Shape analysis in the light of simplicial depth estimators

In this paper we present the maximum simplicial depth estimator and compare it to the ordinary least square estimator in examples from 2D shape analysis focusing on bivariate and multivariate allometrical problems from zoology. We compare two types of estimators derived under different subsets of parametric space on the basis of the linear regression model, θ = (θ1, θ2) ∈ R2 and θ = (θ1, θ2, θ3) ∈ R3, where θ3 = 0. In applications where outliers in the xor y-axis direction occur in the data and residuals from ordinary least-square (OLS) linear regression model are not normally distributed, we recommend the use of the maximum simplicial depth estimators. For generalising the median to higher-dimensional settings, a variety of different maximum depth estimators have been introduced. They extend, for example, the halfplane location depth of Tukey. Let Z1, ..., ZN be independent and identically distributed bivariate random variables, Zn ∈ Z ⊂ R , n = 1, 2, ..., N . For given observations z = (z1, ..., zN), we write zn = (yn, tn). The halfplane location depth of an arbitrary point θ ∈ R relative to z is defined as dl (θ, z) = min∀H # {n : zn ∈ H} , n = 1, ..., N , where H ranges over all closed halfplanes of which the boundary line passes through θ. The deepest location is defined as θ̂dl = argmax ∀θ dl (θ, z) and often called the Tukey median. This depth concept was transferred to regression by e.g. Wellmann et al. (2007), and we discuss it in the following paragraphs. We assume that the bivariate random variables Z1, ..., ZN are independent and identically distributed, that the variables Zn have values in Z ⊂ R, n = 1, 2, ..., N , and that there is a known family of probability measures P = {P (Z1,...,ZN) θ : θ ∈ Θ} with Θ = R . For given observations z1, ..., zN ∈ Z , we always write z = (z1, ..., zN ) and zn = (yn, tn). Let x be the function x : R → R, x(tn) = (1, tn, ..., t n ) T and set X = {x(tn), n = 1, 2, ..., N}. We model the relationship between yn and tn by the linear regression model given by yn = x (tn) T θ + εn, (1) where θ = (θ1, ..., θq) T ∈ R. The parameter space Θ = R is divided up into domains with constant depth of θ with respect to z and X . For given observations z let Dom(z) be the set of all those domains with constant depth of θ with respect to z and X . For each subset { zn1 , ..., znq+1 } of q+1 observations with pairwise different explanatory variables tn1, ..., tnq+1 , Dom (( zn1 , ..., znq+1 )) contains exactly one bounded domain with constant depth. The closure of this domain is a simplex, called S ( zn1 , ..., znq+1 ) . A maximal simplicial depth estimator for given observations z1, ..., zN ∈ Z with respect to a subset K ⊂ R is defined to be a parameter θ̂S ∈ argmax θ∈K dS(θ, z), where the simplicial depth dS of θ within z is defined as the fraction of simplices that contain the parameter θ as an interior point (that is, dS(θ, z) = ( N q+1 )−1 # { {n1,...,nq+1}⊂{1,...,N}:θ∈int(S(zn1 ,...,znq+1)) } , where int denotes the interior of the set and # the cardinality of a set; for details see Wellmann et al., 2007). It is interesting to see that the regression lines with maximum simplicial depth depend on the dimension q of the space in which the interesting parameters are embedded.