Responding to eruptive transitions during the 2020-2021 eruption of La Soufrière volcano, St. Vincent.

A critical challenge during volcanic emergencies is responding to rapid changes in eruptive behaviour. Actionable advice, essential in times of rising uncertainty, demands the rapid synthesis and communication of multiple datasets with prognoses. The 2020-2021 eruption of La Soufrière volcano exemplifies these challenges: a series of explosions from 9-22 April 2021 was preceded by three months of effusive activity, which commenced with a remarkably low level of detected unrest. Here we show how the development of an evolving conceptual model, and the expression of uncertainties via both elicitation and scenarios associated with this model, were key to anticipating this transition. This not only required input from multiple monitoring datasets but contextualisation via state-of-the-art hazard assessments, and evidence-based knowledge of critical decision-making timescales and community needs. In addition, we share strategies employed as a consequence of constraints on recognising and responding to eruptive transitions in a resource-constrained setting, which may guide similarly challenged volcano observatories worldwide.

Central limit theorem for renewal theory for several patterns.

We prove a joint central limit theorem for the vector of counts of nonoverlapping occurrences of m given words as competing renewals. Our underlying model is an i.i.d. sequence over a finite alphabet. The motivation involves restriction enzymes in DNA sequences. We give a simple explicit formula for the limit covariance. This is in terms of the matrix of overlap-matching polynomials, following works of Guibas and Odlyzko (1980), of Breen et al. (1985), and of Biggins and Cannings (1987). The corresponding central limit theorem for counts of overlapping occurrences, rather than competing renewals, was derived by Lundstrom (1990). The above is a special case of a general situation of competing renewals in which occurrences of each type individually form a renewal process, and the individual processes interact in such a way that occurrences of either of two given types also form a renewal process. There is a simple expression for the limit covariance in this general case, involving only the means and variances for each type.

Poisson process approximation for sequence repeats, and sequencing by hybridization.

Sequencing by hybridization is a tool to determine a DNA sequence from the unordered list of all l-tuples contained in this sequence; typical numbers for l are l = 8, 10, 12. For theoretical purposes we assume that the multiset of all l-tuples is known. This multiset determines the DNA sequence uniquely if none of the so-called Ukkonen transformations are possible. These transformations require repeats of (l-1)-tuples in the sequence, with these repeats occurring in certain spatial patterns. We model DNA as an i.i.d. sequence. We first prove Poisson process approximations for the process of indicators of all leftmost long repeats allowing self-overlap and for the process of indicators of all left-most long repeats without self-overlap. Using the Chen-Stein method, we get bounds on the error of these approximations. As a corollary, we approximate the distribution of longest repeats. In the second step we analyze the spatial patterns of the repeats. Finally we combine these two steps to prove an approximation for the probability that a random sequence is uniquely recoverable from its list of l-tuples. For all our results we give some numerical examples including error bounds.

Genomic mapping by anchoring random clones: a mathematical analysis.

A complete physical map of the DNA of an organism, consisting of overlapping clones spanning the genome, is an extremely useful tool for genomic analysis. Various methods for the construction of such physical maps are available. One approach is to assemble the physical map by "fingerprinting" a large number of random clones and inferring overlap between clones with sufficiently similar fingerprints. E.S. Lander and M.S. Waterman (1988, Genomics 2:231-239) have recently provided a mathematical analysis of such physical mapping schemes, useful for planning such a project. Another approach is to assemble the physical map by "anchoring" a large number of random clones--that is, by taking random short regions called anchors and identifying the clones containing each anchor. Here, we provide a mathematical analysis of such a physical mapping scheme.

Tutorial on large deviations for the binomial distribution.

We present, in an easy to use form, the large deviation theory of the binomial distribution: how to approximate the probability of k or more successes in n independent trials, each with success probability p, when the specified fraction of successes, a identical to k/n, satisfies 0 less than p less than a less than 1.

Phase transitions in sequence matches and nucleic acid structure.

Analyses of phase transitions in biopolymers have previously been restricted to studies of average behavior along macromolecules. Extremal properties, such as longest helical region, can now be studied with a new family of probability distributions [Arratia, R., Gordon, L. & Waterman, M. S. (1986) Ann. Stat. 14, 971-993]. Not only is such extremal behavior analyzed with great precision, but new phase transitions are determined. One phase transition occurs when behavior of the free energy of the longest helical region abruptly changes from proportional to sequence length. The annealing of two single-stranded molecules and the melting of a double helix are both considered. These results, initially suggested by studies of optimal matching of random DNA sequences [Smith, T. F., Waterman, M. S. & Burks, C. (1985) Nucleic Acids Res. 13, 645-656], also have importance for significance tests in comparison of nucleic acid or protein sequences.