Single-Pollutant Air Quality Indices: An Illustrative Discussion

We begin this section by discussing the EPA single-pollutant index. In the EPA framework, the range of [0,500] of 1-hour and 8-hour concentrations for each pollutant is split up into six sub-intervals [0,50], [51,100], [101,150], [151,200], [201,300] and [301, сю). These sub-intervals indicate the following categories of emission respectively: “good," "moderate," "unhealthy for sensitive groups," "unhealthy," "very unhealthy" and "hazardous," as specified in a table (http://www.epa.gov/ttn/naaqs/standards/nox/fr/20100209.pdf p. 65).

Let S_p be the set of all pollutants considered by the EPA. That is, S_f, = {CO,NO2O3, SO2, PM}. The index for pollutant j, I^^PA can now be calculated using linear interpolation between the six sub-interval borders as follows:

where C|^;,M stands for the level of emission of pollutant j e Sp, BPf^PA and BP[ q^A are respectively the breakpoint that is greater than or equal to C^EPA and the breakpoint that is less than or equal to C^PJ>A and lf^PA and l[^PA are respectively the AQI values associated with BPf^pA and BPf^PA. We can interpret I^EPA as an affine transformation of a normalized value of the excess (C^PPA - BP[q^A) concentration of pollutant j over the corresponding pre-specified lower bound BP[£^A. An increase in the value of [^P:PA indicates a higher level of pollution generated by pollutant j. We denote the vector of I^EPA values, where j € S_p, by ^EPA. As stated earlier, in the EPA framework an overall air quality indicator AQI^PPA is defined by AQIq^>a —> maXj£s_p{lf^PA}, the maximax index of air quality. (A very interesting application of the EPA methodology for determination of air quality in Tehran can be found in Motesaddi et al. (2017). See also Nigam et al. (2015) for a comparative study assessing the status of air quality in Nagpur using this index and some other indices.)

By concentrating on the pollutant that emits the maximum pollution, the quantifier ignores all those pollutants that may also be harmful to human well-being. Evidently, the choice of the maximax criterion is one possible way of determining overall AQI. However, this method of determining global AQI ignores pollutants whose health impacts may also be bad. For instance, suppose in a population the highest AQI is associated with a pollutant that corresponds to the hazardous category. But it is noted as well that the society is afflicted by some more pollutants that relate to the categories very unhealthy and unhealthy. Consequently, it is sensible to argue that a global AQI should incorporate the impacts of such pollutants also. In Section 7.4 of the paper we analyze some such recommendations made in the literature.

Mayer et al. (2002, 2004) suggested an impact-related index based on the daily integral air quality. For each pollutant j e Sp, a linear interpolation between single index classes is defined as follows:

where С$$_т (j) stands for the daily maximum one-hour concentration of j e {NO2O3, SO2}, for CO and PM10 indicates respectively the daily highest eight-hour running mean concentration and the daily mean concentration; Cfjp(j) and C'f^A(j) are respectively the upper and lower thresholds of concentrations of pollutant j (specified in a table with six classes); and TO) and Iff(j) are respectively the AQI values corresponding to Cfj^A(j) add Clo(/)- The global air quality is then defined as the maximum among the five pollutants considered.

One common property of all single-pollutant indices is scale invariance, that is, that they are expressed as unit-free real numbers. To explain in greater details, let us consider lf^PA as an illustrative example. First, note that all contaminant concentrations are unit-free real numbers. For instance, lead is conveyed as micrograms per gram. As a result, all the six components of I^^PA are unit-free. This in turn makes I^PPA scale invariant. The scale invariance property enables us to make inter-pollutant comparisons of pollutions. This postulate is necessary to make statements like whether the impact of one pollutant on air quality is worse than that of another.

As stated in the introduction, our discussion here is not exhaustive, rather it is illustrative. For alternatives and variants of the above measures, see the review articles by Plaia and Ruggieri (2011) and Mandal and Gorai (2014).

Axioms for a Composite Air Quality Index

Our discussion in this section will be quite general in the sense that the axioms will be presented in terms of an arbitrary number of pollutants n. Let Q = {1,2,..,«} be the set of pollutants. For any j e Q, let C_; e R+ be level of concentration of the pollutant j, where R^]+ is the non-negative part of the real line R¹. We write £• € Dh for the vector of endowments associated with the pollutant j, where D'b is the strictly positive part of the R®, the qj dimensional Euclidean space. For instance, for I^PPA defined in (7.1), Щ = {lf^PA, lf^PA, BP[q^A, BP^ppa). Hence for any j e Q, the corresponding AQI Ij is a non-negative real-valued function defined on R¹. x D'b, that is, Ij : R!|_ x D'b —> R^l+. For any (C_;, £₍) e R+ x D'b, I,(C_t, £₍) indicates the extent of air quality that relates to the concentration of the pollutant j.

