We propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data. The method aims at irregularly spaced longitudinal data, where the number of repeated measurements available per subject is small. In contrast, classical functional data analysis requires a large number of regularly spaced measurements per subject. We assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random (subject-specific) trajectory plus measurement errors. Basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories, and the estimation of the variance of the measurement errors. The eigenfunction basis is estimated from the data, and functional principal components score estimates are obtained by a conditioning step. This conditional estimation method is conceptually simple and straightforward to implement...

/pdf/functional-data-analysis-for-sparse-longitudinal-data-5ajkgdiw77.pdf

Functional Data Analysis for Sparse Longitudinal Data

https://faculty.missouri.edu/~glaserr/3700s14/Antioxidant_Carotenoids_2005.pdf

Carotenoid actions and their relation to health and disease.

Antioxidant activity has been assessed in many ways. The limitation of many newer methods is the frequent lack of an actual substrate in the procedure. The combination of all approaches with the many test methods available explains the large variety of ways in which results of antioxidant testing are reported. The measurement of antioxidant activities, especially of antioxidants that are mixtures, multifunctional or are acting in complex multiphase systems, cannot be evaluated satisfactorily by a simple antioxidant test without due regard to the many variables influencing the results. Several test procedures may be required to evaluate such antioxidant activities. A general method of reporting antioxidant activity independent of the test procedure is proposed.

http://ucce.ucdavis.edu/files/datastore/608-46.pdf

Methods for testing antioxidant activity

Carotenoids are natural pigments which are synthesized by plants and are responsible for the bright colors of various fruits and vegetables. There are several dozen carotenoids in the foods that we eat, and most of these carotenoids have antioxidant activity. β-carotene has been best studied since, in most countries it is the most common carotenoid in fruits and vegetables. However, in the U.S., lycopene from tomatoes now is consumed in approximately the same amount as β-carotene. Antioxidants (including carotenoids) have been studied for their ability to prevent chronic disease. β-carotene and others carotenoids have antioxidant properties in vitro and in animal models. Mixtures of carotenoids or associations with others antioxidants (e.g. vitamin E) can increase their activity against free radicals. The use of animals models for studying carotenoids is limited since most of the animals do not absorb or metabolize carotenoids similarly to humans.Epidemiologic studies have shown an inverse relationship be...

Beta-carotene and other carotenoids as antioxidants.

Carotenoids form one of the most important classes of plant pigments and play a crucial role in defining the quality parameters of fruit and vegetables. Their role in the plant is to act as accessory pigments for light harvesting and in the prevention of photo-oxidative damage, as well as acting as attractants for pollinators. Their function as antioxidants in the plant shows interesting parallels with their potential role as antioxidants in foods and humans. Carotenoids are products of the isoprenoid biosynthetic pathway. The enzymes leading to carotenoid biosynthesis have all been characterised, and more recently the genes encoding these enzymes have been cloned from bacteria, fungi and plants. New information on enzyme activities and the factors leading to the regulation of the pathway is reviewed. Vitamin A deficiency is a widespread problem in the developing world, causing blindness, particularly in the young. This has driven research into finding ways of introducing provitamin A carotenoids into staple crops, and this has recently been achieved in rice and canola through genetic manipulation. The fact that carotenoids show protective activity in vitro and in vivo against a variety of degenerative disease end points has also give impetus to studying whether increasing intakes of the commonly consumed carotenoids would have public health benefits in the developed world. Human intervention studies have been undertaken using supplements of β-carotene rather than utilising foods with enhanced carotenoid levels, but no potential benefit has been shown. Indeed, there is evidence of an increased health risk from the consumption of β-carotene supplements. These observations suggest that the threshold between the beneficial and adverse effects of some carotenoids is low and provides a strong stimulus to further understanding the functional effects of specific carotenoids. Specific needs for future research are identified in the review.

© 2000 Society of Chemical Industry

The potential for the improvement of carotenoid levels in foods and the likely systemic effects

We present the application of a nonparametric method to performing functional principal component analysis for functional curve data that consist of measurements of a random trajectory for a sample of subjects. This design typically consists of an irregular grid of time points on which repeated measurements are taken for a number of subjects. We introduce shrinkage estimates for the functional principal component scores that serve as the random effects in the model. Scatterplot smoothing methods are used to estimate the mean function and covariance surface of this model. We propose improved estimation in the neighborhood of and at the diagonal of the covariance surface, where the measurement errors are reflected. The presence of additive measurement errors motivates shrinkage estimates for the functional principal component scores. Shrinkage estimates are developed through best linear prediction and in a generalized version, aiming at minimizing one-curve-leave-out prediction error. The estimation of individual trajectories combines data obtained from that individual as well as all other individuals. We apply our methods to new data regarding the analysis of the level of 14C-folate in plasma as a function of time since dosing of healthy adults with a small tracer dose of 14C-folic acid. A time transformation was incorporated to handle design irregularity concerning the time points on which the measurements were taken. The proposed methodology, incorporating shrinkage and data-adaptive features, is seen to be well suited for describing population kinetics of 14C-folate-specific activity and random effects, and can also be applied to other functional data analysis problems.

/pdf/shrinkage-estimation-for-functional-principal-component-57wkrycc5c.pdf

Shrinkage Estimation for Functional Principal Component Scores with Application to the Population Kinetics of Plasma Folate

Estimating the concentration of beta-carotene required for maximal protection of low-density lipoproteins in women

This noninvasive method for collection and analysis of a wide range of aldehydes and ketones in human breath may enable assessment of lipid peroxidation and metabolic status in vivo. Breath samples are drawn through silica cartridges impregnated with 2,4-dinitrophenylhydrazine, which traps carbonyls as their hydrazone derivatives. The hydrazone derivatives are eluted from the cartridges with acetonitrile, separated by reversed-phase HPLC, and quantified spectrophotometrically. Using this method, we have measured formaldehyde, acetaldehyde, acetone, propanal, 2-butanone, butanal, pentanal, and hexanal. Recoveries of carbonyls added to Douglas bags were > 90%, except for 2-butanone, which was 86.2%. The overall CVs for sampling plus analyzing duplicate aliquots of breath were < 11%. The results indicate that this protocol can be used to monitor changes of carbonyl production by analyzing expired air, which may, with further study, indicate physiological and pathological status.

Protocol for collection and HPLC analysis of volatile carbonyl compounds in breath.

Solid-phase extraction permits the parallel processing of samples in large numbers. We have applied this technique to the isolation of retinol isotopomers from plasma of humans participating in a study of vitamin A stable isotope dilution. The isotopomers were analyzed by gas chromatography/mass spectrometry. The extraction involves the separation of retinol from its aqueous matrix with a C18 silica-based sorbent followed by removal of lipid contaminants with an aminopropyl silica-based sorbent. Overall recovery of retinol from plasma was 47.2% +/- 1.8%. Purity of the retinol isolated from plasma is comparable with that obtained with a single HPLC method. This method permits the preparation of 32 samples per day by one analyst. Elimination of the need for HPLC permits sample preparation in the field with a minimum of equipment and technical skill.

S. R. Dueker

Papers

Shrinkage Estimation for Functional Principal Component Scores with Application to the Population Kinetics of Plasma Folate

Estimating the concentration of beta-carotene required for maximal protection of low-density lipoproteins in women

Protocol for collection and HPLC analysis of volatile carbonyl compounds in breath.

Solid-phase extraction protocol for isolating retinol-d4 and retinol from plasma for parallel processing for epidemiological studies.

Solid-PhaseExtractionProtocolfor IsolatingRetinol-d4 and RetinolfromPlasmafor ParallelProcessing for Epidemiological Studies