A sample taken from the monitoring data reports of some environmental management departments is first classified by an ideal gray near function cluster analysis. The level of the sample is then determined by a gray correlation analysis and comprehensive assessment conclusions are drawn according to the degree of correlation between the classification of the sample and the values â€‹â€‹given in GB3095-2012.

### The classification of the sample to be assessed

#### Creation of the evaluation index sequence matrix for the selected sample

Permit S. be a sequence of clustering objects, d. S. = {S.1, S.2…, S.m}; xis a sequence of air influencing variables, ie x= {x1, x2…, xn}; xI k is the original monitoring data for S.I (I = 1, 2 …, m ) and xk (k = 1, 2 …, n ); I and m represent the number of objects that are taken into account in clustering; k and nare the number of influencing indices, which are the pollutants mentioned above. Accordingly, the following matrix can be set up (Eq. 1).

$$S = begin {array} {* {20} c} {s_ {1}} {s_ {2}} ldots {s_ {m}} end {array} Left[ {begin{array}{*{20}c} {x_{11} } & {x_{12} } & ldots & {x_{1n} } {x_{21} } & {x_{22} } & ldots & {x_{2n} } ldots & ldots & ldots & ldots {x_{m1} } & {x_{m2} } & ldots & {x_{mn} } end{array} } right]$$

(1)

#### Establishing the matrix of ideal-valued gray function clusters

Permit x0= {x01, x02 …, x0n} be the sequence of ideal values â€‹â€‹that corresponds to each influential index. The principle for determining the ideal value is as follows (Eqs. 2, 3, 4).

The first situation: the larger the influence index (xk) the better the air quality; in this case the ideal value

$$x_ {0k} = max left {{x_ {ik}, i = 1,2, ldots, m} right }, k = 1,2, ldots, n.$$

(2)

The second situation: the smaller the influence index (xk) the better the air quality; in this case the ideal value

$$x_ {0k} = min left {{x_ {ik}, i = 1,2, ldots, m} right }, k = 1,2, ldots, n.$$

(3)

Third, air quality is best when the influence index (xk) indicates a moderate value and is the ideal value

$$x_ {0k} = { text {M}}.$$

(4)

According to the ideal value x0k (Eq. 2, 3 or Eq. 4) and the original monitored data (xI k), the gray key function value YesI k is calculated using (Eq. 5).

$$y_ {ik} = frac {{x_ {ok}}} {{x_ {ik}}} ; left ({i = 1,2, ldots, m; k = 1,2, ldots , n} right)$$

(5)

Where xI k are the original monitored data and x0k is the ideal value that corresponds to the k-th influence index. Also is the function value YesI k is dimensionless, and YesI kâ€‰âˆˆâ€‰[0,1]. YesI k denotes the degree of correlation of S.I and S.0for the kth index. Especially the bigger ones YesI k is the closer S.I has ideal value S.0, and the smaller one YesI k is the further S.I the end S.0.

Thus the following gray closed matrix is Yes can be determined (Eq. 6).

$$Y = left[ {begin{array}{*{20}c} {y_{11} } & {y_{12} } & ldots & {y_{1n} } {y_{21} } & {y_{22} } & ldots & {y_{2n} } begin{gathered} ldots hfill y_{m1} hfill end{gathered} & begin{gathered} ldots hfill y_{m2} hfill end{gathered} & begin{gathered} ldots hfill ldots hfill end{gathered} & begin{gathered} ldots hfill y_{mn} hfill end{gathered} {y_{01} } & {y_{02} } & {…} & {y_{0n} } end{array} } right]$$

(6)

In this case, Yes is the gray key function value. Aside from that, (Yes01, Yes02…, Yes0n) = (1,1 …, 1)1 Ã—n is the ideal sequence, and the greater YesI k is the better S.I is; the biggest YesI k is equal to 1.

