The multivariate L1-median and associated data depth

This paper gives three related results: (i) a new, simple, fast, monotonically converging algorithm for deriving the L1-median of a data cloud in ℝd, a problem that can be traced to Fermat and has fascinated applied mathematicians for over three centuries; (ii) a new general definition for depth functions, as functions of multivariate medians, so that different definitions of medians will, correspondingly, give rise to different dept functions; and (iii) a simple closed-form formula of the L1-depth function for a given data cloud in ℝd.

Statistical analysis of shape of objects based on landmark data.

Two objects with homologous landmarks are said to be of the same shape if the configurations of landmarks of one object can be exactly matched with that of the other by translation, rotation/reflection, and scaling. The observations on an object are coordinates of its landmarks with reference to a set of orthogonal coordinate axes in an appropriate dimensional space. The origin, choice of units, and orientation of the coordinate axes with respect to an object may be different from object to object. In such a case, how do we quantify the shape of an object, find the mean and variation of shape in a population of objects, compare the mean shapes in two or more different populations, and discriminate between objects belonging to two or more different shape distributions. We develop some methods that are invariant to translation, rotation, and scaling of the observations on each object and thereby provide generalizations of multivariate methods for shape analysis.

A natural experiment on plant acclimation: lifetime stomatal frequency response of an individual tree to annual atmospheric CO2 increase.

Carbon dioxide (CO2) has been increasing in atmospheric concentration since the Industrial Revolution. A decreasing number of stomata on leaves of land plants still provides the only morphological evidence that this man-made increase has already affected the biosphere. The current rate of CO2 responsiveness in individual long-lived species cannot be accurately determined from field studies or by controlled-environment experiments. However, the required long-term data sets can be obtained from continuous records of buried leaves from living trees in wetland ecosystems. Fine-resolution analysis of the lifetime leaf record of an individual birch (Betula pendula) indicates a gradual reduction of stomatal frequency as a phenotypic acclimation to CO2 increase. During the past four decades, CO2 increments of 1 part per million by volume resulted in a stomatal density decline of approximately 0.6%. It may be hypothesized that this plastic stomatal frequency response of deciduous tree species has evolved in conjunction with the overall Cenozoic reduction of atmospheric CO2 concentrations.