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Invariant measurement with raters and rating scales : Rasch models for rater-mediated assessments
\"The purpose of this book is to present methods for developing, evaluating and maintaining rater-mediated assessment systems. Rater-mediated assessments involve ratings that are assigned by raters to persons responding to constructed-response items (e.g., written essays and teacher portfolios) and other types of performance assessments. This book addresses the following topics: (1) introduction to the principles of invariant measurement, (2) application of the principles of invariant measurement to rater-mediated assessments, (3) description of the lens model for rater judgments, (4) integration of principles of invariant measurement with the lens model of cognitive processes of raters, (5) illustration of substantive and psychometric issues related to rater-mediated assessments in terms of validity, reliability, and fairness, and (6) discussion of theoretical and practical issues related to rater-mediated assessment systems. Invariant measurement is fast becoming the dominant paradigm for assessment systems around the world, and this book provides an invaluable resource for graduate students, measurement practitioners, substantive theorists in the human sciences, and other individuals interested in invariant measurement when judgments are obtained with rating scales\"-- Provided by publisher.
Book
Spectral Properties of Ruelle Transfer Operators for Regular Gibbs Measures and Decay of Correlations for Contact Anosov Flows
In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact
Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general
class of Gibbs measures. The approach invented in 1997 by Dolgopyat in “On decay of correlations in Anosov flows” and further developed
in Stoyanov (2011) is substantially refined here, allowing to deal with much more general situations than before, although we still
restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which produces the desired estimates
whenever the Gibbs measure admits a Pesin set with exponentially small tails, that is a Pesin set whose preimages along the flow have
measures decaying exponentially fast. We call such Gibbs measures regular. Recent results in Gouëzel and Stoyanov (2019) prove existence
of such Pesin sets for hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures determined by Hölder continuous
potentials. The strong spectral estimates for Ruelle operators and well-established techniques lead to exponential decay of correlations
for Hölder continuous observables, as well as to some other consequences such as: (a) existence of a non-zero analytic continuation of
the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit
Theorem with an exponentially small error.
eBook
Invariant Measurement
This introductory text describes the principles of invariant measurement, how invariant measurement can be achieved with Rasch models, and how to use invariant measurement to solve measurement problems in the social, behavioral, and health sciences. Rasch models are used throughout but a comparison of Rasch models to other item response theory (IRT) models is also provided.
Written with students in mind, the manuscript was class tested to help maximize accessibility. Chapters open with an introduction and close with a summary and discussion. Numerous examples and exercises demonstrate the main issues addressed in each chapter. Key terms are defined when first introduced and in an end-of-text glossary. All of the book's analyses were conducted with the Facets program. The data sets used in the book, sample syntax files for running the Facets program, Excel files for creating item and person response functions, links to related websites, and other material are available at www.GeorgeEngelhard.com.
Highlights include:
A strong philosophical and methodological approach to measurement in the human sciences
Demonstrations of how measurement problems can be addressed using invariant measurement
Practical illustrations of how to create and evaluate scales using invariant measurement
A history of measurement based on test-score and scaling traditions
Previously unpublished work in analyzing rating data, the detection and measurement of rater errors, and the evaluation of rater accuracy
A review of estimation methods, model-data fit, indices used to evaluate the quality of rater-mediated assessments, rater error and bias, and rater accuracy.
Intended as a supplementary text for graduate or advanced undergraduate courses on measurement or test theory, item response theory, scaling theory, psychometrics, advanced measurement techniques, research methods,
eBook
The Periodic and Limiting Behaviors of Invariant Measures for 3D Globally Modified Navier–Stokes Equations
This paper is concerned with the invariant probability measures supported on pullback attractors for 3D globally modified Navier–Stokes equations with delay. We first construct a family of invariant probability measures as well as periodic invariant probability measures when the forcing term is periodic in time. We then present some general results on the limiting behaviors of invariant measures and periodic invariant measures for non-autonomous dynamical systems, and further apply these results to the 3D globally modified Navier–Stokes equations with delay. More precisely, we prove that any limit of a sequence of invariant measures (respectively, periodic invariant measures) of such equations must be an invariant measure (respectively, a periodic invariant measure) of the corresponding limiting equations as parameter
N
tends to some positive number or the delay parameter approaches to zero.
