Identification of taxa can significantly be assisted by statistical classification based on trait measurements either individually or by phylogenetic (clustering) methods. In this article, we present a general Bayesian approach for classifying species individually based on measurements of a mixture of continuous and ordinal traits, and any type of covariates. The trait vector is derived from a latent variable with a multivariate Gaussian distribution. Decision rules based on supervised learning are presented that estimate model parameters through blocked Gibbs sampling. These decision regions allow for uncertainty (partial rejection), so that not necessarily one specific category (taxon) is output when new subjects are classified, but rather a set of categories including the most probable taxa. This type of discriminant analysis employs reward functions with a set-valued input argument, so that an optimal Bayes classifier can be defined. We also present a way of safeguarding against outlying new observations, using an analogue of a p-value within our Bayesian setting. We refer to our Bayesian set-valued classifier as the Karlsson–Hössjer method, and it is illustrated on an original ornithological data set of birds. We also incorporate model selection through cross-validation, exemplified on another original data set of birds. 

In many applications there is ambiguity about which (if any) of a finite number N of hypotheses that best fits an observation. It is of interest then to possibly output a whole set of categories, that is, a scenario where the size of the classified set of categories ranges from 0 to N. Empty sets correspond to an outlier, sets of size 1 represent a firm decision that singles out one hypothesis, sets of size N correspond to a rejection to classify, whereas sets of sizes 2,…,N−1 represent a partial rejection to classify, where some hypotheses are excluded from further analysis. In this paper, we review and unify several proposed methods of Bayesian set-valued classification, where the objective is to find the optimal Bayesian classifier that maximizes the expected reward. We study a large class of reward functions with rewards for sets that include the true category, whereas additive or multiplicative penalties are incurred for sets depending on their size. For models with one homogeneous block of hypotheses, we provide general expressions for the accompanying Bayesian classifier, several of which extend previous results in the literature. Then, we derive novel results for the more general setting when hypotheses are partitioned into blocks, where ambiguity within and between blocks are of different severity. We also discuss how well-known methods of classification, such as conformal prediction, indifference zones, and hierarchical classification, fit into our framework. Finally, set-valued classification is illustrated using an ornithological data set, with taxa partitioned into blocks and parameters estimated using MCMC. The associated reward function’s tuning parameters are chosen through cross-validation.

In this article, we survey and unify a large class or L -functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of L -moments (G. Sillito, Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample, Biometrika 56 (1969), no. 3, 641–650.), L -skewness, and L -kurtosis (J. R. M. Hosking, L-moments: analysis and estimation of distributions using linear combinations or order statistics, J. R. Stat. Soc. Ser. B Stat. Methodol. 52 (1990), no. 1, 105–124.) and introduce order numbers for a large class of L -functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2), respectively, and describe how a family of L -functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre, or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follows a transformed linear model, with a link function that determines the appropriate class of L -functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a framework for asymptotic theory of estimates of L -functionals and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context, a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion.

It is well established that migratory birds in general have advanced their arrival times in spring, and in this paper we investigate potential ways of enhancing the level of detail in future phenological analyses. We perform single as well as multiple species analyses, using linear models on empirical quantiles, non-parametric quantile regression and likelihood-based parametric quantile regression with asymmetric Laplace distributed error terms. We conclude that non-parametric quantile regression appears most suited for single as well as multiple species analyses.