Orthogonality Assumption Research Articles

Small sample adjustment for inference without assuming orthogonality in a mixed model for repeated measures analysis

ABSTRACT The mixed model for repeated measures (MMRM) analysis is sometimes used as a primary statistical analysis for a longitudinal randomized clinical trial. When the MMRM analysis is implemented in ordinary statistical software, the standard error of the treatment effect is estimated by assuming orthogonality between the fixed effects and covariance parameters, based on the characteristics of the normal distribution. However, orthogonality does not hold unless the normality assumption of the error distribution holds, and/or the missing data are derived from the missing completely at random structure. Therefore, assuming orthogonality in the MMRM analysis is not preferable. However, without the assumption of orthogonality, the small-sample bias in the standard error of the treatment effect is significant. Nonetheless, there is no method to improve small-sample performance. Furthermore, there is no software that can easily implement inferences on treatment effects without assuming orthogonality. Hence, we propose two small-sample adjustment methods inflating standard errors that are reasonable in ideal situations and achieve empirical conservatism even in general situations. We also provide an R package to implement these inference processes. The simulation results show that one of the proposed small-sample adjustment methods performs particularly well in terms of underestimation bias of standard errors; consequently, the proposed method is recommended. When using the MMRM analysis, our proposed method is recommended if the sample size is not large and between-group heteroscedasticity is expected.

Uncertainty principles for short‐time free metaplectic transformation

This study devotes to extend Heisenberg's uncertainty inequalities in free metaplectic transformation (FMT) domains into short‐time free metaplectic transformation (STFMT) domains. We disclose an equivalence relation between spreads in time‐STFMT and time domains, as well as FMT‐STFMT and FMT domains. We use them to set up an inequality relation between the uncertainty product in time‐STFMT and FMT‐STFMT domains and that in time and FMT domains and an inequality relation between the uncertainty product in two FMT‐STFMT domains and that in two FMT domains. We deduce uncertainty inequalities of real‐valued functions and complex‐valued window functions for the STFMT and uncertainty inequalities of complex‐valued (window) functions for the orthogonal STFMT, the orthonormal STFMT, and the STFMT without the assumption of orthogonality, respectively. To formulate the attainable lower bounds, we also propose some novel uncertainty inequalities of complex‐valued (window) functions for the orthogonal FMT and the FMT without the assumption of orthogonality, respectively.

Unravelling spatiotemporal patterns in global sea surface temperatures with dynamic mode decomposition

AbstractExtracting global sea surface temperature (SST) patterns is essential for understanding climate variability in both regional and global perspective. Traditional methods, limited by assumptions of orthogonality, hinder our ability to fully capture the dynamics of global SST, particularly in the complex Pacific basin. This study introduces dynamic mode decomposition (DMD), a powerful method from fluid dynamics, to extract global SST patterns since 1900. Our analysis shows that DMD successfully captures well‐known patterns like global warming and the Atlantic Multidecadal Oscillation (AMO). More importantly, it reveals three Pacific SST modes that are spatially non‐orthogonal but exhibit distinct characteristics. Spectral analysis shows that these Pacific modes have distinct periodicities: Pacific Multi‐decadal Mode (PMDM, ~50 years), Quasi‐decadal Mode (PQDM, 8–16 years) and Interannual Mode (PIAM, 2–8 years). This highlights DMD's strength in not only capturing known patterns but also separating modes with similar spatial features but unique dynamics. Further analysis reveals significant differences in their impacts on global air temperature. During positive phases, all three modes induce tropical warming, with PIAM exerting the strongest influence and PMDM the weakest. In the Northern Hemisphere, both PQDM and PIAM drive mid‐latitude cooling and high‐latitude warming, with a more pronounced effect from PQDM. PMDM, in contrast, triggers widespread cooling across the Northern extratropics. In the Southern Hemisphere, PMDM's influence on temperature exhibits a distinct latitudinal banding pattern. PIAM and PQDM, on the other hand, exhibit a regionalized impact, primarily concentrated in the Amundsen and Ross Seas, with PQDM's influence extending to the Antarctic continent.

