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Forecasting Peak Daily Ozone Levels—I. A Regression
with Time Series Errors Model Having a Principal
Component Trigger to Fit 1991 Ozone Levels
Pao-Wen Grace Liu a & Richard Johnson b
a Bureau of Air Management, Wisconsin Department of Natural Resources , Madison ,
Wisconsin , USA
b Department of Statistics , University of Wisconsin , Madison , Wisconsin , USA
Published online: 27 Dec 2011.
To cite this article: Pao-Wen Grace Liu & Richard Johnson (2002) Forecasting Peak Daily Ozone Levels—I. A Regression
with Time Series Errors Model Having a Principal Component Trigger to Fit 1991 Ozone Levels, Journal of the Air & Waste
Management Association, 52:9, 1064-1074, DOI: 10.1080/10473289.2002.10470841
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Liu and Johnson
TECHNICAL PAPER
ISSN 1047-3289 J. Air & Waste Manage. Assoc. 52:1064-1074
Copyright 2002 Air & Waste Management Association
Forecasting Peak Daily Ozone Levels—I. A Regression with
Time Series Errors Model Having a Principal Component
Trigger to Fit 1991 Ozone Levels
Pao-Wen Grace Liu
Bureau of Air Management, Wisconsin Department of Natural Resources, Madison, Wisconsin
Richard Johnson
Department of Statistics, University of Wisconsin, Madison, Wisconsin
ABSTRACT
INTRODUCTION
This research was motivated by the need to warn the
population of Milwaukee, WI, on high-ozone days. A sta-
tistical model for the peak daily 1-hr ozone level is pro-
posed. A Regression with Time Series Errors (RTSE) model,
which includes a principal component (PC) trigger, is
the basis for forecasting the peak daily 1-hr ozone level.
The RTSE model, with a PC trigger, is first employed
to estimate daily peak ozone measured at the University
of Wisconsin, Milwaukee-North (UWM-N), during the
1991 ozone season. The RTSE model uses peak daily tem-
perature, morning vector average wind direction, and
the PC trigger as predictor variables. The PC trigger was
designed to summarize atmospheric circumstances when
peak ozone was greater than 100 parts per billion (ppb).
It is verified that the RTSE model, with a PC trigger, sig-
nificantly improves the prediction of peak daily ozone,
particularly peak ozone greater than 100 ppb. In com-
parison with the RTSE model without the PC trigger, the
RTSE model with a PC trigger raised the R2 from 0.680 to
0.809.1 It is suggested that the RTSE model, with the PC
trigger, is an adequate statistical model that has the po-
tential for real-time ozone forecasting.
The purpose of this research is to design, test, and imple-
ment a technique to forecast next-day peak daily 1-hr
ambient ozone levels. A pilot model is developed. The
application of this pilot model to real-time forecasting in
Milwaukee, WI, will be detailed in a companion paper.2
Ozone in the stratosphere, about 15–55 km altitude,
plays a critical role in protecting humans from harmful UV
radiation. However, ozone in the troposphere between the
Earth’s surface and 10–15 km altitude is a harmful pollutant
that causes human health problems. The ground-level or
ambient ozone is known as a secondary pollutant and is
produced from photochemical reactions under certain me-
teorological conditions. Anthropogenic and biogenic vola-
tile organic compounds (VOCs) and nitrogen oxides (NOx)
are found to be major precursors of ozone pollution.
During the past several decades, ambient ozone has
been a serious environmental problem in Milwaukee. The
Federal Clean Air Act Amendments of 1977 and 1990 re-
sulted in the designation of southeastern Wisconsin as a
severe 1-hr ozone nonattainment area. The Wisconsin De-
partment of Natural Resources (WDNR) submitted ozone
State Implementation Plans (SIPs) in 1979, 1983, and 2000
to demonstrate attainment of the 1-hr ozone standard.3
Reformulated gasoline, improved motor vehicle emissions
control, and control of industrial service have been very
effective in reducing ozone concentration in eastern Wis-
consin. Additionally, voluntary measures such as Ozone
Action Days (OADs), cooperatively conducted by the Wis-
consin State government and industries, have helped re-
duce emissions of ozone precursors. The downward trend
of the temperature-adjusted ozone over the period 1980–
1995 verifies the effectiveness of the ozone control pro-
grams.4 The number of days of violating the 1-hr ozone
standard also has been gradually reduced.4 However, Mil-
waukee County is still designated as one of the six severe
1-hr ozone nonattainment counties (see Figure 1).
IMPLICATIONS
Once this statistical model is shown to adequately fit
historical data, it can be used to make real-time fore-
casts of peak ozone. Milwaukee has been designated
as a severe 1-hr ozone nonattainment area for approxi-
mately 10 years. Early warning on days with peak high
ozone can help people avoid exposures to high ozone,
which causes or exacerbates human respiratory prob-
lems. We developed an ozone model, RTSE and a PC
trigger, that solves some problems of underprediction
of high ozone and also provides the potential to make
actual real-time ozone forecasts.
