Linear Covariance Models Research Articles

Monitoring the process mean under the Bayesian approach with application to hard bake process

This study introduces the Bayesian adaptive exponentially weighted moving average (AEWMA) control chart within the framework of measurement error, examining two separate loss functions: the squared error loss function and the linex loss function. We conduct an analysis of the posterior and posterior predictive distributions utilizing a conjugate prior. In the presence of measurement error (ME), we employ a linear covariate model to assess the control chart's effectiveness. Additionally, we explore the impacts of measurement error by investigating multiple measurements and a method involving linearly increasing variance. We conduct a Monte Carlo simulation study to assess the control chart's performance under ME, examining its run length profile. Subsequently, we offer a specific numerical instance related to the hard-bake process in semiconductor manufacturing, serving to verify the functionality and practical application of the suggested Bayesian AEWMA control chart when confronted with ME.

Logarithmic Voronoi cells for Gaussian models

We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML degree one and linear covariance models the two sets coincide. In particular, they are equal for both directed and undirected graphical models. We introduce decomposition theory of logarithmic Voronoi cells for the latter family. We also study covariance models, for which logarithmic Voronoi cells are, in general, strictly contained in log-normal spectrahedra. We give an explicit description of logarithmic Voronoi cells for the bivariate correlation model and show that they are semi-algebraic sets. Finally, we state a conjecture that logarithmic Voronoi cells for unrestricted correlation models are not semi-algebraic.

Performance of function-based AEWMA coefficient of variation control chart with measurement errors

A new coefficient of variation (CV) functionally adaptive exponentially weighted moving average control chart is presented by considering the influence of the measurement error (ME) and named the “FAEWMA-ME-CV” control chart. The linear covariate model and the multiple measurements method are used to present the control chart. The designed function is efficient to detect the infrequent process changes in the form of parametric shifts in the process CV. The proficiency of the design is evaluated by analyzing the run-length values which are determined by the Monte Carlo simulation runs shown for the various parametric settings in extensive tables. The analysis depicts that the presence of the ME plays a significant role in the performance of a control chart. Moreover, a real-life dataset explained the application of the proposed FAEWMA-ME-CV chart to support the argument.

Bayesian EWMA control chart with measurement error under different loss functions

AbstractThis paper presents the Bayesian EWMA control chart under two different loss functions (i) squared error loss function (SELF) and (ii) linex loss function (LLF) in the presence of measurement error (ME). We take posterior and posterior predictive distribution under the conjugate prior. We used a linear covariate model in the existence of ME to evaluate the control chart. We also studied the effects of multiple measurements and linear increasing variance methods in the existence of the ME. The average run length and standard deviation of run length are used to measure the performance of the Bayesian EWMA control chart with ME. We conducted the Monte Carlo simulation study to evaluate the performance of Bayesian EWMA control chart with ME. A real‐life data example is also presented to demonstrate the application of the control chart.

Performance of adaptive exponentially weighted moving average control chart in the presence of measurement error

A control chart is one of the important industrial tools used for monitoring the stability of manufacturing processes. The performance of the control charts can be affected in the presence of measurement error, which also leads to erroneous conclusions. In this article, we examined the effect of measurement error on an adaptive exponentially weighted moving average (AEWMA) control chart by using simple random sampling (SRS) and ranked set sampling (RSS) schemes. The linear covariate model is used to evaluate the control chart in the presence of measurement error. The case of multiple measurements and linear increasing variance is also investigated in the presence of measurement error. The average run length (ARL) and the standard deviation of run length (SDRL) are used as the performance measurement tool. Our results from the simulations and real data application demonstrated that measurement error affects the performance of control charts and multiple measurements are recommended to be used to avoid the effects of measurement error.

Estimating linear covariance models with numerical nonlinear algebra

Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g. Toeplitz, sparse, trees) that are of interest in statistics. We study the maximum likelihood degree and its dual analogue, and we introduce a new software package LinearCovarianceModels.jl for solving the score equations. All local maxima can thus be computed reliably. In addition we identify several scenarios for which the estimator is a rational function.

10-Year Longitudinal Changes in Fitness Parameters in Physiotherapy Students.

