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Modelling PM2.5 with Fuzzy Exponential Membership

Written By

Danni Guo

Submitted: 15 October 2014 Published: 21 October 2015

DOI: 10.5772/59617

From the Edited Volume

Current Air Quality Issues

Edited by Farhad Nejadkoorki

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1. Introduction

Air quality is a public and environmental issue that concerns people, whether in terms of global climate change or for health and the quality of life. The Environmental Protection Agency (EPA) regulates six primary air pollutants: Ozone, Particulate matter, Carbon Monoxide, Nitrogen Oxides, Sulfur Dioxide, and Lead [3]. Particulate matter (PM) refers to solid particles and liquid droplets found in air. Many manmade and natural sources produce PM directly, or produce pollutants that react in the atmosphere to form PM [9]. PM2.5 are small particles or particulate matter, that are less than 2.5 micrometers in diameter. PM2.5 can be produced by combustion from motor vehicles (esp. diesel powered buses and trucks), power plants, residential wood burning, forest fires, agricultural burning, and industrial processes [15]. They can also be formed in the air through when gases (air poullutants) and orgnaic compounds are transformed through chemical reactions.

These tiny particles PM2.5 can cause health hazards for people and also for the environment. People with heart diesease and lung problems including asthma, and also the elderly and children, are particularily vulnerable and at high risk when exposed to high levels of PM2.5. Due to the tiny size of these PM2.5 particles, they can penetrate to the deepest parts of the lungs, which is very dangerous to the human health. The California Air Resources Board (CARB) scentifically conducts studies and reports on the impacts of air pollutant exposures on public health, and the studies shows the negative impacts of PM2.5 which is known to cause premature death [2]. Scientific studies has repeatedly found links between particulate matter and many health problems of people who has been exposed to high levels of PM2.5, including asthma, bronchitis, respiratory problems, including shortness of breath and painful breathing, and premature deaths [3].

One must also be aware of the effects of high levels of PM2.5 on the environment as well. Since PM2.5 are tiny, they can be carried by wind and travel great distances, so that it can cause problems for areas downwind far from the actual source of air pollution. They have adverse effect on urban areas, agriculture, and the natural envrinonment. High levels of PM2.5 can results in visibility problems, urban haze, and acid rain [3].

The U.S. Environmental Protection Agency has established standards requiring the annual average of the PM2.5 to be not more than 15 micrograms per cubic meter [3]. The State of California monitors and reports on their air pollutants carefully, setting very high standards for their air quality (μg/m3). From 1999-2011, there are 113 station locations monitoring PM2.5. The site design originally planned was well spread statistically. See Figure 1. However, in reality, it is too costly in terms of time, finance, and manpower to keep all the 113 sites to be monitoring and recording every single year. Each year, only a part of the 113 sites were actually sampled, and each year at different locations.

Figure 1.

Complete 113 PM2.5 Observational Sites in the California State

Comparisons of PM2.5 between the years are difficult, due to "missing data" at sample sites [6, 9]. A site that does not have a recorded PM2.5 value is referred to as "missing value", and since there are no patterns so that serious problems would twist the kriging map constructions.

Figure 2.

PM2.5 Samples Collected in California in 1999 (11 sites) and 2011 (52 sites)

Observing the dataset in Figure 2, the worst (in 1999) only 11 sites (9.73% of 113 sites) were used and at the best (in 2009) 65 sites (57.52% of 113 sites) were used. Over 13 years, 1469 annual arithmatic means (μg/m3) should be recorded, but actually, 556 annual arithmatic means (μg/m3) were reported, which occupied 37.85%. Sitewise looking, only one site, Site 2596 (Placer County APCD), was collected data annually and had 13 recored annual arithmatic means (μg/m3), while 16 sites had one annual arithmatic mean (μg/m3) only. The comparisons of PM2.5 annual arithmatic means (μg/m3) between years for a given site or between sites for a given year, i.e., the investigations of PM2.5 annual arithmatic means (μg/m3) patterns will be an extremely difficult task due to data incompleteness. Therefore it is logical to engage fuzzy theory for treating the "missing" or scarce data.


