Nonzero Covariance Research Articles

M-estimation for linear models with exchangeable errors

This paper focuses on the estimation of parameters in linear models with fixed regressors and exchangeable errors, especially, these errors are assumed to be a subset of an infinite exchangeable sequence. We propose a modified M-estimation to estimate the slope parameter, which outperforms the traditional M-estimation when the random errors are exchangeable. The modified M-estimation utilizes a design point centralization technique, leading to a more suitable estimator with improved theoretical properties. To establish the asymptotic properties of the modified M-estimator, we also develop a weighted central limit theorem for exchangeable sequences, which is of independent interest. Regarding the intercept, we prove that there is no consistent estimator for the intercept when the exchangeable errors have non-zero covariance, and we propose an unbiased estimator for the intercept as a viable compromise. Based on our theoretical findings, we develop a simulation-based approach to approximate the confidence set of the slope, and we establish a prediction model specifically designed for linear models with exchangeable errors. Additionally, we illustrate the finite sample performance of the proposed method through various numerical simulations and a real data application.

An Enhanced Direct Position Determination of Mixed Circular and Non-Circular Sources Using Moving Virtual Interpolation Array.

In this study, a moving single-station direct position determination (DPD) algorithm based on virtual interpolated arrays is proposed. Existing moving single-station algorithms face challenges such as the incomplete utilization of sparse array apertures and insufficient consideration of mixed circular and non-circular signals. To address these issues, we propose an enhanced gridless DPD algorithm, suitable for multiple mixed circular and non-circular sources. Through constructing a non-zero unconjugated covariance matrix from the non-circular components of the mixed signals, the data dimensionality is expanded, and the gridless method is used to fill the voids in the coarray, significantly improving localization performance. Additionally, a unitary transformation method is applied to reduce computational complexity. This method transforms complex operations into real operations by applying unitary transformations to steering vectors and subspaces. Simulation results demonstrate that the proposed algorithm offers significant advantages in terms of array degrees of freedom and localization accuracy.

Improving the fusion of global soil moisture datasets from SMAP, SMOS, ASCAT, and MERRA2 by considering the non-zero error covariance

Surface soil moisture (SSM) estimates from different sources have distinct error characteristics. Multi-source data combination is an efficient way to obtain improved SSM data with reduced uncertainties. Previous data merging studies based on the linear weight averaging scheme rarely considered the impacts of data error covariance (EC) and usually needed a reference dataset, which can lead to suboptimal merging weights. This study applied the quadruple collocation (QC) to estimate EC and combine four SSM datasets simultaneously without the need for a reference. Specifically, two passive microwave satellite datasets (the L3 Soil Moisture Active Passive (SMAP)-V7 and the L3 Soil Moisture and Ocean Salinity -INRA-CESBIO (SMOS-IC)-V2), one active microwave dataset from the Advanced Scatterometer (ASCAT), and one model dataset from the Modern-Era Retrospective analysis for Research and Application, Version 2 (MERRA2) were combined. Generally, QC-based data combination reduced SSM data uncertainties with significantly reduced unbiased Root Mean Square Error (ubRMSE) scores against in situ observations and globally decreased fMSE scores. Moreover, in situ evaluation showed that the QC-based fusion products exhibited better skills than the Tripe Collocation (TC)-based products without considering EC. There were statistically significant differences in Pearson correlation coefficients and ubRMSE metric between the QC and TC -based products. Ignoring the EC between SMAPV7 and SMOS-ICV2 caused overestimations in their relative contributions to fusion data and degraded fusion accuracy. Specifically, the QC-based merging weight was reduced averagely by 0.27 (0.28) for SMAP (IC) when their error cross-correlation increased roughly from −0.42 to 0.9. This study can provide guidance for the generation of improved merged datasets at a global scale.

An Efficient MUSIC Algorithm Enhanced by Iteratively Estimating Signal Subspace and Its Applications in Spatial Colored Noise

The classical multiple signal classification (MUSIC) algorithms mainly have two limitations. One is an insufficient number of snapshots, which usually causes an ill-posed sample covariance matrix in many real applications. The other limitation is the intense space-colored and time-white noise, which also breaks the separability between signal and noise subspaces. In the case of the insufficient sample, there are few signal components in the non-zero delay sample covariance matrix (SCM), where the space-colored and time-white noise components are suppressed by the temporal method. A set of non-zero delay sample covariance matrices are constructed, and a nonlinear object function is formulated. Hence, the sufficient non-zero delay SCMs ensure that enough signal components are used for signal subspace estimation. Then, the constrained optimization problem is converted into an unconstrained one by exploiting the Lagrange multiplier method. The nonlinear equation is solved by Newton’s method iteratively. Moreover, a proper initial value of the new algorithm is given, which can improve the convergence of the iterative algorithm. In this paper, the noise subspace is removed by the pre-projection technique in every iteration step. Then, an improved signal subspace is obtained, and a more efficient MUSIC algorithm is proposed. Experimental results show that the proposed algorithm achieves significantly better performance than the existing methods.

