he geometry of statistical efficiency

We will place certain parts of the theory of statistical efficiency into the author’s operator trigonometry (1967), thereby providing new geometrical understanding of statistical efficiency. Important earlier results of Bloomfield and Watson, Durbin and Kendall, Rao and Rao, will be so interpreted. For example, worse case relative least squares efficiency corresponds to and is achieved by the maximal turning antieigenvectors of the covariance matrix. Some little-known historical perspectives will also be exposed. The overall view will be emphasized

State Space Approaches for Modeling Activities in Video Streams

The objective is to discern events and behavior in activities using video sequences, which conform to common human experience. It has several applications such as recognition, temporal segmentation, video indexing and anomaly detection. Activity modeling offers compelling challenges to computational vision systems at several levels ranging from low-level vision tasks for detection and segmentation to high-level models for extracting perceptually salient information. With a focus on the latter, the following approaches are presented: event detection in discrete state space, epitomic representation in continuous state space, temporal segmentation using mixed state models, key frame detection using antieigenvalues and spatio-temporal activity volumes. Significant changes in motion properties are said to be events. We present an event probability sequence representation in which the probability of event occurrence is computed using stable changes at the state level of the discrete state hidden Markov model that generates the observed trajectories. Reliance on a trained model however, can be a limitation. A data-driven antieigenvalue-based approach is proposed for detecting changes. Antieigenvalues are sensitive to turnings whereas eigenvalues capture directions of maximum variance in the data. In both these approaches, events are assumed to be instantaneous quantities. This is relaxed using an epitomic representation in continuous state space. Video sequences are segmented using a sliding window within which the dynamics of each object is assumed to be linear. The system matrix, initial state value and the input signal statistics are said to form an epitome. The system matrices are decomposed using the Iwasawa matrix decomposition to isolate the effect of rotation, scaling and projection of the state vector. It is used to compute physically meaningful distances between epitomes. Epitomes reveal dominant primitives of activities that have an abstracted interpretation. A mixed state approach for activities is presented in which higher-level primitives of behavior is encoded in the discrete state component and observed dynamics in the continuous state component. The effectiveness of mixed state models is demonstrated using temporal segmentation. In addition to motion trajectories, the volume carved out in an xyt cube by a moving object is characterized using Morse functions

Author index to volume 49

Pericas, Enri

Approximations of Antieigenvalue and Antieigenvalue-Type Quantities

We will extend the definition of antieigenvalue of an operator to antieigenvalue-type quantities, in the first section of this paper, in such a way that the relations between antieigenvalue-type quantities and their corresponding Kantorovich-type inequalities are analogous to those of antieigenvalue and Kantorovich inequality. In the second section, we approximate several antieigenvalue-type quantities for arbitrary accretive operators. Each antieigenvalue-type quantity is approximated in terms of the same quantity for normal matrices. In particular, we show that for an arbitrary accretive operator, each antieigenvalue-type quantity is the limit of the same quantity for a sequence of finite-dimensional normal matrices

Level Curves of the Angle Function of a Positive Definite Symmetric Matrix

Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(-1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following question: if A and B are commuting positive definite symmetric matrices such that p(A) + p(B) is obtuse, what is the minimal p(S) such that {A, SBS^(-1)} indefinite? In this dissertation we will describe the level curves of the angle function mapping a unit vector x to the angle between x and Ax for a 3 by 3 PDS matrix A, and discuss their interaction with those of a second such matrix

Geophysical evolution of planetary interiors and surfaces : Moon & Mars

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 127-138).The interiors and surfaces of the terrestrial planetary bodies provide us a unique opportunity to gain insight into planetary evolution, particularly in the early stages subsequent to accretion. Both Mars and the Moon are characterized by well-preserved and ancient surfaces, that preserve a record of geological and geophysical processes that have operated both at the surface and in the interior. With accessibility to orbital and landed spacecraft, the Moon and Mars have a unique qualitative and quantitative role in understanding and constraining the evolution of solid planets in our Solar System, as well as the timing of its many major events. In this thesis I use gravity and topography data to investigate aspects of the surface and interior evolution of the Moon and Mars that include aspects of major processes: impact, volcanism, erosion and internal dynamics.by Alexander Joseph Evans.Ph.D