Excitonic Configuration Interaction: Going Beyond the Frenkel Exciton Model.

We present the excitonic configuration interaction (ECI) method─a fragment-based analogue of the CI method for electronic structure calculations of multichromophoric systems. It can also be viewed as a generalization of the exciton approach, with the following properties: (i) It constructs the effective Hamiltonian exclusively from monomer calculations. (ii) It employs the strong orthogonality assumption and is exact within McWeeny's group function theory, thus requiring only one-electron density matrices of the monomer states. (iii) It is agnostic of the monomer electronic structure method, allowing us to use/combine different methods. (iv) It includes an embedding point charge scheme (called excitonic Hartree-Fock, EHF) to improve the accuracy of the monomer states, but such that the effective full-system Hamiltonian is not explicitly dependent on the embedding. (v) It is systematically improvable, by expanding the set of monomer states and by including configurations where two or more monomers are excited (defining the ECIS, ECISD, etc., methods). The performance of ECI is assessed by computing the absorption spectrum of two exemplary multichromophoric systems, using CIS as the monomer electronic structure method. The accuracy of ECI significantly depends on the chosen embedding charges and the ECI expansion. The most accurate assessed combinations─ECIS or ECISD with EHF embedding─yield spectra that agree qualitatively and quantitatively with full-system direct calculations, with deviations of the excitation energies below 0.1 eV. We also show that ECISD based on CIS monomer calculations can predict states where two monomers are excited simultaneously (e.g., triplet-triplet double-local excitations) that are inaccessible in a full-system CIS calculation.

Separation of fast and slow shear waves and prediction of fracture parameters based on non-orthogonal assumptions

SUMMARY Natural fractures play a significant role in oil and gas reservoirs. Accurate predictions of fracture parameters are vital in reservoir prediction and oil and gas development. The birefringent phenomenon of shear waves in fractured media makes shear wave splitting (SWS) analysis an important tool in formulating fracture predictions. The traditional SWS analysis method is based on an orthogonal assumption of fast and slow shear waves. However, in an orthotropic medium composed of a background vertical transversely isotropic medium and a set of vertical fractures, fast and slow shear waves are not necessarily orthogonal. This causes the traditional SWS analysis method to fail. To solve this problem, we proposed an SWS analysis algorithm with a non-orthogonal assumption of fast and slow shear waves in this study. First, we introduced a parameter (difference angle) to characterize the angle between slow shear waves and the normal polarization directions of the fast shear waves. Subsequently, based on the traditional two-parameter scanning algorithm, a parameter was added to facilitate three-parameter scanning. In addition, we derived an expression for the two-parameter scanning objective function using the non-orthogonal assumption. Two-parameter scanning can accurately extract fast and slow wave time delay data, but it cannot determine an accurate fast shear wave polarization direction. Therefore, we optimized the three-parameter scanning algorithm as follows: first, we used two-parameter scanning to obtain the fast and slow wave time delays and then performed further scanning to determine the polarization direction of the fast shear wave and difference angle. The optimization algorithm significantly improved the computational efficiency. Subsequently, we tested the accuracy of this method using synthetic single-trace and three-component vertical seismic profile data. We demonstrated the implementation process of the three-parameter scanning method using actual data, separated fast and slow shear waves, and predicted fracture parameters. The final fracture parameters were verified.