Introduction The aim of this study is to examine changes in the chosen morphological characteristics, motor conditioning, and coordination skills in physiotherapy undergraduates at a medical university in Bydgoszcz in the first decade of the millennium. We assume prevalence of a secular trend in values of morphological characteristics; however, characteristics of motor skills, particularly fitness levels, will remain relatively unchanged. Materials and Methods The study included young people studying physiotherapy at the medical university of Bydgoszcz 2001-2010. Basic morphological features, including waist and hip circumferences, were measured, and BMI and WHR were calculated. The Cooper test and marching over a distance of two kilometers were employed to determine endurance; right and left spinning coordination was assessed by Starosta test. Calculated Mollison indicators were used in the evaluation of dimorphic differences in subsequent observation time. Statistics. Data are presented as mean with the standard deviation. Groupwise comparison was performed with Student's t-test and linear relationships with Pearson's r. Linear covariance models (ANCOVA) were built as theoretical models. Statistical significance was set at alfa = 0.05. Based on the correlation between the data and the corresponding normal score used, the Shapiro-Wilk test is the best choice for testing the normality of data. Variables were excluded if they exceeded the tolerance level for multicollinearity. Analysis was performed with Statistica 10. Results The study demonstrated a rising number of male students (from 13,3% to 39,2%, chi2 = 10, 13; p = 0,001) and a decrease in age of students overall but no significant changes (from 22, 25 ± 0, 75 to 20, 42 ± 1, 7; F = 32, 9; p < 0,001) in their morphological characteristics and their dimorphic differences (average results for males: BMI, from 22, 25 ± 1, 94 to 26, 27 ± 3, 56; WHR index, from 0, 78 ± 0, 06 to 0, 85 ± 0, 06; average results for females: BMI, from 20, 79 ± 2, 11 to 22, 19 ± 3, 83; WHR index, from 0, 7 ± 0, 04 to 0, 75 ± 0, 15). An overall improvement in endurance was found; however, coordination, especially amongst women, had deteriorated (turn right, from 377,1 ± 48, 1 to 343,1 ± 23, 3; turn left, from 375,0 ± 5, 61 to 345,6 ± 43, 1). We observed multidirectional changes in the analyzed motor skills and most of them were statistically significant. Conclusions Longitudinal study of physiotherapy students revealed no differences in morphological and dimorphic characteristics and multimodal changes in assessed motor skills, thus highlighting a need for further research into the identification of their causes. Moreover, a requirement for entry motor skills assessment and curriculum reorientation towards delivery of a broader scope of physical education was suggested.

Performance of maximum EWMA control chart in the presence of measurement error using auxiliary information

EWMA and Max-EWMA charts are considered efficient for individual as well as joint monitoring of mean and variance shifts in the production process. However, measurement error is affecting the efficiency of these charts. In this study we propose a maximum exponentially weighted moving average with measurement error using auxiliary information control chart and name it Max-EWMAMEAI control chart. The efficiency of this chart is highlighted and the effect of measurement error is shown by the values of and calculated through simulations using linear covariate model. The case of linearly increasing variance is examined and multiple measurements technique has been applied to reduce the error effect. A real life example is also included to support the simulation results.

Parameter Estimation Effect of the Homogeneously Weighted Moving Average Chart to Monitor the Mean of Autocorrelated Observations With Measurement Errors

In statistical process monitoring, the usual assumption when designing monitoring schemes is that process parameters are known and have perfect measurements with independent and identically distributed observations. However, in real-life situation, these assumptions rarely hold. Hence, in this paper, the Phase II performance of the homogenously weighted moving average (HWMA) $\bar {X}$ monitoring scheme under the combined effect of autocorrelation and measurement errors is investigated when the unknown process parameters are estimated from an in-control Phase I dataset. Two models are considered, i.e. the first-order autoregressive model for within-sample autocorrelation and the linear covariate model for (constant and linearly increasing variance) measurement system error. Sampling strategies based on skipping some observations as well as mixing different subgroup samples and taking multiple measurements are implemented to reduce the negative effect of autocorrelation and measurement errors. Since the latter sampling strategies incur costs, as an alternative, increasing the slope coefficient of the linear covariate model compensate the negative effect of measurement errors. The new HWMA $\bar {X}$ scheme is shown to have some interesting detection abilities as compared to its competitors. A real-life example is used to illustrate the implementation of the proposed monitoring scheme.