1064 Journal of the Air & Waste Management Association
Volume 52 September 2002

Liu and Johnson
Wisconsin has developed a subjective forecast system
operated by the state meteorologists. Beginning in the
summer of 1995, southeastern Wisconsin has been par-
ticipating in a voluntary effort to help reduce ground-
level ozone concentrations in the southern Lake Michigan
region. The Lake Michigan Ozone Region includes south-
eastern Wisconsin, northeastern Illinois, northwestern
Indiana, and western Michigan (see Figure 2). In 1995,
the WDNR joined with the public media and private en-
terprises in beginning a program of prior-day notification
for those days in eastern Wisconsin that are forecast to have
weather conditions favorable for the production of high
ozone. Such a day is called an OAD. However, the OAD
program is established based on a qualitative description
of the weather, and no specific ozone level or range of val-
ues is produced by this forecast. The goal of the OAD pro-
gram is to be able to notify the public in time to reduce
traffic emission and to avoid ozone exposure.
Figure 1. Ozone 1-hr nonattainment areas in southeastern Wisconsin
in 1998.
Photochemical models, such as the urban airshed
model (UAM), are difficult to calibrate and require inten-
sive computer resources.14 Simulation of a single ozone
event requires a considerable amount of data input (in-
cluding emission, meteorological, and chemistry data),
and it is time-consuming.5,15 Particularly, the UAM model
was designed for regulatory purposes to validate strate-
gies of reducing ozone precursors. Chang and Cardelino16
conclude that the numerical processing barrier of apply-
ing UAM to forecast peak ozone has been removed by
using low-cost and powerful workstations. However, in
Wisconsin, the UAM was used only to demonstrate how
to attain the 1-hr ozone standard in the Lake Michigan
area—not to make immediate forecasts.17 WDNR research
Many different techniques have been used to analyze
ozone data during the past 20 years. A common problem is
that these errors tend to be much larger for high ozone,
and those high concentrations are underpredicted.5-8 Also,
these models were verified with archived data only. Most
of the forecasts can be fairly accurate during the days that
ozone did not exceed the 1-hr standard and temperatures
were not dramatically high.6,9,10 When these models were
constructed, they were based on entire seasons, which con-
tain mostly average-ozone days. Consequently, those mod-
els failed to predict the high-ozone days that are usually
only ~5% of the entire season.11 Therefore, it is essential to
develop a model with improved accuracy at the higher
ozone levels that can be used to forecast ozone in real time.
The objective of this research is to build a pilot ozone
model that can successfully improve the prediction of high
ozone concentrations and that has the potential to make
real-time forecasts. A special case of the Box-Jenkins transfer
function model, a Regression with Time Series Errors (RTSE)
model, is employed along with a principal component (PC)
analysis to construct the pilot ozone model.12,13 The pur-
pose of the PC analysis is to create a trigger for effectively
estimating high ozone concentrations. The proposed ozone
model is called the RTSE model with a PC trigger.
STATEMENT OF THE PROBLEM
Figure 1 indicates the severity of the ozone exceedance
problem in Wisconsin. Six counties, including Milwau-
kee, are designated as severe nonattainment counties and
the two other counties are designated moderate and rural
transport nonattainment. Though during the past 10
years, the 1-hr ozone nonattainment area has been re-
duced from 11 to eight counties, Milwaukee County is
still designated as a severe 1-hr ozone nonattainment area.
Figure 2. The Lake Michigan Ozone Study area. Counties shaded
dark are in nonattainment areas in Wisconsin, Illinois, Indiana, and
Michigan of the 1-hr NAAQS for ozone, 1991.
Volume 52 September 2002
Journal of the Air & Waste Management Association 1065

Liu and Johnson
emphasizes that the UAM systematically underpredicts
high ozone values.18
on to increase the prediction of peak ozone above the
baseline prediction.
Mathematical/statistical models, such as regression
models, have difficulty predicting extreme ozone values.5-8
Underpredicting and overpredicting frequently are ob-
served among the ozone modeling studies. Most ozone
modeling focuses on estimating the long-term ozone trend
or implementing emission-control policy.9,10,19,20
RTSE Model
An RTSE model is advantageous for two reasons. First, the
time dependence of ozone and meteorological variables
are modeled. Air pollution, particularly ozone, is highly
correlated over time.26,27 Second, an RTSE model can in-
corporate reduced dimensional summaries of multivari-
ate meteorological variables. Indeed, both meteorology
variables and ozone precursors have a great influence on
ozone formation and both have been considered in ozone
modeling.8,9,14,28-30
A frequently observed problem in regression analysis
is that some of the employed regression models seriously
violate a required assumption. The models were designed
without noticing, mentioning, or fulfilling the fundamen-
tal assumption that the errors in the model have to be
independent and identically distributed (i.i.d. assump-
tion). In addition, most of the regression studies used all
of the ozone data from an entire season to construct their
forecast models. During an entire ozone season, the peak
daily ozone values vary greatly from 30 to 200 parts per
billion (ppb), but the extremely high ozone episodes ap-
pear to be always less than 5% of the entire season.11 Only
three studies improved their forecasts by using data strati-
fied according to ozone level.7,8,21
The command proc arima in the SAS statistical soft-
ware package was used to fit and analyze our time-series
models to build an RTSE model.31 In the usual notation,
let B be the backward shift operator so that φ1BYt = φ1Yt-1
and φ2B2Yt = φ2Yt-2. The Greek letters denote parameters to
be estimated. One example of an RTSE model is
(1)
Box-Jenkins12 Auto-Regressive Integrated Moving
Average (ARIMA) time-series models provide another
method to predict series of daily ozone. Time-series mod-
els relax the assumption of independence and are con-
sistent with the nature of serially correlated air pollutants.
