A semi-variance approach to visualising phylogenetic autocorrelation

Comparing traits across species has been a hallmark of biological research for centuries. While inter-specific comparisons can be highly informative, phylogenetic inertia can bias estimates if not properly accounted for in comparative analyses. In response, researchers typically treat phylogenetic inertia as a form of autocorrelation that can be detected, modelled, and corrected for. Despite the range of methods available for quantifying the strength of phylogenetic autocorrelation, no tools exist for visualising these autocorrelation structures. Here we derive variogram methods suitable for phylogenic data, and show how they can be used to straightforwardly visualise phylogenetic autocorrelation. We then demonstrate their utility for three empirical examples: sexual size dimorphism (SSD) in the Musteloidea, maximum per capita rate of population growth, r, in the Carnivora, and brain size in the Artiodactyla. When modelling musteloid SSD, the empirical variogram showed a tendency for the variance in SSD to stabilise over time, a characteristic feature of Ornstein-Uhlenbeck (OU) evolution. In agreement with this visual assessment, model selection identified the OU model as the best fit to the data. In contrast, the infinitely diffusive Brownian Motion (BM) model did not capture the asymptotic behaviour of the variogram and was less supported than the OU model. Phylogenetic variograms proved equally useful in understanding why an OU model was selected when modelling r in the Carnivora, and why BM was the selected evolutionary model for brain size in the Artiodactyla. Because the variograms of the various evolutionary processes each have different theoretical profiles, comparing fitted semi-variance functions against empirical semi-variograms can serve as a useful diagnostic tool, allowing researchers to understand why any given evolutionary model might be selected over another, which features are well captured by the model, and which are not. This allows for fitted models to be compared against the empirical variogram, facilitating model identification prior to subsequent analyses. We therefore recommend that any phylogenetic analysis begin with a non-parametric estimate of the autocorrelation structure of the data that can be visualized. The methods developed in this work are openly available in the new R package ctpm.

Determination of robust optimum plot size and shape – a model-based approach

SummaryDetermination of optimum plot size has been regarded as an important and useful area of study for agriculturists and statisticians since the first remarkable contribution on this problem came to light in a paper by Smith (1938). As we explore the scientific literature relating to this problem, we may note a number of contributions, including those of Modjeska and Rawlings (1983), Webster and Burgess (1984), Sethi (1985), Zhang et al. (1990, 1994), Bhatti et al.(1991), Fagroud and Meirvenne (2002), etc. In Pal et al. (2007), a general method was presented by means of which the optimum plot size can be determined through a systematic analytical procedure. The importance of the procedure stems from the fact that even with Fisherian blocking, the correlation among the residuals is not eliminated (as such the residuals remain correlated). The method is based on an application of an empirical variogram constructed on real-life data sets (obtained from uniformity trials) wherein the data are serially correlated. This paper presents a deep and extensive investigation (involving theoretical exploration of the effect of different plot sizes and shapes in discovering the point – actually the minimum radius of curvature of the variogram at that point – beyond which the theoretical variogram assumes stationary values with further increase in lags) in the case of the most commonly employed model (incorporating a correlation structure) assumed to represent real-life data situations (uniformity trial or designed experiments, RBD/LSD).

Applying Kriging algorithm based on Matlab environment to interpolate porosity and permeability values of lower Miocene sandstone reservoir, ST Xam oil field

The paper presents the Kriging technique based on Matlab environment applied to interpolate the value of all points in the interpolation range from porosity values obtained from 13 wells of lower Miocene reservoir, ST Xam oil field. The MATLAB function meshgrids are used to create the interpolated cell (cell-Kriging) instead of point discrete interpolation. After selecting the Variogram model with nugget values and the correlation threshold (in scope), the next step is Kriging porosity values which regression permeability values. Finally, displays the values in the cells and interpolated coordinates X, Y, respectively. With input data the first mission is to analyze this set, select the necessary parameters and removal of useless data, and assess the scope of application of each type of data. Then combine the document with wellogging interpretation results to determine reservoirs and the layered in which filter out the corresponding data averaging and conducting. Based on the selected average value of the corresponding products in each well for each subclass, calculate the results of an empirical Variogram model as the basis for Kriging weighted matrix. The last work is to calculate error and evaluate the reliability of the Kriging results. The error of porosity model are minor and distributed apropriately with kriging range. However the results are numerous correlation. The permeability experiment results are collected just from 03 points, therefore the ultimate solution is recurred porosity from porosity Kriging results.