Singularity of Data Analytic Operations

Statistical data by their very nature are indeterminate in the sense that if one repeated the process of collecting the data the new data set would be somewhat different from the original. Therefore, a statistical method, a map $\Phi$ taking a data set $x$ to a point in some space F, should be stable at $x$: Small perturbations in $x$ should result in a small change in $\Phi(x)$. Otherwise, $\Phi$ is useless at $x$ or -- and this is important -- near $x$. So one doesn't want $\Phi$ to have "singularities," data sets $x$ s.t.\ the the limit of $\Phi(y)$ as $y$ approaches $x$ doesn't exist. (Yes, the same issue arises elsewhere in applied math.) However, broad classes of statistical methods have topological obstructions of continuity: They must have singularities. We show why and give lower bounds on the Hausdorff dimension, even Hausdorff measure, of the set of singularities of such data maps. There seem to be numerous examples. We apply mainly topological methods to study the (topological) singularities of functions defined (on dense subsets of) "data spaces" and taking values in spaces with nontrivial homology. At least in this book, data spaces are usually compact manifolds. The purpose is to gain insight into the numerical conditioning of statistical description, data summarization, and inference and learning methods. We prove general results that can often be used to bound below the dimension of the singular set. We apply our topological results to develop lower bounds on Hausdorff measure of the singular set. We apply these methods to the study of plane fitting and measuring location of data on spheres. \emph{This is not a "final" version, merely another attempt.}Comment: 325 pages, 8 figure

An Algorithm for Unconstrained Quadratically Penalized Convex Optimization

A descent algorithm, "Quasi-Quadratic Minimization with Memory" (QQMM), is proposed for unconstrained minimization of the sum, $F$, of a non-negative convex function, $V$, and a quadratic form. Such problems come up in regularized estimation in machine learning and statistics. In addition to values of $F$, QQMM requires the (sub)gradient of $V$. Two features of QQMM help keep low the number of evaluations of the objective function it needs. First, QQMM provides good control over stopping the iterative search. This feature makes QQMM well adapted to statistical problems because in such problems the objective function is based on random data and therefore stopping early is sensible. Secondly, QQMM uses a complex method for determining trial minimizers of $F$. After a description of the problem and algorithm a simulation study comparing QQMM to the popular BFGS optimization algorithm is described. The simulation study and other experiments suggest that QQMM is generally substantially faster than BFGS in the problem domain for which it was designed. A QQMM-BFGS hybrid is also generally substantially faster than BFGS but does better than QQMM when QQMM is very slow.Comment: Submitted to the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Collaborative Freight Transportation to Improve Efficiency and Sustainability

Collaborative distribution offers the potential for substantial improvements in freight transportation. As collaboration increases, more loads are available for sharing among transportation service providers, leading to more fully loaded trailers that travel fewer miles and reduce the cost per load on average. In this study, we develop approaches to analyze improvements in key performance measures as collaboration increases in freight transportation. For the data sets analyzed, improvements include a 34% increase in trailer fullness, a 29% reduction in average costs per load, and a 25% decrease in average miles per load. Based on this analysis, collaboration provides substantial improvements for transportation service providers and opportunities for increased driver retention. Drivers would benefit from a better quality of life, more local routes, and more time home with their families. In addition to the economic and social benefit, the environmental benefit include reducing the miles driven and the resulting CO2 emissions

