Correlated Error Structure Research Articles

A Simulation Study of the Effects of Additive, Multiplicative, Correlated, and Uncorrelated Errors on Principal Component Analysis

ABSTRACTMeasurement errors are ubiquitous in all experimental sciences. Depending on the particular experimental platform used to acquire data, different types of errors are introduced, amounting to an admixture of additive and multiplicative error components that can be uncorrelated or correlated. In this paper, we investigate the effect of different types of experimental error on the recovery of the subspace with principal component analysis (PCA) using numerical simulations. Specifically, we assessed how different error characteristics (variance, correlation, and correlation structure), loading structures, and data distributions influence the accuracy to estimate an error‐free (true) subspace from sampled data with PCA. Quality was assessed in terms of the mean squared reconstruction error and the congruence to the error‐free loadings, using the pseudorank and adjusting for rotational ambiguity. Analysis of variance reveals that the error variance, error correlation structure, and their interaction with the loading structure are the factors mostly affecting quality of loading estimation from sampled data. We advocate for the need to characterize and assess the nature of measurement error and the need to adapt formulations of PCA that can explicitly take into account error structures in the model fitting.

Exploring the Interdependence of Vertical Extrapolation Uncertainties in Repowering Wind Farms

Assessing a wind farm’s annual energy production (AEP) involves modelling the wind resource and the wind-to-power conversion at the site. The greenfield pre-construction phase generally comprises the installation of wind measurement devices. For repowering projects, the wind data from the pre-construction phase of the existing farm can be used as wind input to assess the energy yield of the repowered wind farm. Indeed, one study demonstrates that when the modelling error correlations are known, the AEP prediction uncertainty of the repowered farm can be reduced by combining the energy production records of the existing farm with the AEP assessment for both farms. Previous studies have successfully identified the correlation structure for certain errors, especially for horizontal flow modelling, but not for vertical flow modelling. However, vertical extrapolation is essential, as the wind measurement heights are generally lower than the hub height on the repowered farm. This paper bridges this research gap and demonstrates that the correlation structure of errors in vertical profile modelling is Gaussian, with parameters dependent on shear values and heights. The distribution is validated against site data from simple to moderately complex sites in France.

An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry

All sensing systems have some inherent error. Often, these errors are systematic, and observations taken within a similar region of space and time can have correlated error structure. However, the data from these systems are frequently assumed to have completely independent and uncorrelated error. This work introduces a correlated error model for GNSS reflectometry (GNSS-R) using observations from NASA’s Cyclone Global Navigation Satellite System (CYGNSS). We validate our model against near-simultaneous observations between two CYGNSS satellites and double-difference our results with modeled observables to extract the correlated error structure due to the observing system itself. Our results are useful to catalog for future GNSS-R missions and can be applied to construct an error covariance matrix for weather data assimilation.

Improved estimation of functional enrichment in SNP heritability using feasible generalized least squares

Functional enrichment results typically implicate tissue or cell-type-specific biological pathways in disease pathogenesis and as therapeutic targets. We propose generalized linkage disequilibrium score regression (g-LDSC) that requires only genome-wide association studies (GWASs) summary-level data to estimate functional enrichment. The method adopts the same assumptions and regression model formulation as stratified linkage disequilibrium score regression (s-LDSC). Although s-LDSC only partially uses LD information, our method uses the whole LD matrix, which accounts for possible correlated error structure via a feasible generalized least-squares estimation. We demonstrate through simulation studies under various scenarios that g-LDSC provides more precise estimates of functional enrichment than s-LDSC, regardless of model misspecification. In an application to GWAS summary statistics of 15 traits from the UK Biobank, estimates of functional enrichment using g-LDSC were lower and more realistic than those obtained from s-LDSC. In addition, g-LDSC detected more significantly enriched functional annotations among 24 functional annotations for the 15 traits than s-LDSC (118 vs. 51).

