# Discriminant Analysis of Water Inrush Sources in the Weibei Coalfield, Shaanxi Province, China

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{+}+ Na

^{+}, Mg

^{2+}, NH

_{4}

^{+}, Cl

^{−}, SO

_{4}

^{2−}, and pH, have significant discrimination ability and good effect and can effectively distinguish the three main water inrush aquifers in the Weibei mining area. Then, the clustering stepwise discriminant analysis method was used to select 24 water samples and 14 trace element indicators from the conventional water chemistry test results. Based on principal component analysis, a principal component analysis discriminant model of trace elements was established for the four main aquifers. The accuracy and misjudgment rate of the Bayes multi-class linear discriminant using conventional ions as explanatory variables were 64.3% and 35.7%, respectively, showing a poor discriminant effect. On this basis, seven characteristic trace elements were analyzed according to Bayes multi-class linear discriminant analysis, the mutual influence and restriction relationship regarding the migration of these seven trace elements in the groundwater system of the mining area was determined, and the modified Bayes multi-class linear discriminant analysis model of trace elements for the water inrush source was established, which was more accurate than the conventional ion Bayes multi-class linear discriminant analysis model. The accuracy rate reached 92.9%. This research is of great significance for mine water-source identification and water-inrush prevention guidance.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Data Acquisition

#### 2.2.1. Occurrence of the Four Aquifers

#### Quaternary Loose Rock Aquifers

#### Permian Sandstone Fractured Aquifers

#### Carboniferous and Ordovician Sandstone (Limestone) Fractured Aquifers

#### 2.2.2. Sampling

^{2+}ion content was very low, which did not meet the characteristics of Austrian gray water. The samples were rejected, and 38 groups of valid test water samples were collected in the Weibei mining area, including 2 sampling points for Quaternary aquifers, 10 sampling points for Permian sandstone fractured aquifers, and 7 sampling points for Limestone water of Taiyuan Formation. There were 11 sampling points for Austrian grey water and 8 sampling points for other surface water and old-air mixed water.

#### 2.2.3. Test Methodology

#### 2.3. Methods

#### 2.3.1. Hierarchical Clustering Stepwise Discriminant Analysis

_{ij}”. If any sample is regarded as a class, and the similarity between classes can be represented by the Euclidean distance D

_{E}, then:

_{il}, X

_{jl}represent each index of the sample, respectively.

_{ij}between each pair of n samples, find the two classes with the smallest distance, merge them into a new class, recalculate the distance between the new class and other types of distance, and then merge the two categories with the smallest distance and repeat the above process until all samples are clustered into one category. All the water sample points (there are n) of the water inrush aquifer in the mining area whose Euclidean distance D

_{E}is less than a certain man-made specified value, P, were selected for participation in the stepwise discriminant analysis, and n water sample points were divided with m indicators into four categories. The stepwise discriminant analysis selected the indicators ${x}_{1},{x}_{2},\dots ,{x}_{n}$ with significant discriminative effects on the four categories from its m indicators to form the discriminant function:

_{0(An)}, C

_{1(An)}, C

_{2(An)}, …, C

_{n(An)}represent the discriminant coefficient of the An-th class. q

_{(An)}represents the event probability of the An-th class.

#### 2.3.2. F-Test

_{1}represents the number of samples of the first type. n

_{2}represents the number of samples of the second type. p represents the number of discriminant variables. ${\mathrm{D}}_{1,2}^{2}$ represents the generalized Mahalanobis distance, and its value can be expressed as:

#### 2.3.3. Principal Component Analysis

_{i}= 1 (i = 1, 2, …, t), and consider its multi-class linear transformation:

