Robust estimation using the Huber function with a data-dependent tuning constant

Robust estimation often relies on a dispersion function that is more slowly varying at large values than the square function. However, the choice of tuning constant in dispersion functions may impact the estimation efficiency to a great extent. For a given family of dispersion functions such as the Huber family, we suggest obtaining the "best" tuning constant from the data so that the asymptotic efficiency is maximized. This data-driven approach can automatically adjust the value of the tuning constant to provide the necessary resistance against outliers. Simulation studies show that substantial efficiency can be gained by this data-dependent approach compared with the traditional approach in which the tuning constant is fixed. We briefly illustrate the proposed method using two datasets.

Iterative estimating equations: Linear convergence and asymptotic properties

We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size increases to infinity. Furthermore, we show that the limiting estimator is consistent and asymptotically efficient, as expected. The method applies to semiparametric regression models with unspecified covariances among the observations. In the special case of linear models, the procedure reduces to iterative reweighted least squares. Finite sample performance of the procedure is studied by simulations, and compared with other methods. A numerical example from a medical study is considered to illustrate the application of the method.

Unbiased estimating equations from working correlation models for irregularly timed repeated measures

The method of generalized estimating equation-, (GEEs) has been criticized recently for a failure to protect against misspecification of working correlation models, which in some cases leads to loss of efficiency or infeasibility of solutions. However, the feasibility and efficiency of GEE methods can be enhanced considerably by using flexible families of working correlation models. We propose two ways of constructing unbiased estimating equations from general correlation models for irregularly timed repeated measures to supplement and enhance GEE. The supplementary estimating equations are obtained by differentiation of the Cholesky decomposition of the working correlation, or as score equations for decoupled Gaussian pseudolikelihood. The estimating equations are solved with computational effort equivalent to that required for a first-order GEE. Full details and analytic expressions are developed for a generalized Markovian model that was evaluated through simulation. Large-sample ".sandwich" standard errors for working correlation parameter estimates are derived and shown to have good performance. The proposed estimating functions are further illustrated in an analysis of repeated measures of pulmonary function in children.

Working correlation structure misspecification, estimation and covariate design: Implications for generalised estimating equations performance

The method of generalised estimating equations for regression modelling of clustered outcomes allows for specification of a working matrix that is intended to approximate the true correlation matrix of the observations. We investigate the asymptotic relative efficiency of the generalised estimating equation for the mean parameters when the correlation parameters are estimated by various methods. The asymptotic relative efficiency depends on three-features of the analysis, namely (i) the discrepancy between the working correlation structure and the unobservable true correlation structure, (ii) the method by which the correlation parameters are estimated and (iii) the 'design', by which we refer to both the structures of the predictor matrices within clusters and distribution of cluster sizes. Analytical and numerical studies of realistic data-analysis scenarios show that choice of working covariance model has a substantial impact on regression estimator efficiency. Protection against avoidable loss of efficiency associated with covariance misspecification is obtained when a 'Gaussian estimation' pseudolikelihood procedure is used with an AR(1) structure.

Assignment of fundamental vibrations of. ,N'-dimethylthiourea,

The infrared spectra of symmetric N,N′-dimethylthiourea (s-DMTU) and its N-deuterated (s-DMTU-d2) species have been measured. The fundamental frequencies have been assigned by comparison with the assignments in structurally related molecules and the infrared band shifts on N-deuteration, S-methylation, available Raman data and with the aid of theoretical band assignments from normal coordinate treatments for s-DMTU-d0 and -d2. A force field is derived for s-DMTU by transferring the force constants chiefly from N-methylthiourea and the subsequent refinement of the force constants by a least squares procedure.

Approximate solutions of Fredholm integral equations of the second kind

This note is concerned with the problem of determining approximate solutions of Fredholm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equations is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail. (c) 2009 Elsevier Inc. All rights reserved.

Identification of parameters in unconfined aquifers

A method is presented for identification of parameters in unconfined aquifers from pumping tests, based on the optimisation of the objective function using the least squares approach. Four parameters are to be evaluated, namely: The hydraulic conductivity in the radial and the vertical directions, the storage coefficient and the specific yield. The sensitivity analysis technique is used for solving the optimisation problem. Besides eliminating the subjectivity involved in the graphical procedure, the method takes into account the field data at all time intervals without classifying them into small and large time intervals and does not use the approximation that the ratio of the storage coefficient to the specific yield tends to zero. Two illustrative examples are presented and it is found that the parameter estimates from the computational and graphical procedures differ fairly significantly.

Estimating equations for parameters in stochastic growth models from tag-recapture data

James (1991, Biometrics 47, 1519-1530) constructed unbiased estimating functions for estimating the two parameters in the von Bertalanffy growth curve from tag-recapture data. This paper provides unbiased estimating functions for a class of growth models that incorporate stochastic components and explanatory variables. a simulation study using seasonal growth models indicates that the proposed method works well while the least-squares methods that are commonly used in the literature may produce substantially biased estimates. The proposed model and method are also applied to real data from tagged rack lobsters to assess the possible seasonal effect on growth.

