Growth of the Brownian forest

Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton--Watson forests with i.i.d. exponential branch lengths. We give elementary proofs of this composition rule and explain how it is intimately linked with Williams' decomposition for Brownian motion with drift.Comment: Published at http://dx.doi.org/10.1214/009117905000000422 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Horizontal Tunnelability Graph is Dual to Level Set Trees

Time series data, reflecting phenomena like climate patterns and stock prices, offer key insights for prediction and trend analysis. Contemporary research has independently developed disparate geometric approaches to time series analysis. These include tree methods, visibility algorithms, as well as persistence-based barcodes common to topological data analysis. This thesis enhances time series analysis by innovatively combining these perspectives through our concept of horizontal tunnelability. We prove that the level set tree gotten from its Harris Path (a time series), is dual to the time series' horizontal tunnelability graph, itself a subgraph of the more common horizontal visibility graph. This technique extends previous work by relating Merge, Chiral Merge, and Level Set Trees together along with visibility and persistence methodologies. Our method promises significant computational advantages and illuminates the tying threads between previously unconnected work. To facilitate its implementation, we provide accompanying empirical code and discuss its advantages

Topological Properties of Epidemic Aftershock Processes

Earthquakes in seismological catalogs and acoustic emission events in lab experiments can be statistically described as point events in linear Hawkes processes, where the spatiotemporal rate is a linear superposition of background intensity and aftershock clusters triggered by preceding activity. Traditionally, statistical seismology interpreted these models as the outcome of epidemic branching processes, where one-to-one causal links can be established between mainshocks and aftershocks. Declustering techniques are used to infer the underlying triggering trees and relate their topological properties with epidemic branching models. Here, we review how the standard Epidemic Type Aftershock Sequence (ETAS) model extends from the Galton-Watson branching processes and bridges two extreme cases: Poisson and scale-free power law trees. We report the statistical laws expected in triggering trees regarding some topological properties. We find that the statistics of such topological properties depend exclusively on two parameters of the standard ETAS model: the average branching ratio nb and the ratio between exponents α and b characterizing the production of aftershocks and the distribution of magnitudes, respectively. In particular, the classification of clusters into bursts and swarms proposed by Zaliapin and Ben-Zion (2013b, https://doi.org/10.1002/jgrb.50178) appears naturally in the aftershock sequences of the standard ETAS model depending on nb and α/b. On the other hand swarms can also appear by false causal connections between independent events in nontectonic seismogenic episodes. From these results, one can use the memory-less Galton-Watson as a null model for empirical triggering processes and assess the validity of the ETAS hypothesis to reproduce the statistics of natural and artificial catalogs

The Weight Function in the Subtree Kernel is Decisive

Tree data are ubiquitous because they model a large variety of situations, e.g., the architecture of plants, the secondary structure of RNA, or the hierarchy of XML files. Nevertheless, the analysis of these non-Euclidean data is difficult per se. In this paper, we focus on the subtree kernel that is a convolution kernel for tree data introduced by Vishwanathan and Smola in the early 2000's. More precisely, we investigate the influence of the weight function from a theoretical perspective and in real data applications. We establish on a 2-classes stochastic model that the performance of the subtree kernel is improved when the weight of leaves vanishes, which motivates the definition of a new weight function, learned from the data and not fixed by the user as usually done. To this end, we define a unified framework for computing the subtree kernel from ordered or unordered trees, that is particularly suitable for tuning parameters. We show through eight real data classification problems the great efficiency of our approach, in particular for small datasets, which also states the high importance of the weight function. Finally, a visualization tool of the significant features is derived.Comment: 36 page

Coding multitype forests: application to the law of the total population of branching forests

By extending the breadth first search algorithm to any $d$-type critical or subcritical irreducible branching tree, we show that such trees may be encoded through $d$ independent, integer valued, $d$-dimensional random walks. An application of this coding together with a multivariate extension of the Ballot Theorem allow us to give an explicit form of the law of the total progeny, jointly with the number of subtrees of each type, in terms of the offspring distribution of the branching process

Title By Registration: instituting modern property law and creating racial value in the settler colony

The transformation in prevailing conceptualizations of property and the drive to render land as fungible as possible, the desire to commoditize land that had been pursued in earnest since the seventeenth century in England, was realized in the space of the settler colony decades before it would be implemented in the United Kingdom. The author explores how the commodity logic of abstraction that subtended new property logics during this time, reflected in the Torrens system of title by registration, was accompanied by a racial logic of abstraction that rendered the land of the Native, or Savage vacant and ripe for appropriation. By way of conclusion, the author speculates on the ways in which the imposition of English property law in the settler colony influenced the development of modern property law in England