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An experimental demonstration of level attraction with coupled pendulums
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 91 (No. 8) |
| Halaman | : | 585-594 |
| Abstrak | : | We have experimentally demonstrated dissipative coupling in a double pendulum system through observation, which shows three distinctly different patterns of motion over the accessible parameter space. The described dissipative coupling apparatus is easy to manufacture and budget-friendly. The theoretical calculations are also suitable for the undergraduate level. Our experiment can serve as a novel demonstration for ubiquitous dynamic coupling effects encountered in many disparate physical systems. Unlike the well-known spring-coupled pendulums, our experiment employs Lenz's effect to couple the pendulums through electromagnetic damping, which, to the best of our knowledge, has not been demonstrated in the classroom. Our pendulums exhibit level attraction behaviour between two modes, induced by the dissipative coupling. This stands in contrast to the traditionally taught concept of level repulsion (avoided crossing) with spring-coupled pendulums. This experiment showcases distinctly different time domain dynamics of the dissipatively coupled pendulums over the parameter space, characterized by different oscillation patterns, damping rates, and relative phase between the two pendulums, which is a valuable lesson elucidating the dynamics of synchronization in linear systems for undergraduate students. |
An alternative way to solve the small oscillations problem
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 91 (No. 8) |
| Halaman | : | 579-594 |
| Abstrak | : | An alternative approach to the n-dimensional small oscillations problem is presented. This method is based on the finding of n new independent constants of motion to get the n eigenfrequencies and the n normal coordinates of the problem. These constants of motion exist and may be explicitly constructed for any small oscillations problem. Three examples are presented. One of them involves solving a five-dimensional small oscillations problem whose solution is usually obtained by finding the roots of a quintic algebraic equation. The approach constructed here is especially suited to deal with high-dimensional problems. Applications to small oscillations as well as to high-degree algebraic equation solutions are discussed. |
Studentized Sensitivity Analysis for the Sample Average Treatment Effect in Paired Observational Studies
| Pengarang | : | Colin B. Fogarty |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 115 (No. 531) |
| Halaman | : | 1518-1530 |
| Abstrak | : | A fundamental limitation of causal inference in observational studies is that perceived evidence for an effect might instead be explained by factors not accounted for in the primary analysis. Methods for assessing the sensitivity of a study’s conclusions to unmeasured confounding have been established under the assumption that the treatment effect is constant across all individuals. In the potential presence of unmeasured confounding, it has been argued that certain patterns of effect heterogeneity may conspire with unobserved covariates to render the performed sensitivity analysis inadequate. We present a new method for conducting a sensitivity analysis for the sample average treatment effect in the presence of effect heterogeneity in paired observational studies. Our recommended procedure, called the studentized sensitivity analysis, represents an extension of recent work on studentized permutation tests to the case of observational studies, where randomizations are no longer drawn uniformly. The method naturally extends conventional tests for the sample average treatment effect in paired experiments to the case of unknown, but bounded, probabilities of assignment to treatment. In so doing, we illustrate that concerns about certain sensitivity analyses operating under the presumption of constant effects are largely unwarranted. |
Bayesian Inference for Sequential Treatments Under Latent Sequential Ignorability
| Pengarang | : | Federico Ricciardi |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 115 (No. 531) |
| Halaman | : | 1498-1517 |
| Abstrak | : | We focus on causal inference for longitudinal treatments, where units are assigned to treatments at multiple time points, aiming to assess the effect of different treatment sequences on an outcome observed at a final point. A common assumption in similar studies is sequential ignorability (SI): treatment assignment at each time point is assumed independent of future potential outcomes given past observed outcomes and covariates. SI is questionable when treatment participation depends on individual choices, and treatment assignment may depend on unobservable quantities associated with future outcomes. We rely on principal stratification to formulate a relaxed version of SI: latent sequential ignorability (LSI) assumes that treatment assignment is conditionally independent on future potential outcomes given past treatments, covariates, and principal stratum membership, a latent variable defined by the joint value of observed and missing intermediate outcomes. We evaluate SI and LSI, using theoretical arguments and simulation studies to investigate the performance of the two assumptions when one holds and inference is conducted under both. Simulations show that when SI does not hold, inference performed under SI leads to misleading conclusions. Conversely, LSI generally leads to correct posterior distributions, irrespective of which assumption holds. |
Statistical Inference for Covariate-Adaptive Randomization Procedures
| Pengarang | : | Wei Ma |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 115 (No. 531) |
| Halaman | : | 1488-1497 |
| Abstrak | : | Covariate-adaptive randomization (CAR) procedures are frequently used in comparative studies to increase the covariate balance across treatment groups. However, because randomization inevitably uses the covariate information when forming balanced treatment groups, the validity of classical statistical methods after such randomization is often unclear. In this article, we derive the theoretical properties of statistical methods based on general CAR under the linear model framework. More importantly, we explicitly unveil the relationship between covariate-adaptive and inference properties by deriving the asymptotic representations of the corresponding estimators. We apply the proposed general theory to various randomization procedures such as complete randomization, rerandomization, pairwise sequential randomization, and Atkinson’s DA-biased coin design and compare their performance analytically. Based on the theoretical results, we then propose a new approach to obtain valid and more powerful tests. These results open a door to understand and analyze experiments based on CAR. Simulation studies provide further evidence of the advantages of the proposed framework and the theoretical results. Supplementary materials for this article are available online. |
