Archive for continued fractions

structure and uncertainty, Bristol, Sept. 27

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , on September 28, 2012 by xi'an

The last sessions at the SuSTain workshop. were equally riveting but I alas had to leave early to get a noon flight—as it happens, while I expected to get home early enough to work, run, cook, and do maths with my daughter, my taxi got stuck in an endless traffic jam and I only had time for the maths!—, hence missing the talks by Chris Holmes—second time after Kyoto!—, Sofia Massa, and Arnoldo Frigessi… I am glad I managed to get Michael Newton’s and Forrest Crawford’s talks, though, as Michael presented a highly pedagogical entry to computational concepts related to system biology (a potential candidate for an MCMSki IV talk?) and Forrest discussed some birth-and-death processes, including the Yule process, that allowed for closed form expressions of their Laplace transform via continued fractions. (Continued fractions, one of my favourite mathematical objects!!! Rarely appearing in statistics, though…) I have to check on Forrest’s recent papers to understand how widely this approach applies to philogenetic trees, but this opens a fairly interesting alternative to ABC!

This was a highly enjoyable meeting, first and foremost due to the quality of the talks and of their scheduling, but also by the pleasure of seeing again many friends of many years—notice how I carefully avoided using “old friends”!—, by the relaxed and open atmosphere of the workshop—in the terrific location of Goldney Hall—and of course of unofficially celebrating Peter Green’s deeds and contributions to the field, the profession, and the statistics group in Bristol! Deeds and contributions so far, as I am sure he will keep contributing in many ways in the coming years and decades, as already shown by his committed involvement in the very recent creation of BayesComp. I thus most gladly join the other participants of this workshop both to thank him most sincerely for those many and multifaceted contributions and to wish him all the best for those coming decades!

As an aside, I also enjoyed being “back” in Bristol once again, as I do like the city, the surrounding Somerset countryside, the nearby South Wales, and the wide running possibilities (from the Downs to the Mendip Hills!). While I sampled many great hotels in Bristol and Clifton over the years, I now rank the Avon Gorges Hotel where I stayed this time quite high in the list, both for its convenient (running!) location and its top-quality facilities (incl. high-speed WiFi!)

Évariste Galois in the métro…

Posted in Books, pictures, Travel, University life with tags , , , , , on December 18, 2011 by xi'an

This morning, while waiting for a very late métro, I took those pictures from the platform.

These posters were in Bourg-la-Reine, where Évariste Galois was born in 1811 and which now celebrates the 200th anniversary of his birth. There are other celebrations around, at IHP for instance.

Numerical analysis for statisticians

Posted in Books, R, Statistics, University life with tags , , , , , , , , , on August 26, 2011 by xi'an

“In the end, it really is just a matter of choosing the relevant parts of mathematics and ignoring the rest. Of course, the hard part is deciding what is irrelevant.”

Somehow, I had missed the first edition of this book and thus I started reading it this afternoon with a newcomer’s eyes (obviously, I will not comment on the differences with the first edition, sketched by the author in the Preface). Past the initial surprise of discovering it was a mathematics book rather than an algorithmic book, I became engrossed into my reading and could not let it go! Numerical Analysis for Statisticians, by Kenneth Lange, is a wonderful book. It provides most of the necessary background in calculus and some algebra to conduct rigorous numerical analyses of statistical problems. This includes expansions, eigen-analysis, optimisation, integration, approximation theory, and simulation, in less than 600 pages. It may be due to the fact that I was reading the book in my garden, with the background noise of the wind in tree leaves, but I cannot find any solid fact to grumble about! Not even about  the MCMC chapters! I simply enjoyed Numerical Analysis for Statisticians from beginning till end.

“Many fine textbooks (…) are hardly substitutes for a theoretical treatment emphasizing mathematical motivations and derivations. However, students do need exposure to real computing and thoughtful numerical exercises. Mastery of theory is enhanced by the nitty gritty of coding.” 

From the above, it may sound as if Numerical Analysis for Statisticians does not fulfill its purpose and is too much of a mathematical book. Be assured this is not the case: the contents are firmly grounded in calculus (analysis) but the (numerical) algorithms are only one code away. An illustration (among many) is found in Section 8.4: Finding a Single Eigenvalue, where Kenneth Lange shows how the Raleigh quotient algorithm of the previous section can be exploited to this aim, when supplemented with a good initial guess based on Gerschgorin’s circle theorem. This is brilliantly executed in two pages and the code is just one keyboard away. The EM algorithm is immersed into a larger M[&]M perspective. Problems are numerous and mostly of high standards, meaning one (including me) has to sit and think about them. References are kept to a minimum, they are mostly (highly recommended) books, plus a few papers primarily exploited in the problem sections. (When reading the Preface, I found that “John Kimmel, [his] long suffering editor, exhibited extraordinary patience in encouraging [him] to get on with this project”. The quality of Numerical Analysis for Statisticians is also a testimony to John’s editorial acumen!)

“Every advance in computer architecture and software tempts statisticians to tackle numerically harder problems. To do so intelligently requires a good working knowledge of numerical analysis. This book equips students to craft their own software and to understand the advantages and disadvantages of different numerical methods. Issues of numerical stability, accurate approximation, computational complexity, and mathematical modeling share the limelight in a broad yet rigorous overview of those parts of numerical analysis most relevant to statisticians.”

While I am reacting so enthusiastically to the book (imagine, there is even a full chapter on continued fractions!), it may be that my French math background is biasing my evaluation and that graduate students over the World would find the book too hard. However, I do not think so: the style of Numerical Analysis for Statisticians is very fluid and the rigorous mathematics are mostly at the level of undergraduate calculus. The more advanced topics like wavelets, Fourier transforms and Hilbert spaces are very well-introduced and do not require prerequisites in complex calculus or functional analysis. (Although I take no joy in this, even measure theory does not appear to be a prerequisite!) On the other hand, there is a prerequisite for a good background in statistics. This book will clearly involve a lot of work from the reader, but the respect shown by Kenneth Lange to those readers will sufficiently motivate them to keep them going till assimilation of those essential notions. Numerical Analysis for Statisticians is also recommended for more senior researchers and not only for building one or two courses on the bases of statistical computing. It contains most of the math bases that we need, even if we do not know we need them! Truly an essential book.

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