Variance Matrices Research Articles

X-ray scattering processes and chemometrics for differentiating complex samples using conventional EDXRF equipment

Mild variations in organic matrices, which are investigated in this work, are caused by alterations in X-ray Raman scattering. The multivariate approaches, principal component analysis (PCA) and hierarchical cluster analysis (HCA), are applied to visualize these effects. Conventional energy-dispersive X-ray fluorescence equipment is used, where organic compounds produce intense scattering of the X-ray source. X-ray Raman processes, before obtained only for solid samples using synchrotron radiation, are indirectly visualized here through PCA scores and HCA cluster analysis, since they alter the Compton and Rayleigh scattering. As a result, their influences can be seen in known sample characteristics, as those associated with gender and melanin in dog hairs, and the differentiation in coconut varieties. Chemometrics has shown that, despite their complexity, natural samples can be easily classified.

Estimation of genetic parameters for test day records of dairy traits in the first three lactations

Application of test-day models for the genetic evaluation of dairy populations requires the solution of large mixed model equations. The size of the (co)variance matrices required with such models can be reduced through the use of its first eigenvectors. Here, the first two eigenvectors of (co)variance matrices estimated for dairy traits in first lactation were used as covariables to jointly estimate genetic parameters of the first three lactations. These eigenvectors appear to be similar across traits and have a biological interpretation, one being related to the level of production and the other to persistency. Furthermore, they explain more than 95% of the total genetic variation. Variances and heritabilities obtained with this model were consistent with previous studies. High correlations were found among production levels in different lactations. Persistency measures were less correlated. Genetic correlations between second and third lactations were close to one, indicating that these can be considered as the same trait. Genetic correlations within lactation were high except between extreme parts of the lactation. This study shows that the use of eigenvectors can reduce the rank of (co)variance matrices for the test-day model and can provide consistent genetic parameters.

The effect of temperature and wing morphology on quantitative genetic variation in the cricket Gryllus firmus, with an appendix examining the statistical properties of the Jackknife–manova method of matrix comparison

We investigated the effect of temperature and wing morphology on the quantitative genetic variances and covariances of five size-related traits in the sand cricket, Gryllus firmus. Micropterous and macropterous crickets were reared in the laboratory at 24, 28 and 32 degrees C. Quantitative genetic parameters were estimated using a nested full-sib family design, and (co)variance matrices were compared using the T method, Flury hierarchy and Jackknife-manova method. The results revealed that the mean phenotypic value of each trait varied significantly among temperatures and wing morphs, but temperature reaction norms were not similar across all traits. Micropterous individuals were always smaller than macropterous individuals while expressing more phenotypic variation, a finding discussed in terms of canalization and life-history trade-offs. We observed little variation between the matrices of among-family (co)variation corresponding to each combination of temperature and wing morphology, with only one matrix of six differing in structure from the others. The implications of this result are discussed with respect to the prediction of evolutionary trajectories.

On the difference between ML and REML estimators in the modelling of multivariate longitudinal data

A random effects model is examined in the multivariate setting where more than one characteristics are measured at each time point. ML and REML estimators are obtained under the restriction that estimates of variance matrices being at least p.s.d. It is shown that REML has greater probability of giving full rank estimates of variance components matrices but as regards the efficiency in the estimation of the location parameter, correct specification of the number of random effects is needed. In general, REML provides larger estimates of variance of model parameters than ML.

Analysis of beam propagation and pulse propagation with variance matrix treatment

Applying the analogies between beam propagation and pulse propagation, so-called space and time variance matrices are presented, respectively. Straightforwardly, characteristics of paraxial light beam propagation and pulse propagation are described in space and time variance matrices, respectively.

Optimization of Linear Multivariable Systems with Structured Perturbations and Prescribed Closed-Loop Eigenvalues

In this article the conventional linear quadratic regulator (LQR) problem is modified to include a third term penalizing state trajectory deviations. The modification is motivated by the tendency of system parameters to assume different values rather than their nominal ones due to components aging, number truncation, and other factors. The proposed optimal controller not only results in an explicit solution to the modified LQR problem, but also allows the designer to assign the closed-loop eigenvalues to prespecified locations. The proposed controller permits the designer to consider undesired variations in both system matrices (A, B) and also to assign the closed-loop eigenvalues to some desired locations. This eigenvalues assignment is achieved without imposing any restriction either on their nature, multiplicity, or their open- or closed-loop locations in the complex plane.