Though the problems of underpredictions and
overpredictions are still observed, some research has
emphasized that Box-Jenkins multivariate time-series
models are superior to their univariate time-series model
in predicting ozone concentrations.22-24
A rather severe weakness in existing forecasting pro-
cedures has been indicated. As a result, the RTSE model
with a PC trigger was developed to solve the significant
underestimation of high ozone concentrations. This
method also is proposed because of its capability of
quantitative forecasting and its inexpensive operations.
where Yt stands for the response variable (ozone), and the
predictor variables X1,t and X2,t could be temperature and
wind speed. The noise Nt arises from an innovation series
at, which consists of a sequence of i.i.d. assumption ran-
dom shocks. The noise term Nt in eq 1 follows an AR(2)
time-series model, as opposed to classic regression mod-
els, as detailed in the appendix.12
The process of developing an RTSE model is presented
as a flow chart in Figure 3. The first three major steps are
sequentially displayed in the corresponding box.
(1) Identify the structure of Nt (eq 1);
(2) Conduct a regression with time series errors model.
The Nt could be similar to that in eq 1; and
(3) Append future inputs to the original data set to
make forecasts. This is a crucial step in making
the RTSE flexible enough to incorporate updated
values for the predictor variables obtained from
other forecasting methods.
METHODS
Forecasting systems combining a nonlinear regression
with an ozone-conducive criteria have been recommended
to improve forecasting accuracy.8,21,25 Our research will use
Box-Jenkins multivariate time-series models in combina-
tion with PC analysis.12,13 The special form of the Box-
Jenkins transfer function model, an RTSE model, is
combined with PC analysis. In our proposed RTSE model,
the PC trigger, in addition to meteorological and NOx pre-
dictors, is one of the predictors. Particularly, the PC trig-
ger in the RTSE model is intended to predict ozone above
100 ppb. For regular ozone days, the RTSE model fore-
casts the ozone values as a baseline, and the PC trigger is
turned off. For high-ozone days, the PC trigger is turned
Identify the Structure of Noise Nt. As a benchmark, the ozone
series is first fit with respect to its own history and with-
out using any predictor variables. Then it can be deter-
mined if the inclusion of any of the candidate predictors
listed in Table 1 results in substantial improvements. Ac-
cording to Milionis and Davies,26 with no predictor vari-
able, the model structure of the noise Nt should resemble
that of the response variable. The stochastic structure of
Nt quantifies the amount of dependence in the ozone se-
ries. If the predictor variables (X1,t and X2,t) in eq 1 are not
important, the ARIMA model in eq 1 reduces to Nt only.
1066 Journal of the Air & Waste Management Association
Volume 52 September 2002

Liu and Johnson
parameters. In addition to being parsimonious, a good
ARIMA model has to be stationary. A stationary series has
mean, variance, and autocorrelation coefficients that are
essentially constant through time.12,32 Nonstationary data
often can be transformed to stationary data by using
differencing or transformations. Transformation, such as
taking a logarithm, can transform data to have constant
variance over time.
Conduct an RTSE Model. The three stages of building an
RTSE model are similar to those of developing an ARIMA
model: identification, estimation, and diagnostic check-
ing. The identification and estimation stages may refer to
Step 2 in Figure 3, and the diagnostic-checking stage re-
fers to the decision process “check model’s adequacy” in
Figure 3. This step resulted in a tentative model by deter-
mining its significant parameters. The parameters are
estimated in an iterative way with software that simulta-
neously estimates both the parameters of the regression
and the Nt to achieve a minimum residual variance (or
residual mean square). SAS software can achieve this pur-
pose and was employed in this study.31,33
The diamond shape in Figure 3 indicates that diag-
nostic checking is a decision process that decides whether
to further revise the model. If the tentative model is not
adequate, another model structure should be identified.
This process should be iterated until an adequate model
is achieved. Other predictor values and other noise struc-
tures can be examined. The Ljung-Box statistics and re-
sidual autocorrelation functions are employed primarily
for checking the adequacy of any candidate model.12,32,34
Figure 3. Process of building an RTSE model.
While this noise could be an initial approximation, it is
likely to change when predictor variables are introduced.
After introducing predictor variables, Nt must be updated.