_{1}) = ${\mathrm{I}}_{\mathrm{i}}^{\prime}$∑I

_{i}, Cov(γ

_{i}, γ

_{j})= ${\mathrm{I}}_{\mathrm{i}}^{\prime}$∑I

_{j}(i, j = 1, 2, …, t). If you want to use γ

_{1}to replace the original t trace element variables X

_{1}, X

_{2}, …, X

_{t}, this requires γ

_{1}to reflect the information of the original t variables as best as possible. According to statistical analysis theory, the larger the Var(γ

_{1}), the more information γ

_{1}contains. Therefore, I

_{1}should be calculated under constraint conditions so that Var(γ

_{1}) reaches the maximum value. At this time, γ

_{1}is called the first principal component. If a principal component cannot represent the trace element information of the aquifer, as reflected by the original t variables, consider using γ

_{2}. To effectively represent the information of the original variable, the existing trace element information of γ

_{1}does not need to appear in γ

_{2}, which should have Cov(γ

_{1}, γ

_{2}) = 0. Therefore, to find γ

_{2}is to find I

_{2}under the constraints so that Var(γ

_{2}) reaches the maximum value. The required γ

_{2}is called the second principal component. Similarly, the third principal component, the fourth principal component, etc., can be defined. Generally, the i-th principal component γ

_{i}= $\mathrm{X}$ of X refers to finding I

_{i}under the constraints and Cov(γ

_{i},γ

_{k}) = 0 (k < i), so that Var(γ

_{i}) reaches a maximum.

_{i}= I′ (i = 1, 2, …, p). Among them, I

_{i}is the unit eigenvector of the corresponding λ

_{i}; at this time, Var(γ

_{i}) = λ

_{i}(i = 1, 2, …, t). Generally, let λ

_{1}, λ

_{2}, …, λ

_{t}≥ 0 is the obtained eigenroot and I

_{1}, I

_{2}, …, I

_{t}are the corresponding eigenvectors. Then, it is called: ${\lambda}_{i}/{\displaystyle {\sum}_{i=1}^{p}{\lambda}_{i}}$, which is the contribution rate of the i-th principal component γ

_{i}(i = 1, 2, …, t), and its size reflects the information of X

_{1}, X

_{2}, …, X

_{t}: $\sum}_{i=1}^{m}{\lambda}_{i}}/{\displaystyle {\sum}_{i=1}^{t}{\lambda}_{i$.

_{1}is the largest; it extracts X

_{1}, X

_{2}, …, X

_{t}with the largest amount of information and the strongest comprehensive ability, so it can be reflected by the first principal component. The trace element information is used to comprehensively analyze the hydrogeochemical characteristics of trace elements in the Weibei mining area [31].

#### 2.3.4. Bayes Criterion

_{g}). If each sample measures t variables (${x}_{1},{x}_{2},\dots ,{x}_{t}$), then each sample can be regarded as a point in the p-dimensional space {R} [32]. n samples form a p-dimensional sample space {R}. An unknown sample $X({x}_{1},{x}_{2},\dots ,{x}_{t})$ is also regarded as a point in the p-dimensional space to see which subspace it falls into or which subspace has the highest probability; then, it can be classified as one of the G mothers [32].

_{h}parent is wrongly classified into the A

_{g}parent, the loss caused is recorded as L(g/h), and it is agreed that when h = g, L(g/h) = 0; when h ≠ g, L(g/h) > 0.

_{g}parent to A

_{h}is denoted as $P\left\{g/h\right\}$, then, when the probability distribution density ${f}_{g}\left(x\right)$ of G parents is known, we have $P\left\{g/h\right\}={\displaystyle {\int}_{{R}_{h}}^{}{f}_{g}\left(x\right)}dx$. The average loss caused by misclassifications of the sample originally belonging to the A

_{g}parent into the A

_{h}parent is:

_{h}parent is wrongly classified into the A

_{g}parent, the loss is recorded as L(h/g), and the same is obtained:

_{g}. Therefore, when the class G prior probability q

_{g}is given to the parent, the principle of dividing the space {R} to minimize the average loss of misclassification is called the Bayes criterion or attribution criteria. That is to say, the individual with the largest posterior probability belonging to the A

_{g}parent is assigned to the A

_{g}parent. The maximum a posteriori probability is equivalent to the maximum q

_{g}f

_{g}(x), so the discriminant function of any individual x can be obtained.