Omnivariant Generalized Least Squares regression: Theory, geochronological applications, and making the case for reconciled Δ47 calibrations

Least-squares regression methods are mathematically powerful, conceptually and computationally simple, and widely used in many fields. However, none of the commonly-used flavors of least-squares regression, such as York regression or Generalized Least Squares (GLS), take into account the full set of covariances between all observed (x, y) values. Here we describe the Omnivariant Generalized Least Squares (OGLS) method to fit a model of the form y = f(x), accounting for the full error correlation structure of the (x, y) data, based on a first-order linear propagation of the uncertainties in all variables into errors in y residuals, followed by minimizing the vector of y residuals with respect to the Mahalanobis norm defined by its covariance matrix. This approach may be described as a generalization of both York regression and GLS. It is mathematically exact for straight-line fits, and is also suitable for many non-linear models. Here we describe the principles of OGLS regression and discuss its properties, caveats, and practical use, and provide two consistent open-source implementations in Python and R. To illustrate how various fields of geochronology and stable-isotope geochemistry may benefit from this new method, we discuss how OGLS may specifically apply to 40Ar/39Ar dating and how it provides robust mathematical evidence that Δ47 carbonate calibrations in the recently defined I-CDES metrological scale are statistically indistinguishable, effectively solving long-standing methodological discrepancies.

A criterion and incremental design construction for simultaneous kriging predictions

In this paper, we further investigate the problem of selecting a set of design points for universal kriging, which is a widely used technique for spatial data analysis. Our goal is to select the design points in order to make simultaneous predictions of the random variable of interest at a finite number of unsampled locations with maximum precision. Specifically, we consider as response a correlated random field given by a linear model with an unknown parameter vector and a spatial error correlation structure. We propose a new design criterion that aims at simultaneously minimizing the variation of the prediction errors at various points. We also present various efficient techniques for incrementally building designs for that criterion scaling well for high dimensions. Thus the method is particularly suitable for big data applications in areas of spatial data analysis such as mining, hydrogeology, natural resource monitoring, and environmental sciences or equivalently for any computer simulation experiments. We have demonstrated the effectiveness of the proposed designs through two illustrative examples: one by simulation and another based on real data from Upper Austria.

Estimation of Heritability under Correlated Errors Using the Full-Sib Model

In plant and animal breeding, sometimes observations are not independently distributed. There may exist a correlated relationship between the observations. In the presence of highly correlated observations, the classical premise of independence between observations is violated. Plant and animal breeders are particularly interested to study the genetic components for different important traits. In general, for estimating heritability, a random component in the model must adhere to specific assumptions, such as random components, including errors, having a normal distribution, and being identically independently distributed. However, in many real-world situations, all of the assumptions are not fulfilled. In this study, correlated error structures are considered errors that are associated to estimate heritability for the full-sib model. The number of immediately preceding observations in an autoregressive series that are used to predict the value at the current observation is defined as the order of the autoregressive models. First-order and second-order autoregressive models i.e., AR(1) and AR(2) error structures, have been considered. In the case of the full-sib model, theoretical derivation of Expected Mean sum square (EMS) considering AR(1) structure has been obtained. A numerical explanation is provided for the derived EMS considering AR(1) structure. The predicted mean squares error (MSE) is obtained after including the AR(1) error structures in the model, and heritability is estimated using the resulting equations. It is noticed that correlated errors have a major influence on heritability estimation. Different correlation patterns, such as AR(1) and AR(2), can be inferred to change heritability estimates and MSE values. To attain better results, several combinations are offered for various scenarios.