PSI-17 Genetic analysis of weaning weight in Hungarian Simmental cattle

Abstract The aim of the study was to partition the total phenotypic variation in the weaning weight of Hungarian Simmental calves into their various causal components. The data used was provided by the Association of Hungarian Simmental Breeders. The dataset comprised of the weaning weight records of 44,278 calves (sire = 879, dam = 14,811) born from 1975 to 2020. A total of six models were fitted to the weaning weight data. Herd, birth year, calving order and sex were included as fixed effects in the models. Model 1 had direct genetic effect as the only random effect. Model 2 had a permanent maternal environment as an additional random effect. Model 3 had both direct and maternal genetic effects, with their covariance is being zero. Model 4 was similar to Model 3 but with non-zero direct-maternal genetic covariance. Model 5 had direct, maternal genetic and permanent environmental effects and a zero direct-maternal genetic covariance. Model 6 was similar to model 5 but the direct-maternal genetic effect was assumed to be correlated. Variance components and genetic parameters were estimated using restricted maximum likelihood method with the Wombat software. The best fit model was determined using the Log likelihood ratio test. Inclusion of direct maternal genetic covariance increased the variance components estimates dramatically which resulted in a corresponding increase in the direct and maternal heritability estimates. The best fitted model (Model 4) had direct and maternal genetic effects as the only random effects with a non-zero direct-maternal genetic covariance. The direct heritability, maternal heritability and direct-maternal genetic correlation estimate of the best model was 0.57, 0.16 and -0.78, respectively. Our result suggests the problem of (co)sampling variation in the partitioning of additive genetic effect into direct and maternal components.

Uncertainty in Measured Raindrop Size Distributions from Four Types of Collocated Instruments

Four types (2D-video disdrometer: 2DVD; precipitation occurrence sensor system: POSS; micro-rain radar: MRR; and Joss–Waldvogel disdrometer: JWD) of sixteen instruments were collocated within a square area of 400 m2 from 16 April to 8 May 2008 for intercomparison of drop size distribution (DSD) of rain. This unique dataset was used to study the inherent measurement uncertainty due to the diversity of the measuring principles and sampling sizes of the four types of instruments. The DSD intercomparison shows generally good agreement among them, except that the POSS and MRR had higher concentrations of small raindrops (<1.0 mm) and offered a better chance to observe big raindrops (>5.2 mm). The measurement uncertainty ( σ ) was obtained quantitatively after considering the zero or non-zero measurement error covariance between two instruments of the same type. The results indicate the measurement uncertainties were found to be neither independent nor identical among the same type of instruments. The MRR is relatively accurate (lower σ ) due to large sampling volumes and accurate measurement of the Doppler power spectrum. The JWD is the least accurate due to the small sampling volumes. The σ decreases rapidly with increasing time-averaging window. The 2DVD shows the best accuracy of R in longer averaging time, but this is not true for Z due to the small sampling volume. The MRR outperformed other instruments for Z for entire averaging time due to its measuring principle.

Non-local Observations and Information Transfer in Data Assimilation

Nonlocal observations are observations that cannot be allocated one specific spatial location. Examples are observations that are spatial averages of linear or nonlinear functions of system variables. In conventional data assimilation such as (ensemble) Kalman Filters and variational methods information transfer between observations and model variables is governed by covariance matrices that are either preset or determined from the dynamical evolution of the system. In many science fields the covariance structures have limited spatial extent, and this paper discusses what happens when this spatial extent is smaller then the support of the observation operator that maps state space to observations space. It is shown that information is carried beyond the physical information in the prior covariance structures by the nonlocal observational constraints, building an information bridge (or information channel) that has not been studied before: the posterior covariance can have nonzero covariance structures where the prior has a covariance of zero. It is shown that in standard data-assimilation techniques that enforce a covariance structure and limit information transfer to that structure the order in which local and nonlocal observations are assimilated can have a large influence on the analysis. Local observations should be assimilated first. This relates directly to localisation used in Ensemble Kalman Filters and Smoothers, but also to variational methods with a prescribed covariance structure where observations are assimilated in batches. This suggests that the emphasis on covariance modelling should shift away from the prior covariance and towards the modelling of the covariances between model and observation space. Furthermore, it is shown that local observations with non-locally correlated observation errors behave in the same way as uncorrelated observations that are nonlocal. Several theoretical results are illustrated with simple numerical examples. The significance of the information bridge provided by nonlocal observations is highlighted further through discussions of temporally nonlocal observations, and new ideas on targeted observations.