_{g}(g = 1, 2, …, G) obeys the normal distribution N (a

_{g}, ∑) (g = 1, 2, …, G) [33], and its probability density function is:

_{g}and ∑ are the mean vector and covariance matrix of the parent A

_{g}, respectively.

_{g}of the parent and the parent parameters are known, and the parent covariance matrix is not significantly different (when the statistics are equal) [33], the G discriminant functions can be obtained as follows:

## 3. Results

#### 3.1. Conventional Hydrochemical Characteristics and Discrimination Model

#### 3.1.1. Relationship between Conventional Ions and Total Dissolved Solids (TDS) in Karst Water

^{+}, Na

^{+}, Ca

^{2+}, Mg

^{2+}, Cl

^{−}, SO

_{4}

^{2−}, CO

_{3}

^{2−}, HCO

_{3}

^{−}) in water.

^{2+}, Mg

^{2+}, HCO

_{3}

^{−}, and TDS can determine the hydraulic connection between the Hancheng hydrological subunit and the Tongchuan-Pucheng-Heyang hydrological subunit. TDS are about 2000 mg/L, which is the boundary between Hancheng and Tongpu Hehe hydrological subunits (Figure 5, Figure 6 and Figure 7).

^{2+}, Mg

^{2+}, and HCO

_{3}

^{−}of Carboniferous water and Ordovician limestone water in the Tongchuan-Pucheng-Heyang hydrological subunit and TDS shows that the dissolution of the Carboniferous aquifer and Ordovician limestone aquifer in the Tongchuan-Pucheng-Heyang hydrological subunit is not significant. That is, when TDS are below 2000 mg/L, the changes in Ca

^{2+}, Mg

^{2+}, and HCO

_{3}

^{−}ions are not significant. The variation range is concentrated at about 100 mg/L, indicating that the dissolution of calcite, dolomite, and other minerals in the Tongchuan-Pucheng-Heyang hydrological subunit is close to saturation. That is to say, the Tongchuan-Pucheng-Heyang hydrological subunit has good exchange conditions with surface water and atmospheric water, and the karst groundwater in this area has poor solubility to the karst minerals in this area. The hydrogeological conditions of the Hancheng hydrological subunit and Tongchuan-Pucheng-Heyang hydrological subunit are obviously different. In addition, the hydrological subunit in the current mining area of Hancheng is located in the middle- and deep-detention areas, with a relatively closed environment and a poor exchange environment with surface water and atmospheric precipitation.

_{ca}

^{2+}> K

_{(K}

^{+}

_{+Na}

^{+}

_{)}> K

_{Mg}

^{2+}, so the migration ability of cations in karst water is Ca

^{2+}> K

^{+}+Na

^{+}> Mg

^{2+}. From Figure 8, the Tongchuan-Pucheng-Heyang hydrological unit and the Hancheng hydrogeological unit are considered separately. According to the slope of the conventional ion trend line, K

_{K}

^{+}

_{+Na}

^{+}> K

_{ca}

^{2+}> K

_{Mg}

^{2+}in the Tongchuan-Pucheng-Heyang, indicating that the runoff supply of surface water and atmospheric precipitation in this hydrogeological unit is good, and the calcium ion of the Tongchuan-Pucheng-Heyang hydrological unit is saturated. In Hancheng, K

_{ca}

^{2+}> K

_{(K}

^{+}

_{+Na}

^{+}

_{)}> K

_{Mg}

^{2+}, so the migration ability of cations in karst water is Ca

^{2+}> K

^{+}+ Na

^{+}> Mg

^{2+}.