Improved Breitung and Roling estimator for mixed-frequency models with application to forecasting inflation rates

Instead of applying the commonly used parametric Almon or Beta lag distribution of MIDAS, Breitung and Roling (J Forecast 34:588–603, 2015) suggested a nonparametric smoothed least-squares shrinkage estimator (henceforth SLS1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${SLS}_{1}$$\\end{document}) for estimating mixed-frequency models. This SLS1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${SLS}_{1}$$\\end{document} approach ensures a flexible smooth trending lag distribution. However, even if the biasing parameter in SLS1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${SLS}_{1}$$\\end{document} solves the overparameterization problem, the cost is a decreased goodness-of-fit. Therefore, we suggest a modification of this shrinkage regression into a two-parameter smoothed least-squares estimator (SLS2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${SLS}_{2}$$\\end{document}). This estimator solves the overparameterization problem, and it has superior properties since it ensures that the orthogonality assumption between residuals and the predicted dependent variable holds, which leads to an increased goodness-of-fit. Our theoretical comparisons, supported by simulations, demonstrate that the increase in goodness-of-fit of the proposed two-parameter estimator also leads to a decrease in the mean square error of SLS2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${SLS}_{2},$$\\end{document} compared to that of SLS1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${SLS}_{1}$$\\end{document}. Empirical results, where the inflation rate is forecasted based on the oil returns, demonstrate that our proposed SLS2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${SLS}_{2}$$\\end{document} estimator for mixed-frequency models provides better estimates in terms of decreased MSE and improved R2, which in turn leads to better forecasts.

Bayesian Φ q -optimal designs for multi-factor additive non linear models with heteroscedastic errors

This article considers Bayesian Φ q -optimal designs for multi-factor additive non linear models where model errors are heteroscedastic. For additive non linear models with a constant term, a sufficient condition is given in order to derive Bayesian Φ q -optimal product designs, which are achieved from univariate optimal designs with respect to every marginal model with a single factor. However, in the case of ignoring a constant term, an additional assumption of orthogonality is proposed to ensure that optimal designs can be found. Then, the corresponding optimal product designs can be built with the help of the equivalence theorem for the Bayesian Φ q -optimality criterion. Several examples are given to illustrate the effectiveness of theoretical results on optimal product designs.

Interference mitigation in intentional jammers aided non-uniform heterogeneous cellular networks.

Coverage and capacity are optimized in fifth generation (5G) networks by small base station (SBS) distribution in the coverage realm of macro base station (MBS). However, system performance is significantly reduced by inter-cell interference (ICI) because of the orthogonal frequency division multiple access assumption. In addition to ICI, this work considers intentional jammers' interference (IJI) due to the presence of jammers. These Jammers try to inject undesirable energies into the legitimate communication band, which significantly degrade uplink (UL) signal-to-interference ratio (SIR). To reduce ICI and IJI, in this work, we employ SBS muting, where the SBSs near MBS are switched off. To further mitigate ICI and IJI, we use one of the effective interference management schemes a.k.a reverse frequency allocation (RFA). We presume that due to mitigation in ICI and IJI, the UL coverage performance of the proposed network model can be further improved.

Bifactor and Bifactor S-1 Model Estimations with Non-Reverse-Coded Data

The bifactor model is an extension of Spearman’s two-factor theory. The bifactor model has a strict assumption, which is named orthogonality. The bifactor S-1 model was developed by stretching the orthogonality assumption of the bifactor model. The bifactor S-1 model, contrary to the bifactor model, allows correlation between specific factors and enables items that do not form a common specific factor to be loaded only on the general factor. In psychology, data are mostly multidimensional due to the nature of psychological constructs. The Positive and Negative Affect Schedule (PANAS) which is one of the psychological tests and has two dimensions named positive affect and negative affect. In the literature studies on PANAS, negative affect dimensions were not reverse coded while implementing the bifactor model. Therefore, negative path coefficients were revealed. The purpose of this study is to ascertain whether or not the items in the negative affect factor should be reverse coded in the PANAS. Within the scope of the current study, bifactor and bifactor S-1 model analyses were implemented for the two data sets, which were reverse coded and non-reverse coded. As a result of this study, with reverse-coded data, the bifactor S-1 model was seen as the better model for the PANAS. Additionally, in the modeling of unique variances of items with specific factors, the bifactor S-1 model performed well and also resolved the problem of negative loading on the general factor. The point to take into consideration, which should be noted by researchers who will study the PANAS, is that negative items should be reverse coded.

