Collinearity refers to the non independence of predictor variables, usually in a regression-type analysis. It is a common feature of any descriptive ecological data set and can be a problem for parameter estimation because it inflates the variance of regression parameters and hence potentially leads to the wrong identification of relevant predictors in a statistical model. Collinearity is a severe problem when a model is trained on data from one region or time, and predicted to another with a different or unknown structure of collinearity. To demonstrate the reach of the problem of collinearity in ecology, we show how relationships among predictors differ between biomes, change over spatial scales and through time. Across disciplines, different approaches to addressing collinearity problems have been developed, ranging from clustering of predictors, threshold-based pre-selection, through latent variable methods, to shrinkage and regularisation. Using simulated data with five predictor-response relationships of increasing complexity and eight levels of collinearity we compared ways to address collinearity with standard multiple regression and machine-learning approaches. We assessed the performance of each approach by testing its impact on prediction to new data. In the extreme, we tested whether the methods were able to identify the true underlying relationship in a training dataset with strong collinearity by evaluating its performance on a test dataset without any collinearity. We found that methods specifically designed for collinearity, such as latent variable methods and tree based models, did not outperform the traditional GLM and threshold-based pre-selection. Our results highlight the value of GLM in combination with penalised methods (particularly ridge) and threshold-based pre-selection when omitted variables are considered in the final interpretation. However, all approaches tested yielded degraded predictions under change in collinearity structure and the ‘folk lore’-thresholds of correlation coefficients between predictor variables of |r| >0.7 was an appropriate indicator for when collinearity begins to severely distort model estimation and subsequent prediction. The use of ecological understanding of the system in pre-analysis variable selection and the choice of the least sensitive statistical approaches reduce the problems of collinearity, but cannot ultimately solve them.

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1600-0587.2012.07348.x

Collinearity: a review of methods to deal with it and a simulation study evaluating their performance

A family of probabilistic time series models is developed to analyze the time evolution of topics in large document collections. The approach is to use state space models on the natural parameters of the multinomial distributions that represent the topics. Variational approximations based on Kalman filters and nonparametric wavelet regression are developed to carry out approximate posterior inference over the latent topics. In addition to giving quantitative, predictive models of a sequential corpus, dynamic topic models provide a qualitative window into the contents of a large document collection. The models are demonstrated by analyzing the OCR'ed archives of the journal Science from 1880 through 2000.

Dynamic topic models

to the DLM: The First-Order Polynomial Model.- to the DLM: The Dynamic Regression Model.- The Dynamic Linear Model.- Univariate Time Series DLM Theory.- Model Specification and Design.- Polynomial Trend Models.- Seasonal Models.- Regression, Autoregression, and Related Models.- Illustrations and Extensions of Standard DLMs.- Intervention and Monitoring.- Multi-Process Models.- Non-Linear Dynamic Models: Analytic and Numerical Approximations.- Exponential Family Dynamic Models.- Simulation-Based Methods in Dynamic Models.- Multivariate Modelling and Forecasting.- Distribution Theory and Linear Algebra.

Bayesian Forecasting and Dynamic Models

Topic models, such as latent Dirichlet allocation (LDA), can be useful tools for the statistical analysis of document collections and other discrete data. The LDA model assumes that the words of each document arise from a mixture of topics, each of which is a distribution over the vocabulary. A limitation of LDA is the inability to model topic correlation even though, for example, a document about genetics is more likely to also be about disease than X-ray astronomy. This limitation stems from the use of the Dirichlet distribution to model the variability among the topic proportions. In this paper we develop the correlated topic model (CTM), where the topic proportions exhibit correlation via the logistic normal distribution [J. Roy. Statist. Soc. Ser. B 44 (1982) 139--177]. We derive a fast variational inference algorithm for approximate posterior inference in this model, which is complicated by the fact that the logistic normal is not conjugate to the multinomial. We apply the CTM to the articles from Science published from 1990--1999, a data set that comprises 57M words. The CTM gives a better fit of the data than LDA, and we demonstrate its use as an exploratory tool of large document collections.

A correlated topic model of Science

. SoilGrids produces maps of soil properties for the entire globe at medium spatial resolution (250 m cell size) using state-of-the-art machine learning methods to generate the necessary models. It takes as inputs soil observations from about 240 000 locations worldwide and over 400 global environmental covariates describing vegetation, terrain morphology, climate, geology and hydrology. The aim of this work was the production of global maps of soil properties, with cross-validation, hyper-parameter selection and quantification of spatially explicit uncertainty, as implemented in the SoilGrids version 2.0 product incorporating state-of-the-art practices and adapting them for global digital soil mapping with legacy data. The paper presents the evaluation of the global predictions produced for soil organic carbon content, total nitrogen, coarse fragments, pH (water), cation exchange capacity, bulk density and texture fractions at six standard depths (up to 200 cm). The quantitative evaluation showed metrics in line with previous global, continental and large-region studies. The qualitative evaluation showed that coarse-scale patterns are well reproduced. The spatial uncertainty at global scale highlighted the need for more soil observations, especially in high-latitude regions.

/pdf/soilgrids-2-0-producing-soil-information-for-the-globe-with-egkdd930cv.pdf

SoilGrids 2.0: producing soil information for the globe with quantified spatial uncertainty

The statistical analysis of compositional data

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The statistical analysis of compositional data