#### 3.1.2. Conventional Hydrochemical Discrimination Model for Main Water Inrush Aquifers

#### Cluster Analysis of Aquifer System

^{+}+ Na

^{+}, Ca

^{2+}, Mg

^{2+}, NH

_{4}

^{+}, Cl

^{−}, SO

_{4}

^{2−}, HCO

_{3}

^{−}, PH value, total hardness, and TDS, and these selected water samples can better reflect the hydrogeochemical sampling features at the point.

#### Gradual Discriminant Analysis of Aquifer

- (1)
- Discriminant analysis

^{+}+ Na

^{+}] represents the sum of the concentrations of potassium ions and sodium ions (mg/L); [Mg

^{2+}] represents the concentration of magnesium ions (mg/L); [NH

_{4}

^{+}] represents the concentration of ammonium ions (mg/L); [Cl

^{−}] represents chloride ion concentration (mg/L); [SO

_{4}

^{2−}] represents sulfate ion concentration (mg/L).

^{+}+ Na

^{+}, Mg

^{2+}, NH

_{4}

^{+}, Cl

^{−}, SO

_{4}

^{2−}, and pH were substituted into each discriminant function formula, respectively, to calculate the Y value of each type. The largest Y value was taken, and the sample was classified into this type.

- (2)
- Discrimination effect test

_{0.05(2, 17)}, the difference between classes is significant, indicating that the six selected variables, K

^{+}+ Na

^{+}, Mg

^{2+}, NH

_{4}

^{+}, Cl

^{−}, SO

_{4}

^{2−}, and pH have significant discriminative ability and good effect, and can effectively discriminate the three main water inrush aquifers in the Weibei mining area.

#### 3.2. Hydrogeochemical Characteristics and Discriminant Model of Trace Elements

#### 3.2.1. Systematic Cluster Analysis of Trace Elements in Aquifer

#### Cluster Analysis of Aquifer System

#### Gradual Discriminant Analysis of Aquifer

- (1)
- Discriminant analysis

- (2)
- Discrimination effect test

_{1,3}and F

_{2,3}are both greater than F

_{0.05(2,12)}, indicating that there are significant differences between the first class and the third class, and between the second class and the third class and the discriminant effect is better. However, F

_{1,2}< F

_{0.05(2, 12)}, indicating that there is no significant difference between the first and second types, the total discriminant significance rate is 67%, and the discriminant effect is poor (Table 6).

#### 3.2.2. Principal Component Analysis of Trace Elements

#### Analysis of Eigenvalues and Cumulative Variance Contribution Rate

#### Interpretation of Principal Components of Groundwater

#### Discrimination of Principal Components of Hydrogeochemistry of Aquifers

#### 3.2.3. Analysis of Trace Element Content in the Mining Area

#### Selection of Characteristic Trace Elements

^{−}, SO

_{4}

^{2−}, HCO

_{3}

^{−}, K

^{+}, Mg

^{2+}, and Ca

^{2+}as analysis variables, the systematic clustering method is used for analysis, and the clustering results are shown in Figure 16. These trace elements are highly correlated with groundwater conventional ions and show a good correlation with each other. The migration of these seven trace elements in the groundwater system of the mining area is not isolated, but they interact with and restrict each other. These seven trace elements can be selected as characteristic trace elements to identify the type of water inrush source and analyze the hydrogeochemical characteristics of the mining area.

#### Relationship between Characteristic Trace Elements and Aquifers

#### 3.2.4. Bayes Multi-Class LDA Model of Characteristic Trace Elements

#### Bayes Multi-Class LDA Model of Characteristic Trace Elements

#### Conventional Ion Bayes Multi-Class LDA Model

^{−}, SO

_{4}

^{2−}, HCO

_{3}

^{−}, K

^{+}, Mg

^{2+}, and Ca

^{2+}in the mining area were taken as explanatory variables. According to the principle of the Bayes multi-class LDA model, the Bayes multi-class LDA function was obtained. The estimated values of their function coefficients are shown in Table 10. The retrospective discriminant results are shown in Table 11.