Closed-Form Formulae for European Options Under Three-Factor Models

In this paper, we derive new closed-form valuations to European options under three-factor hybrid models that include stochastic interest rates and stochastic volatility and incorporate a nonzero covariance structure between factors. We make novel use of the empirically proven 3/2 stochastic volatility model with a time-dependent drift in which we are free to choose the moving reversion target. This model has been shown by many authors to empirically outperform other volatility models in maximising model fit. We also improve the valuation of European options under the Heston volatility and Cox, Ingersoll, Ross interest rate model, recently published in the literature, by replacing open-form infinite series with closed-form analytic expressions. For completeness, we also add a fuller covariance structure in this setting and detail closed-form valuations for options. The inclusion of nonzero covariances amongst the factors can significantly improve option pricing by allowing for a wider variety of market behaviour. The solutions are derived by firstly formulating the price of a European call option in terms of the corresponding characteristic function of the underlying price and then determining a partial differential equation for the characteristic function. By including empirically proven models into our analysis, the options formulae could provide more realistic prices for investors and practitioners.

Disentangling the role of variance and covariance information in portfolio selection problems

The covariation among financial asset returns is often a key ingredient used in the construction of optimal portfolios. Estimating covariances from data, however, is challenging due to the potential influence of estimation error, specially in high-dimensional problems, which can impact negatively the performance of the resulting portfolios. We address this question by putting forward a simple approach to disentangle the role of variance and covariance information in the case of mean-variance efficient portfolios. Specifically, mean-variance portfolios can be represented as a two-fund rule: one fund is a fully invested portfolio that depends on diagonal covariance elements, whereas the other is a long-short, self financed portfolio associated with the presence of non-zero off-diagonal covariance elements. We characterize the contribution of each of these two components to the overall performance in terms of out-of-sample returns, risk, risk-adjusted returns and turnover. Finally, we provide an empirical illustration of the proposed portfolio decomposition using both simulated and real market data.

Price-cost markups in UK industry

ABSTRACTThis article tests for the presence of monopolistic price markups across UK industrial sectors by testing for a non-zero covariance between the Solow residual and various instruments. This restriction derives from profit maximising conditions for a representative imperfectly competitive firm under the assumption of constant returns to scale. We find evidence of significant markups in the Manufacturing and Services sectors which together account for more than 90% of total employment within the UK.

Quantitative PET Imaging in Drug Development: Estimation of Target Occupancy.

Positron emission tomography, an imaging tool using radiolabeled tracers in humans and preclinical species, has been widely used in recent years in drug development, particularly in the central nervous system. One important goal of PET in drug development is assessing the occupancy of various molecular targets (e.g., receptors, transporters, enzymes) by exogenous drugs. The current linear mathematical approaches used to determine occupancy using PET imaging experiments are presented. These algorithms use results from multiple regions with different target content in two scans, a baseline (pre-drug) scan and a post-drug scan. New mathematical estimation approaches to determine target occupancy, using maximum likelihood, are presented. A major challenge in these methods is the proper definition of the covariance matrix of the regional binding measures, accounting for different variance of the individual regional measures and their nonzero covariance, factors that have been ignored by conventional methods. The novel methods are compared to standard methods using simulation and real human occupancy data. The simulation data showed the expected reduction in variance and bias using the proper maximum likelihood methods, when the assumptions of the estimation method matched those in simulation. Between-method differences for data from human occupancy studies were less obvious, in part due to small dataset sizes. These maximum likelihood methods form the basis for development of improved PET covariance models, in order to minimize bias and variance in PET occupancy studies.

Panel kink regression with an unknown threshold

In this paper, we extend the kink regression model with an unknown threshold in Hansen (2017) to the panel data framework, where the cross-sectional dimension (N) goes to infinity and the time period (T) is fixed. Following the literature of threshold regressions, we propose an estimator based on the within-group transformation. Under fixed threshold effect assumption, we establish that the slope and threshold estimators are jointly normally distributed with the same convergence rate OpN−1∕2 and a non-zero asymptotic covariance. We also suggest a sup-Wald test for the presence of kink effect, and derive its limiting distribution. A bootstrap procedure is proposed to obtain the bootstrap p-values to improve the finite sample performance of the test. Monte Carlo simulations show that the FE estimator and the sup-Wald test perform quite well in estimating the unknown parameters and testing for kink effect, respectively.

