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We suggest a new approach for classification based on nonparametricly estimated likelihoods. Due to the scarcity of data in high dimensions, full nonparametric estimation of the likelihood functions for each population is impractical. Instead, we propose to build a class of estimated nonparametric candidate likelihood models based on a Markov property and to provide multiple likelihood estimates that are useful for guiding a classification algorithm. Our density estimates require only estimates of one and two-dimensional marginal distributions, which can effectively get around the curse of dimensionality problem. A classification algorithm based on those estimated likelihoods is presented. A modification to it utilizing variable selection of differences in log of estimated marginal densities is also suggested to specifically handle high dimensional data.

The use of receptor modeling is now a widely accepted approach to model air pollution data. The resulting estimates of pollution source profiles have error and frequently the uncertainties are obtained under an assumption of independence. In addition traditional Bootstrap approaches are very computationally intensive. We present an intuitive Jackknife alternative that is much less computationally intensive and in simulation examples and actual data seems to demonstrate that it provides wider confidence intervals and larger standard errors for receptor model profile estimates than does the Bootstrap done under the assumption of independence.

There has been some debate whether inverse or classical calibration methods are superior when there are multivariate predictors and some of them are missing. In this paper, we compare these two methods in the case where the design is not completely known. We develop some general results in the multivariate case and carry out extensive simulations in a univariate model with partly known regressors and several error distributions. These simulations reveal that the methods perform differently, depending on the specifics of the model. Neither method, however, turns out to be consistently superior to the other.

The relationship of the concentration of air pollutants to wind direction has been determined by nonparametric regression using a Gaussian kernel. The results are smooth curves with error bars that allow for the accurate determination of the wind direction where the concentration peaks, and thus, the location of nearby sources. Equations for this method and associated confidence intervals are given. A nonsubjective method is given to estimate the only adjustable parameter. A test of the method was carried out using cyclohexane data from 1997 at two sites near a heavy industrial region in Houston, Texas, USA. According to published emissions inventories, 70% of the cyclohexane emissions are from one source. Nonparametric regression correctly identified the direction of this source from each site. The location of the source determined by triangulation of these directions was

This paper presents and compares a new algorithm for finding the number of factors in a data analytic model. After we describe the new method, called NUMFACT, we compare it with standard methods for finding the number of factors to use in a model. The standard methods that we compare NUMFACT with are Malinowski's indicator function, Wold's cross-validation approach, Bartlett's test, scree plots, the rule-of-one, and using the number of factors (eigenvectors) needed to explain 90% of the trace of a correlation matrix. Using a diverse set of real applications, NUMFACT is shown to be the clear method of choice.

A new stepwise approach to variable selection for spectroscopy that includes chemical information and attempts to test several spectral regions producing high ranking coefficients has been developed to improve on currently available methods. Existing selection techniques can, in general, be placed into two groups: the first, time-consuming optimization approaches that ignore available information about sample chemistry and require considerable expertise to arrive at appropriate solutions (e.g. genetic algorithms), and the second, stepwise procedures that tend to select many variables in the same area containing redundant information. The algorithm described here is a fast stepwise procedure that uses multiple ranking chains to identify several spectral regions correlated with known sample properties. The multiple-chain approach allows the generation of a final ranking vector that moves quickly away from the initial selection point, testing several areas exhibiting correlation between spectra and composition early in the stepping procedure. Quantitative evidence of the success of this approach as applied to Raman spectroscopy is given in terms of processing speed, number of selected variables, and prediction error in comparison with other selection methods. In this respect, the procedure described here may be considered as a significant evolutionary step in variable selection algorithms.

This paper gives methods that use measurements from calibrated instruments in an effective and understandable manner. While some chemometric methods such as partial least squares might be considered, the procedures that we use are more transparent. In this paper two simple methods are proposed that use standard and saddlepoint approximations to combine nonlinear estimates from different regions of the instrument response. The asymptotic accuracy of the approximations is discussed. A worked example is given. A simulation study is also reported that supports our recommendations.

One important issue in chemometrics is to detect interactions among several factors. In this paper, we propose methods that detect interactions using low dimensional smoothers. Two methods are investigated and compared with usual least squared methods via Monte Carlo simulations. In addition, we show, using real data, how the methods affect our decisions.

This paper represents an ongoing receptor modeling research of airborne species in El Paso, Texas. It represents a six month collaboration between the authors. It extends the case study reported by Spiegelman and Dattner in 1992. For completeness the background material is reviewed.