^{−}, SO

_{4}

^{2−}, K

^{+}+ Na

^{+}and Ca

^{2+}are similar, while the absolute values of the coefficients of Mg

^{2+}are large. Among the Bayes multi-class LDA function with conventional ions as explanatory variables, these are generally balanced. The accuracy of Bayes multi-class linear discrimination analysis with conventional ions as explanatory variables was 64.3%, and the error rate was 35.7%. The discrimination effect was poor.

#### Bayes Multi-Class LDA Model of Characteristic Trace Elements Corrected by Conventional Ions

## 4. Discussion

^{2+}and TDS, which is helpful for understanding the karst water system. The water cycle is very important. In addition, most of the conventional ion content in the Weibei mining area increases with the increase of TDS, which shows the accuracy and reliability of using TDS as a comprehensive index to simulate the groundwater concentration gradient field to analyze its cycle characteristics [35]. The karst water system of the Tongchuan-Pucheng-Heyang hydrogeological unit is closely related to the overlying shallow aquifer, surface water, and atmospheric precipitation. The dissolution of limestone and dolomite reaches saturation in the recharge area, and the karst groundwater in this area has poor solubility in the karst minerals in this area. In addition, the currently exploited Hancheng hydrogeological unit is the middle and deep detention area of the karst water system, with a relatively closed environment and poor exchange environment with surface water and atmospheric precipitation.

^{+}+ Na

^{+}, Ca

^{2+}, Mg

^{2+}, NH

_{4}

^{+}, Cl

^{−}, SO

_{4}

^{2−}, HCO

_{3}

^{−}, pH value, total hardness, and TDS, and total hardness. The model was found to be able to better distinguish the source of water inrush through the Mahalanobian distance test. Through F test, it is considered that the six selected variables K

^{+}+ Na

^{+}, Mg

^{2+}, NH

_{4}

^{+}, Cl

^{−}, SO

_{4}

^{2−}, and pH have significant discrimination ability and good effect, and effectively identify the three main water-inrush aquifers in Weibei mining area.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | |

q | Unit water inflow |

K | the permeability coefficient |

m | the number of indicators |

n | the number of samples |

D_{E} | Euclidean distance |

i, j | the sample serial number |

d_{ij} | the similarity of any two points in the m-dimensional space |

X_{il}, X_{jl} | each sample index |

Y_{(An)} | the discriminant function of the An-th class |

Cn_{(An)} | discriminant coefficient of the An-th class |

q_{(An)} | the event probability of the An-th class |

Y | the discriminant value |

F_{1,2} | the test value between test classes 1 and 2 |

n_{1} | the number of samples of the first type |

n_{2} | the number of samples of the second type |

P | the number of discriminant variables |

${\mathrm{D}}_{1,2}^{2}$ | the generalized Mahalanobis distance |

S | the covariance matrix between variables |

${\stackrel{-}{\mathrm{X}}}_{\left(1\right)}$ | the sample mean vector of the first category |

${\stackrel{-}{\mathrm{X}}}_{\left(2\right)}$ | the sample mean vector of the second category |

Z | the degrees of freedom |

${\mathrm{F}}_{\mathrm{\alpha}}$ | the F critical value at the significance level α |

$X=({X}_{1},{X}_{2},\dots ,{X}_{p})$ | the P-dimensional random variable |

I | the constant vector |

t | the number of element variables |

G | the number of precursors |

A_{g} | the sample of the G precursors |

A_{h} | the sample of the H precursors |

{R} | the p-dimensional space |

L(g/h) | the loss caused by misclassification of samples originally belonging to the A_{h} parent into the A_{g} parent |

P{g/h} | the probability caused by misclassification of samples originally belonging to the A_{g} parent into the A_{h} parent |