Since for any j e Q, L is a non-negative real number, the vector I = (I,h, ■ ■ ■,In) is an element of R'|, that is, f e R’|. A composite AQI f is a nonnegative real-valued function whose arguments are single-pollutant AQIs. More precisely, f : R'{_ -t R^l+. For any I € R'|, F(I) gives us a comprehensive picture of overall air quality in a population in terms of a real number. Essential to this formulation is the axiom "Independence of Irrelevant Information," which says that in measuring global air quality all information other than air quality levels of individual pollutants are ignored. This postulate parallels Sen's (1977) "welfarism," which says that all non-utility features should be ignored in the welfare evaluation of a population.

We now propose some axioms for the general composite AQI F. The first axiom we put forward is symmetry. It makes the AQI anonymous in a particular way. This means that the AQI f should remain invariant under reordering of the single-pollutant indices. For instance, if we have only two pollutants, say, CO and РМю, we want to make the indicator insensitive to whether CO or PM₁₀ is labeled pollutant 1 or 2 in the determination of overall AQI. An analogous axiom is extensively used in the literature on welfare evaluation. We state the postulate formally as:

Symmetry: For all I e R+, F(I) = F(IP), where P is any n x n permutation matrix, an n x n matrix with entries 0 and 1 and each of whose rows and columns sums to 1.

The next axiom we consider is monotonicity. Monotonicity says that if the AQI value of any pollutant increases, say, because of an increase in its concentration, then global pollution increases. Since pollution is a social bad, instead of being a good, an increase in the magnitude of pollution, as conveyed by the corresponding pollutant's AQI, should indicate an upward trend of overall pollution. This desideratum may be regarded as the air quality counterpart of the "Strong Pareto Principle," a requirement under which social welfare increases if any individual's utility goes up. This quite natural axiom is formalized below as follows.

Monotonicity: For all 1 = {I,h, ■ ■ • Ли) € R”₊ and for any i e Q,

F(I) < F(I,I,_i, I, + c, I_i+1, where c > 0 and (h,+

c,i_i+€ Rⁿ+.

In the monotonicity axiom, we are concerned only with AQI of only one contaminant. However, it is likely that AQIs of one or more contaminants change simultaneously. To understand this, consider two different contaminants of which the AQI of the former is higher than that of the latter. Given this, since pollution is a social bad, it is quite logical to argue that an increase in the AQI of the former, identified with higher pollution, will have more detrimental effect on human well-being than that caused by an equal increase in the AQI of the latter with lower pollution. Our next axiom, which may be regarded as the air-quality sister of the poverty "monotonicity sensitivity axiom" of Kakwani (1980), puts forward this idea (see also, Lasso de la Vega and Urrutia (2012) and Chakravarty (2018)). Since two pollutants are directly considered in the formulation of this axiom, it also represents the view that air pollution is a relative phenomenon. The axiom can be analytically stated as:

Monotonicity Sensitivity: For all I = (h,l2, ■ ■ • ,P-i,T, Р+ъ • • ■ h₊1.....M € R'| and for any i,j e Q, where i,j € Q, if I,- > Ij, then:

where

To illustrate the above postulate more elaborately, suppose 2, and 2_; match up respectively with the groupings hazardous and very unhealthy. Then, an equal increase in 2, and 2, means that the society's vulnerability to both the hazardous and more unhealthy headings goes up. Evidently, the disastrous impact of the former on human living is likely to be more than that of the latter. Our monotonicity sensitivity axiom claims this.

The final axiom we introduce for a composite AQI is normalization, which expresses the composite AQI F as an average of pollutant-wise AQI. Technically,

Normalization: If 2 e R'+ is such that 2, = c > 0 for all i e Q, then F(2) = c.

However, normalization should not be used to prefer one composite AQI over another. This is because it may be possible to convert a nonnormalized index to its normalized counterpart by taking suitable transformation so that all the other properties of the original nonnormalized index remain unaltered. This is clearly established in the next section.