Spiegelman, C.H., 1992. Plotting aids for multivariate calibration and chemostatistics. Chemometrics and Intelligent Laboratory Systems , 15: 29–38. There are few published procedures for plotting multivariate calibration data. In this paper I give some new plotting techniques that have been useful in my research and consulting. There are plots for detecting matrix effects, and other violations of Beer's law, plots for checking the selectivity of channels and frequencies, and plots that help diagnose the importance of frequencies within a peak.

Cline, D.B.H. and Spiegelman, C.H., 1991. Bias correcting confidence intervals for a nearly common property. Chemometrics and Intelligent Laboratory System, 11: 131–136. Confidence intervals are an important tool. Realistic confidence intervals account for both random errors and systematic errors (bias). We improve the usual method for combining random and systematic errors. The new methods are simple and often result in increased accuracy for confidence interval levels.

Spiegelman, C.H., Watters, R.L. and Hungwu, L., 1991. A statistical method for calibrating flame emission spectrometry which takes account of errors in the calibration standards. Chemometrics and Intelligent Laboratory Systems , 11: 121–130. The determination of potassium in sample solutions using flame emission spectrometry (FES) requires that the calibration function be estimated. Calibration standard solutions of potassium are made in the laboratory and nebulized into the FES instrument. A total of 240 data points were collected from chemical analyses. However the data come from only eight different values of the standards and are highly correlated within each standard. In the final analysis a sample size of eight was chosen to estimate the calibration curve. The main goal of this paper is to estimate the calibration function based on these data and then measure the amount of potassium in samples using this calibration function. The secondary goal is to show some of the important exploratory data analysis that should be done in any calibration. Since both the underlying theory of emission spectrometry and the scatter plot of data points suggest a linear relationship between the emission intensity and potassium concentration, a linear regression model is applied to fit these data and the residuals are examined based on regression assumptions. Data transformation is then attempted to stabilize the nonconstant variance of the residuals due to the fact that residuals fail to meet the assumptions. However, because the suggested transformation of taking the logarithm or 1/4 power of both x and y is hard to interpret, and because the log transformation would require an addition of an arbitrary constant to the standard values, we proceeded with the untransformed data. Outlier detection was used to find possible outliers. Ten consecutive observations (obs. 201–210) in the data set are potential outliers for they have absolute studentized residuals bigger than 2.7. However, influential observation techniques indicate that their effect on the estimation of the calibration curve is not great. In order to help compensate for the error in the calibration standards, we expand the calibration interval estimates. This compensation is important and helps to avoid the rather ad-hoc deletion of unusually influential data from the analysis. We think that a plausible explanation for the outliers is error in the calibration standards. In recognition of the heterogeneity of variance indicated by our data, we perform a weighted least squares type of confidence interval estimation for our calibration curve. The coefficient and standard error estimates are quite close for all weighted cases; in contrast, the unweighted case yields different values for the standard errors. If heteroscedasticity (nonconstant variance of the observations) is ignored, confidence intervals will be too wide at the low end and too narrow at the high end of the calibration curve. At both ends of the calibration curve, each resulting multiple-use calibration confidence interval is somewhat wider than the corresponding single-use calibration confidence interval. Finally, since the measurement errors that have a known finite bound in working standards have been taken into account, the increase in confidence intervals relative to presumed exact standards is about 0.1%.

Ordinary least squares (OLS) is one of the most commonly used criteria for fitting data to models and for estimating parameters. Orthogonal distance regression (ODR) extends least squares data fitting to problems with independent variables that are not known exactly. In this paper, we present the results of an empirical study designed to compare OLS to ODR for fitting both linear and non-linear models when there are errors in the independent variables. The results indicate that, for the data and performance criteria considered, ODR never performs appreciably worse than OLS and often performs considerably better.

A statistical model is investigated that expresses observations, such as blood lead levels, as an additive function of true levels and random measurement errors. Both empirical results (obtained from a series of computer simulation experiments) and theoretical results indicate how certain summary statistics for the observations vary in response to random measurement errors. Such results are applied to a very large data base of pediatric blood lead levels collected in New York City during 1970–1976, and they indicate that the observed trends in geometric mean blood lead levels are not significantly altered by the possible presence of measurement errors.

Eberhardt, K.R., Reeve, C.P. and Spiegelman, C.H., 1989. A minimax approach to combining means, with practical examples. Chemometrics and Intelligent Laboratory Systems , 5: 129–148. This paper describes a method for combining sample means that accounts for bias in those means. It compares the unweighted mean, the weighted mean using reciprocal estimated variances for weights, and a minimax weighted mean. When the individual means are subject to nontrivial biases we show that the minimax estimator can lead to important decreases in mean squared error and confidence interval width. Our recommendations are based on statistical theory and on simulations based on three Standard Reference Material data sets.