The benefits of incorporating utility dependencies in finite mixture probit models

We propose an application of a new finite mixture multinomial conditional probit (FM-MNCP) model that accommodates preference heterogeneity and explicitly accounts for utility dependencies between choice alternatives considering both local and background contrast effects. The latter is accomplished by using a one-factor structure for segment-specific covariance matrices allowing for nonzero off-diagonal covariance elements. We compare the model to a finite mixture multinomial independent probit (FM-MNIP) model that as well accommodates heterogeneity but assumes independence. That way, we address the potential benefits of a model that additionally accounts for dependencies over a model that accommodates heterogeneity only. Our model comparison is based on empirical data for smoothies and is assessed in terms of fit, holdout validation, and market share predictions. One of the main findings of our empirical study is that allowing for utility dependencies may counterbalance the effects of considering heterogeneity, and vice versa. Additional findings from a simulation study indicate that the FM-MNCP model outperforms the FM-MNIP model with respect to parameter recovery.

Planar quantum squeezing in spin-1/2 systems

Planar quantum squeezed (PQS) states, i.e. quantum states which are squeezed in two orthogonal spin components on a plane, have recently attracted attention due to their applications in atomic interferometry and quantum information [, ]. In this paper we present an application of the framework described by Puentes et al [] for planar quantum squeezing via quantum nondemolition (QND) measurement, for the particular case of spin-1/2 systems and nonzero covariance between orthogonal spin components. Our regime, consisting of spin-1/2 entangled planar-squeezed (EPQS) states, is of interest as it can present higher precision for quantum parameter estimation and quantum magnetometry at finite intervals. We show that entangled planar-squeezed states (EPQS) can be used to reconstruct a specific quantum parameter, such as a phase or a magnetic field, within a finite interval with higher precision than PQS states, and without iterative procedures. EPQS is of interest in cases where limited prior knowledge about the specific subinterval location for a phase is available, but it does not require knowledge of the exact phase a priori, as in the case of squeezing on a single spin component.

LARS NETWORK FILTRATION IN THE STUDY OF EEG BRAIN CONNECTIVITY.

In a brain network, weak and non-significant edge weights between nodes signal spurious connections and are often discarded by thresholding the weights. Traditional practice of thresholding edge weights at an arbitrary value can be problematic. Network filtration provides an alternative by summarizing the changes in the network topology with respect to a broad range of thresholds. A well established network filtration approach depends on the graphical-LASSO (least absolute shrinkage and selection operator) model, where a sequence of binary networks are obtained based on non-zero sparse inverse covariance (IC) estimates of partial correlations at a range of sparsity parameters. The limitation of the graphical-LASSO network model is that it relies on the structural information rather than actual entries of the sparse IC matrices and therefore can only yield approximate dynamic topological changes in the network. In the current study, we propose a new network filtration approach based on least angle regression (LARS) that gives exact filtration values at which the filtration changes and apply it to study brain connectivity in response to emotional stimuli across different age groups via electroencephalographic (EEG) data.

Assessing dynamic spectral causality by lagged adaptive directed transfer function and instantaneous effect factor.

It is of significance to assess the dynamic spectral causality among physiological signals. Several practical estimators adapted from spectral Granger causality have been exploited to track dynamic causality based on the framework of time-varying multivariate autoregressive (tvMVAR) models. The nonzero covariance of the model's residuals has been used to describe the instantaneous effect phenomenon in some causality estimators. However, for the situations with Gaussian residuals in some autoregressive models, it is challenging to distinguish the directed instantaneous causality if the sufficient prior information about the "causal ordering" is missing. Here, we propose a new algorithm to assess the time-varying causal ordering of tvMVAR model under the assumption that the signals follow the same acyclic causal ordering for all time lags and to estimate the instantaneous effect factor (IEF) value in order to track the dynamic directed instantaneous connectivity. The time-lagged adaptive directed transfer function (ADTF) is also estimated to assess the lagged causality after removing the instantaneous effect. In this study, we first investigated the performance of the causal-ordering estimation algorithm and the accuracy of IEF value. Then, we presented the results of IEF and time-lagged ADTF method by comparing with the conventional ADTF method through simulations of various propagation models. Statistical analysis results suggest that the new algorithm could accurately estimate the causal ordering and give a good estimation of the IEF values in the Gaussian residual conditions. Meanwhile, the time-lagged ADTF approach is also more accurate in estimating the time-lagged dynamic interactions in a complex nervous system after extracting the instantaneous effect. In addition to the simulation studies, we applied the proposed method to estimate the dynamic spectral causality on real visual evoked potential (VEP) data in a human subject. Its usefulness in time-variant spectral causality assessment was demonstrated through the mutual causality investigation of brain activity during the VEP experiments.

