Gap bootstrap methods for massive data sets with an application to transportation engineering

Abstract

LAHIRI, SPIEGELMAN, APPIAH AND RILETT and then use some analytical considerations to put the individual pieces together, thereby alleviating the computational issues associated with large data sets to a great extent.The class of problems we consider here is the estimation of standard errors of estimators of population parameters based on massive multivariate data sets that may have heterogeneous distributions. A primary example is the origin-destination (OD) model in transportation engineering. In an OD model, which motivates this work and which is described in detail in Section 2 below, the data represent traffic volumes at a number of origins and destinations collected over short intervals of time (e.g., 5 minute intervals) daily, over a long period (several months), thereby leading to a massive data set. Here, the main goals of statistical analysis are (i) uncertainty quantification associated with the estimation of the parameters in the OD model and (ii) to improve prediction of traffic volumes at the origins and the destinations over a given stretch of the highway. Other examples of massive data sets having the required structural property include (i) receptor modeling in environmental monitoring, where spatio-temporal data are collected for many pollution receptors over a long time, and (ii) toxicological models for dietary intakes and drugs, where blood levels of a large number of toxins and organic compounds are monitored in repeated samples for a large number of patients. The key feature of these data sets is the presence of "gaps" which allow one to partition the original data set into smaller subsets with nice properties.The "largeness" and potential inhomogeneity of such data sets present challenges for estimated model uncertainty evaluation. The standard propagation of error formula or the delta method relies on assumptions of independence and identical distributions, stationarity (for space-time data) or other kinds of uniformity which, in most instances, are not appropriate for such data sets. Alternatively, one may try to apply the bootstrap and other resampling methods to assess the uncertainty. It is known that the ordinary bootstrap method typically underestimates the standard error for parameters when the data are dependent (positively correlated). The block bootstrap has become a popular tool for dealing with dependent data. By using blocks, the local dependence structure in the data is maintained and, hence, the resulting estimates from the block bootstrap tend to be less biased than those from the traditional (i.i.d.) bootstrap. For more details, see Lahiri (1999Lahiri ( , 2003. However, computational complexity of naive block bootstrap methods increases significantly with the size of the data sets, as the given estimator has to be computed repeatedly based on resamples that have the same size as the original data set. In this paper, we propose two resampling methods, generally both referred to as Gap Bootstraps, that exploit the "gap" in the dependence structure of such large-scale data se...

A Scalable Bootstrap for Massive Data

“…For instance, one might assume that BLB subsets can be chosen so that the data within them are stationary. However, after bootstrapping each subset as prescribed by the BLB, the results could not be combined via simple averaging due to intersubset non-stationarity; rather, a more complex procedure would be required, as discussed in the related work of Lahiri et al (2012).…”

Resampling Methods for Estimating Travel Time Uncertainty: Application of the Gap Bootstrap

“…Gap bootstrapping is a procedure that is appropriate for datasets that can be partitioned into approximately exchangeable partitions ( 32 ). The only previous applications of the gap bootstrap evaluate uncertainties in point estimates of origin–destination split proportions ( 32, 33 ).…”

“…Standard historical statistical methods do not apply and any modeling with this data must account for this characteristic behavior. For such data sets, Lahiri et al found that the gap bootstrap method outperformed other commonly used resampling methods in closeness to the true standard errors based on Monte-Carlo simulation ( 32 ).…”

Section on the special year for mathematics of planet earth (MPE 2013)

“…This reprint differs from the original in pagination and typographic detail. 1 2 T. GNEITING Lahiri et al (2012) set out to solve problems of prediction and estimation, respectively, that arise in transportation engineering. The challenges posed by a planet at risk have been a major driver in the development of statistical theory and methodology, and the papers in this special section document the use of state of the art techniques in addressing critical real world problems.…”

“…State of the art applied statistical work is inevitably computational. Developments of note in this context include the advent of the integrated nested Laplace approximation [INLA; Rue, Martino and Chopin (2009)] technique in Bayesian computing, which Illian, Sørbye and Rue (2012) make available for fitting complex spatial point processes, and the ubiquity of massive data sets, which require the adaptation of classical techniques, as explored by Lahiri et al (2012) in the case of bootstrap resampling.…”