Variance Estimation for the Instrumental Variables Approach to Measurement Error in Generalized Linear Models

This paper derives and gives explicit formulas for a derived sandwich variance estimate. This variance estimate is appropriate for generalized linear additive measurement error models fitted using instrumental variables. We also generalize the known results for linear regression. As such, this article explains the theoretical justification for the sandwich estimate of variance utilized in the software for measurement error developed under the Small Business Innovation Research Grant (SBIR) by StataCorp. The results admit estimation of variance matrices for measurement error models where there is an instrument for the unknown covariate.

Amorphous Calcium Phosphate-Based Bioactive Polymeric Composites for Mineralized Tissue Regeneration.

Amorphous calcium phosphate (ACP), a postulated precursor in the formation of biological hydroxyapatite, has been evaluated as a filler phase in bioactive polymeric composites that utilize dental monomers to form the matrix phase on polymerization. In addition to excellent biocompatibility, these composites provided sustained release of calcium and phosphate ions into simulated saliva milieus. In an effort to enhance the physicochemical and mechanical properties and extend the utility of remineralizing ACP composites to a greater variety of dental applications, we have focused on: a) hybridizing ACP by introducing silica and/or zirconia, b) assessing the efficacy of potential coupling agents, c) investigating the effects of chemical structure and compositional variation of the resin matrices on the mechanical strength and ion-releasing properties of the composites, and d) improving the intrinsic adhesiveness of composites by using bifunctional monomers with an affinity for tooth structure in resin formulations. Si- and Zr-modified ACPs along with several monomer systems are found useful in formulating composites with improved mechanical and remineralizing properties. Structure-property studies have proven helpful in advancing our understanding of the remineralizing behavior of these bioactive composites. It is expected that this knowledge base will direct future research and lead to clinically valuable products, especially therapeutic materials appropriate for the healing or even regeneration of defective teeth and bone structures.

Norm bounds for summation of two normal matrices

A sharp upper bound is obtained for ∥ A+i B∥, where A and B are n× n Hermitian matrices satisfying a 1 I⩽ A⩽ a 2 I and b 1 I⩽ B⩽ b 2 I. Similarly, an optimal bound is obtained for ∥ U+ V∥, where U and V are n× n unitary matrices with any specified spectra; the study leads to some surprising phenomena of discontinuity concerning the spectral variation of unitary matrices. Moreover, it is proven that for two (non-commuting) normal matrices A and B with spectra σ( A) and σ( B), the optimal norm bound for A+ B equals min λ∈ C max α∈σ(A)|α+λ|+ max β∈σ(B)|β−λ| . Extensionsof the results to infinite dimensional cases are also considered.

Comparing alternative random regression models to analyse first lactation daily milk yield data in Holstein–Friesian cattle

Test-day (TD) milk yields from Spanish Holstein cows were analysed in three independent data sets (35 615, 35 209 and 27 272 TD records, respectively) with a set of random regression models. Wilmink and Ali-Schaeffer lactation functions and, Legendre polynomials (RRL) of varying order (up to six coefficients) on additive genetic (AG) and permanent environmental (PE) effects were used. The analysis of the eigenvalues and eigenvectors of the AG and PE random regression (co)variance matrices revealed the possibility of reducing the dimension of the RRL submodels, particularly for the AG effects. Lactational submodels provided the largest daily AG variance estimates at the onset of lactation, as well as low or even negative genetic correlations between peripheral TD. Polynomials of higher order (four or above) showed oscillatory patterns with larger variances and lower genetic correlations predicted for the extremes of lactation. Model performance was assessed using a broad range of criteria. The results showed a strong consistency among data sets in terms of models ranking. Lactational models showed a worse performance than the RRL models with the same number of parameters. For RRL models, all criteria except the Bayesian information criterion, favoured the most complex model. This criterion selected a model with 2–3 coefficients for the AG effects and 5–6 coefficients for the PE effects.

