Mahalanobis distances for ecological niche modelling and outlier detection: implications of sample size, error, and bias for selecting and parameterising a multivariate location and scatter method

Abstract

The Mahalanobis distance is a statistical technique that has been used in statistics and data science for data classification and outlier detection, and in ecology to quantify species-environment relationships in habitat and ecological niche models. Mahalanobis distances are based on the location and scatter of a multivariate normal distribution, and can measure how distant any point in space is from the centre of this kind of distribution. Three different methods for calculating the multivariate location and scatter are commonly used: the sample mean and variance-covariance, the minimum covariance determinant, and the minimum volume ellipsoid. The minimum covariance determinant and minimum volume ellipsoid were developed to be robust to outliers by minimising the multivariate location and scatter for a subset of the full sample, with the proportion of the full sample forming the subset being controlled by a user-defined parameter. This outlier robustness means the minimum covariance determinant and the minimum volume ellipsoid are highly relevant for ecological niche analyses, which are usually based on natural history observations that are likely to contain errors. However, natural history observations will also contain extreme bias, to which the minimum covariance determinant and the minimum volume ellipsoid will also be sensitive. To provide guidance for selecting and parameterising a multivariate location and scatter method, a series of virtual ecological niche modelling experiments were conducted to demonstrate the performance of each multivariate location and scatter method under different levels of sample size, errors, and bias. The results show that there is no optimal modelling approach, and that choices need to be made based on the individual data and question. The sample mean and variance-covariance method will perform best on very small sample sizes if the data are free of error and bias. At larger sample sizes the minimum covariance determinant and minimum volume ellipsoid methods perform as well or better, but only if they are appropriately parameterised. Modellers who are more concerned about the prevalence of errors should retain a smaller proportion of the full data set, while modellers more concerned about the prevalence of bias should retain a larger proportion of the full data set. I conclude that Mahalanobis distances are a useful niche modelling technique, but only for questions relating to the fundamental niche of a species where the assumption of multivariate normality is reasonable. Users of the minimum covariance determinant and minimum volume ellipsoid methods must also clearly report their parameterisations so that the results can be interpreted correctly.

A new method to estimate the ecological niche throughn-dimensional hypervolumes that combines convex hulls and elliptical envelopes

“…Recent studies have suggested that modeling approaches based on elliptical envelopes may be more appropriate when the goal is to estimate the fundamental niche of a species (Soberón and Peterson, 2020; Etherington, 2021); while the realized niche may be modeled through more complex shapes (Jiménez-Valverde et al, 2008). On the other hand, it is well known that the CH approach is sensitive to outliers, which may be common in presence-only samples even after going through standard data cleaning procedures.…”

“…From the long list of methods to estimate convex hypervolumes used in SDM and ENM, the following stand out: ( 1 ) convex hulls or polyhedra surrounding all the occurrences (Broennimann et al, 2007; Cornwell et al, 2006; Dallas et al, 2017; Walker and Cocks, 1991); ( 2 ) elliptical envelopes which may contain all or a portion of the occurrences (Swanson et al, 2015; Etherington, 2021), and ( 3 ) hyper-rectangles or box-shaped approaches (Busby, 1991; Nix et al, 1986). It has been shown that there is no clear “best” way to delineate hypervolumes (Blonder, 2018; Merow et al, 2014) and the choice of an appropriate method will depend on the goals of the analysis and data limitations (Blonder et al, 2018; Etherington, 2021; Peterson et al, 2011). The convex hull (CH) approach uses all observations to build the minimum convex polyhedron that contains them.…”

“…To define the niche of a species, the idea is to delineate a hypervolume as a geometrical shape within a multidimensional Euclidean space (or hyperspace) that can be used to describe the topology of the ecological system under study and its spatial relationship with other species (Blonder, 2018). In this context, extant methods to fit hypervolumes have being criticized because of their poor performance in high-dimensional spaces, or the high variability of their outputs according to the parameter choice (Junker et al, 2016; Etherington, 2021). Thus, there is a need to explore and compare different methods to estimate n -dimensional hypervolumes used to model the realized niche of a species.…”

“…The EE approaches are parametric and they assume a symmetrical response to environmental limits, although there are situations in which this geometry does not fit the data well and therefore does not adequately represents a species niche (Austin, 1987). Additionally, the EE approaches are closely related to the Mahalanobis distance model which delineates ellipsoids that represent the geometric shape of the data points that are at a certain distance from an n -dimensional vector and incorporates the correlation between environmental variables through the estimation of a matrix that describes the orientation and extent of the niche (Farber and Kadmon, 2003; Clark et al, 1993; Etherington, 2021). Under this approach, the calculation of intersections between n -dimensional ellipsoids (which is often desired in community ecology), require the application of approximation methods that involve great computational efforts (Qiao et al, 2016; Rabiei and Saleeby, 2021; Swanson et al, 2015).…”

Application of improved and optimized fuzzy neural network in classification evaluation of top coal cavability

“…This paper uses the average value within the grade to fill the missing parameters for the data with missing parameters considering the scarcity of sample data. At the same time, to minimize the influence of outliers on the model, the Mahalanobis distance method 27 is used to detect outliers, and samples detected as outliers at a confidence level of 0.5% are eliminated. After processing, 60 groups of data samples are obtained, as shown in attached Table 2 .…”