Multiple testing and variable selection along the path of the least angle regression

Abstract We investigate multiple testing and variable selection using the Least Angle Regression (LARS) algorithm in high dimensions under the assumption of Gaussian noise. LARS is known to produce a piecewise affine solution path with change points referred to as the knots of the LARS path. The key to our results is an expression in closed form of the exact joint law of a $K$-tuple of knots conditional on the variables selected by LARS, the so-called post-selection joint law of the LARS knots. Numerical experiments demonstrate the perfect fit of our findings. This paper makes three main contributions. First, we build testing procedures on variables entering the model along the LARS path in the general design case when the noise level can be unknown. These testing procedures are referred to as the Generalized $t$-Spacing tests and we prove that they have an exact non-asymptotic level (i.e. the Type I error is exactly controlled). This extends work of [ 31] where the spacing test works for consecutive knots and known variance. Second, we introduce a new exact multiple testing procedure after model selection in the general design case when the noise level may be unknown. We prove that this testing procedure has exact non-asymptotic level for general design and unknown noise level. Third, we prove exact control of the false discovery rate under orthogonal design assumption. Monte-Carlo simulations and a real data experiment are provided to illustrate our results in this case. Of independent interest, we introduce an equivalent formulation of the LARS algorithm based on a recursive function.

Are Fiscal Multipliers Estimated with Proxy‐SVARs Robust?*

AbstractWe estimate government spending and tax multipliers in the United States via a flexible proxy‐SVAR model, where identification is achieved by combining fiscal and non‐fiscal instruments with additional parametric restrictions. We find that, while the spending multiplier is robustly estimated to be larger than one across different models, the estimate of the tax multiplier is sensitive to the combination of instruments that we use. We unveil that the key factor behind these heterogeneous estimates is the assumption of orthogonality between total factor productivity (our main non‐fiscal proxy) and the tax shock. If this assumption is imposed, the tax multiplier is around one. If it is not imposed, the tax multiplier is three times as large. Our results point to the need of accounting for the large uncertainty surrounding the tax multiplier for the design of optimal fiscal policies.

The unbearable lightness of equilibria in a low interest rate environment

Structural models with no solution are incoherent, and those with multiple solutions are incomplete. We show that models with occasionally binding constraints are not generically coherent. Coherency requires restrictions on the parameters or on the support of the distribution of the shocks. In presence of multiple shocks, the support restrictions cannot be independent from each other, so the assumption of orthogonality of structural shocks is incompatible with coherency. Models whose coherency is based on support restrictions are generically incomplete, admitting a very large number of minimum state variable solutions.

How structural is unemployment in the United States?

AbstractAggregate matching efficiency (AME) is constructed and endogenized for assessing driving forces behind U.S. unemployment fluctuations. Working with AME sidesteps the problematic orthogonal assumptions on sources of mismatch in the previous studies. Matching efficiency and labor supply shocks, which capture potential variations in structural factors, are identified in a time‐varying structural vector autoregression. Empirical results suggest that the bulk of U.S. unemployment fluctuations is cyclical rather than structural. Meanwhile, the contribution of aggregate shocks to unemployment is substantially time‐varying. In particular, the surge of unemployment rate during the Great Recession is mainly resulting from a shortfall in aggregate demand.