Measurement errors in Gaussian mixture models using high-dimensional air pollution constituents data

BACKGROUND AND AIM: An active area of current interest in air pollution epidemiology is characterizing the impact of air pollution constituents on health outcomes. Statistical analysis of air pollution data suffers from bias due to measurement error of these exposures, and this bias is exacerbated when the exposures are high-dimensional. It is well known that exposure to fine particles (PM2.5) can cause adverse health effects. However, the independent effects of PM2.5 constituents have not been as well investigated. METHODS: In this work, we develop a novel approach to correcting for the bias due to measurement errors associated with multiple correlated air pollutants with a complex correlated error structure through the use of Gaussian mixture models under the main / external validation design framework. The clusters are corrected for measurement error and then analyzed in a Cox model to investigate the impacts of different air pollutant profiles on time to even outcomes, such as all-cause mortality and cancer incidence. RESULTS: Extensive simulation studies are carried out to investigate the performance of the proposed method, and to explore sensitivity to departures from Gaussian assumptions. The method is illustrated in a study of air pollutants on all-cause mortality in the Nurses’ Health Study, with exposures validated in the RIOPA and MESA studies.

Estimation of hurst exponent for sequential monitoring of clinical trials with covariate adaptive randomization

BackgroundClassical Brownian motion (BM) has been commonly used in monitoring clinical trials including those with covariate adaptive randomization (CAR). Independent increment property is commonly assumed in the sequential monitoring process of the clinical trials with CAR designs. However, in reality, correlation may exist in the error terms of the underlying model, resulting in dependent increment in the sequential monitoring process. MethodsWe conducted simulations for estimating the Hurst exponent to evaluate the stochastic property in the covariate adaptive randomized clinical trials under two scenarios: 1. CAR designs with independent and identically distributed error terms. 2. CAR designs with correlated error terms. The theoretical properties of covariate adaptive randomized clinical trials with correlated error structure were investigated. A test statistic including the covariance pattern of the error terms was proposed. ConclusionIn our study, the sequential test statistics under CAR procedure is shown to be asymptotically Brownian motion when the error structure is correctly specified. Further, Brownian motion is a special case of fractional Brownian motion when Hurst exponent equals to 0.5. Our simulations are consistent with the theoretical asymptotic results.

Graph decomposition methods for variance balanced block designs with correlated errors

Block designs have an important historical footing in the design of experiments. Let θ be the treatment mean vector in the statistical model with fixed block effects. It is desirable to have the covariance of the least squares estimator of the treatment effects, Cov(θ̂), be a completely symmetric matrix. When this occurs, we have a variance balanced design (VBD). It is known that a pairwise balanced design can be converted into a VBD by choosing blocks with a multiplicity inversely proportional to the block size. This assumes that experimental errors are uncorrelated. However, this need not be the case, especially in ecological experiments where blocks have spatial correlation or laboratory settings with temporal correlation. This paper proposes the use of graph decompositions as a generalization of balanced incomplete block designs for the construction of VBDs in the presence of correlated errors. A general result is obtained to construct VBDs for various error correlation structures. Several applications are given to illustrate the relationship between graph decompositions and VBDs.

A systematic study on the effect of different error structures on pseudo‐univariate and multivariate figures of merit

AbstractIn this study, the effect of different error structures on psedounivariate and multivariate analytical figures of merit in simulated data of hyphenated chromatographic systems was investigated. Different error structures (e.g., homoscedastic, heteroscedastic, and correlated) were investigated. For this purpose, five components systems at five concentration levels with three replicates were simulated. Different types of error were added to the data. Multivariate Curve Resolution‐Alternating Least Squares (MCR‐ALS) and Maximum Likelihood Principal Component Analysis (MLPCA‐MCR‐ALS) methods were used. After resolution, pseudo‐univariate and multivariate analytical figures of merit were calculated and compared. As it expected, the detection limit for noisy datasets is higher than the noise‐free datasets, whereas the slopes of the calibration curves are not significantly different. Comparing the results generally showed that the detection limit values in multivariable mode were better than the univariate mode. The LODs of data (pseudo‐univariate and multivariate) with homoscedastic and correlated error structure by MCR‐ALS and MLPCA‐MCR‐ALS were the same. The analysis of data with heteroscedastic error structure by MLPCA‐MCR‐ALS had a lower detection limit than analysis with MCR‐ALS. The figures of merit obtained from WLS and OLS regression in heteroscedastic datasets were compared and better LODs were obtained after WLS method.