Auxiliary Information Based Mixed EWMA–CUSUM Mean Control Chart with Measurement Error

Measurement error is a usually met distortion factor in real-world applications that influences the outcome of a process. The performance of the control charts can be affected in the presence of measurement error which also leads to erroneous conclusions regarding the average run length. In this paper, we proposed regression estimator-based mixed exponentially weighted moving average-cumulative sum (MxEMCUS) control chart to monitor the process mean and investigate the effect of measurement error on shift detection ability of the proposed control chart. The performance of MxEMCUS control chart is evaluated in the cases of mean shift with and without measurement error. The linear covariate model is used to evaluate the control chart in the presence of measurement error. The case of multiple measurements and linear increasing variance is also investigated in the presence of measurement error. The average run length and standard deviation of run length are used as the performance measuring tool. Based on simulation results, it is concluded that in the presence of measurement error, proposed chart has better performance as compared to other existing charts. An example with real data set is given to demonstrate the implementation of MxEMCUS control chart.

Estimation of covariance matrix of multivariate longitudinal data using modified Choleksky and hypersphere decompositions.

Linear models are typically used to analyze multivariate longitudinal data. With these models, estimating the covariance matrix is not easy because the covariance matrix should account for complex correlated structures: the correlation between responses at each time point, the correlation within separate responses over time, and the cross-correlation between different responses at different times. In addition, the estimated covariance matrix should satisfy the positive definiteness condition, and it may be heteroscedastic. However, in practice, the structure of the covariance matrix is assumed to be homoscedastic and highly parsimonious, such as exchangeable or autoregressive with order one. These assumptions are too strong and result in inefficient estimates of the effects of covariates. Several studies have been conducted to solve these restrictions using modified Cholesky decomposition (MCD) and linear covariance models. However, modeling the correlation between responses at each time point is not easy because there is no natural ordering of the responses. In this paper, we use MCD and hypersphere decomposition to model the complex correlation structures for multivariate longitudinal data. We observe that the estimated covariance matrix using the decompositions is positive-definite and can be heteroscedastic and that it is also interpretable. The proposed methods are illustrated using data from a nonalcoholic fatty liver disease study.

A Non-Linear Model for Censored and Mismeasured Time Varying Covariates in Survival Models, with Applications in Human Immunodeficiency Virus and Acquired Immune Deficiency Syndrome Studies

Summary In survival regression analysis, when the time-dependent covariates are censored and measured with errors, a joint model is often considered for the longitudinal covariate data and the survival data. Typically, an empirical linear (mixed) model is assumed for the time-dependent covariates. However, such an empirical linear covariate model may be inappropriate for the (unobserved) censored covariate values that may behave quite differently from the observed covariate process. In applications such as human immunodeficiency virus–acquired immune deficiency syndrome studies, a mechanistic non-linear model can be derived for the covariate process on the basis of the underlying data generation mechanisms and such a non-linear covariate model may provide better ‘predictions’ for the censored and mismeasured covariate values. We propose a joint Cox and non-linear mixed effect model to model survival data with censored and mismeasured time varying covariates. We use likelihood methods for inference, implemented by the Monte Carlo EM algorithm. The models and methods are evaluated by simulations. An acquired immune deficiency syndrome data set is analysed in detail, where the time-dependent covariate is a viral load which may be censored because of a lower detection limit and may also be measured with errors. The results based on linear and non-linear covariate models are compared and new insights are gained.

Modeling the Cholesky factors of covariance matrices of multivariate longitudinal data.

