais] [Relié]

When I was working at Oak Ridge National Lab in the late 1970s, I worked with other scientists and statisticians on data and model validation. Toby Mitchell a specialist in experimental design was developing sampling techniques to use in model validation. He has since passed on. One of the techniques he used was Latin hypercube sampling. The authors of this text are from Italy and Canada. They have computer science and mathematical backgrounds but are not statisticians. Yet once you start reading book you will see that they appreciate both the deterministic and the stochastic aspects of modeling.

What they do, they call Global Sensitivity Analysis. They are very bright and are lucid in their explanations and description of philosophical issues. This is not something that those of us who do statistical modeling are very familiar with but it is important to know. It is especially gratifying to see that these authors are always wary of modeling assumptions and look for novel ways to test them. They point out that validating models is complicated. Many models that we construct are complex and even when we are aware of this and test aspects of the model. we often take some things for granted and accept aspects of the model as a given. I really enjoyed reading the afterword where these issues are well brought out.

Specific methods include th elementary effects method discussed in Chapter 3 that is based on the work of Max Morris in a 1991 paper in Technometrics. In Chapter 4, they cover variance-based methods which relies on the work of Sobol and others. They illustrate the applications of these methods with an infection dynamics model.

The main idea is to determine which factors affect the output variables as well as which interaction effects play a role. In the last chapter the authors make recommendations as to when to apply each technique.

At the beginning of the sections they raise questions that they answer in the section. This approach and the problem sets followed by complete and clear solutions makes the text readable and enjoyable even for novices.

To begin with, I purchased this book when I was about halfway through my Ph.D. (which I have now completed, and presently doing a post-doc at NASA). My research is on the various issues involved in uncertainty quantification in the context of engineering systems. In today's world, nothing is certain - it is important to quantify the amount of uncertainty in each and everything. This book introduced me to an entirely new topic in the domain of uncertainty quantification - the topic of global sensitivity analysis. I knew what I wanted to learn before buying this book, and the book taught that precisely. I have even used what I learnt in my research and I probably have a couple of publications based on my learning.

Most people tend to think of "sensitivity analysis" as something related to mathematical derivatives. But, for non-linear functions, it is important to specify "the point" at which the derivative must be evaluated, because the derivative itself is a non-constant function if the original function is non-linear. Because the derivative is measured at a "particular point", the entire analysis becomes "local" in some sense. How is this connected to uncertainty quantification?

Lets say there are 5 independent quantities which are used to compute a dependent quantity using a mathematical model. For example, in a stringed instrument such as a violin or guitar, the frequency we hear is dependent on length of vibration "l", string mass "m", and tension "T" (technically, its simply tension and linear density, but never mind). If the three independent quantities have uncertainty, then there is some uncertainty in the frequency. We can compute the mean and standard deviation of the frequency using Taylor's series approximation of the formula for the frequency of a vibrating string; these are then called as the first-order mean and first-order standard deviation in the uncertainty quantification literature; but analysis is local. For a truly global analysis, we must consider multiple samples of "l", "m", and "T" and compute the entire probability distribution of frequency, using Monte Carlo simulation or something more advanced.

Now, what is "global sensitivity analysis"? It basically analyzes how much of the uncertainty in the dependent variable comes from each of the independent variables. Of course, we will be happy if somebody says that the string-tension contributes to 50% of the uncertainty in frequency, length 30%, and mass 20%. The book starts off with the aim of doing this, and goes on to explain why it is not possible to get such numbers (that sum to 100) because of the so-called interaction between the independent variables. There are so many researchers working in the topic of uncertainty quantification, but very few of them deal with sensitivity analysis. Whenever we propagate uncertainty from independent to dependent variables, the authors of the present book write that sensitivity analysis must be performed to complement uncertainty propagation. Sure, it is computationally demanding, but it provides a lot of information.

The book is simply superb. When I first read the original paper of Sobol (on whose work this book is based on, and in turn dedicated to him), I did not understand how exactly the so-called sensitivity indices can be computed in practice (I am an engineer and bothered about implementation as equally as the theory). But this book was an amazing teacher. The tutorial method of writing adopted by the authors is extremely helpful. The chapter organization, the questions-to-solve at the end of each chapter, etc. are all brilliantly done. This is the only book I purchased during my Ph.D. and it was some buy!! Now, I am using these global sensitivity methods for my post-doc research as well.