Factor structure, measurement invariance and psychometric properties of the Quality of Life Scale WHOQOL-BREF in the Ecuadorian context

BackgroundThe short version of the World Health Organization Quality of Life questionnaire (WHOQOL-BREF) is a popular instrument used to assess quality of life. The objective of this study was to evaluate the following psychometric properties: structural validity, convergent validity, internal consistency, and measurement invariance across sex of the WHOQOL-BREF in a sample of Ecuadorian adults.MethodsWe used a sample of undergraduates (n = 987) to assess the WHOQOL-BREF original four-factor structure, a model with correlated factors, a hierarchical model, and two models resulting from the exploratory factor analysis and exploratory graph analysis. All the models were evaluated using confirmatory factor analysis.ResultsThe results of the exploratory factor analysis and exploratory graph analysis suggest that the items are organized into four factors, although differently from the original version and the orthogonality assumption is not maintained. The confirmatory factor analysis shows that the original WHOQOL-BREF structure with correlated factors presents adequate psychometric properties. However, we propose a four-factor structure that has the best psychometric properties and adequate internal consistency. The results of the measurement invariance show that strict and strong invariance is achieved between men and women. Convergent validity analysis reveals moderate correlations with self-esteem, resilience, and social support.ConclusionsDespite the original version of the WHOQOL-BREF with correlated factors has acceptable psychometric properties in the Ecuadorian context, we propose a version with a different organization of its items, which is consistent with the findings of other investigations.

Kinkyreg: Instrument-free inference for linear regression models with endogenous regressors

In models with endogenous regressors, a standard regression approach is to exploit just-identifying or overidentifying orthogonality conditions by using instrumental variables. In just-identified models, the identifying orthogonality assumptions cannot be tested without the imposition of other nontestable assumptions. While formal testing of overidentifying restrictions is possible, its interpretation still hinges on the validity of an initial set of untestable just-identifying orthogonality conditions. We present the kinkyreg command for kinky least-squares inference, which adopts an alternative approach to identification. By exploiting nonorthogonality conditions in the form of bounds on the admissible degree of endogeneity, feasible test procedures can be constructed that do not require instrumental variables. The kinky least-squares confidence bands can be more informative than confidence intervals obtained from instrumental-variables estimation, especially when the instruments are weak. Moreover, the approach facilitates a sensitivity analysis for standard instrumental-variables inference. In particular, it allows the user to assess the validity of previously untestable just-identifying exclusion restrictions. Further instrument-free tests include linear hypotheses, functional form, heteroskedasticity, and serial correlation tests.

An optimal sensor/actuator placement method for flexible structures considering spatially varying disturbances

Disturbances acting on flexible structures at spatially varying locations instead of fixed points may lead to deteriorated vibration control performance. To tackle this problem, this article presents an optimal sensor/actuator placement method, in which the closed-loop spatial norm is employed as the optimization criterion. In addition, a new way to calculate the spatial norm is proposed, which is independent of the modal orthogonality assumption in previous research. An optimization framework is established to optimize sensor/actuator placement by minimizing the closed-loop spatial norm using the genetic algorithm. Comprehensive numerical simulations are implemented on a fixed-fixed plate to validate the proposed method. Results show that magnitude of vibrations is reduced and decays faster after the optimization, which indicates that the proposed method markedly improves control performance when spatially varying disturbances exist.

Mutual singularities of overlapping attractor and repeller.

We apply the concepts of relative dimensions and mutual singularities to characterize the fractal properties of overlapping attractor and repeller in chaotic dynamical systems. We consider one analytically solvable example (a generalized baker's map); two other examples, the Anosov-Möbius and the Chirikov-Möbius maps, which possess fractal attractor and repeller on a two-dimensional torus, are explored numerically. We demonstrate that although for these maps the stable and unstable directions are not orthogonal to each other, the relative Rényi and Kullback-Leibler dimensions as well as the mutual singularity spectra for the attractor and repeller can be well approximated under orthogonality assumption of two fractals.

Narrowband multi-source direction-of-arrival estimation in the spherical harmonic domain.