${f}_{g}\left(x\right)$ | the class G probability distribution density |

W_{h} | the average loss caused by misclassifying the sample originally belonging to the A_{g} parent into the A_{h} parent |

${q}_{h}$ | the class H prior probability |

$P(h,g)$ | the probability caused by the misclassification of samples originally belonging to the A_{h} parent into the A_{g} parent |

W_{R} | the average loss of misclassification of the G-type parent |

W_{g} | the average loss caused by misclassifying the sample originally belonging to the A_{h} parent into the A_{g} parent |

${f}_{h}(x)$ | the class H probability distribution density |

q_{g} | the class G prior probability |

a_{g} | the mean vector of A_{g} |

f_{x}(x) | the probability density function of A_{g} |

q_{g}f_{g}(x) | the G discriminant functions |

y_{g}(x) | the multi-class LDA functions of the normal parent under the Bayes criterion |

c_{ig} | the multi-class LDA coefficient |

K^{+} | the potassium ion concentration |

Na^{+} | the sodium ion concentration |

Mg^{2+} | the magnesium ion concentration |

NH_{4}^{+} | the ammonium ion concentration |

Cl^{−} | the chloride ion concentration |

SO_{4}^{2−} | the sulfate ion concentration |

Li | the lithium element concentration |

Sc | he scandium element concentration |

V | the vanadium element concentration |

Cr | the chromium element concentration |

Ni | the nickel element concentration |

Cu | the copper element concentration |

Zn | the zinc element concentration |

Rb | the rubidium element concentration |

Mo | the molybdenum element concentration |

Sb | the antimony element concentration |

Cs | the cesium element concentration |

Ba | the barium element concentration |

U | the uranium element concentration |

Sr | the strontium element Concentration |

Greek symbols | |

$\gamma $_{i} | replace the original p trace element variables X_{1}, X_{2}, …, X_{p} |

$\lambda $ | the obtained eigenroot |

Abbreviations | |

LDA | linear discriminant analysis |

TDS | total dissolved solid |

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Diggings | Colliery | Thickness (m) | q (L/s·m) | K (m/d) | Predominantly Lithological |
---|---|---|---|---|---|

Tongchuan | Xujiagou | 100~130 | 0.0654 | 0.626 | Clay, sub-clay, sandy clay |

PuBai | Zhujiahe | 0~152.22 | 0.008~0.47 | 0.013~18.7 | Loess, sub-clay, sub-sand |

Chenghe | Dongjiahe | 1.79~134.41 | 0.01~0.1 | 0.0073~1.55 | Loess, gravel-bearing sandstone |

Wang Cun | 0~160 | 0.119~0.264 | 6.33 | Conglomerate, clay, sub-sand, sand and silt | |

Hancheng | Elephant Mountain | 0~100 | 0.068~21.11 | 1.3~4.45 | Sand, sub-sand |

Mulberry Tree Ping | 0~100 | 1.93~6.73 | Silt, sub-clay |

Aquifer Segments | Thickness (m) | Hydrogeological Features |
---|---|---|

Shiqianfeng formation aquifer | 0~100 | The lithology is mainly sandstone, followed by mudstone and sandy mudstone, and the aquifer is mainly lower middle- and coarse-grained sandstone, about 20 m thick, with fracture development, including fracture diving, q = 0.1~0.8 L/s·m, K = 2.06~33.77 m/d. |

Upper shihe formation aquifer | 0~300 | The lithology is mainly purple variegated, yellow–green sandy mudstone and siltstone, interspersed with medium–coarse-grained sandstone and thin mudstone layers. q = 0.0004~1.14 L/s·m, K = 0.0009~3.89 m/d. |

Lower shihe formation aquifer | 17.56~230.86 | q = 0.00084~0.473 L/s·m, K = 0.00443~1.96 m/d |

Shanxi formation | 18.44~100.68 | Composed of light gray, gray–green, yellow–green sandstone, siltstone, dark gray sandy mudstone, mudstone, and No. 2 and No. 3 coal seams. q = 0.0001~0.08 L/s·m, K = 0.00036~0.231 m/d |