A Support Vector Regression Approach for Three–Level Longitudinal Data


 
 Background: Longitudinal data structure is frequently observed in health science. This introduces correlation to the data that needs to be handled in modelling process. Recently, machine learning approaches have been introduced in the context of longitudinal data for prediction of the response variable purpose. In this paper a mixed-effects least squares support vector regression model is presented for three-level longitudinal data. In the proposed model, multiple random-effect terms are used for considering the existing correlation structures in longitudinal data. The proposed model is flexible in modelling (non-)linear and complex relationships between predictors and response, while it takes into account the hierarchical structure of data and is computationally efficient. 
 Methods Both random intercept and random trend models with a special correlation structure of errors are illustrated. A real data example on human Brucellosis rate is analysed and two simulation studies are performed to illustrate the proposed model. The fitting and generalisation performance of the proposed model are investigated and compared with the ordinary least squares support vector regression and linear mixed-effects models. 
 Results: Based on the human Brucellosis rate example and two simulation studies, the proposed models had the best performance in generalisation. Also, the fitting performances of the proposed models were better than that of the classic models. 
 Conclusion: Our study revealed that in the presence of nonlinear relationship between covariates and outcome, the proposed MLS-SVR model has the best fitting and generalisation performance and can capture correlation of the data.
 

Identification of Leverage Points in Principal Component Regression and r-k Class Estimators with AR(1) Error Structure

The determination of leverage observations have been frequently investigated through ordinary least squares and some biased estimators proposed under the multicollinearity problem in the linear regression models. Recently, the identification of leverage and influential observations have been also popular on the general linear regression models with correlated error structure. This paper proposes a new projection matrix and a new quasiprojection matrix to determination of leverage observations for principal component regression and r-k class estimators, respectively, in general linear regression model with first-order autoregressive error structure. Some useful properties of these matrices are presented. Leverage observations obtained by generalized least squares and ridge regression estimators available in the literature have been compared with proposed principal component regression and r-k class estimators over a simulation study and a numerical example. In the literature, the first leverage is considered separately due to the first-order autoregressive error structure. Therefore, the behaviours of first leverages obtained by principal component regression and r-k class estimators has been also investigated according to the autocorrelation coefficient and biasing parameter through applications. The results showed that the leverage of the first observation obtained by principal component regression and r-k estimators is smaller than that obtained by generalized least squares and ridge regression estimators. In addition, as the autocorrelation coefficient goes to -1, the leverage of the first transformed observation decreases for PCR and r-k class estimators, while its increases while the autocorrelation coefficient goes to 1.

Statistical inference for the partially linear single-index model of panel data with serially correlated error structure

Due to its flexibility, a partially linear single-index model arises in many contemporary scientific endeavors. In this paper, we set foot on its inference under settings of panel data and a serially correlated error component structure. By combining the local polynomial technique with the bias-corrected generalized estimating equations, we propose a feasible weighted generalized estimating equation estimation (GEE-FW) for unknown regression parameters. The GEE-FW is shown to be asymptotically normal and more efficient than the unweighted generalized estimating equation estimation based on working independence. Moreover, a two-stage local linear generalized estimating equation estimation (GEE-TS) of the unknown link function is also proposed, which takes the contemporaneous correlation into account and achieves an increase in its estimated efficiency. The asymptotic property of the GEE-TS is established as well and we show that it is asymptotically more efficient than the one which ignores the contemporaneous correlation. Conducted numerical simulation studies and actual data analysis show that the proposed estimation methods are effective and have good performance in both theory and application.

A Note on Second Order Slope Rotatable Designs under Intra-class Correlated Errors Using Pairwise Balanced Designs

In this paper, a study on second order slope rotatable designs under intra-class correlation structure of errors using pairwise balanced designs is suggested. Further, the variance function of the estimated slopes for different values of intra-class correlated coefficient ‘ ’ for 6 ≤ v ≤ 15 (v- number of factors) are studied.

