Composite Air Quality Indices: A Brief Illuminating Discussion
The brief illustrational discussion presented in this section will enable us to justify the axioms proposed in the earlier section from a broader perspective in the sense that many of the existing composite measures are shown to satisfy at least one of the axioms. In fact, this will also provide a background for the logical comparison of our axiom-based quantitative evaluators with several alternative analytical proposals.
We begin our analysis with a discussion of the symmetric mean of order в of EPA single-pollutant indices, under the general assumption that there are n such contaminants. Formally, for any IEPA e 2?'|, this composite AQI is defined as
where в > 1 is a parameter. This symmetric, normalized index satisfies monotonicity and monotonicity sensitivity for all в > 1. In fact, monotonicity sensitivity holds if and only if 9 > 1. As the value of в increases from 1, this aggregator assigns higher weights to the pollutants that are more harmful to human living conditions. Since air pollution is socially highly undesirable, this characteristic is a quite sensible property.
As в —> ос, Iq approaches maxyesJ){2;Ep/'}, the EPA maximax global index. Hence Iq may be regarded as a simple parametric generalization of the EPA global AQI. In the next section of the paper we axiomatize this generalized ЕРА index using a set of intuitive postulates, some of which have been introduced in Section 7.3.
Swamee and Tyagi (1999) proposed the use of the symmetric aggregate of order в of single dimensional indices s,- = sj ^)’" as an indicator of global air quality, where q, is the level of concentration of pollutant i e Q, qs is a threshold concentration limit for the pollutant, ss is a scale factor and m is a constant. Formally, for any seD", the overall air quality quantifier is defined as
where D" is the strictly positive part of the и-dimensional Euclidean space and в is a constant. Positivity of all components of s is necessary for the symmetric composite AQI Iq to be well defined. This is because for в < 0, IfJ becomes undefined if s, = 0 for one or more i e Q. Evidently, as in Iq, for simultaneous satisfaction of monotonicity and monotonicity sensitivity we need в > 1. Monotonicity sensitivity is violated for в < 1, although it is monotonic. However, for в < 1, as the value of в reduces, the less harmful pollutants are given more importance in the aggregation. For в = 0 the normalized coun-
terpart ) of Iq becomes the geometric mean of individual indices and as в -¥ —oo it approaches minj€Q{s;}, the maximin composite AQI. (See the next section for further discussion and characterization.)
Kyrkilis et al. (2007) suggested a variant of Iq under the assumptions that the constant m takes on the value 1 and в > 1. As a result, this symmetric composite AQI is unambiguously monotonic and monotonically sensitive. As
l
в -> oo, the normalized ordinally equivalent sister (%^-) of the proposed index approaches maxj€Q{s;}, the maximax global AQI. (See also Khanna (2000) and Zhou et al. (2006).)
The global AQI suggested by Gurjar et al. (2008) relies on the combined effects of three pollutants, NO2, SO2 and PM, using the WHO guidelines as norms. The objective of the recommended global AQI was to measure air quality in megacities. More precisely, this overall AQI is a summary measure of deviations of concentrations of these three pollutants from the corresponding WHO norms. In the general case for n pollutants, it is formally defined as
where the vector q = (q 1,q2,...,qn) € D" of atmospheric concentrations of n pollutants is arbitrary and q, > 0 is the WHO recommended threshold for pollutant 1,1 < i < n. The initial formulation of this overall AQI was expressed as the sum of Igiq) and a similar expression for atmospheric emission (per year,
per capita etc.) along with substitution of £ by ^ for the overall sum. Since data on the WHO guidelines for atmospheric emissions were not available, the second component was given up. Assuming that (®^Д) may be regarded as the AQI for pollutant i e Q, the global AQI Iq can certainly be regarded as a composite AQI. This symmetric, monotonic composite AQI takes on the value 0 if the WHO guidelines are respected across contaminants. The straight-line iso-pollution contour associated with the composite AQI Iq makes the underlying pollutants perfect substitutes. (A modified version of the EPA global AQI that can be interpreted as a summary measure of single-pollutant indices, where breakpoints are based on the EC directives and the WHO guidelines, was suggested by Murena (2004).)
The Oak Ridge National Laboratory suggested a methodology for measuring air quality when the number of pollutants may be greater than 1. The Oak Ridge AQI (ORAQI, for short) is defined as
where q, and q* denote respectively the monitored/predicted concentration and the national ambient air quality standard for the ilh pollutant; and a and b are constants. Given positivity of qsjr a and b, an increase in the value of <7, increases IqR. Monotonicity sensitivity is satisfied if b > 1. Under positivity of q*, a and b, the indicator takes on the value zero if each <7, is zero (see Thom 1976).
An index based on mortality relative risk values for different contaminants was developed by Cairncross et al. (2007). The global AQI f^(q) is defined as the sum of transformed mortality risks. Formally,
where q e Rn+, whose zth component q, is the time-averaged concentration of pollutant i e Q, is arbitrary and a, is a number directly proportional to the incremental relative risk coefficient (RR, - 1) corresponding to pollutant i € Q. (These coefficients for different pollutants were computed in the paper of Cairncross et al. (2007).) This composite AQI is monotonic under the assumption that д, > 0 for all i e Q. It is normalized if Xw=i«/ = 1- Evidently, this AQI, which is a summary measure of pollution-based incremental relative risk of daily mortality, can be treated as a composite indicator. Thus, the measure looks at the harmfulness of air pollution from a different perspective.
Sharma et al. (1991) suggested a measure with the objective of determining depreciation in air quality. The index is bounded between -10 and 0, where the lower bound —10 is achieved if air quality is the worst with respect to pollutants that are taken into account. On the other hand, the upper bound
0 indicates the best air quality. This quantifier of air quality is formally defined as
where I e [0,1]" is the non-negative vector of air quality indices for the n contaminators considered and [0,1]" is the и-fold Cartesian product of the closed set [0,1]. Each /, varies between 0 and 1, which represent respectively the worst and the best air qualities due to the it It pollutant. The positive scalar CW, is the composite weight assigned to the ith pollutant index. It is defined as
where TW„ the total weight of the ith pollutant is given by the sum of the esthetic, biophysical impact and health weights of the ith pollutant. Each of these three-component weights of TW, is assigned a value lying between
1 and 5 by an expert group.
Given QV,s, I® takes on the minimum value -10 if /, is 0 for each pollutant i. This indicates the highest level of depreciation. In contrast, the minimum depreciation level, zero, is demonstrated if I, = 1 for all i. If not all components of the vector I e [0,1]" are zero, then will signify non-zero depreciation.