2. Literature review and methodology

2.1. Fuzzy theory and membership kriging approach

Zadeh's fuzzy theory [22, 23] poineered a new mathematical branch. His membership approaches were quickly spreading and merging into many other mathematical branches, for example, engineering, business, economics, etc. and generating huge impacts in mathematical theories and applications. But it is aware that associated with Zadeh's fuzzy mathematical achivements, researchers gradually discovered three fundamental issues: self-duality dilemma, variable dilemma, and membership dilemma. Guo et al. [8] discussed those dilemmas in detail and pointed out Liu's credibility measure theory [11] is a solid mathematical treatment to address fuzzy phenomenon modelling. The credibility measure, similar to probability measure, assumes self-duality. Consequently, parallel to probability theory, a fuzzy variable and its (credibility) distribution can be defined. Furthermore, the membership function of a fuzzy variable can also be specified by its credibility distribution. Without any doubts, credibility measure theory is applying to practical situations sucessfully, say, Peng and Liu [14] considered parallel machine scheduling problems with processing times, Zheng and Liu [24] studied a fuzzy vehicle routing optimization problem, Guo et al. [2008] proposed credibility distribution grade kriging for investigation California State PM10 spatial patterns, Wang et al. [20, 21] investigated a fuzzy inventory model without backordering, Sampath and Deepa [16] developed sampling plans containing fuzziness and randomness, and others. Nevertheless, the investigations in statistical estimation and hypothesis testing problems with fuzzy credibility distribution are very slowly progressed, for example, Li et al. [10], Sampath and Ramya [17, 18] in worked with fuzzy normal distributions, and studied the exponential credibility distribution function [1].

With statistically well-designed scheme, the collected spatial data can be analyzed and presented by kriging maps. If we are facing spatial data with "missing" or scarce fuzziness, it is impossible to construct kriging maps. It is noticed that the air pollutants were measured in different sites each year, even the site design originally planned was well spread statistically. We call a site that does not have a recorded PM2.5 concentration as "missing value" site, as continued from Guo's research [4, 5]. To address the basic requirements in constructing kriging maps, Guo first proposed membership kriging in Zadeh's sense, see the MSc thesis [4] in which the linear, quadratic and hyperbolic tangent membership functions were used. Later Guo [5, 7] had developed the membership kriging under credibility theory. Following the membership kriging route of treating uncertainty, Shada et al. [19] and Zoraghein et al. [25] made considerable contributions in their papers. In this chapter, we will integrate exponential membership and kriging, to fill in the "missing data" based on existing sample data, and making a comparison of PM2.5 concentrations in California from 1999-2011.

2.2. Fuzzy exponential distribution

It is necessary to introduce the basics of Liu's fuzzy credibility theory [11]. Let Θ be a nonempty set and M a σ- algebra over Θ. Elements of M are called events. Cr{A} denotes a number or grade associated with event A, called credibility (measure or grade). Credibility measure Cr{} satisfies the axioms normality, monotonicity, self-duality and maximality:

Cr{Θ}=1Cr{A}Cr{B} for ABCr{A}+Cr{Ac}=1 for any event ACr{iAi}=supi(Cr{Ai}) for any events Ai with supi(Cr{Ai})<0.5E1

Definition 1: The set function Cr{} on M is called a credibility measure if it follows Axiom Normality, Axiom Monotonicity, Axiom Self-Duality and Axiom Maximality shown in Equation 1. The triplet (Θ, M ,Cr) is called a credibility space.

Definition 2: A measurable function,η, mapping from a credibility space (Θ, M,Cr) to a set of real numbers ᾣ 𝔹 ⊂ ℝ.

Definition 3: The credibility distribution Ψ of a fuzzy variable η on the credibility measure space (Θ, M,Cr) is Ψ: ℝ [0,1], where Ψ(x) = Cr{θ∈Θ|η) x}, x ∈ ℝ.

Definition 4: The function μ of a fuzzy variable η on the credibility measure space (Θ, M,Cr) is called as a membership function: μ(x)=(2Cr{η=x})1,  x.

Theorem 5: (Credibility Inversion Theorem) Let η be a fuzzy variable on the credibility measure space (Θ, M,Cr) with membership function μ. Then for any set B of real numbers,


Corollary 6: Let η be a fuzzy variable on a credibility measure space (Θ, M,Cr) with membership function μ. Then the credibility distribution Ψ is

Ψ(x)=12(supyxμ(y)+1supy>xμ(y)), for x.E3

It is obvious the concept of credibility measure is very abstract although the credibility measure has normality, monotonicity, self-duality and maximality mathematical properties. The credibility measure loses the intuitive feature as Zadeh's membership orientation. Credibility Inversion Theorem and its corollary have just revealed the deep link between an abstract measure and intuitive membership. Such a link definitely paves the way for real-life applications. For example, the trapezoidal fuzzy variable has membership function μ:


where c, c1, c2 are all positive200, c > c1,c > c2, c2 > c1.