An improved Fabens method for estimation of growth parameters in the vonBertalanffy model with individual asymptotes

In the analysis of tagging data, it has been found that the least-squares method, based on the increment function known as the Fabens method, produces biased estimates because individual variability in growth is not allowed for. This paper modifies the Fabens method to account for individual variability in the length asymptote. Significance tests using t-statistics or log-likelihood ratio statistics may be applied to show the level of individual variability. Simulation results indicate that the modified method reduces the biases in the estimates to negligible proportions. Tagging data from tiger prawns (Penaeus esculentus and Penaeus semisulcatus) and rock lobster (Panulirus ornatus) are analysed as an illustration.

Application of commercially available, low-cost, miniaturised NIR spectrometers to the assessment of the sugar content of intact fruit

Recent decreases in costs, and improvements in performance, of silicon array detectors open a range of potential applications of relevance to plant physiologists, associated with spectral analysis in the visible and short-wave near infra-red (far-red) spectrum. The performance characteristics of three commercially available ‘miniature’ spectrometers based on silicon array detectors operating in the 650–1050-nm spectral region (MMS1 from Zeiss, S2000 from Ocean Optics, and FICS from Oriel, operated with a Larry detector) were compared with respect to the application of non-invasive prediction of sugar content of fruit using near infra-red spectroscopy (NIRS). The FICS–Larry gave the best wavelength resolution; however, the narrow slit and small pixel size of the charge-coupled device detector resulted in a very low sensitivity, and this instrumentation was not considered further. Wavelength resolution was poor with the MMS1 relative to the S2000 (e.g. full width at half maximum of the 912 nm Hg peak, 13 and 2 nm for the MMS1 and S2000, respectively), but the large pixel height of the array used in the MMS1 gave it sensitivity comparable to the S2000. The signal-to-signal standard error ratio of spectra was greater by an order of magnitude with the MMS1, relative to the S2000, at both near saturation and low light levels. Calibrations were developed using reflectance spectra of filter paper soaked in range of concentrations (0–20% w/v) of sucrose, using a modified partial least squares procedure. Calibrations developed with the MMS1 were superior to those developed using the S2000 (e.g. coefficient of correlation of 0.90 and 0.62, and standard error of cross-validation of 1.9 and 5.4%, respectively), indicating the importance of high signal to noise ratio over wavelength resolution to calibration accuracy. The design of a bench top assembly using the MMS1 for the non-invasive assessment of mesocarp sugar content of (intact) melon fruit is reported in terms of light source and angle between detector and light source, and optimisation of math treatment (derivative condition and smoothing function).

Influence of environmental and instrumental variables on the non-invasive prediction of Brix in pineapple using near infrared spectroscopy

The Brix content of pineapple fruit can be non-invasively predicted from the second derivative of near infrared reflectance spectra. Correlations obtained using a NIRSystems 6500 spectrophotometer through multiple linear regression and modified partial least squares analyses using a post-dispersive configuration were comparable with that from a pre-dispersive configuration in terms of accuracy (e.g. coefficient of determination, R2, 0.73; standard error of cross validation, SECV, 1.01°Brix). The effective depth of sample assessed was slightly greater using the post-dispersive technique (about 20 mm for pineapple fruit), as expected in relation to the higher incident light intensity, relative to the pre-dispersive configuration. The effect of such environmental variables as temperature, humidity and external light, and instrumental variables such as the number of scans averaged to form a spectrum, were considered with respect to the accuracy and precision of the measurement of absorbance at 876 nm, as a key term in the calibration for Brix, and predicted Brix. The application of post-dispersive near infrared technology to in-line assessment of intact fruit in a packing shed environment is discussed.

Non-invasive assessment of pineapple and mango fruit quality using near infra-red spectroscopy

The potential of near infra-red (NIR) spectroscopy for non-invasive measurement of fruit quality of pineapple (Ananas comosus var. Smooth Cayenne) and mango (Magnifera indica var. Kensington) fruit was assessed. A remote reflectance fibre optic probe, placed in contact with the fruit skin surface in a light-proof box, was used to deliver monochromatic light to the fruit, and to collect NIR reflectance spectra (760–2500 nm). The probe illuminated and collected reflected radiation from an area of about 16 cm2. The NIR spectral attributes were correlated with pineapple juice Brix and with mango flesh dry matter (DM) measured from fruit flesh directly underlying the scanned area. The highest correlations for both fruit were found using the second derivative of the spectra (d2 log 1/R) and an additive calibration equation. Multiple linear regression (MLR) on pineapple fruit spectra (n = 85) gave a calibration equation using d2 log 1/R at wavelengths of 866, 760, 1232 and 832 nm with a multiple coefficient of determination (R2) of 0.75, and a standard error of calibration (SEC) of 1.21 °Brix. Modified partial least squares (MPLS) regression analysis yielded a calibration equation with R2 = 0.91, SEC = 0.69, and a standard error of cross validation (SECV) of 1.09 oBrix. For mango, MLR gave a calibration equation using d2 log 1/R at 904, 872, 1660 and 1516 nm with R2 = 0.90, and SEC = 0.85% DM and a bias of 0.39. Using MPLS analysis, a calibration equation with R2 = 0.98, SEC = 0.54 and SECV = 1.19 was obtained. We conclude that NIR technology offers the potential to assess fruit sweetness in intact whole pineapple and DM in mango fruit, respectively, to within 1° Brix and 1% DM, and could be used for the grading of fruit in fruit packing sheds.

