Archive for stochastic processes
Generalized Poisson difference autoregressive processes [on-line]
Posted in Books, Statistics, University life with tags autoregressive model, Bayesian inference, forecasting, integer sequence, integer time-series, International Journal of Forecasting, MCMC, Poisson difference, publication, stochastic processes, Università Ca' Foscari Venezia, Venice, visiting position on January 21, 2024 by xi'an
Martin Hairer gets Breakthrough Prize (and $3M)
Posted in Books, University life with tags algebraic geometry, Breakthrough Prize in Mathematics, Fields Institute, Fields medal, Glauber dynamics, Martin Hairer, PDEs, stochastic processes, Taylor expansion, University of Warwic on September 14, 2020 by xi'an
Just heard the news that Fields Medallist Martin Hairer (formerly U of Warwick) got the 2021 Breakthrough Prize in Mathematics for his unification theory of stochastic partial differential equations, which he likens to a form of Taylor expansion in the massive Inventiones paper describing this breakthrough. (Looking at the previous winners of the prize, who also made its selection committee, this represents a break from focussing primarily on algebraic geometry! If not from sticking to male recipients…)
We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semi-linear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the
Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of 3-dimensional ferromagnets near their critical temperature.
from least squares to signal processing and particle filtering
Posted in Books, Kids, Statistics, University life with tags Andrei Kolmogorov, auxiliary particle filter, Carl Friedrich Gauss, importance sampling, Kalman filter, R.E. Kálmán, Shannon, signal processing, stochastic processes on June 6, 2017 by xi'an
Nozer Singpurwalla, Nick. Polson, and Refik Soyer have just arXived a remarkable survey on the history of signal processing, from Gauß, Yule, Kolmogorov and Wiener, to Ragazzini, Shanon, Kálmán [who, I was surprised to learn, died in Gainesville last year!], Gibbs sampling, and the particle filters of the 1990’s.
Sunday morning reading
Posted in Books, Kids, University life with tags Brexit, Nature, predatory publishing, random walk, refereeing, stochastic processes on June 30, 2016 by xi'an
A very interesting issue of Nature I read this morning while having breakfast. A post-brexit read of a pre-brexit issue. Apart from the several articles arguing against Brexit and its dire consequences on British science [but preaching to the converted for which percentage of the Brexit voters does read Nature?!], a short vignette on the differences between fields for the average time spent for refereeing a paper (maths takes twice as long as social sciences and academics older than 65 half the time of researchers under 36!). A letter calling for action against predatory publishers. And the first maths paper published since I started reading Nature on an almost-regular basis: it studies mean first-passage time for non-Markov random walks. Which are specified as time-homogeneous increments. It is sort of a weird maths paper in that I do not see where the maths novelty stands and why the paper only contains half a dozen formulas… Maybe not a maths paper after all.
métro static
Posted in Kids, Statistics, Travel with tags Bayesian tests of hypotheses, Birmingham, concentration inequalities, signal processing, SPA 2015, stochastic processes, University of Oxford on July 19, 2015 by xi'an
[In the train shuttle at Birmingham airport, two young guys, maybe back from SPA 2015, discussing signal processing:]
– In Bayesian statistics, they use a different approach to testing hypotheses… You see, they put priors on the different hypotheses…
– But in the end it all boils down to concentration inequalities…
