Random Tessellation Forests

Space partitioning methods such as random forests and the Mondrian process are powerful machine learning methods for multi-dimensional and relational data, and are based on recursively cutting a domain. The flexibility of these methods is often limited by the requirement that the cuts be axis aligned. The Ostomachion process and the self-consistent binary space partitioning-tree process were recently introduced as generalizations of the Mondrian process for space partitioning with non-axis aligned cuts in the two dimensional plane. Motivated by the need for a multi-dimensional partitioning tree with non-axis aligned cuts, we propose the Random Tessellation Process (RTP), a framework that includes the Mondrian process and the binary space partitioning-tree process as special cases. We derive a sequential Monte Carlo algorithm for inference, and provide random forest methods. Our process is self-consistent and can relax axis-aligned constraints, allowing complex inter-dimensional dependence to be captured. We present a simulation study, and analyse gene expression data of brain tissue, showing improved accuracies over other methods.Comment: 11 pages, 4 figure

Minimax Rates for High-Dimensional Random Tessellation Forests

Random forests are a popular class of algorithms used for regression and classification. The algorithm introduced by Breiman in 2001 and many of its variants are ensembles of randomized decision trees built from axis-aligned partitions of the feature space. One such variant, called Mondrian forests, was proposed to handle the online setting and is the first class of random forests for which minimax rates were obtained in arbitrary dimension. However, the restriction to axis-aligned splits fails to capture dependencies between features, and random forests that use oblique splits have shown improved empirical performance for many tasks. In this work, we show that a large class of random forests with general split directions also achieve minimax rates in arbitrary dimension. This class includes STIT forests, a generalization of Mondrian forests to arbitrary split directions, as well as random forests derived from Poisson hyperplane tessellations. These are the first results showing that random forest variants with oblique splits can obtain minimax optimality in arbitrary dimension. Our proof technique relies on the novel application of the theory of stationary random tessellations in stochastic geometry to statistical learning theory.Comment: 20 page

Assessing the recreation values at risk from wildfire: an exploratory analysis

The levels of participation in various types of outdoor recreation in forested areas are substantial. Studies have shown that over 18.5 million days, representing approximately 80% of recreation user days, were spent by Canadians in recreational activities in forested lands. Furthermore, recreation has significant social and economic value that should be reflected in management decisions if sustainable forest management is to be achieved. The importance of recreation in forests has resulted in the selection of measures of recreation participation as one of the relevant indicators of sustainable forest management reporting in Canada. This suggests that recreation areas should be an important component of the values of forest at risk due to loss from wildfire. However, the presence of recreationists, who are considered to be the highest values at risk, dispersed on the fire prone landscape presents some issues for fire management agencies. These issues include the possibility of recreationists perishing in a wildfire and/or the possibility of fire starts as a result of recreation activities which are projected to increase into the future. For fire management agencies that strive to suppress all wildfires, the latter issue is particularly challenging when faced with resource constraints. Thus, a move away from suppression of all wildfires to suppression based on protecting highest values at risk is needed. An explicit incorporation of recreation values is advantageous in that these values are closely linked to the presence of recreationists. Therefore, during fire events, directing resources to high value recreation areas fulfill a fire management goal of protecting highest values at risk as well as identifying areas of the landscape where the suppression efforts are to be directed.Resource /Energy Economics and Policy,

Spectral radius of finite and infinite planar graphs and of graphs of bounded genus

It is well known that the spectral radius of a tree whose maximum degree is $D$ cannot exceed $2\sqrt{D-1}$. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the spectral radius $\rho(G)$ of a planar graph $G$ of maximum vertex degree $D\ge 4$ satisfies $\sqrt{D}\le \rho(G)\le \sqrt{8D-16}+7.75$. This result is best possible up to the additive constant--we construct an (infinite) planar graph of maximum degree $D$, whose spectral radius is $\sqrt{8D-16}$. This generalizes and improves several previous results and solves an open problem proposed by Tom Hayes. Similar bounds are derived for graphs of bounded genus. For every $k$, these bounds can be improved by excluding $K_{2,k}$ as a subgraph. In particular, the upper bound is strengthened for 5-connected graphs. All our results hold for finite as well as for infinite graphs. At the end we enhance the graph decomposition method introduced in the first part of the paper and apply it to tessellations of the hyperbolic plane. We derive bounds on the spectral radius that are close to the true value, and even in the simplest case of regular tessellations of type $\{p,q\}$ we derive an essential improvement over known results, obtaining exact estimates in the first order term and non-trivial estimates for the second order asymptotics

Physically Based Tree Rendering

This project produced and rendered physically based models of several tree species in real time. This was accomplished by procedurally generating a branch hierarchy and leaves within some bounding volumes according to rules that define a tree species. Multiple trees can be rendered at interactive frame rates greater than 45 frames per second. This enables the production of more realistic forests in games and visual applications procedurally and also frees up disk space for other resources because large model files of trees do not need to be stored on disk

Prediction of Enzyme Mutant Activity Using Computational Mutagenesis and Incremental Transduction

Wet laboratory mutagenesis to determine enzyme activity changes is expensive and time consuming. This paper expands on standard one-shot learning by proposing an incremental transductive method (T2bRF) for the prediction of enzyme mutant activity during mutagenesis using Delaunay tessellation and 4-body statistical potentials for representation. Incremental learning is in tune with both eScience and actual experimentation, as it accounts for cumulative annotation effects of enzyme mutant activity over time. The experimental results reported, using cross-validation, show that overall the incremental transductive method proposed, using random forest as base classifier, yields better results compared to one-shot learning methods. T2bRF is shown to yield 90% on T4 and LAC (and 86% on HIV-1). This is significantly better than state-of-the-art competing methods, whose performance yield is at 80% or less using the same datasets