A good ARIMA model is typically statistically adequate
and parsimonious for a fitting given realization of a time
series.31 The principal of parsimony is characterized by
fitting a set of data with the smallest number of estimated
Append Future Inputs to the Original Data Set to Make Fore-
casts. When making forecasts with the RTSE model, the
future values of predictors, such as the forecast tempera-
ture and forecast wind speed, must be obtained from out-
side sources. Future values of predictors need to be
Table 1. Response variable and predictor variables in the ozone model development.
Significant
Response Variable and Candidate Predictor Variables
Notation
Predictor Variable
Ozone
Natural logged peak daily 1-hr ozone concentration (ppb)
lnO3
Tp
Temperature
Peak daily 1-hr temperature (ºF)
Tp
Solar radiation
Average solar radiation from 5:00 a.m. to 8:00 p.m. (MJ/m2)
24-hr averaged dew-point temperature (ºF)
SR
Dew-point temperature
NOx
Dpt
Averaged NOx from 6:00 a.m. to 9:00 a.m. (ppb)
NOx
Morning wind speed
Noon-afternoon wind speed
Morning wind direction
Noon-afternoon wind direction
PC
Vector averaged wind speed from 5:00 a.m. to 10:00 a.m. (mph)
Vector averaged wind speed from 10:00 a.m. to 7:00 p.m. (mph)
Vector averaged wind direction from 5:00 a.m. to 10:00 a.m. (degree)
Vector averaged wind direction from 10:00 a.m. to 7:00 p.m. (degree)
0 or 0.033Tp – 0.052NOx + 0.888WD510 + 0.456WD1019
WS510
WS1019
WD510
WD1019
PC
WD510
PC
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Journal of the Air & Waste Management Association 1067

Liu and Johnson
appended to the original data set. The method for how
the future values will be appended to the original data set
and incorporated into the forecasts is detailed in a com-
panion paper.2
drought summer of 1988 and the associated anomalous
weather conditions.35
The comparison of the two candidate first PCs is listed
in Table 2. The notation for the variables is given in Table
1. Because PC II explained more variance than PC I, PC II
was selected for the forecast model in this study. PC II
explained 85.9% of the variability of the weather and NOx
during the days when ozone was above 100 ppb from
1993–1998, while PC I only explained 77% of the vari-
ability for the high-ozone days during 1987–1998. Because
explaining maximum variance in one data set is no guar-
antee that the component will be the best predictor, a
couple of alternative PCs were also investigated. The first
PC of PC II was replaced with the second PC. There was a
slight deterioration in predicting ability. We also tried
obtaining PC II with temperature and morning vector
average wind direction (WD510) excluded. However, our
visual inspection and their larger estimated variance both
indicate that neither the first nor the second PC of the
changed PC II can predict high-ozone days well.
For parsimony, PC II was simplified as eq 2. It still
explains the variance as well as that in Table 2. Therefore,
PC II is determined to be the PC trigger in the proposed
RTSE model.
PC Analysis
To improve the prediction of extremely high ozone concen-
tration (ozone greater than 100 ppb), a PC analysis is per-
formed. The PC in this study is a linear combination of the
weather and NOx variables during high-ozone days, and it is
anticipated to demonstrate the levels of NOx and weather
conditions associated with high ozone. The resultant PC is
employed as a trigger and actually is included as one of the
predictors in the RTSE model. For days when observed ozone
is greater than 100 ppb, the PC trigger is turned on. This
trigger is calculated according to eq 2. For days with ozone
below 100 ppb, the PC trigger is turned off and set to 0.
Theory. The general objectives of a PC analysis are data
reduction and data interpretation.13 For example, start-
ing with a large number of p variables, PC analysis re-
duces to k variables, which can reproduce almost as much
as the original p variables. The original data set consisting
of n measurements on p variables is reduced to summary
data consisting of n measurements on k principal compo-
nents. Each PC is a linear combination of the original p
variables. The first of the k components reproduces the
most variability and is called the first PC. The first PC of
NOx and the meteorological variables were obtained us-
ing sas princorm, which is the command in the SAS sta-
tistical software package.34
PCt = 0.033 Tpt – 0.052 NOxt +
(2)
0.888 WD510t + 0.456 WD1019t
Database
Monitoring Site Location. The monitoring site was the
WDNR monitoring site located at the University of Wis-
consin, Milwaukee-North (UWM-N). This monitoring site
was determined for the following reasons: (1) the Mil-
waukee area has been designated as a severe non-
attainment area for the 1-hr National Ambient Air Qual-
ity Standards (NAAQS) for ozone since 1991, and (2) the
air quality data at the UWM-N site are complete and of
excellent quality compared with the other monitors.
The UWM-N monitoring station is at an elevation of
750 ft and is located at 43º 04.47’ N and 87º 53.07’ W.
Several large multistoried buildings are on the north side
PC Development. Two PC analyses were conducted from
two different databases. The first candidate PC was com-
puted from days that had ozone above 100 ppb from 1987–
1998. The second candidate was generated from those days
with ozone levels greater than 100 ppb during 1993–1998.