**Table 3.**Hydrogeological characteristics of fractured aquifers in Carboniferous and Ordovician sandstone (limestone).

Aquifer Segments | Thickness (m) | Hydrogeological Features |
---|---|---|

Taiyuan formation | 5~105 | The lower part is mainly quartz sandstone and sandstone, interspersed with siltstone and mudstone, and the middle part is composed of quartz sandstone, siltstone, limestone, and coal seam, q = 0.000052~0.0316 L/s·m, K = 0.003~1.649 m/d |

Benxi formation | 0~41.01 | The lithology is mainly gray clumpy clay mudstone, gray mudstone, sandy mudstone, and gray quartz sandstone, q = 0.0002~0.154 L/s·m, K = 0.00041~0.07 m/d |

Ordovician | q = 0.00015~124 L/s·m, K = 0.000077~12.41 m/d |

Detect Items | Method | Instrument |
---|---|---|

PH | Electrode method | PP-50-P11 acidity meter |

HCO_{3}^{−}, CO_{3}^{2−}, OH^{−}, Cl^{−}, SO_{4}^{2−}, K^{+}, Na^{+}, Ca^{2+}, Mg^{2+}, NH_{4}^{+} | Titration method (DZ/T0064-93, DZ55-87) | Digital titrators 1620506 |

Li, Sc, Ti, V, Mn, Cr, Co, Ni, Cu, Zn, Rb, Mo, Sb, Cs, Ba, U, Sr | DZ/T 0064.80-1993 | ICP-MS 2000 |

Class | Interclass F Value | F_{0.05(2, 17)} |
---|---|---|

2 and 1 | 5.457 | 3.59 |

3 and 1 | 26.206 | |

3 and 2 | 7.589 |

Class | Interclass F Value | F_{0.05(2, 12)} |
---|---|---|

2 and 1 | 2.626 | 3.88 |

3 and 1 | 20.081 | |

3 and 2 | 5.709 |

Principal Component | Characteristic Value | Cumulative Variance Contribution Rate |
---|---|---|

Z_{1} | 6.756 | 39.739 |

Z_{2} | 3.051 | 57.688 |

Z_{3} | 1.483 | 66.409 |

Z_{4} | 1.187 | 73.393 |

Z_{5} | 0.988 | 79.204 |

Z_{6} | 0.905 | 84.527 |

Z_{7} | 0.766 | 89.035 |

Z_{8} | 0.666 | 92.953 |

Z_{9} | 0.489 | 95.832 |

Z_{10} | 0.271 | 97.424 |

**Table 8.**Estimation of coefficients (C

_{0g}, C

_{1g}, C

_{2g}, …, C

_{pg}) of Bayes multi-class LDA function for trace elements.

Variable | Group | ||
---|---|---|---|

x | Y_{1}(x) Permian | Y_{2}(x) Taiyuan | Y_{3}(x) Ordovician |

Li | −0.034 | −0.003 | −0.043 |

Ni | −0.278 | −0.290 | −0.175 |

Rb | 0.069 | 0.151 | 0.178 |

Mo | 0.057 | 0.081 | 0.023 |

Ba | 0.015 | 0.039 | 0.014 |

Sr | 0.002 | 0.002 | 0.002 |

Cr | 0.289 | 0.257 | 0.187 |

(constant) | −16.597 | −18.215 | −9.189 |

Original Classification | New Classification | |||
---|---|---|---|---|

1 (Permian) | 2 (Taiyuan) | 3 (Ordovician) | All | |

1 (Permian) | 8 | 1 | 1 | 10 |

2 (Taiyuan) | 1 | 6 | 0 | 7 |

3 (Ordovician) | 0 | 0 | 11 | 11 |

All | 9 | 7 | 12 | 28 |

**Table 10.**Estimation of coefficients (C

_{0g}, C

_{1g}, C

_{2g}, …, C

_{pg}) of Bayes multi-class LDA function for conventional ions.

