An exploration of two infinite families of snarks

Thesis (M.S.) University of Alaska Fairbanks, 2019In this paper, we generalize a single example of a snark that admits a drawing with even rotational symmetry into two infinite families using a voltage graph construction techniques derived from cyclic Pseudo-Loupekine snarks. We expose an enforced chirality in coloring the underlying 5-pole that generated the known example, and use this fact to show that the infinite families are in fact snarks. We explore the construction of these families in terms of the blowup construction. We show that a graph in either family with rotational symmetry of order m has automorphism group of order m2m⁺¹. The oddness of graphs in both families is determined exactly, and shown to increase linearly with the order of rotational symmetry.Chapter 1: Introduction -- 1.1 General Graph Theory -- Chapter 2: Introduction to Snarks -- 2.1 History -- 2.2 Motivation -- 2.3 Loupekine Snarks and k-poles -- 2.4 Conditions on Triviality -- Chapter 3: The Construction of Two Families of Snarks -- 3.1 Voltage Graphs and Lifts -- 3.2 The Family of Snarks, Fm -- 3.3 A Second Family of Snarks, Rm -- Chapter 4: Results -- 4.1 Proof that the graphs Fm and Rm are Snarks -- 4.2 Interpreting Fm and Rm as Blowup Graphs -- 4.3 Automorphism Group -- 4.4 Oddness -- Chapter 5: Conclusions and Open Questions -- References

Estimating Abundance from Counts in Large Data Sets of Irregularly-Spaced Plots using Spatial Basis Functions

Monitoring plant and animal populations is an important goal for both academic research and management of natural resources. Successful management of populations often depends on obtaining estimates of their mean or total over a region. The basic problem considered in this paper is the estimation of a total from a sample of plots containing count data, but the plot placements are spatially irregular and non randomized. Our application had counts from thousands of irregularly-spaced aerial photo images. We used change-of-support methods to model counts in images as a realization of an inhomogeneous Poisson process that used spatial basis functions to model the spatial intensity surface. The method was very fast and took only a few seconds for thousands of images. The fitted intensity surface was integrated to provide an estimate from all unsampled areas, which is added to the observed counts. The proposed method also provides a finite area correction factor to variance estimation. The intensity surface from an inhomogeneous Poisson process tends to be too smooth for locally clustered points, typical of animal distributions, so we introduce several new overdispersion estimators due to poor performance of the classic one. We used simulated data to examine estimation bias and to investigate several variance estimators with overdispersion. A real example is given of harbor seal counts from aerial surveys in an Alaskan glacial fjord.Comment: 37 pages, 7 figures, 4 tables, keywords: sampling, change-of-support, spatial point processes, intensity function, random effects, Poisson process, overdispersio

Statistical analysis of spatial pattern in ecological data

A general statistical framework is proposed for unifying definitions of pattern and process in ecology and Statistics and Probability; This statistical framework allows several pattern techniques used in ecology, namely nested ANOVA, two term local variance, and paired quadrat variance, to be compared the variogram as a function of aggregation. There are two unbiased estimators of the variogram under aggregation, and they are compared in a simulation study. Which is better depends on the process autocorrelation;Another ecological quantity of interest is average patch size in a transect of data. The effects of three factors: (1) the signal-to-noise ratio, (2) the expected sizes of the patches relative to the plot size, and 3) the distribution of patch sizes, are determined for three estimators of average patch size: (1) two term local variance, (2) a moving two-sample t-test, and (3) a Bayesian approach using simulated annealing, in a 3 x 2 x 3 factorial simulation experiment. All three factors are important to the performance of the methods, and the Bayesian approach is the method which is recommended. An example from grassland vegetation is included;Besides estimation, spatial prediction is important in ecology. For spatial prediction, it has been usual to predict one variable at a time (e.g. kriging or cokriging). It is often desirable to predict the joint spatial abundance of ecological variables. Simultaneous spatial prediction of several variables is developed using covariances and variograms and cross-variograms. It is shown that the multivariable spatial predictor is the same as cokriging one variable at a time. However, multivariable spatial prediction yields the mean-squared-prediction-error matrix, and so allows construction of joint multivariable prediction regions