The high-ozone days usually were separated in time, so
we treated the day-to-day variation in their atmospheric
conditions as independent. From 1983 to 1990, improved
emission control measures resulted in a steady decrease
in ozone precursor emissions.35 This
Table 2. PC analysis.
steady decrease continued throughout
the 1990s largely as a result of new con-
trol measures implemented under the
Analysis
Database Year Domain
Proportion of Explained Variance
1990 amendments to the federal Clean
Analysis Ia
Air Act. The 1987–1998 database pre-
PC I = 0.025 Tp – 0.058 NOx – 0.004 Dpt + 0.890 WD510 + 0.014 WS510 + 0.450 WD1019
sents a variety of environments for
high-ozone days, compared with the
1993–1998 database, in which many
of the emission controls have already
been implemented. Also, the 1993–
1998 database excludes the extreme
1987–1998
0.770
Analysis II
PC II = 0.033 Tp – 0.052 NOx – 0.014 Dpt + 0.888 WD510 + 0.014 WS510 + 0.456 WD1019 + 0.009 WS1019
1993–1998
0.859
aThe notations in the component are described in Table 1.
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Volume 52 September 2002

Liu and Johnson
of this monitoring station. A residential area surrounds
the south side of the station. East of the station is a street
with heavy traffic. The west side of the station is sur-
rounded with either residential or urban areas.
The ozone-conducive weather conditions in the Lake
Michigan Ozone Problem Area (see Figure 2) are usually
associated with high temperature, high relative humid-
ity, and southwesterly, southerly, or southeasterly winds
along the Lake Michigan shore.37 In most locations, peak
daily temperature is highly correlated with ozone.15,38,39
High dew-point temperatures are strongly associated with
stagnating anticyclones and provide the moisture favor-
able for photochemical reactions.15,38,39 Generally, low
wind speeds and high solar radiation are significant con-
tributors to high ozone.15,38,39 In addition to the weather
components, NOx serves as an ozone precursor resulting
from traffic emissions during the morning rush hour.36
In a synoptic meteorological situation, southeastern
Wisconsin witnesses warm, humid, and weak southerly to
southwesterly surface winds in the morning of high-ozone
days. Fresh emissions from the morning rush hour and fac-
tories blow eastward over Lake Michigan. During the late
spring and summer months, the surface waters of Lake
Michigan are noticeably colder (by as much as 20 ºF) than
the nearby land. Consequently, air parcels containing both
regional ozone and ozone precursors sink toward the sur-
face under clear skies.40 Thus, as the subsiding air (rich in
VOCs, NOx, and increasing in ozone) descends to near Lake
Michigan’s surface, it slowly begins to spread horizontally
toward the land in all directions. If these lake-based winds
are strong enough to overcome the prevailing (land-based)
synoptic-scale winds, they cause the onshore penetration
of the ozone-laden “lake breeze” that usually occurs by late
morning to mid-afternoon.
Database Determination. Variables analyzed in this research
include ozone, NOx, and meteorological variables. The
temporal domain is the 1991 ozone season. The ozone
season officially is defined by the WDNR as April 15
through October 15, which takes into account all pos-
sible high-ozone days in a year. All of the ozone and NOx
observations were retrieved from the U.S. Environmental
Protection Agency’s (EPA) Aerometric Information Re-
trieval System (AIRS). Meteorological data were obtained
through the State Climatology office and consisted of
surface weather observations made by the National
Weather Service (NWS) at the Milwaukee General Mitchell
International Airport.
The 1991 ozone season is appropriate for this ozone
study because
(1) 1991 is the calendar year having the greatest num-
ber of ozone 1-hr NAAQS exceedances for any year in
Wisconsin during the 1990s. There were a relatively large
number of exceedances observed in 1991 at UWM-N.
From 1991 to 1999, 11 readings at UWM-N exceeded
the 1-hr ozone standard (124 ppb when rounding is
considered). However, six of them were observed in
1991. In 1991, 15 ozone readings were greater than
100 ppb and eight were greater than 120 ppb.
(2) During the summer of 1991, the Lake Michigan
Ozone Study (LMOS) was conducted by the states of
Wisconsin, Illinois, Michigan, and Indiana (see Fig-
ure 2). The LMOS field study yielded a full, rich, and
accurate measurement database, which included
data collected aboard ships in Lake Michigan, by
aircraft, and at special ground-based monitoring
sites. For 1991, considerable relevant information is
available from the LMOS. Above all, the data quality
of 1991 is reliable.
A back trajectory analysis indicated that almost 100%
of all high-ozone days in Wisconsin occurred on days
when the surface winds had a considerable southerly com-
ponent.3 Two general airflow patterns contribute to
elevated ozone concentrations in eastern Wisconsin: a
southerly-to-southeasterly path and a southwesterly
direction, which help the ozone-laden lake breeze to oc-
cur. Consequently, wind variables are represented as two
components to explain the lake breeze effect on ozone
formation. The first wind component for the hourly peri-
ods of 5:00 a.m.–10:00 a.m. underscores precursor emis-
sion transport from the Milwaukee area to the lake in the
mornings, and the second wind component for 10:00 a.m.–
7:00 p.m. underscores ozone transport from the lake to
the Milwaukee area in the late morning and afternoons.