Modeling the covariance matrix of multivariate longitudinal data is more challenging as compared to its univariate counterpart due to the presence of correlations among multiple responses. The modified Cholesky block decomposition reduces the task of covariance modeling into parsimonious modeling of its two matrix factors: the regression coefficient matrices and the innovation covariance matrices. These parameters are statistically interpretable, however ensuring positive-definiteness of several (innovation) covariance matrices presents itself as a new challenge. We address this problem using a subclass of Anderson's (1973) linear covariance models and model several covariance matrices using linear combinations of known positive-definite basis matrices with unknown non-negative scalar coefficients. A novelty of this approach is that positive-definiteness is guaranteed by construction; it removes a drawback of Anderson's model and hence makes linear covariance models more realistic and viable in practice. Maximum likelihood estimates are computed using a simple iterative majorization-minimization algorithm. The estimators are shown to be asymptotically normal and consistent. Simulation and a data example illustrate the applicability of the proposed method in providing good models for the covariance structure of a multivariate longitudinal data.

Genome-wide investigation of mRNA lifetime determinants in Escherichia coli cells cultured at different growth rates.

BackgroundChanges to mRNA lifetime adjust mRNA concentration, facilitating the adaptation of growth rate to changes in growth conditions. However, the mechanisms regulating mRNA lifetime are poorly understood at the genome-wide scale and have not been investigated in bacteria growing at different rates.ResultsWe used linear covariance models and the best model selected according to the Akaike information criterion to identify and rank intrinsic and extrinsic general transcript parameters correlated with mRNA lifetime, using mRNA half-life datasets for E. coli, obtained at four growth rates. The principal parameter correlated with mRNA stability was mRNA concentration, the mRNAs most concentrated in the cells being the least stable. However, sequence-related features (codon adaptation index (CAI), ORF length, GC content, polycistronic mRNA), gene function and essentiality also affected mRNA lifetime at all growth rates. We also identified sequence motifs within the 5′UTRs potentially related to mRNA stability. Growth rate-dependent effects were confined to particular functional categories (e.g. carbohydrate and nucleotide metabolism). Finally, mRNA stability was less strongly correlated with the amount of protein produced than mRNA concentration and CAI.ConclusionsThis study provides the most complete genome-wide analysis to date of the general factors correlated with mRNA lifetime in E. coli. We have generalized for the entire population of transcripts or excluded determinants previously defined as regulators of stability for some particular mRNAs and identified new, unexpected general indicators. These results will pave the way for discussions of the underlying mechanisms and their interaction with the growth physiology of bacteria.Electronic supplementary materialThe online version of this article (doi:10.1186/s12864-015-1482-8) contains supplementary material, which is available to authorized users.

The Effect of Measurement Error on X̃-R̃ Fuzzy Control Charts

Control charts are tools used for monitoring manufacturing processes. Fuzzy Set Theory has found its way in control charts and new types of fuzzy control charts, with different capabilities, has been introduced. In this paper, a process in which the result of the measuring of each piece is imprecise is studied, and a X̃-R̃ fuzzy control chart is used for monitoring. The aim is to study the effect of measurement error on the effectiveness of the fuzzy control chart to detect out of control situations. The model used in this research is a linear covariate model. ARL parameters are used to study the performance of the fuzzy control chart when the parameters of covariate model is increased or decreased.

Design of pharmacokinetic studies for latent covariates

Latent covariates are covariates that are known to exist but are either observable but unavailable or unobservable at the time of the clinical study. Designs to account for latent covariates must incorporate both uncertainty in the prevalence of the covariate and the data-type of the covariate. The informativeness of the covariate will then depend on whether the covariate data is continuous, ordinal or nominal. In this work we consider designs for latent covariates that may either directly influence the parameter of interest, or indirectly via actions on an observable covariate which then influences the parameter of interest. We consider a motivating example based on the effect of a genetic polymorphism on the influence of a continuous covariate (age) on drug clearance (CL). The polymorphism could take the case of a haplotype with many variant alleles, or a copy number variation in genes with different phenotypic expressions which could be treated as continuous data, or as a bi- or tri-allelic single nucleotide polymorphism that could form either an ordinal or nominal covariate on drug CL. The aim of this study was to investigate designs for clinical studies for latent covariates that accommodate both unknown prevalence and unknown data-type. Initially, the informativeness of a covariate was explored using linear regression assuming the three data-types continuous, ordinal and nominal. The linear covariate model was then considered within a nonlinear mixed effects modelling framework. Two simulation scenarios were considered: (1) the influence of the latent covariate directly on the parameter of interest and (2) the influence of the latent covariate on an observable non-latent continuous covariate, which was assumed to follow a normal or stratified distribution, and the effect of this covariate on the parameter of interest. A power analysis for population PK modelling (1) where the latent covariate had direct influence on the parameter also showed similar behaviour to the linear regression solution. When the influence of the latent covariate was mediated via another observable non-latent continuous covariate, the power for the continuous model was highest but the power of the ordinal model was indistinguishable from that of the nominal model. Stratification of the observable non-latent continuous covariate did not appreciably change the power to identify the latent covariate from that when we assumed the observable covariate conformed to a normal distribution. It was found that parameter estimation is generally at least 1.5 to 7 fold more precise for continuous models than for categorical models.