With the help of Equation 3, the fuzzy credibility distribution is thus,


Liu [12, 13], Wang and Tian [21] and [1] studied the exponential fuzzy distribution with a membership function, denoted as Exp(m):


The support of an exponential membership function is set [0,+∞), the nonnegative part of the real line, ℝ. The expected value and second moment of exponential fuzzy variable are




where the parameter m > 0 determining the mean and variance of the exponential fuzzy variable. Thus it is intuitive to reveal how the shape of membership curve affected by the possible values of the parameter m > 0.

x m=12 m=15 m=22 m=50
0 1.0000 1.0000 1.0000 1.0000
1 0.9466 0.9573 0.9709 0.9872
2 0.8935 0.9147 0.9418 0.9744
3 0.8410 0.8724 0.9128 0.9615
4 0.7894 0.8306 0.8839 0.9487
5 0.7390 0.7894 0.8553 0.9360
6 0.6899 0.7490 0.8269 0.9232
7 0.6424 0.7094 0.7987 0.9105
8 0.5968 0.6707 0.7709 0.8978
9 0.5530 0.6332 0.7435 0.8851
10 0.5113 0.5968 0.7165 0.8724
11 0.4717 0.5616 0.6899 0.8598
12 0.4342 0.5277 0.6638 0.8473
13 0.3990 0.4952 0.6382 0.8348
14 0.3660 0.4640 0.6132 0.8223
15 0.3351 0.4342 0.5887 0.8100
16 0.3063 0.4059 0.5647 0.7976
17 0.2796 0.3789 0.5414 0.7854
18 0.2549 0.3533 0.5187 0.7732
19 0.2320 0.3292 0.4966 0.7610
20 0.2110 0.3063 0.4752 0.7490
21 0.1917 0.2848 0.4544 0.7370
22 0.1739 0.2645 0.4342 0.7251
23 0.1577 0.2455 0.4147 0.7133
24 0.1428 0.2277 0.3959 0.7016
25 0.1293 0.2110 0.3777 0.6899
26 0.1170 0.1954 0.3602 0.6784
27 0.1057 0.1808 0.3433 0.6669
28 0.0955 0.1672 0.3270 0.6556
29 0.0863 0.1546 0.3114 0.6443
30 0.0779 0.1428 0.2964 0.6332

Table 1.

Impacts of m in the Shape of Exponential Membership Curve

The value choice of parameter m is not aimless. m = 22 corresponds to the PM2.5 annual arithematic mean 11.85 (μg/m3), while m = 50 corresponds to the PM2.5 annual arithematic mean 25.89 (μg/m3). Therefore, the one-parameter exponential fuzzy variable may well cope to the modelling requirements of the California PM2.5 annual arithematic mean dataset. Furthermore, using the one-parameter exponential fuzzy variable it may develop a delicate scheme of testing credibility hypothesis.

2.3. Credibility hypothesis testing with exponential membership

Similiar to the popular Neyman-Pearson Lemma in probability theory, likelihood ratio L0/L1, constant k, and critical region C of size α, are involved in the testing hypothesis H0: θ = θ0 against alternative hypothesis H1: θ = θ1. The likelihood is defined as the product of the densities for given sampled population. Hence we can call Neyman-Pearson testing criterion as likelihood ratio criterion. Inevitably, Type I error and Type II error concepts are also engaged in describing testing procedure. For hypothesis testing under credibility theory, Sampath and Ramya [17] proposed a membership ratio criterion. The membership criterion applies to any forms of credibility distributions, but exponential credibility distribution has its unique advantage. Therefore, the remaining descriptions will be focused on credibility hypothesis testing under exponential membership function [1].

Definition 7: Credibility hypothesis is a statement describing the possible rejection of a null hypothesis, H0: μ = μ0 with the credibility distribution of a fuzzy variable against an alternative hypothesis H1: μ = μ1 with another credibility distribution of a fuzzy variable.

Definition 8: Credibility hypothesis testing is the rule describing reject or not reject a null hypothesis if the calculated values sampled from the fuzzy distribution defined by null hypothesis.

Definition 9: Credibility rejection region is the subset of the support under a fuzzy distribution, denoted as C, on which the null credibility hypothesis is rejected H0: μ = μ0, i.e., C={ηΘ|H0 is rejected}

Definition 10: Type I error is the mistake by rejecting the null credibility hypothesis H0: μ = μ0 when it is true and Type II error the mistake by not rejecting the null credibility hypothesis H0: μ = μ0 when it is false.