A Simple Method for Weighted Bending of Genetic (Co)variance Matrices

Due to computational demand, elements of genetic correlation matrices may have been estimated separately and then combined together into a single correlation matrix at a later stage. Because these matrices should be positive definite (PD) a statistical method commonly known as “bending” is used to make them positive definite. The conventional bending method ignores the reliability of different correlations and may subject any of them to change in order to make a positive definite correlation matrix. A simple method to obtain a weighted bended matrix to be used in animal breeding applications is proposed, and the implementation of the method is demonstrated by an example.

Constraints on Random Effects and Mixed Linear Model Predictions

In Mixed Linear Models theory one is assumed to know the structure of random effects covariance matrix. The suggestions are sometimes contradictious, especially if the model includes interactions between fixed effects and random effects. This paper presents conditions under which two different random effects' variance matrices will yield equal estimation and prediction results. Some examples and simulation results are given also.

Seasonal and spatial variations of cross-correlation matrices used by stochastic weather generators

We examine seasonal and spatial variations of stochastic-weather-generator (SWG) parameters and their impact on simulated weather sequences. Using daily weather observations from 29 stations across the contiguous United States, we estimate monthly station-specific parameters that are compared with the constant parameters that frequently are used in SWG applications. A WGEN- type SWG is then used to generate a 100 yr record of daily maximum and minimum air temperature and daily total solar radiation at each station. These sequences are compared to sequences generated with constant parameters. While the means and standard deviations of the generated sequences are in agreement, the SWG with station-specific parameters preserves relationships between variables. This is evident in both the lag-0 and lag-1 cross-correlations between generated variables and derived variables, such as diurnal temperature range. These results suggest that literature-based SWG parameters may be appropriate for applications where monthly values of the means and stan- dard deviations of generated variables are of interest. For applications that require proper simulation of relationships between variables, station-specific parameterizations are recommended.

Variance Estimation for the Instrumental Variables Approach to Measurement Error in Generalized Linear Models

T his paper derives and gives explicit formulas for a derived sandwich variance estimate. This variance estimate is appropriate for generalized linear additive measurement error models “tted using instrumental variables. We also generalize the known results for linear regression. As such, this article explains the theoretical justi“cation for the sandwich estimate of variance utilized in the software for measurement error developed under the Small Business Innovation Research Grant (SBIR) by StataCorp. The results admit estimation of variance matrices for measurement error models where there is an instrument for the un- known covariate.

Population divergence of genetic (co)variance matrices in a subdivided plant species, Brassica cretica

Abstract The present study of Brassica cretica had two objectives. First, we compared estimates of population structure (Qst) for seven phenotypic characters with the corresponding measures for allozyme markers (Fst) to evaluate the supposition that genetic drift is a major determinant of the evolutionary history of this species. Secondly, we compared the genetic (co)variance (G) matrices of five populations to examine whether a long history of population isolation is associated with large, consistent differences in the genetic (co)variance structure. Differences between estimates of Fst and Qst were too small to be declared significant, indicating that stochastic processes have played a major role in the structuring of quantitative variation in this species. Comparison of populations using the common principal component (CPC) method rejected the hypothesis that the G matrices differed by a simple constant of proportionality: most of the variation involved principal component structure rather than the eigenvalues. However, there was strong evidence for proportionality in comparisons using the method of percentage reduction in mean-square error (MSE), at least when characters with unusually high (co)variance estimates were included in the analyses. Although the CPC and MSE methods provide different, but complementary, views of G matrix variation, we urge caution in the use of proportionality as an indicator of whether genetic drift is responsible for divergence in the G matrix.

Optimal design of nonlinear experiments with parameter constraints

The nonlinear regression model with N observations yi=η(xi,θ) +ei, and with the parameter θ subject to q nonlinear constraints Cj (θ)=0; j=1, …,q, is considered. As an example, the spline regression with unknown nodes is taken. Expressions for the variances (variance matrices) of the LSE are discussed. Because of the complexity of these expressions, and the singularity of the variance matrix of the LSE for θ, the optimality criteria and their properties, in particular the convexity and the equivalence theorem are considered from different aspects. Also the possibility of restriction to designs with limited values of measures of nonlinearity is mentioned.