Linear covariance models to examine the determinants of protein levels in Lactococcus lactis

When confronted to environmental changes, microorganisms adjust protein levels in order to adapt their growth and metabolic performances. Biological mechanisms involved in protein regulation are extremely complex and still poorly understood. This study aims at the identification, by statistical modelling, of the major determinants of protein concentrations in a bacterial model Lactococcus lactis. Protein concentrations were predicted by covariance models taking into account various quantitative and qualitative parameters. Best models were selected thanks to Akaïke Information Criterion. For protein estimation, we found that the sequence-related feature Codon Adaptative Index was a more influential parameter than the transcript amount, suggesting the control by genetic determinism is stronger than by metabolic adaptation. In addition, protein length, aromaticity but also their biological functions, were proved to have unexpected influences on protein concentrations. These protein determinants were for the first time demonstrated to be not constant and depended on the adaptation process, the main difference between permanent and transient adaptations being detected for regulatory protein concentrations. With the growing accumulation of omics data this statistical method appears to be a valuable tool to understand biological networks and their regulations. This approach was applied to study the translation of proteins but can be extended to other metabolic processes and is also adaptable to other microorganisms.

Genetic Models for Carcass Traits with Different Slaughter Endpoints in Selected Hanwoo Herds I. Linear Covariance Models

Carcass characteristics data of Hanwoo (N = 1,084) were collected from two stations of the National Livestock Institute of Animal Science (NIAS), Korea and records from thirteen individual cow-calf operators were analyzed to estimate variance and covariance components and the effect of different slaughter endpoints. Carcass traits analyzed were cold carcass weight (CWT, kg), REA (rib eye area, cm 2 ), back fat thickness (mm) and marbling score (1-7). Four different models were examined. All models included sex and contemporary group as fixed effects and the animal's direct genetic potential and environment as random effects. The first model fitted a linear covariate of age at slaughter. The second model fitted both linear and quadratic covariates of age at slaughter. The third model fitted a linear covariate of body weight at slaughter. The fourth model fitted both linear covariates of age at slaughter and body weight at slaughter. Variance components were estimated using the REML procedure with Gibb's sampler. Heritability estimate of CWT was in the range of 0.08-0.11 depending on the model applied. Heritability estimates of BF, REA and MS were in the ranges of 0.23-0.28, 0.19-0.26, and 0.44-0.45, respectively. Genetic correlations between CWT and BF, between CWT and REA, and between CWT and MS were in the ranges of -0.33 - -0.14, 0.73-0.84, and -0.01-0.11, respectively. Genetic correlations between REA and BF, between MS and BF and between REA and MS were in the ranges of -0.82 ∼ -0.72, 0.04-0.28 and -0.08 ∼ -0.02, respectively. Variance and covariance components estimated varied by model with different slaughter endpoints. Body weight endpoint was more effective for direct selection in favor of yield traits and body weight endpoints affected more of the correlated response to selection for the traits of yield and quality of edible portion of beef.

A Linear Covariate Model of Warranty Attitudes and Behaviors

The new Magnuson-Moss Warranty and Federal Trade Commission Improvement Act requires businesses to reappraise their current warranty practices. They now face standards for what a warranty must contain and how easy it is to read. The authors test a model relating consumer socioeconomic characteristics and attitudes to warranty behaviors. The methodological and substantive implications of the findings are explored.

A Linear Covariate Model of Warranty Attitudes and Behaviors

A Linear Covariate Model of Warranty Attitudes and Behaviors