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Sinusoidal Modeling Applied to Spatially Variant Tropospheric Ozone Air Pollution

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SINUSOIDAL MODELING APPLIED TO SPATIALLY VARIANT TROPOSPHERIC OZONE AIR POLLUTION By Nicholas Z. Muller and Peter C. B. Phillips January 2006 COWLES FOUNDATION DISCUSSION PAPER NO. 1548 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/Sin usoidal M odeling A pplied to Spatially VariantTropospheric Ozone A ir P ollution1Nichola s Z. MullerSc hool of Forestry and Env ironm ental StudiesYa le Un iversity1773 Huntington Tpke.Trumbull, CT 06611USA(203) [email protected] C. B. PhillipsCo wles Foundation, Yale UniversityUniversit y of York & Universit y of AucklandDecember 4th,20051Phillips ackno wledges support from the NSF under Grant No. SES 04-142254. Mr. Muller wouldlike to thank the Glaser Progress Foundation for supporting this researc h .AbstractThis paper demonstrates how parsimonious models of sinusoidal functions can be used to fitspatially variant time series in which there is considerable variation of a periodic type. A typ-ical shortcoming of such tools relates to the difficulty in capturing idiosyncratic variation inperiodic models. The strategy developed here addresses this deficiency. While previous workhas sought to overcome the shortcoming by augmenting sinusoids with other techniques, thepresent approach employs station-specific sinusoids to supplement a common regional compo-nent, which succeeds in capturing local idiosyncratic behavior in a parsimonious manner. Theexperiments conducted herein reveal that a semi-parametric approach enables such modelsto fit spatially varying time series with periodic behavior in a remarkably tight fashion. Themethods are applied to a panel data set consisting of hourly air pollution measurements. Theaugmented sinusoidal models produce an excellent fit to these data at three different levels ofspatial detail.JEL Classification :C22&C23Key words and phrases: Air Pollution, Idiosyncratic component, Regional variation, Semi-parametric model, Sinusoidal function, Spatial-temporal data, Tropospheric Ozone.1IntroductionModels based on sinusoidal functions can adequately fit time series that exhibit strong periodicbeha vior (Bloomfield, 2000). However, such models usually encounter difficulties emulatingtime series with cyclical behavior that deviates from a fixed periodic structure (Lewis and Ray,1997). In such cases, some alternative approaches hav e been proposed to augment sinusoidalmodels to improve sample period fit and prediction. For instance, Campbell and Walker (1977)employ a model that includes both a deterministic sinusoid and a second-order autoregressivecomponent to describe annual lynx trappings. Dixon and Tawn (1998) construct a modelof sea-lev el estimation that consists of a sinusoidal component governing tidal oscillations, alinear model capturing long-term trends, and weather-dependent model to estimate surge.The present article develops a new set of statistical tools that are designed to model spa-tially varying time series which display some systematic periodic behavior and also manifestcharacteristics that are station-specific to individual locations. The methodological innova-tion is to use sinusoidal functions to represent spatiotemporal variation in a semiparametricmanner. The technique involv es first fitting a finite linear com bination of sinusoidal functionsto capture the spatially common periodic features of a certain series. This common periodicelement may be regarded as parametric and will usually be quite parsimonious. Once thisparametric model of common features is determined, it is augmented with a nonparametriccomponent to model idiosyncratic local spatial features, again using sinusoidal functions inthe form of a sieve approximation (e.g. Grenander, 1981). This nonparametric model is fittedusing local residuals from the common model. Com bining the nonparametric and parametriccomponents into a single semiparametric framework provides a mec h anism for capturing ele-ments of common variation in spatiotemporal behavior while having the flexibility to emulatea substantial degree of local variation. The advantages of this approach are two-fold. First,the initial sinusoidal specification extracts the common near-periodic element in a comp lexspatiotemporal process using just a few parameters. Second, the no nparametric componenttailors the more rigid common periodic structure to local patterns of variation. This approachresolves a principal drawback of sinusoidal modeling that is cited in the literature (lack of flexi-1bility) and enables the investigator to find common elements of spatiotemporal variation in thedata in a parametric manner that increases statistical efficiency. The new approach appearsto have broad applicability to spatiotemporal data that manifest some common periodicitybut substantial local variations about the common cycle.We apply this machinery to a panel data set consisting of air pollution measurements in thecontiguous United States. Specifically, the data involve measurement s of tropospheric ozone(O3) from the U.S. Environmental Protection Agency’s (USEPA) air pollution monitoringnetwork (USEPA 1). This common pollutant exhibits a characteristic unimodal diurnal shapewhen plotted against the hours in a day (see Figure 1). To this daily structure we fitthemodels outlined above. The modeling approach adopted is well suited to this statisticalproblem and its various policy applications. First, hourly measurements of O3do exhibit afairly regular periodic structure, which suggests a parametric sinusoidal fit will be generallywell suited to the data. Additionally, the specificshapeofthetimeseriesvariationitselfvarieswidely across space. These data therefore provide a suitable context for the application ofour semiparametric approach. Second, the O3data set is a rich collection of nearly 4 millionobservations collected in 1996, providing an interesting spatiotemporal setting to test theperformance of these new tools.Finally,thisapplicationisinanareaofimmediate policy relevance. Since troposphericO3produces a variety of deleterious effects on human health (Bell et al., 2004) and welfare,the USEPA has designated O3as a criteria air pollutant. This classification stipulates thatO3is subject to hourly measurement in order to assess regulatory com pliance across bothtime and space. The network of monitors calibrated to measure O3consists of


Sinusoidal Modeling Applied to Spatially Variant Tropospheric Ozone Air Pollution

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