Eric Chicken

Statistical Process Monitoring of Nonlinear Profiles Using Wavelets

Many modern industrial processes are capable of generating rich and complex data records that do not readily permit the use of traditional statistical process-control techniques. For example, a “single observation” from a process might consist of n pairs of (x, y) data that can be described as y = f (x) when the process is in control. Such data structures or relationships between y and x are called profiles. Examples of profiles include calibration curves in chemical processing, oxide thickness across wafer surfaces in semiconductor manufacturing, and radar signals of military targets. In this paper, a semiparametric wavelet method is proposed for monitoring for changes in sequences of nonlinear profiles. Based on a likelihood ratio test involving a changepoint model, the method uses the spatial-adaptivity properties of wavelets to accurately detect profile changes taking nearly limitless functional forms. The method is used to differentiate between different radar profiles and its performance is assessed with Monte Carlo simulation. The results presented indicate the method can quickly detect a wide variety of changes from a given, in-control profile.

Block thresholding and wavelet estimation for nonequispaced samples

For samples with the design points occurring as a Poisson process or having a uniform distribution, the wavelet method of block thresholding can be applied directly to the data as though it was equispaced without sacrificing adaptivity or optimality. When the underlying true function is in certain Besov and Holder classes, the resulting estimator achieves the minimax rate of convergence. Simulation results are examined.

Coastal carbonate aquifer sensitivity to tides

Regional groundwater flow in karstified carbonate aquifers typically occurs via a connected system of conduits, rather than in a porous matrix. This feature makes such aquifers difficult to characterize and quantify and they are among the most poorly modeled of all physical systems. Current models drastically overpredict travel times of pollutants, and current statistical methods cannot reliably determine the salient properties of these aquifers using surface-flow data. Better understanding of such aquifers is needed, since they are widespread [Veni, 2002], typically more productive than other types of aquifers, and vulnerable to contamination.

Block-dependent thresholding in wavelet regression

Nonparametric regression via wavelets is usually implemented under the assumptions of dyadic sample size, equally spaced and fixed sample points, and independent and identically distributed normal errors. An estimator is proposed which, through the use of linear transformations and block thresholding, can simultaneously achieve both global and local optimal rates of convergence even for data that does not possess the above three assumptions. Additionally, the estimator exhibits fast computation time and is spatially adaptive over large classes of Besov and Hölder functions. The thresholds are dependent on the varying levels of noise in each block of wavelet coefficients, rather than on a single estimate of the noise as is usually done. This block-dependent method is compared against term-by-term wavelet methods with noise-dependent thresholding via theoretical asymptotic convergence rates as well as by simulations and comparisons on a well-known data set. Simulation results show that this block-dependent estimator is superior in terms of reconstruction error to term-by-term wavelet estimators and universal-type block estimators.

Nonparametric Changepoint Estimation for Sequential Nonlinear Profile Monitoring

We consider changepoint detection in a process in which the observed points are profiles: large sets of functionally related points (x,y). Few changepoint detection methods have been proposed that do not rely in some capacity on the assumption that the observational errors are normally distributed. In this paper, a nonparametric distribution-free wavelet method is proposed for monitoring for changes in sequences of nonlinear profiles. No assumptions are made on the nature or form of the changes between the profiles other than finite square-integrability, and no distributional assumption is made on the noise. Using only the magnitudes and location maps of thresholded wavelet coefficients, our method uses the spatial adaptivity property of wavelets to accurately detect profile changes when the signal is obscured with a variety of non-Gaussian errors. The proposed method outperforms existing (and much more complex) methods under various conditions of non-Gaussianity. The method does not rely on estimates designed for normally distributed errors, yet it is robust enough to work reasonably well under Gaussian conditions. The efficiency of the proposed method, including comparisons with existing profile monitoring methods, is shown via simulation. We also apply the proposed method to vertical density profile data, a common real data set used in profile monitoring. Copyright

Performance and Prediction for Varying Survival Time Scales

The Cox proportional hazards model is widely used for analyzing associations between risk factors and occurrences of events. One of the essential requirements of defining Cox proportional hazards model is the choice of a unique and well-defined time scale. Two time scales are generally used in epidemiological studies: time-on-study and chronological age. The former is the most frequently used time scale, both in clinical studies and longitudinal observation studies. However, there is no general consensus on which time scale is the most appropriate for a given question or study. In this article, we address the question of robustness of the results using one time scale when the other is actually the correct one. We use three criteria to measure the performances of these models through simulations: magnitude of the bias of the regression coefficients, mean square errors, and the measure of overall predictive discrimination of the models. We conclude that the time-on-study models are more robust to misspecification of the underlying time scale.

Baseline age effect on parameter estimates in Cox models

The Cox proportional hazards model is widely used in time-to-event analysis. Two time scales are used in practice: time-on-study and chronological age. The former is the most frequently used time scale in clinical studies and longitudinal observation studies. However, there is no general consensus about which time scale is the best. It has been asserted that if the cumulative baseline hazard is exponential or if the age-at-entry is independent of the covariate, then the two models are equivalent. We show that neither of these conditions leads to equivalency. Variability in the age-at-entry of individuals in the study causes the models to differ significantly. This is shown both analytically and through a simulation study. Additionally, we show that the time-on-study model is more robust to changes in age-at-entry than the chronological age model.

Relationships between Gender, Cognitive Ability, Preference, and Calculus Performance

In this research, we examined the relationships between gender, spatial ability, verbal-logical reasoning ability, calculus performance, and preference for visual or analytic processing. Data were collected from 150 calculus students at four high schools in two school districts. The results suggest that spatial and verbal-logical reasoning abilities are important factors of calculus performance. Preferred mode of processing was unrelated to spatial and verbal-logical reasoning abilities suggesting that cognitive abilities did not predict students’ preference for visual or analytic processing. There were no significant differences between the two sexes in cognitive abilities, preferred mode of processing, and calculus performance.

Time Scales in Epidemiological Analysis: An Empirical Comparison

The Cox proportional hazards model is routinely used to analyze time-to-event data. To use this model requires the definition of a unique well-defined time scale. Most often, observation time is used as the time scale for both clinical and observational studies. Recently after a suggestion that it may be a more appropriate scale, chronological age has begun to appear as the time scale used in some reports. There appears to be no general consensus about which time scale is appropriate for any given analysis. It has been suggested that if the baseline hazard is exponential or if the age-at-entry is independent of covariates used in the model, then the two time scales provide similar results. In this report we provide an empirical examination of the results using the two different time scales using a large collection of data sets to examine the relationship between systolic blood pressure and coronary heart disease death (CHD death). We demonstrate, in this empirical example that the two time-scales can lead to differing results even when these two conditions appear to hold.

Visual thinking and gender differences in high school calculus

This study sought to examine calculus students’ mathematical performances and preferences for visual or analytic thinking regarding derivative and antiderivative tasks presented graphically. It extends previous studies by investigating factors mediating calculus students’ mathematical performances and their preferred modes of thinking. Data were collected from 183 Advanced Placement calculus students in five high schools. Students’ visual preferences were not influenced by gender. Statistically significant differences in visual preference scores were found among high- and low-performing students. Thus, the results suggest that stronger preference for visual thinking was associated with higher mathematical performances.