Efficiently Inferring the Demographic History of Many Populations With Allele Count Data
| Pengarang | : | Jack Kamm |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 115 (No. 531) |
| Halaman | : | 1472-1487 |
| Abstrak | : | The sample frequency spectrum (SFS), or histogram of allele counts, is an important summary statistic in evolutionary biology, and is often used to infer the history of population size changes, migrations, and other demographic events affecting a set of populations. The expected multipopulation SFS under a given demographic model can be efficiently computed when the populations in the model are related by a tree, scaling to hundreds of populations. Admixture, back-migration, and introgression are common natural processes that violate the assumption of a tree-like population history, however, and until now the expected SFS could be computed for only a handful of populations when the demographic history is not a tree. In this article, we present a new method for efficiently computing the expected SFS and linear functionals of it, for demographies described by general directed acyclic graphs. This method can scale to more populations than previously possible for complex demographic histories including admixture. We apply our method to an 8-population SFS to estimate the timing and strength of a proposed “basal Eurasian” admixture event in human history. We implement and release our method in a new open-source software package momi2. |
Nonparametric Estimation of Multivariate Mixtures
| Pengarang | : | Chaowen Zheng |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 115 (No. 531) |
| Halaman | : | 1456-1471 |
| Abstrak | : | A multivariate mixture model is determined by three elements: the number of components, the mixing proportions, and the component distributions. Assuming that the number of components is given and that each mixture component has independent marginal distributions, we propose a nonparametric method to estimate the component distributions. The basic idea is to convert the estimation of component density functions to a problem of estimating the coordinates of the component density functions with respect to a good set of basis functions. Specifically, we construct a set of basis functions by using conditional density functions and try to recover the coordinates of component density functions with respect to this set of basis functions. Furthermore, we show that our estimator for the component density functions is consistent. Numerical studies are used to compare our algorithm with other existing nonparametric methods of estimating component distributions under the assumption of conditionally independent marginals. |
Simple Local Polynomial Density Estimators
| Pengarang | : | Matias D. Cattaneo |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 115 (No. 531) |
| Halaman | : | 1449-1455 |
| Abstrak | : | This article introduces an intuitive and easy-to-implement nonparametric density estimator based on local polynomial techniques. The estimator is fully boundary adaptive and automatic, but does not require prebinning or any other transformation of the data. We study the main asymptotic properties of the estimator, and use these results to provide principled estimation, inference, and bandwidth selection methods. As a substantive application of our results, we develop a novel discontinuity in density testing procedure, an important problem in regression discontinuity designs and other program evaluation settings. An illustrative empirical application is given. Two companion Stata and R software packages are provided. |
Principal Boundary on Riemannian Manifolds
| Pengarang | : | Zhigang Yao |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 115 (No. 531) |
| Halaman | : | 1435-1448 |
| Abstrak | : | We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic metric on the manifolds. Motivated by finding an optimal boundary between the two classes, we invent a novel approach—the principal boundary. From the perspective of classification, the principal boundary is defined as an optimal curve that moves in between the principal flows traced out from two classes of data, and at any point on the boundary, it maximizes the margin between the two classes. We estimate the boundary in quality with its direction, supervised by the two principal flows. We show that the principal boundary yields the usual decision boundary found by the support vector machine in the sense that locally, the two boundaries coincide. Some optimality and convergence properties of the random principal boundary and its population counterpart are also shown. We illustrate how to find, use, and interpret the principal boundary with an application in real data. Supplementary materials for this article are available online. |
Statistical Analysis of Functions on Surfaces, With an Application to Medical Imaging
| Pengarang | : | Eardi Lila |
| Nama Majalah/Jurnal | : | Journal of the American Statistical Association |
| Volume / Edisi | : | 115 (No. 531) |
| Halaman | : | 1420-1434 |
| Abstrak | : | Abstract–In functional data analysis, data are commonly assumed to be smooth functions on a fixed interval of the real line. In this work, we introduce a comprehensive framework for the analysis of functional data, whose domain is a two-dimensional manifold and the domain itself is subject to variability from sample to sample. We formulate a statistical model for such data, here called functions on surfaces, which enables a joint representation of the geometric and functional aspects, and propose an associated estimation framework. We assess the validity of the framework by performing a simulation study and we finally apply it to the analysis of neuroimaging data of cortical thickness, acquired from the brains of different subjects, and thus lying on domains with different geometries. Supplementary materials for this article are available online. |