Synthesis and structure of mixed ligand isonitroso-β-ketoiminenickel(II) complexes. Crystal structure of (1-ethoxycarbonyl-2-npropylimino-1-propanoneoximato) (1-methoxycarbonyl-2-imino-1-propanoneoximato)nickel(II)

Mixed ligand complexes of the type Ni(R-AB)(AC') and Ni(R-AC)(AB') where AB/AC denote N-bonded isonitroso- [3-ketoimino ligands, AB'/AC' denote the corresponding Obonded ligands and R = Me, Et, n-Pr are synthesised and characterised. The complexes are neutral with square planar geometry around nickel(II). The bonding isomerism of the isonitroso group is discussed on the basis of i.r. and 1H n.m.r. studies. The crystal structure of the title complex, Ni(n-Pr-IEAI)(IMAI') has been determined from diffractometer data by Patterson and Fourier methods and refined by least squares to R = 0.088 for 2209 observed reflections. Unit cell constants are: a = 11.945(2), b = 22.436(7), c = 13.248(5) ~, [3 = 95.13(2) ~ The space group is P2Jc with Z = 8. Niekel(II) has a square planar coordination of two imine nitrogens, an isonitroso-nitrogen (from n-Pr-IEAI) and another isonitrosooxygen (from IMAI').

Assessment of internal quality attributes of mandarin fruit. 1. NIR calibration model development

The utility of near infrared spectroscopy as a non-invasive technique for the assessment of internal eating quality parameters of mandarin fruit (Citrus reticulata cv. Imperial) was assessed. The calibration procedure for the attributes of TSS (total soluble solids) and DM (dry matter) was optimised with respect to a reference sampling technique, scan averaging, spectral window, data pre-treatment (in terms of derivative treatment and scatter correction routine) and regression procedure. The recommended procedure involved sampling of an equatorial position on the fruit with 1 scan per spectrum, and modified partial least squares model development on a 720–950-nm window, pre-treated as first derivative absorbance data (gap size of 4 data points) with standard normal variance and detrend scatter correction. Calibration model performance for the attributes of TSS and DM content was encouraging (typical Rc2 of >0.75 and 0.90, respectively; typical root mean squared standard error of calibration of <0.4 and 0.6%, respectively), whereas that for juiciness and total acidity was unacceptable. The robustness of the TSS and DM calibrations across new populations of fruit is documented in a companion study.

Robustness of NIR calibrations for assessing fruit quality

Predictive models based on near infra-red spectroscopy for the assessment of fruit internal quality attributes must exhibit a degree of robustness across the parameters of variety, district and time to be of practical use in fruit grading. At the time this thesis was initiated, while there were a number of published reports on the development of near infra-red based calibration models for the assessment of internal quality attributes of intact fruit, there were no reports of the reliability ("robustness") of such models across time, cultivars or growing regions. As existing published reports varied in instrumentation employed, a re-analysis of existing data was not possible. An instrument platform, based on partial transmittance optics, a halogen light source and (Zeiss MMS 1) detector operating in the short wavelength near infra-red region was developed for use in the assessment of intact fruit. This platform was used to assess populations of macadamia kernels, melons and mandarin fruit for total soluble solids, dry matter and oil concentration. Calibration procedures were optimised and robustness assessed across growing areas, time of harvest, season and variety. In general, global modified partial least squares regression (MPLS) calibration models based on derivatised absorbance data were better than either multiple linear regression or `local' MPLS models in the prediction of independent validation populations . Robustness was most affected by growing season, relative to the growing district or variety . Various calibration updating procedures were evaluated in terms of calibration robustness. Random selection of samples from the validation population for addition to the calibration population was equivalent to or better than other methods of sample addition (methods based on the Mahalanobis distance of samples from either the centroid of the population or neighbourhood samples). In these exercises the global Mahalanobis distance (GH) was calculated using the scores and loadings from the calibration population on the independent validation population. In practice, it is recommended that model predictive performance be monitored in terms of predicted sample GH, with model updating using as few as 10 samples from the new population undertaken when the average GH value exceeds 1 .0 .