Following Bloomfield et al.,9 for each of the hourly
periods, the two components of wind, v and u, are com-
puted as the summation of the products of hourly wind
speed times the cosine function of wind direction and
as hourly wind speed times the sine function of wind
direction, respectively. Then the v and u components
are calculated by the vector average wind speed and
wind direction specific to the 5:00 a.m.–10:00 a.m. and
Predictor Variable Determination. The response variable and
predictor variables are listed in Table 1. The variables used
in this research were determined based on a comprehen-
sive literature review.7,8,36 Those candidate predictor vari-
ables generally include the four most important variables
pointed out by Comrie:15 daily maximum temperature,
average dew-point temperature, average daily wind speed,
and daily total sunshine. Daily values in this study were
determined from hourly averages rather than from exist-
ing values that have been published. The 24-hr diurnal
variations between ozone and the candidate predictor
variables are displayed in Figure 4. June 26, 1991, was
chosen to represent a typical high-ozone day.
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Journal of the Air & Waste Management Association 1069

Liu and Johnson
Figure 4. Hourly variation for ozone and the candidate predictor variables. Black dots with solid lines are ozone values. The dashed lines are
(a) temperature, (b) dew-point temperature, (c) solar radiation, (d) NOx, (e) wind direction, and (f) wind speed.
10:00 a.m.–7:00 p.m. time periods. One limitation on
our predictor variables is that the hourly average of
VOCs is not available. From a chemical standpoint, the
production of ozone is a function of solar radiation and
of the concentrations of the NOx and VOCs emissions.
Seinfeld and Pandis41 emphasized that the ratio of NOx/
VOCs is also significant. Though NOx and VOCs emis-
sions are precursors of ozone pollution, several studies
cited that emissions are not the dominant factor on
ozone variation. Comrie and Yarnal39 believed that much
of the ozone variation is weather-related and emphasized
the strong dependency of photochemical reactions on
meteorological conditions. Also, Luria et al.42 conclude
that emissions from many major anthropogenic sources
do not fluctuate significantly from day to day during
weekdays. They highlighted that the dominant factors
in ozone production are atmospheric conditions. Even
though, for some studies, emissions were originally
considered as possible predictor variables, they were
finally discarded as not statistically significant.23 There-
fore, it is unlikely that the unavailability of emission
data will prevent useful ozone predictions.
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Liu and Johnson
Table 3. Model statistics of the RTSE model with a PC trigger.
RESULTS
The RTSE Model with and without a PC Trigger
Temperature, WD510, and PC trigger were the resultant
significant predictors out of the nine candidate predictors
in the RTSE model (see Table 1). The idea of turning the PC
trigger on and off is described in the flow chart in Figure 5.
The PC trigger was employed in the RTSE model as one of
the predictor variables, and its calculation is subject to
whether the ozone level is greater than 100 ppb. The
Pearson sample correlation between Opre and Oobs, or its
square R2 = r2(Opre, Oobs), is used as a descriptive measure of
the closeness of fit. Root Mean Square Error (RMSE) also is
used as the performance statistic to evaluate the model.
Three time-series parameters show significant as well
as temperature, WD510, current-day PC, and previous-
day PC triggers in the RTSE model with a PC trigger.
Estimate of
Parameters
Parametersa
Standard Errors
T Ratio
Tpt
0.02
–0.0003
0.001
–0.0006
0.20
0.002
0.0002
0.0002
0.0002
0.080
11.41
–1.94
6.02
WD510t
PCt
PCt-1
θ3
φ1
φ5
–2.81
2.55
0.47
0.065
7.26
0.28
0.065
4.35
aPCt-1 denotes the B back shift so that BPCt = PCt-1. Greek letters denote the parameters
of the noise term Nt (eq 1).
0.899 between fitted and observed ozone. The square cor-
relation coefficient (R2) is 0.809. The scatter plot of the
observed and fitted ozone is displayed in Figure 6.
Eq 3 is the form used for forecasting in a related pa-
per.2 The PC trigger will be calculated only when a high-
ozone day is predicted. The RTSE model with a PC trigger
shows a significant capability of predicting extremely high
ozone. The improvement in estimating high ozone, which
results from adding a PC trigger to the RTSE model, can
be highlighted in comparison with an RTSE model with-
out a PC trigger. Figures 7 and 8 show the significant dif-
ference between the two RTSE models with and without
the PC trigger.
lnO3t = 0.52 + 0.02Tpt – 0.0003WD510t +
(0.001 + 0.0006B)PCt +
(3)
(1 – 0.20B3) at/(1 – 0.47B – 0.28B5)
The statistics of this model are addressed in Table 3. After
predicting on the ln-scale, each forecast was transformed
to the original scale by using eq 4, and r, R2, and RMSE
were calculated based on the original scale.
1
Opre = exp(Opre + S.D.× S.D.)
(4)
2
where S.D., the standard deviation of the residuals,
equals 0.223.
The RTSE model without PC resulted in only one sig-
nificant variable, temperature, which is frequently recog-
nized as the most important variable in ozone modeling.