Definition 11: Level of credibility significance is the maximal credibility to make a Type I error in testing a credibility hypothesis H0: μ = μ0, denoted as α.

Definition 12: The best credibility rejection region of credibility significance level α, C*, if this region possesses the maximal power (measured by credibility) under alternative hypothesis K with all possible credibility rejection regions of level of credibility significance α, i.e.,


where C is any region satisfying the condition Cr{ηC|H}α. The power of the credibility hypothesis testing is Cr{ηC*}.

With the exponential membership function having single parameter m > 0, the best credibility rejection region of credibility significance level α, C* should be an interval so that we name it as best credibility rejection interval of credibility significance level α.

Theorem 13: For testing the null credibility hypothesis H0: m = m1 against the alternative credibility hypothesis K: m = m2, (m1 < m2), under exponential fuzzy distributions μ(x) = Exp(m) as Equation 6 specified, the membership ratio criterion is engaged. The criterion states that given credibility significance level α < 0.5, the best credibility rejection interval C*:

C*=[6m1πln(1αα),+), α(0,0.5).E10

The credibility of credibility rejection interval C* under alternative hypothesis is greater than α.

α 0.10 0.15 0.20 0.25 0.30
x0 37.56361 29.65463 18.78181 14.48535 10.58305

Table 2.

x0 and α under H0: m1=21.926

Table 2 illustrates relationship between the best credibility rejection interval boundary x0 for selected credibility significance level α < 0.5. For example, let α = 0.20, m1 = 21.93, then the best credibility rejection interval C*= [18.78, +∞). We have to point out that the choice of credibility significance level α in credibility hypothesis testing should not follow the "thumb rule" the significance level α = 0.05 in probability hypothesis testing. Although the two significance levels have the same role in hypothesis testing, nevertheless, the practical meanings of credibility significance level α and the significance level α are quite different. From Table 2 and California PM2.5 distribution patterns, it is is logical and practical selecting the credibility significance level α = 0.25, which gives x0 =14.485.


3. Analysis and results

3.1. Exponential fuzzy membership kriging

Now having examined the methodology, we can now calculate the membership grades. But first let us have a quick overlook on California PM2.5 1999-2011 records.

Year Number of Sites Annual Maximum Annual Minimum Annual Average
1999 11 23.4190 8.3274 13.6146
2000 22 23.9229 7.7971 14.3465
2001 25 25.0446 4.1492 12.5643
2002 41 25.8332 6.3356 15.9160
2003 44 24.8348 4.3802 12.6993
2004 43 18.1962 4.4146 11.3249
2005 40 22.4178 4.8117 12.0750
2006 47 21.5991 5.1022 11.4211
2007 55 25.2073 3.2883 11.6240
2008 59 21.2357 6.7716 12.3861
2009 65 21.2423 3.3373 10.7292
2010 52 17.1908 2.9665 9.2255
2011 52 18.0950 3.3682 10.7934
Total 556

Table 3.

California PM2.5 1999-2011 Annual PM2.5 Concentrations

From Table 3, we can see that the annual average PM2.5 range between 9.2255 and 15.9160. Most of the averages are wondering about 11.0 and 12.7. Therefore, it is very reasonable to estimate X¯ =11.85.

Exponential membership grade kriging scheme:

  1. Calculating overall exponential membership parameter m. Based on Equation 7, we can estimate the exponential membership parameter m. Then in this paper, m = 21.93 is used for membership grade smoothing, kriging, and hypothesis testing.

  2. Calculating every exponential membership grade eij = 2.0/(1+exp(πxij/m 6)) for site j given year i. For any site, if "missing" value, xij = 1, otherwise, xij = eij (> 0).

  3. Performing membership grade interpolation. (i) Equal weights, for any given site j, if xij = 1, which is a "missing" value, let xije^ij where e^ij=(ei1,j+ei+1,j)/2 (conditioning on the availability of the nearest neighbours ei-1,j and ei+1,j < 1); (ii) Unequal weights, for a given site j, if the neighbour years are quite far, say, N between the gap, then, we may take linear interpolation for filling those "missing" value. that is given eij and ei+N,j < 1, let xi+l,jeij+l*(ei+N,jeij)/N, l = 1,2,..., N-1.

  4. Performing membership grade extrapolation, including forward and backward extrapolation. Equal weights (1/3) are used mostly. Unequal weights are also used.

