Books to Read While the Algae Grow in Your Fur, January 2019

Attention conservation notice: I have no taste. I also have no qualifications to discuss the history of millenarianism, or really even statistical graphics.

Bärbel Finkenstädt, Leonhard Held and Valerie Isham (eds.), Statistical Methods for Spatio-Temporal Systems

This is an edited volume arising from a conference, with all the virtues and vices that implies. (Several chapters have references to the papers which first published the work expounded in other chapters.) I will, accordingly, review the chapters in order.

Chapter 1: "Spatio-Temporal Point Processes: Methods and Applications" (Diggle). Mostly a precis of case studies from Diggle's (deservedly standard) books on the subject, which I will get around to finishing one of these years.

Chapter 2: "Spatio-Temporal Modelling --- with a View to Biological Growth" (Vedel Jensen, Jónsdóttir, Schmiegel, and Barndorff-Nielsen). This chapter divides into two parts. One is about "ambit stochastics". In a random field $Z(s,t)$, the "ambit" of the space-time point-instant $(s,t)$ is the set of point-instants $(q,u)$, $u < t$, where $Z(q,u)$ is (causally) relevant to $Z(r,t)$. (This is what, in my own work, I've called the "past cone" of $(s,t)$.) Having a regular geometry for the ambit imposes some tractable restrictions on random fields, which are explored here for models of growth-without-decay. The second part of this chapter will only make sense to hardened habituees of Levy processes, and perhaps not even to all of them.

Chapter 3: "Using Transforms to Analyze Space-Time Processes" (Fuentes, Guttorp, and Sampson): A very nice survey of Fourier transform, wavelet transform, and PCA approaches to decomposing spatio-temporal data. There's a good account of some tests for non-stationarity, based on the idea that (essentially) we should get the nearly same transforms for different parts of the data if things really are stationary. (I should think carefully about the assumptions and the implied asymptotic regime here, since the argument makes sense, but it also makes sense that sufficiently slow mean-reversion is indistinguishable from non-stationarity.)

Chapter 4: "Geostatistical Space-Time Models, Stationarity, Seperability, and Full Symmetry" (Gneiting, Genton, and Guttorp): "Geostatistics" here refers to "kriging", or using linear prediction on correlated data. As every schoolchild knows, this boils down to finding the covariance function, $\mathrm{Cov}[Z(s_1, t_1), Z(s_2, t_2)]$. This chapter considers three kinds of symmetry restrictions on the covariance functions: "separability", where $\mathrm{Cov}[Z(s_1, t_1), Z(s_2, t_2)] = C_S(s_1, s_2) C_T(t_1, t_2)$; the weaker notion of "full symmetry", where $\mathrm{Cov}[Z(s_1, t_1), Z(s_2, t_2)] = $\mathrm{Cov}[Z(s_1, t_2), Z(s_2, t_1)]$; and "stationarity", where $\mathrm{Cov}[Z(s_1, t_1), Z(s_2, t_2)] = $\mathrm{Cov}[Z(s_1+q, t_1+h), Z(s_2+q, t_2+h)]$. As the authors explain, while separable covariance functions are often used because of their mathematical tractability, they look really weird; "full symmetry" can do a lot of the same work, at less cost in implausibility.

Chapter 5: "Space-Time Modelling of Rainfall for Continuous Simulations" (Chandler, Isham, Belline, Yang and Northrop): A detailed exposition of two models for rainfall, at different spatio-temporal scales, and how they are both motivated by and connected to data. I appreciate their frankness about things that didn't work, and the difficulties of connecting the different models.

Chapter 6, "A Primer on Space-Time Modeling from a Bayesian Perspective" (Higdon): Here "space-time modeling" means "Gaussian Markov random fields". Does what it says on the label.

All the chapters combine theory with examples --- chapter 2 is perhaps the most mathematically sophisticated one, and also the one where the examples do the least work. The most useful, from my point of view, were Chapters 3 and 4, but that's because I was teaching a class where I did a lot of kriging ad PCA, and (with some regret) no point processes. If you have a professional interest in spatio-temporal statistics, and a fair degree of prior acquaintance, I can recommend this as a useful collection of examples, case studies, and expositions of some detailed topics.

Errata, of a sort: There are supposed to be color plate