Journal Article
Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant Measure
Finitely-additive measures invariant to the action of some groups on a separable infinitedimensional real Hilbert space are constructed. The invariantness of a measure is studied with respect to the group of shifts on a vector of Hilbert space, the orthogonal group and some groups of symplectomorphisms of the Hilbert space equipped with the shift-invariant symplectic form. A considered invariant measure is locally finite, σ finite, but it is not countably additive. The analog of the ergodic decomposition of invariant finitely additivemeasures with respect to some groups are obtained. The set of measures that are invariant with respect to a group is parametrized using the obtained decomposition. The paper describes the spaces of complex-valued functions which are quadratically integrable with respect to constructed invariant measures. This space is used to define the Koopman unitary representation of the group of transformations of the Hilbert space. To define the strong continuity subspaces of a Koopman group, we analyze the spectral properties of its generator.
Journal Article
Convolution and Involution on Function Spaces of Homogeneous Spaces
Let ... be a locally compact group and also let ... be a compact subgroup of ... It is shown that, if ... is a relatively invariant measure on ... then there is a well-defined convolution on ... such that the Banach space ... becomes a Banach algebra. We also find a generalized definition of this convolution for other ...-spaces. Finally, we show that various types of involutions can be considered on ... 2010 Mathematics Subject Classification:43A15, 43A85 (ProQuest: ... denotes formulae omitted.)
Journal Article
Contributions of Mexican mathematicians abroad in pure and applied mathematics : second meeting, Matemáticos Mexicanos en el Mundo, December 15-19, 2014, Centro de Investigación en Matemáticas, Guanajuato, Mexico
This volume contains the proceedings of the Second Workshop of Mexican Mathematicians Abroad (II Reunion de Matematicos Mexicanos en el Mundo), held from December 15-19, 2014, at Centro de Investigacion en Matematicas (CIMAT) in Guanajuato, Mexico.This meeting was the second in a series of ongoing biannual meetings aimed at showcasing the research of Mexican mathematicians based outside of Mexico.The book features articles drawn from eight broad research areas: algebra, analysis, applied mathematics, combinatorics, dynamical systems, geometry, probability theory, and topology. Their topics range from novel applications of non-commutative probability to graph theory, to interactions between dynamical systems and geophysical flows.Several articles survey the fields and problems on which the authors work, highlighting research lines currently underrepresented in Mexico. The research-oriented articles provide either alternative approaches to well-known problems or new advances in active research fields. The wide selection of topics makes the book accessible to advanced graduate students and researchers in mathematics from different fields.This book is published in cooperation with Sociedad Matematica Mexicana.
eBook
Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics
In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences.
Journal Article
Free minimal actions of countable groups with invariant probability measures
We prove that for any countable group
$\\unicode[STIX]{x1D6E4}$
, there exists a free minimal continuous action
$\\unicode[STIX]{x1D6FC}:\\unicode[STIX]{x1D6E4}\\curvearrowright {\\mathcal{C}}$
on the Cantor set admitting an invariant Borel probability measure.
Journal Article
Solvable Stationary Non Equilibrium States
We consider the one dimensional boundary driven harmonic model and its continuous version, both introduced in (Frassek et al. in J Stat Phys 180: 135–171, 2020). By combining duality and integrability the authors of (Frassek and Giardiná in J Math Phys 63: 103301, 2022) obtained the invariant measures in a combinatorial representation. Here we give an integral representation of the invariant measures which turns out to be a convex combination of inhomogeneous product of geometric distributions for the discrete model and a convex combination of inhomogeneous product of exponential distributions for the continuous one. The mean values of the geometric and of the exponential variables are distributed according to the order statistics of i.i.d. uniform random variables on a suitable interval fixed by the boundary sources. The result is obtained solving exactly the stationary condition written in terms of the joint generating function. The method has an interest in itself and can be generalized to study other models. We briefly discuss some applications.
Journal Article