Space–time zero-inflated count models of Harbor seals

Environmental data are spatial, temporal, and often come with many zeros. In this paper, we included space–time random effects in zero-inflated Poisson (ZIP) and ‘hurdle’ models to investigate haulout patterns of harbor seals on glacial ice. The data consisted of counts, for 18 dates on a lattice grid of samples, of harbor seals hauled out on glacial ice in Disenchantment Bay, near Yakutat, Alaska. A hurdle model is similar to a ZIP model except it does not mix zeros from the binary and count processes. Both models can be used for zero-inflated data, and we compared space–time ZIP and hurdle models in a Bayesian hierarchical model. Space–time ZIP and hurdle models were constructed by using spatial conditional autoregressive (CAR) models and temporal first-order autoregressive (AR(1)) models as random effects in ZIP and hurdle regression models. We created maps of smoothed predictions for harbor seal counts based on ice density, other covariates, and spatio-temporal random effects. For both models predictions around the edges appeared to be positively biased. The linex loss function is an asymmetric loss function that penalizes overprediction more than underprediction, and we used it to correct for prediction bias to get the best map for space–time ZIP and hurdle models

A mixed-model moving-average approach to geostatistical modeling in stream networks

Spatial autocorrelation is an intrinsic characteristic in freshwater stream environments where nested watersheds and flow connectivity may produce patterns that are not captured by Euclidean distance. Yet, many common autocovariance functions used in geostatistical models are statistically invalid when Euclidean distance is replaced with hydrologic distance. We use simple worked examples to illustrate a recently developed moving-average approach used to construct two types of valid autocovariance models that are based on hydrologic distances. These models were designed to represent the spatial configuration, longitudinal connectivity, discharge, and flow direction in a stream network. They also exhibit a different covariance structure than Euclidean models and represent a true difference in the way that spatial relationships are represented. Nevertheless, the multi-scale complexities of stream environments may not be fully captured using a model based on one covariance structure. We advocate using a variance component approach, which allows a mixture of autocovariance models (Euclidean and stream models) to be incorporated into a single geostatistical model. As an example, we fit and compare ‘‘mixed models,’’ based on multiple covariance structures, for a biological indicator. The mixed model proves to be a flexible approach because many sources of information can be incorporated into a single model

Modeling growth of mandibles in the Western Arctic caribou herd

We compared growth curves for ramus length and diastema length from two autumn collections of mandibles of male Western Arctic Herd caribou in Alaska. We were primarily interested in determining if growth curves of caribou mandibles differed between caribou born during 1959-1967, after the herd had been high for several years and was probably declining in size, and those born during 1976-1988, when the herd was increasing in size. To compare these growth curves, we used a nonlinear model and used maximum likelihood estimates and likelihood ratio tests. We found that growth rates were similar between periods, but intercepts and variances of growth curves differed. From this we infer that calves were smaller in autumn during the 1960s and that significant compensatory growth did not occur later in life

Evaluation of the spatial linear model, random forest and gradient nearest-neighbour methods for imputing potential productivity and biomass of the Pacific Northwest forests

Increasingly, forest management and conservation plans require spatially explicit information within a management or conservation unit. Forest biomass and potential productivity are critical variables for forest planning and assessment in the Pacific Northwest. Their values are often estimated from ground-measured sample data. For unsampled locations, forest analysts and planners lack forest productivity and biomass values, so values must be predicted. Using simulated data and forest inventory and analysis data collected in Oregon and Washington, we examined the performance of the spatial linear model (SLM), random forest (RF) and gradient nearest neighbour (GNN) for mapping and estimating biomass and potential productivity of Pacific Northwest forests. Simulations of artificial populations and subsamplings of forest biomass and productivity data showed that the SLM had smaller empirical root-mean-squared prediction errors (RMSPE) for a wide variety of data types, with generally less bias and better interval coverage than RFand GNN. These patterns held for both point predictions and for population averages, with the SLM reducing RMSPE by 30.0 and 52.6 per cent over two GNN methods in predicting point estimates for forest biomass and potential productivity