  5. Carrying the filling "missing" cells task until thirteen years 1999 to 2011, each year 113 membership grades are all calculated, {e^ij,i=1,2,,13,j=1,2,,113}.

Now, it is ready to construct exponential membership grades kriging maps and use these 13 maps for comparisons.

Figure 3.

Kriging maps of Exponential membership grades for California 1999-2011

As one can observe from Figure 3, central California regions have very low membership grades, and the rest of Calfornia have higher membership grades.

3.2. Calculated PM2.5 concentrations

It is impossible for the public and governmental officers to accept the membership grade kriging maps. Therefore, kriging maps of every year PM2.5 concentrations (collected and estimated together) must be constructed. The conversion formula is


Kriging maps with "completed" of every year PM2.5 data {xij, i =1,2,...,13, j =1,2,...,113}. Comparison of kriging maps with the "completed" data can now be performed. See Figure 4. A complete dataset is now available, the 113 observation sites now have the full 13 year PM2.5 concentration, containing 1469 data records. The 13 ordinary kriging prediction maps are generated, and one can clearly observe the changes in PM2.5 year by year.

Figure 4.

Kriging maps of completed estimated PM2.5 concentrations for California 1999-2011

One can now observe and compare the year by year changes of PM2.5 concentration, and note the regions of high and low PM2.5 concentration. The dark brown colours represent high PM2.5 concentrations, and light yellow colours represent areas with low PM2.5 concentrations. Note that central California shows to have continual high levels of PM2.5 concentration, year after year.


4. Interpretation and conclusion

In the results section, the dataset is now calculated and completed. However, it is now up to us to interpret the maps, and decide how to best make use of the calculated dataset, so that it provides us with easy to read information. And we can do this through a change map and 13 health safety maps.

Figure 5.

Changes in PM2.5 Concentrations in California between 1999 and 2011

As one can clearly see from Figure 5, that PM2.5 concentration has clearly decreased and air quality has improved remarkably over the years. The blue and green colours show negative changes, and orange shows positive changes. Counties such as Los Angeles and Orange show the highest decrease, and other counties such as Lassen, Plumas, Sierra, Inyo, and Imperial show some increase in PM2.5 concentration. However, a decrease in PM2.5 concentration does not indicate safety in air quality.

In terms of credibility hypothesis testing, say, with credibility significance level α = 0.25, critical point for the best credibility rejection interval is x0 =14.485. The indictor λij is defined as


For comparisons of air quality safety, we generate PM2.5 safety maps with two colours: blue colour if λij = 1, orange colour otherwise, in total 13 safety maps. See Figure 6.

Figure 6.

Air Quality PM2.5 Safety areas in California 1999-2011

One can now observe that over the 13 years period 1999-2011, Stanislaus, Merced, Madera, Fresno, Kings, Tulare, Kern, Los Angeles, San Bernardino, Orange, and Riverside counties are the counties with the highest PM2.5 safety problems. These areas have shown to be unsafe for public health safety, and especially for those with lung and heart problems, and for children and the elderly. These places are also sources of environmental and ecological concerns.

In conclusion, facing the difficult problem of "incomplete" PM2.5 data in California from 1999-2011, we used the interpolation and extrapolation smoothing approaches for "filling" those "missing value" sites. For easy computation, the fuzzy exponential membership function is assumed. The treatment is based on an assumption that the smoothing is performed for a given site rather than over different sites for a given year. Such an assumption is emphasizing the fact: the data recorded are PM2.5 concentration annual arithmetic means and they shouldn't change too dramatically over neighbour years. As to neighbour sites impacting, the membership grade kriging approach is adequate enough for generating smoothed maps. Furthermore, for utilizing credibility hypothesis testing theory, we perform parameter estimation of the fuzzy exponential membership function and in terms of membership ratio criterion for deriving the safety maps under 0.25 credibility significance level. Membership ratio criterion is very similar to likelihood ratio criterion in theoretical development. By comparing those 13 PM2.5 concentration safety maps to 1999-2011 change map, it is quite justifiable to say the safety maps under the credibility hypothesis testing procedure are very intuitive and convenient to the public. Finally, interpreting the 13 safety maps will provide the public with knowledge of air quality in California.



I would like to thank the California Air Resources Board for providing the air quality data used in this paper. This study is supported financially by the National Research Foundation of South Africa (Ref. No. IFR2009090800013).


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Written By

Danni Guo

Submitted: 15 October 2014 Published: 21 October 2015