LMI-based minimax estimation and filtering under unknown covariances

In this paper, we consider a minimax approach to the estimation and filtering problems in the stochastic framework, where covariances of the random factors are completely unknown. The term ‘random factors’ refers either to unknown parameters and measurement noise in the estimation problem or to disturbance process and the initial state of a linear discrete-time dynamic system in the filtering problem. We introduce a notion of the attenuation level of random factors as a performance measure for both a linear unbiased estimate and a filter. This is the worst-case variance of the estimation error normalised by the sum of variances of all random factors over all nonzero covariance matrices. It is shown that this performance measure is equal to the spectral norm of the ‘transfer matrix’ and therefore the minimax estimate and filter can be computed in terms of linear matrix inequalities (LMIs). Moreover, the explicit formulae for both the minimax estimate and the minimal value of the attenuation level are presented in the estimation problem. It turns out that the above attenuation level of random factors coincides with the attenuation level of deterministic factors that is the worst-case normalised squared Euclidian norm of the estimation error over all nonzero sample values of random factors. In addition, we demonstrate that the LMI technique can be applied to derive the optimal robust estimator and filter, when there is a priori information about convex polyhedral sets which unknown covariance matrices of random factors belong to. Two illustrative examples show advantages of the minimax approach proposed.

“Variable localization” in an ensemble Kalman filter: Application to the carbon cycle data assimilation

[1] In ensemble Kalman filter, space localization is used to reduce the impact of long-distance sampling errors in the ensemble estimation of the forecast error covariance. When two variables are not physically correlated, their error covariance is still estimated by the ensemble and, therefore, it is dominated by sampling errors. We introduce a “variable localization” method, zeroing out such covariances between unrelated variables to the problem of assimilating carbon dioxide concentrations into a dynamical model using the local ensemble transform Kalman filter (LETKF) in an observing system simulation experiments (OSSE) framework. A system where meteorological and carbon variables are simultaneously assimilated is used to estimate surface carbon fluxes that are not directly observed. A range of covariance structures are explored for the LETKF, with emphasis on configurations allowing nonzero error covariance between carbon variables and the wind field, which affects transport of atmospheric CO2, but not between CO2 and the other meteorological variables. Such variable localization scheme zeroes out the background error covariance among prognostic variables that are not physically related, thus reducing sampling errors. Results from the identical twin experiments show that the performance in the estimation of surface carbon fluxes obtained using variable localization is much better than that using a standard full covariance approach. The relative improvement increases when the surface fluxes change with time and model error becomes significant.

Accuracy of Pseudo-Inverse Covariance Learning—A Random Matrix Theory Analysis

For many learning problems, estimates of the inverse population covariance are required and often obtained by inverting the sample covariance matrix. Increasingly for modern scientific data sets, the number of sample points is less than the number of features and so the sample covariance is not invertible. In such circumstances, the Moore-Penrose pseudo-inverse sample covariance matrix, constructed from the eigenvectors corresponding to nonzero sample covariance eigenvalues, is often used as an approximation to the inverse population covariance matrix. The reconstruction error of the pseudo-inverse sample covariance matrix in estimating the true inverse covariance can be quantified via the Frobenius norm of the difference between the two. The reconstruction error is dominated by the smallest nonzero sample covariance eigenvalues and diverges as the sample size becomes comparable to the number of features. For high-dimensional data, we use random matrix theory techniques and results to study the reconstruction error for a wide class of population covariance matrices. We also show how bagging and random subspace methods can result in a reduction in the reconstruction error and can be combined to improve the accuracy of classifiers that utilize the pseudo-inverse sample covariance matrix. We test our analysis on both simulated and benchmark data sets.

Teaching Causal Inference in Undergraduate Econometrics

This paper argues that the current way in which the undergraduate introductory econometrics course is taught is neither inline with current empirical practice nor very intuitive. It proposes a shift in focus of the course on causal inference using the Roy-Rubin Causal Model (RRCM). A second theme of the paper is the suggestion to use random regressors from the start to improve the ability of students to intuitively relate to the regression model and to enable the teacher to present many regression pitfalls under the umbrella of the non-zero covariance between regressors and the regression error term. Finally, the paper discusses how to make room for the new material and suggests ways of dealing with the added complexity of the approach.