The resulting correlation coefficient is 0.825, the R2 is
ˆ
σ
The RMSE (
) is 12.2. The RTSE with a PC trigger model
at
indicates a fairly good fit, with a correlation coefficient of
Figure 5. Process of fitting the 1991 ozone season with the RTSE
Figure 6. The RTSE model with the PC trigger, scatter plot of observed
model with a PC trigger.
and fitted ozone.
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Journal of the Air & Waste Management Association 1071

Liu and Johnson
Figure 7. The RTSE model with the PC trigger fitting 1991 ozone
series, Milwaukee.
0.680, and the RMSE is 15.6, on the original scale. The
model is described in eq 5.
lnO3t = 0.70 + 0.03Tpt +
(5)
at/(1 – 0.55B + 0.16B3 – 0.26B5)
Figure 9. The RTSE model without the PC trigger, scatter plot of
observed and fitted ozone.
Figure 8 shows that prediction of ozone using eq 5 has
the common difficulty of being unable to predict high
ozone. This is particularly true for ozone greater than
100 ppb. The scatter plot of the observed and fitted ozone
is shown in Figure 9. Table 4 lists the statistics of the RTSE
model with and without PC for the entire ozone series.
The number of predicted days with ozone above 100 ppb
is apparently different between the two approaches. The
RTSE model with PC predicted 12 out of 15 observed days
with ozone above 100 ppb, but the RTSE model without
PC predicted only five of the 15 days. Box and Jenkins12
suggest using a 50% forecast interval instead of a 95%
forecast interval when making forecasts with time-series
models. The RTSE model with PC has a narrower 50%
forecast interval (8.2 ppb) than that of the RTSE model
without PC trigger (10.5 ppb).
real-time forecast for the summer of 1997. When fore-
casting real-time ozone, the result is limited greatly by
the accuracy of weather forecasts that are major inputs to
the model.46 In other words, Chang and Cardelino16 must
have dealt with significant variability among the weather
forecasts. Most of the remaining studies used different data
sets to validate their models.
CONCLUSION
The proposed RTSE model with a PC trigger was applied
to fit historical data. The results suggest that the problem
of underpredicting high ozone has been reduced. To con-
struct a useful ozone model, the following considerations
should be addressed:
Table 5 is a comparison between this research and
the other relevant studies. Compared with the existing
ozone models, R2 of the RTSE model with PC seems to be
relatively high, although the R2 produced in this study
was based on the same data that were used to build and
validate the model. Chang and Cardelino16 performed a
(1) Ambient air conditions, especially weather con-
ditions that cause extremely high ozone, are
different than those that cause normal ozone
levels. Predicting extremely high ozone must
be conducted differently than predicting nor-
mal ozone levels.
(2) Time-series correlation among ozone and other
predictors cannot be ignored. A multivariate time-
series model can describe persistence among
ozone, ozone precursors, and meteorological fac-
tors. It also helps in predicting next-day ozone
based on appropriate statistical assumptions.
(3) Statistical models have the advantage of easy
operation, and they are relatively inexpensive
and less time-consuming. Quantitative predic-
tions can be generated with statistical models,
and a forecast interval can be provided for each
prediction.
Figure 8. A pure RTSE model without PC trigger fitting 1991 ozone
series, Milwaukee.
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Liu and Johnson
Table 4. Statistics of RTSE model with and without PC trigger.
Predicted O3 ≥
50% Forecast
Interval
Corr. Coeff.
R2
RMSE
100 ppb Daysa
RTSE with PC
0.899
0.825
0.809
0.680
12.2
15.6
12
5
8.2
RTSE without PC
10.5
aThe total number of days when ozone was greater than 100 ppb was 15.
Table 5. Ozone study review based on R2 results.
3. Attainment Demonstration for Ozone for the Year 2007, The Phase 3 At-
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Source
R2
Method
Chang and Cardelino16
Chen et al.43
0.38
0.45
UAM-FM
Multidimensional phase space model
Neural network using unlagged variables
Loess/generalized additive model
Comrie15
0.70
Davis and Speckman44
0.61–0.68
Hubbard and Cobourn8 0.705–0.818 Hi-Lo + baseline hybrid regression model
Liu45
0.680
0.608
0.809
RTSE model without a PC trigger
Bivariate time-series model
RTSE model with a PC trigger
Simpson and Layton24
This study
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casting and Control, 3rd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1994.
13. Johnson, R.; Wichern, D.W. Applied Multivariate Statistical Analysis;
Prentice Hall: Englewood Cliffs, NJ, 1992.
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16. Chang, M.E.; Cardelino, C. Application of the Urban Airshed Model
to Forecasting Next-Day Peak Ozone Concentrations in Atlanta, Geor-
gia; J. Air & Waste Manage. Assoc. 2000, 50, 2010–2024.
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18. Adamski, W. An Analysis of Measured and Predicted Concentrations Aloft
of Ozone and NO2 in the Lake Michigan Airshed during 12–14 July, 1995;
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sources: 1997.
