**3. Analysis and results**

**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 *H*0: *μ* = *μ*0, i.e.,

**Definition 10:** Type I error is the mistake by rejecting the null credibility hypothesis *H*0: *μ* = *μ*<sup>0</sup> when it is true and Type II error the mistake by not rejecting the null credibility hypothesis *H*0:

**Definition 11:** Level of credibility significance is the maximal credibility to make a Type I error

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.,

> h

where *C* is any region satisfying the condition Cr{*η* ∈*C* |*H* }≤*α*. The power of the credibility

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

**Theorem 13:** For testing the null credibility hypothesis *H*0: *m* = *m*<sup>1</sup> against the alternative credibility hypothesis *K*: *m* = *m*2, (*m*1 < *m*2), under exponential fuzzy distributions *μ*(*x*) = Exp(*m*) as Equation 6 specified, the membership ratio criterion is engaged. The criterion states that

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

Table 2 illustrates relationship between the best credibility rejection interval boundary *x*0 for selected credibility significance level *α <* 0.5. For example, let *α* = 0.20, *m*<sup>1</sup> = 21.93, then the best

significance level *α* in credibility hypothesis testing should not follow the "thumb rule" the

a

given credibility significance level α < 0.5, the best credibility rejection interval *C*\*

 a

é ö æ ö - <sup>=</sup> <sup>ê</sup> ç ÷ +¥ Î <sup>÷</sup> <sup>÷</sup> <sup>ê</sup> è ø ë ø

( ) \* <sup>1</sup> 6 1 ln , , 0,0.5 . *<sup>m</sup> <sup>C</sup>* a

Î ³Î *C K CK* (9)

, if this

:

under alternative hypothesis is greater than *α*.

= [18.78, +∞). We have to point out that the choice of credibility

(10)

**Definition 12:** The best credibility rejection region of credibility significance level *α*, *C*\*

{ } { } \* Cr | Cr | ,

in testing a credibility hypothesis *H*0: *μ* = *μ*0, denoted as *α*.

h

as best credibility rejection interval of credibility significance level *α*.

p

The credibility of credibility rejection interval *C* \*

**Table 2.** *x*0 and *α* under *H*0: *m*1=21.926

credibility rejection interval *C*\*

*C* ={*η* ∈*Θ* | *H*<sup>0</sup> is rejected}

316 Current Air Quality Issues

*μ* = *μ*0 when it is false.

hypothesis testing is Cr{*η* ∈*C* \*}.

#### **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.


**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 *<sup>X</sup>*¯ =11.85.

Exponential membership grade kriging scheme:


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

2005 2006 2007

2008 2009 2010

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

2011

230 Figure 3. Kriging maps of Exponential membership grades for California 1999-2011

229

231

234

233 membership grades.

1999 2000 2001

2002 2003 2004

230 Figure 3. Kriging maps of Exponential membership grades for California 1999-2011 **Figure 3.** Kriging maps of Exponential membership grades for California 1999-2011

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

**3.** Performing membership grade interpolation. (i) Equal weights, for any given site *j*, if *xij*

**4.** Performing membership grade extrapolation, including forward and backward extrapo‐

**5.** Carrying the filling "missing" cells task until thirteen years 1999 to 2011, each year 113

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

1999 2000 2001

2002 2003 2004

2005 2006 2007

2008 2009 2010

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

2011

230 Figure 3. Kriging maps of Exponential membership grades for California 1999-2011

229

231

234

233 membership grades.

^ *ij*

lation. Equal weights (1/3) are used mostly. Unequal weights are also used.

 where *e* ^

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

*ij* =(*ei*−1, *<sup>j</sup>* + *ei*+1, *<sup>j</sup>*

, *i* =1, 2, ⋯, 13, *j* =1, 2, ⋯, 113}.

^ *ij*

interpolation for filling those "missing" value. that is given *eij*

) / *N* , *l* = 1,2,..., *N*-1.

1, which is a "missing" value, let *xij* ≜*e*

membership grades are all calculated, {*e*

*xi*+*l*, *<sup>j</sup>* ≜*eij* + *l* \*(*ei*+*<sup>N</sup>* , *<sup>j</sup>* −*eij*

318 Current Air Quality Issues

maps for comparisons.

=

229

231

234

233 membership grades.

) / 2 (conditioning on the

and *ei+N,j* < 1, let

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