13,082 research outputs found
Sufficient dimension reduction based on an ensemble of minimum average variance estimators
We introduce a class of dimension reduction estimators based on an ensemble
of the minimum average variance estimates of functions that characterize the
central subspace, such as the characteristic functions, the Box--Cox
transformations and wavelet basis. The ensemble estimators exhaustively
estimate the central subspace without imposing restrictive conditions on the
predictors, and have the same convergence rate as the minimum average variance
estimates. They are flexible and easy to implement, and allow repeated use of
the available sample, which enhances accuracy. They are applicable to both
univariate and multivariate responses in a unified form. We establish the
consistency and convergence rate of these estimators, and the consistency of a
cross validation criterion for order determination. We compare the ensemble
estimators with other estimators in a wide variety of models, and establish
their competent performance.Comment: Published in at http://dx.doi.org/10.1214/11-AOS950 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes
Given a homogeneous component of an exterior algebra, we characterize those
subspaces in which every nonzero element is decomposable. In geometric terms,
this corresponds to characterizing the projective linear subvarieties of the
Grassmann variety with its Plucker embedding. When the base field is finite, we
consider the more general question of determining the maximum number of points
on sections of Grassmannians by linear subvarieties of a fixed (co)dimension.
This corresponds to a known open problem of determining the complete weight
hierarchy of linear error correcting codes associated to Grassmann varieties.
We recover most of the known results as well as prove some new results. In the
process we obtain, and utilize, a simple generalization of the Griesmer-Wei
bound for arbitrary linear codes.Comment: 16 page
Connectedness modulo a topological property
Let be a topological property. We say that a space is
-connected if there exists no pair and of disjoint
cozero-sets of with non- closure such that the remainder
is contained in a cozero-set of with
closure. If is taken to be "being empty" then -connectedness coincides with connectedness in its usual sense. We
characterize completely regular -connected spaces, with
subject to some mild requirements. Then, we study conditions
under which unions of -connected subspaces of a space are
-connected. Also, we study classes of mappings which preserve
-connectedness. We conclude with a detailed study of the special
case in which is pseudocompactness. In particular, when
is pseudocompactness, we prove that a completely regular space
is -connected if and only if is connected, and that -connectedness is
preserved under perfect open continuous surjections. We leave some problems
open.Comment: 12 page
Complexity, BioComplexity, the Connectionist Conjecture and Ontology of Complexity\ud
This paper develops and integrates major ideas and concepts on complexity and biocomplexity - the connectionist conjecture, universal ontology of complexity, irreducible complexity of totality & inherent randomness, perpetual evolution of information, emergence of criticality and equivalence of symmetry & complexity. This paper introduces the Connectionist Conjecture which states that the one and only representation of Totality is the connectionist one i.e. in terms of nodes and edges. This paper also introduces an idea of Universal Ontology of Complexity and develops concepts in that direction. The paper also develops ideas and concepts on the perpetual evolution of information, irreducibility and computability of totality, all in the context of the Connectionist Conjecture. The paper indicates that the control and communication are the prime functionals that are responsible for the symmetry and complexity of complex phenomenon. The paper takes the stand that the phenomenon of life (including its evolution) is probably the nearest to what we can describe with the term “complexity”. The paper also assumes that signaling and communication within the living world and of the living world with the environment creates the connectionist structure of the biocomplexity. With life and its evolution as the substrate, the paper develops ideas towards the ontology of complexity. The paper introduces new complexity theoretic interpretations of fundamental biomolecular parameters. The paper also develops ideas on the methodology to determine the complexity of “true” complex phenomena.\u
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