Second Order Slope Rotatable Designs under Tri-diagonal Correlation Structure of Errors Using a Pair of Incomplete Block Designs

Box and Hunter [1] introduced the concept of rotatability for response surface designs. The concept of slope-rotatability was introduced by Hader and Park [2] as an analogous to rotatability property, which is an important design criterion for response surface design. Slope-rotatable design is that of which the variance of partial derivative is a function of distance from the design (d). Recently, a few measures of slope-rotatability for a given response surface design was introduced. In this paper, a new method of slope rotatability for second order response surface designs under tri-diagonal correlation structure of errors using a pair of symmetrical unequal block arrangements with two unequal block sizes is studied. Further, a study on the dependence of variance function of the second order response surface at different design points for different values of tri-diagonal correlation coefficient ρ which lies between -0.9 to 0.9 and the distance from centre (d) is suggested.

Loss Reserving Estimation With Correlated Run-Off Triangles in a Quantile Longitudinal Model

In this paper, we consider a loss reserving model for a general insurance portfolio consisting of a number of correlated run-off triangles that can be embedded within the quantile regression model for longitudinal data. The model proposes a combination of the between- and within-subportfolios (run-off triangles) estimating functions for regression parameter estimation, which take into account the correlation and variation of the run-off triangles. The proposed method is robust to the error correlation structure, improves the efficiency of parameter estimators, and is useful for the estimation of the reserve risk margin and value at risk (VaR) in actuarial and finance applications.

Rapid adjustment and post‐processing of temperature forecast trajectories

AbstractModern weather forecasts are commonly issued as consistent multi‐day forecast trajectories with a time resolution of 1–3 hours. Prior to issuing, statistical post‐processing is routinely used to correct systematic errors and misrepresentations of the forecast uncertainty. However, once the forecast has been issued, it is rarely updated before it is replaced in the next forecast cycle of the numerical weather prediction (NWP) model. This paper shows that the error correlation structure within the forecast trajectory can be utilized to substantially improve the forecast between the NWP forecast cycles by applying additional post‐processing steps each time new observations become available. The proposed rapid adjustment is applied to temperature forecast trajectories from the UK Met Office's convective‐scale ensemble MOGREPS‐UK. MOGREPS‐UK is run four times daily and produces hourly forecasts for up to 36 hours ahead. Our results indicate that the rapidly adjusted forecast from the previous NWP forecast cycle outperforms the new forecast for the first few hours of the next cycle, or until the new forecast itself can be rapidly adjusted, suggesting a new strategy for updating the forecast cycle.

Estimation and testing for panel data partially linear single-index models with errors correlated in space and time

We consider a panel data partially linear single-index models (PDPLSIM) with errors correlated in space and time. A serially correlated error structure is adopted for the correlation in time. We propose using a semiparametric minimum average variance estimation (SMAVE) to obtain estimators for both the parameters and unknown link function. We not only establish an asymptotically normal distribution for the estimators of the parameters in the single index and the linear component of the model, but also obtain an asymptotically normal distribution for the nonparametric local linear estimator of the unknown link function. Then, a fitting of spatial and time-wise correlation structures is investigated. Based on the estimators, we propose a generalized F-type test method to deal with testing problems of index parameters of PDPLSIM with errors correlated in space and time. It is shown that under the null hypothesis, the proposed test statistic follows asymptotically a [Formula: see text]-distribution with the scale constant and degrees of freedom being independent of nuisance parameters or functions. Simulated studies and real data examples have been used to illustrate our proposed methodology.

Analysis of panel data partially linear single-index models with serially correlated errors

In this paper, we study a panel data partially linear single-index models (PDPLSIM) with serially correlated errors. A semiparametric minimum average variance estimation method was given to estimate both the unknown parameters and link function. The asymptotic properties of the resulting estimators are established. Then, the serially correlated error structure is investigated. Furthermore, a generalized F-type test method was proposed to check the index parameters. It is shown that under the null hypothesis the proposed test statistic follows asymptotically a χ2-distribution. A numerical simulation and a real data example show that the proposed methods are effective.