Factors Contributing to the Rise of Buprenorphine Misuse: 2008-2013

OBJECTIVE: The purpose of the present study was to examine the motivations underlying the use of buprenorphine outside of therapeutic channels and the factors that might account for the reported rapid increase in buprenorphine misuse in recent years. METHODS: This study used: (1) a mixed methods approach consisting of a structured, self-administered survey (N=10,568) and reflexive, qualitative interviews (N=208) among patients entering substance abuse treatment programs for opioid dependence across the country, centered on opioid misuse patterns and related behaviors; and (2) interviews with 30 law enforcement agencies nationwide about primary diverted drugs in their jurisdictions. RESULTS: Our results demonstrate that the misuse of buprenorphine has increased substantially in the last 5 years, particularly amongst past month heroin users. Our quantitative and qualitative data suggest that the recent increases in buprenorphine misuse are due primarily to the fact that it serves a variety of functions for the opioid-abusing population: to get high, manage withdrawal sickness, as a substitute for more preferred drugs, to treat pain, manage psychiatric issues and as a self-directed effort to wean themselves off opioids. CONCLUSION: The non-therapeutic use of buprenorphinehas risen dramatically in the past five years, particularly in those who also use heroin. However, it appears that buprenorphine is rarely preferred for its inherent euphorigenic properties, but rather serves as a substitute for other drugs, particularly heroin, or as a drug used, preferable to methadone, to self-medicate withdrawal sickness or wean off opioids

The Changing Face of Heroin Use in the United States: A Retrospective Analysis of the Past 50 Years

IMPORTANCE: Over the past several years, there have been a number of mainstream media reports that the abuse of heroin has migrated from low-income urban areas with large minority populations to more affluent suburban and rural areas with primarily white populations. OBJECTIVE: To examine the veracity of these anecdotal reports and define the relationship between the abuse of prescription opioids and the abuse of heroin. DESIGN, SETTING, AND PARTICIPANTS: Using a mixed-methods approach, we analyzed (1) data from an ongoing study that uses structured, self-administered surveys to gather retrospective data on past drug use patterns among patients entering substance abuse treatment programs across the country who received a primary (DSM-IV) diagnosis of heroin use/dependence (n = 2797) and (2) data from unstructured qualitative interviews with a subset of patients (n = 54) who completed the structured interview. MAIN OUTCOMES AND MEASURES: In addition to data on population demographics and current residential location, we used cross-tabulations to assess prevalence rates as a function of the decade of the initiation of abuse for (1) first opioid used (prescription opioid or heroin), (2) sex, (3) race/ethnicity, and (4) age at first use. Respondents indicated in an open-ended format why they chose heroin as their primary drug and the interrelationship between their use of heroin and their use of prescription opioids. RESULTS: Approximately 85% of treatment-seeking patients approached to complete the Survey of Key Informants\u27 Patients Program did so. Respondents who began using heroin in the 1960s were predominantly young men (82.8%; mean age, 16.5 years) whose first opioid of abuse was heroin (80%). However, more recent users were older (mean age, 22.9 years) men and women living in less urban areas (75.2%) who were introduced to opioids through prescription drugs (75.0%). Whites and nonwhites were equally represented in those initiating use prior to the 1980s, but nearly 90% of respondents who began use in the last decade were white. Although the high produced by heroin was described as a significant factor in its selection, it was often used because it was more readily accessible and much less expensive than prescription opioids. CONCLUSION AND RELEVANCE: Our data show that the demographic composition of heroin users entering treatment has shifted over the last 50 years such that heroin use has changed from an inner-city, minority-centered problem to one that has a more widespread geographical distribution, involving primarily white men and women in their late 20s living outside of large urban areas

Patterns of Prescription Opioid Abuse and Comorbidity in an Aging Treatment Population

Very little is known about the impact of age and gender on drug abuse treatment needs. To examine this, we recruited 2,573 opioid-dependent patients, aged from 18 to 75 years, entering treatment across the country from 2008 to 2010 to complete a self-administered survey examining drug use histories and the extent of comorbid psychiatric and physical disorders. Moderate to very severe pain and psychiatric disorders, including polysubstance abuse, were present in a significant fraction of 18- to 24-year-olds, but their severity grew exponentially as a function of age: 75% of those older than 45 years had debilitating pain and psychiatric problems. Women had more pain than men and much worse psychiatric issues in all age groups. Our results indicate that a one-size-fits-all approach to prevention, intervention, and treatment of opioid abuse that ignores the shifting needs of opioid-abusing men and women as they age is destined to fail