A conventional approach to wideband multi-source (MS) direction-of-arrival (DOA) estimation is to perform single source (SS) DOA estimation in time-frequency (TF) bins for which a SS assumption is valid. Such methods use the W-disjoint orthogonality (WDO) assumption due to the speech sparseness. As the number of sources increases, the chance of violating the WDO assumption increases. As shown in the challenging scenarios with multiple simultaneously active sources over a short period of time masking each other, it is possible for a strongly masked source (due to inconsistency of activity or quietness) to be rarely dominant in a TF bin. SS-based DOA estimators fail in the detection or accurate localization of masked sources in such scenarios. Two analytical approaches are proposed for narrowband DOA estimation based on the MS assumption in a bin in the spherical harmonic domain. In the first approach, eigenvalue decomposition is used to decompose a MS scenario into multiple SS scenarios, and a SS-based analytical DOA estimation is performed on each. The second approach analytically estimates two DOAs per bin assuming the presence of two active sources per bin. The evaluation validates the improvement to double accuracy and robustness to sensor noise compared to the baseline methods.

Study of 5G-NR-MIMO Links in the Presence of an Interferer

Many communication systems are based on the Multiple Input, Multiple Output (MIMO) scheme, and Orthogonal Space–time Block Transmit diversity Coding (OSTBC), combined with Maximal Ratio Receive Combining (MRRC), to create an optimal diversity system. A system with optimal diversity fixes and optimizes the channel’s effects under multi-path and Rayleigh fading with maximum energy efficiency; however, the challenge does not end with dealing with the channel destruction of the multi-path impacts. Susceptibility to interference is a significant vulnerability in future wireless mobile networks. The 5th Generation New Radio (5G-NR) technologies bring hundreds of small cells and pieces of User Equipment (UE) per indoor or outdoor local area scenario under a specific Long Term Evolution (LTE)-based station (e-NodeB), or under 5G-NR base-station (g-NodeB). It is necessary to study issues that deal with many interference signals, and smart jammers from advanced communication equipment cause deterioration in the links between the UE, the small cells, and the NodeB. In this paper, we study and present the significant impact and performances of 2×2 Alamouti Phase-Shift Keying (PSK) modulation techniques in the presence of an interferer and a smart jammer. The destructive effects affecting the MIMO array and the advanced diversity technique without closed-loop MIMO are analyzed. The performance is evaluated in terms of Bit Error Rate (BER) vs. Signal to Interference Ratio (SIR). In addition, we proved the impairment of the orthogonal spectrum assumption mathematically.

An FDTD Investigation of Orthogonality and the Backscattering of HF Waves in the Presence of Ionospheric Irregularities

AbstractOne of the most successful instruments used to investigate the Earth's ionosphere is coherent scatter radars such as the Super Dual Auroral Radar Network (SuperDARN). Their method of mapping plasma convection at high latitudes is contingent on irregularities that momentarily exist in the ionosphere and are elongated along the direction of the geomagnetic field. Under these conditions, SuperDARN transmits high‐frequency (HF) waves that are then backscattered from these plasma irregularities. In order for maximum backscatter to occur, the wave vector of the incident wave must be orthogonal to the magnetic field lines. This is called the “orthogonality condition.” Over the years, ray tracing results have generated many assumptions, where and how orthogonality occurs. However, since ray tracing only tracks the primary direction of energy flow and typically does not account for diffraction, it provides an incomplete understanding of the interaction of HF waves with small ionospheric irregularities. This work investigates the orthogonality assumptions by modeling at high‐resolution HF electromagnetic wave interactions with small‐scale irregularities in the ionosphere. Specifically, the finite‐difference time‐domain (FDTD) method is employed to determine where orthogonality occurs in the ionosphere and the number of times the condition is satisfied during the simulation. The results provide insights into the orthogonality condition as a function of source frequency and elevation angle for both unperturbed and perturbed ionospheres. From these results, small‐scale irregularities are observed to sometimes contribute significantly to the production of backscatter. A discussion is provided to highlight where three existing orthogonality condition assumptions are incorrect or misleading.