Spatio‐temporal variation in Markov chain models of subtidal community succession

AbstractIn this paper we ask whether succession in a rocky subtidal community varies in space and time, and if so how much affect that variation has on predictions of community dynamics and structure. We describe succession by Markov chain models based on observed frequencies of species replacements. We use loglinear analysis to detect and quantify spatio‐temporal variation in the transition matrices describing succession. The analysis shows that space and time, but not their interaction, have highly significant effects on transition probabilities. To explore the ecological importance of the spatio‐temporal variability detected in this analysis, we compare the equilibria and the transient dynamics among three Markov chain models: a time‐averaged model that includes the effects of space on succession, a spatially averaged model that include the effects of time, and a constant matrix that averages over the effects of space and time. All three models predicted similar equilibrium composition and similar rates of convergence to equilibrium, as measured by the damping ratio or the subdominant Lyapunov exponent. The predicted equilibria from all three models were very similar to the observed community structure. Thus, although spatial and temporal variation is statistically significant, at least in this system this variation does not prevent homogeneous models from predicting community structure.

Counterexample to a conjecture of Elsner on the spectral variation of matrices

In 1985, Elsner proved that the Hausdorff distance Δ between the spectra of two n× n matrices A and B satisfies Δ σ(A),σ(B) n⩽ ∥A∥+∥B∥ n−1∥A−B∥, where ∥·∥ denotes the operator norm with respect to the Euclidean norm on C n . He further conjectured that the same inequality holds for all operator norms. We disprove this conjecture, and also the weaker conjecture where (∥ A∥+∥ B∥) is replaced by 2max(∥ A∥,∥ B∥).

A Comparison of an Analytical Approach and a Standard Simulation Approach in Bayesian Forecasting Applied to Monthly Data from Insurance of Companies

In this paper, we apply the theory of Bayesian forecasting and dynamic linear models, as presented in West and Harrison (1997), to monthly data from insurance of companies. The total number reported claims of compensation is chosen to be the primary time series of interest. The model is decomposed into a trend block, a seasonal effects block and a regression block with a transformed number of policies as regressor. An essential part of the West and Harrison (1997) approach is to find optimal discount factors for each block and hence avoid the specification of the variance matrices of the error terms in the system equations. The BATS package of Pole et al. (1994) is applied in the analysis. We compare predictions based on this analytical approach with predictions based on a standard simulation approach applying the BUGS package of Spiegelhalter et al. (1995). The motivation for this comparison is to gain knowledge on the quality of predictions based on more or less standard simulation techniques in other applications where an analytical approach is impossible. The predicted values of the two approaches are very similar. The uncertainties, however, in the predictions based on the simulation approach are far larger especially two months or more ahead. This partly indicates the advantages of applying optimal discount factors and partly the disadvantages of at least a standard simulation approach for long term predictions.

Nitrite and nitrate measurement in human urine by capillary electrophoresis.

Nitrite and nitrate have been widely used as markers for nitric oxide (NO) formation in vivo and represent the major NO oxidation products in biological fluids. In the present study, the use of capillary electrophoresis (CE) in the measurement of nitrite and nitrate in human urine is described. Urine samples were electrophoresed in an extended light path fused-silica capillary (104 cm; 75 microm ID) at an applied negative potential of 30 kV, and UV detection at 214 nm. Using electrokinetic sample injection (-6 kV x 20 s), we found that urine concentration, pH, sodium and chloride interfered with nitrite and nitrate detection. The detection of nitrite and nitrate was decreased when hydrodynamic sample injection was used (30 mbar x 60 s). However, basal levels of urinary nitrite (0.25 +/- 0.05 microM) and nitrate (591 +/- 115 microM) were detected and no interference by variations in urine concentration and pH was noted when hydrodynamic sample injection was used. Thus, hydrodynamic sample injection is convenient for the measurement of urinary nitrite and nitrate and avoids the effect of variations in urine matrices and pH on nitrite and nitrate detection.