19. Rao, S.T.; Zurbenko, I.G. Detecting and Tracking Changes in Ozone
Air Quality; J. Air & Waste Manage. Assoc. 1994, 44, 1089–1092.
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22. Jorquera, H.; Perez, R.; Cipriano, A.; Espejo, A.; Letelier, M.V.; Acuna,
G. Forecasting Ozone Daily Maximum Levels at Santiago, Chile;
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The PC trigger described in eq 2 underlines the linear
relationship among the meteorological and NOx condi-
tions for those days with ozone greater than 100 ppb.
Using the PC trigger to improve high ozone predictions is
an easy and useful approach. Adding the PC trigger to the
RTSE model raised the R2 from 0.680 to 0.809. For 15 days
with ozone above 100 ppb, the RTSE model with PC trig-
ger predicted 12 days, while the RTSE model without PC
trigger only predicted five days.
The RTSE model with a PC trigger indicated its po-
tential to predict normal and high ozone levels. The op-
eration of this model to real-time forecasting will be
detailed in a related paper.2 Several PC trigger rules, which
can determine the turning on or turning off the PC trig-
ger, are developed in that paper.2
ACKNOWLEDGMENTS
The authors would like to extend our appreciation to the
Wisconsin Department of Natural Resources for funding
this publication, providing the monitoring data and con-
structive information, and funding this project. Pao-Wen
G. Liu also thanks Professor Paul M. Berthouex, Univer-
sity of Wisconsin, Madison, for his advice and Larry Bruss,
WDNR, for his input in this research.
23. Robeson, S.M.; Steyn, D.G. Evaluation and Comparison of Statistical
Forecast Models for Daily Maximum Ozone Concentrations; Atmos.
Environ. 1990, 24B, 303–312.
24. Simpson, R.W.; Layton, A.P. Forecasting Peak Ozone Levels; Atmos.
Environ. 1983, 17, 1649–1654.
25. Prior, E.J.; Schiess, J.R.; McDougal, D.S. Approach to Forecasting Daily
Maximum Ozone Levels in St. Louis; Environ. Sci. Technol. 1981, 15,
430–436.
26. Milionis, A.E.; Davies, T.D. Regression and Stochastic Models for Air
Pollution—I. Review, Comments and Suggestions; Atmos. Environ.
1994, 28, 2801–2810.
REFERENCES
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2. Liu, P.W.; Johnson, R. Forecast Peak Daily Ozone Levels—II. Using A
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27. Milionis, A.E.; Davies, T.D. Regression and Stochastic Models for Air
Pollution—II. Application of Stochastic Models to Examine the Links
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versions; Atmos. Environ. 1994, 28, 2811–2822.
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that counts down to the day of t – p. Equation A.1 can
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of Nonlinear Regression and Neural Network Models for Ground-Level
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Institute Inc.: Cary, NC, 1991.
1 − φ1B − φ2 B2 − ... − φpBp Xt = at
(A.2)
(
)
where B is the backshift operator [i.e., BkXt = Xt-k, and
(1–Bk)Xt = Xt – Xt-k for k = 1, .., p].
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and Cases; John Wiley & Sons: New York, 1983.
33. Liu, L.; Hudak, G.B.; Box, G.E.P.; Mervin, E.M.; Tiao, G.C. Forecasting
and Time Series Analysis Using the SCA Statistical System; Scientific
Computing Associates: Dekalb, IL, 1992.
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of Order p, q [ARMA(p, q)]
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Xt = φ1Xt−1 + φ2Xt−2 +...+ φpXt−p
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+ at − θ1at−1 − θ2at−2 −...− θqat−q
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(1− φ1B − φ2B2 −...− φpBp)Xt
(A.4)
= (1− θ1B − θ2B2 −...− θqBq)at
In an ARMA(p, q) model, the autoregressive (AR) term
has the order p and the moving average (MA) term has
the order q. The AR term is on the top line of eq A.4 and
the MA term is on the bottom line of eq A.4.
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Concentrations: Nonparametric Short-Term Prediction; Atmos.
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Average Ozone in Houston; Atmos. Environ. 1999, 33 (16), 2487–2500.
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May 2000.
APPENDIX—TIME SERIES AUTOREGRESSIVE
MOVING AVERAGE (ARMA) MODEL
To illustrate typical classes of time-series models cited in
this paper, AR(p) and ARMA(p,q) models are explained as
follows.
About the Authors
Pao-Wen Grace Liu is an air pollution control specialist
at the Bureau of Air Management, Wisconsin Department
of Natural Resources. She received her Ph.D. from the
University of Wisconsin, Madison, by completing this
ozone-forecasting project. Richard Johnson is a professor
of statistics at the University of Wisconsin, Madison. The
corresponding author is Pao-Wen Grace Liu, e-mail:
Pao-Wen.Grace@dnr.state.wi.us.
Autoregressive Process of Order p [AR(p)]
Xt = φ1Xt −1 + φ2 Xt −2 + .... + φpXt −p + at
(A.1)
φ1,...,φp
where at is white noise and
are parameters which
must lie within certain limits. Here p stands for the lag
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