#### The classification of the sample to be assessed

Since the influence of each influence index is different, the weighting of each influence index must be taken into account. Permit P.I be the comprehensive analysis value of S.I. P.I can be expressed as follows (Eq. 7)

$$P_ {i} = sum limits_ {k = 1} ^ {n} {Wy_ {ik}} left ({i = 1,2 ldots, m} right)$$

(7)

Where W. is the weight of each influencing index, and da is the number of indexes k, the number of W. Values â€‹â€‹is too k(W.1, W.2…, W.k). Accordingly, the following equation can be set up (Eq. 8).

$$W_ {k} = frac {{ sum limits_ {i = 1} ^ {m} {X _ {{i { text {k}}}}}}}} {{ sum limits_ {i = 1} ^ {m} { sum limits_ {k = 1} ^ {n} {X_ {ik}}}}} ; left ({k = 1,2 ldots, n} right)$$

(8th)

Based on the actual total analysis value P.I, P.J= (P.1, P.2…, P.m)T. The following equation (Eq. 9) can be used to calculate the gray limit value P.ij from P.I in relation to P.J.

$$P_ {ij} = frac {{ min (p_ {i}, p_ {j})}} {{ max (p_ {i}, p_ {j})}} ; left ({i , j = 1,2 ldots, m} right)$$

(9)

Then,

$$P = left ({P_ {ij}} right) _ {m times m}.$$

(10)

if P. (Eq. 10) satisfies the following three conditions: (1) Reflexivity, where P.ij= 1 (I = J); (2) symmetry, where P.ij= P.ji; and (3) normativity, whereP.ijâ€‰âˆˆâ€‰[0,1], we can find the appropriate threshold from the P.Matrix, cut off the branches with weight values â€‹â€‹less than Î», which corresponds to the similarity coefficient4.5and set the classification (S_ {t} ^ { prime} ) ( T= 1, 2 …, C.) if the level Î» meets the corresponding requirement. (S_ {t} ^ { prime} ) represents any classification of the air in a particular region. The following equations (Eqs. 11, 12) can be set up.

$$S_ {t} ^ { prime} = left ({S_ {1} ^ { prime}, S_ {2} ^ { prime} ldots, S_ {c} ^ { prime}} right ) ^ {{ text {T}}}$$

(11)

$$S_ {tk} ^ { prime} = left ({S_ {t1} ^ { prime}, S_ {t2} ^ { prime} ldots, S_ {tn} ^ { prime}} right )$$

(12)

Where (S_ {t} ^ { prime} ) is the tth classification, (S_ {tk} ^ { prime} ) is the kth index of the tth classification, Tis the number of classifications, andkis the number of influencing indices.

(S_ {tk} ^ { prime} ) can be expressed in the following matrix form (Eq. 13).

$$S_ {tk} ^ { prime} = left[ {begin{array}{*{20}c} {s_{11}^{prime } } & {s_{12}^{prime } } & ldots & {s_{1n}^{prime } } {s_{21}^{prime } } & {s_{22}^{prime } } & ldots & {s_{2n}^{prime } } ldots & ldots & ldots & ldots {s_{cc}^{prime } } & {s_{c2}^{prime } } & ldots & {s_{cn}^{prime } } end{array} } right]$$

(13)

### Correlation degree analysis of the sample to be evaluated

Permit (S_ {t} ^ { prime} ) be the sample to be assessed and let x= ( x1, x2…, xn), this is the influence index rate mentioned above and the rating index used for (S_ {t} ^ { prime} ). Permit ({ text {S}} _ {0} ^ { prime} ) the specified air quality classification inGB3095-2012. Then the equation for the correlation coefficient is as follows (Eq. 14)14th.

$$zeta_ {t} (k) = frac {{ mathop { min} limits_ {t in c} mathop { min} limits_ {k in n} left | {S_ {t} ^ { prime} (k) – { text {S}} _ {0} ^ { prime} (k)} right | + epsilon mathop { max} limits_ {t in c} mathop { max} limits_ {k in n} left | {S_ {t} ^ { prime} (k) – { text {S}} _ {0} ^ { prime} (k)} right |}} {{ left | {S_ {t} ^ { prime} (k) – { text {S}} _ {0} ^ { prime} (k)} right | + epsilon mathop { max} limits_ {t in c} mathop { max} limits_ {k in n} left | {S_ {t} ^ { prime} (k) – { text {S}} _ {0} ^ { prime} (k)} right |}}$$

(14)

WhereÎ¶T ( k) is the correlation coefficient and Îµ is the coefficient of resolution with a general value of 0.54.5.

In addition, the degree of correlation ( R.T) Equation reads as follows (Eq. 15).

$$R_ {t} = frac {1} {n} sum limits_ {k = 1} ^ {n} { zeta_ {t}} (k)$$

(fifteen)

The value ofR.T is calculated using (Eq. 15). The maximum value ofR.T indicates that the sample to be assessed has the highest degree of correlation with the air quality under consideration. Therefore, the sample is classified accordingly.

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