Variable | Group | ||
---|---|---|---|

x | Y_{1}(x) Permian | Y_{2}(x) Taiyuan | Y_{3}(x) Ordovician |

K^{+} + Na^{+} | 0.080 | 0.056 | 0.063 |

Ca^{2+} | 0.089 | 0.053 | 0.094 |

Mg^{2+} | 0.245 | 0.171 | 0.236 |

Cl^{−} | −0.052 | −0.037 | −0.038 |

SO_{4}^{2−} | −0.040 | −0.024 | −0.039 |

(constant) | −9.533 | −6.275 | −7.651 |

**Table 11.**Retrospective discrimination results of Bayes multi-class LDA function for conventional ions.

Original Classification | New Classification | |||
---|---|---|---|---|

1 (Permian) | 2 (Taiyuan) | 3 (Ordovician) | All | |

1 (Permian) | 5 | 2 | 3 | 10 |

2 (Taiyuan) | 2 | 5 | 0 | 7 |

3 (Ordovician) | 1 | 2 | 8 | 11 |

All | 8 | 9 | 11 | 28 |

**Table 12.**Modified Bayes multi-class LDA function coefficient (C

_{0g}, C

_{1g}, C

_{2g}, …, C

_{pg}) estimates.

Variable | Group | ||
---|---|---|---|

x | Y_{1}(x) Permian | Y_{2}(x) Taiyuan | Y_{3}(x) Ordovician |

K^{+} + Na^{+} | −0.036 | −0.208 | −0.088 |

Ca^{2+} | −0.013 | −0.201 | −0.021 |

Mg^{2+} | 0.045 | −0.392 | −0.064 |

Cl^{−} | 0.031 | 0.157 | 0.080 |

SO_{4}^{2−} | 0.013 | 0.104 | 0.023 |

Li | −0.042 | −0.111 | −0.045 |

Ni | −0.471 | −0.864 | −0.601 |

Rb | 0.265 | 0.847 | 0.657 |

Mo | 0.086 | 0.139 | 0.053 |

Ba | 0.048 | 0.161 | 0.064 |

Cr | 0.409 | 0.691 | 0.446 |

Sr | 0.001 | 0.002 | 0.001 |

(constant) | −20.046 | −31.169 | −17.515 |

Original Classification | New Classification | |||
---|---|---|---|---|

1 (Permian) | 2 (Taiyuan) | 3 (Ordovician) | All | |

1 (Permian) | 9 | 0 | 1 | 10 |

2 (Taiyuan) | 0 | 6 | 1 | 7 |

3 (Ordovician) | 0 | 0 | 11 | 11 |

All | 9 | 6 | 11 | 22 |

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## Share and Cite

**MDPI and ACS Style**

Xue, W.; Hou, E.; Zhao, X.; Ye, Y.; Tsangaratos, P.; Ilia, I.; Chen, W. Discriminant Analysis of Water Inrush Sources in the Weibei Coalfield, Shaanxi Province, China. *Water* **2023**, *15*, 453.
https://doi.org/10.3390/w15030453

**AMA Style**

Xue W, Hou E, Zhao X, Ye Y, Tsangaratos P, Ilia I, Chen W. Discriminant Analysis of Water Inrush Sources in the Weibei Coalfield, Shaanxi Province, China. *Water*. 2023; 15(3):453.
https://doi.org/10.3390/w15030453

**Chicago/Turabian Style**

Xue, Weifeng, Enke Hou, Xia Zhao, Yong Ye, Paraskevas Tsangaratos, Ioanna Ilia, and Wei Chen. 2023. "Discriminant Analysis of Water Inrush Sources in the Weibei Coalfield, Shaanxi Province, China" *Water* 15, no. 3: 453.
https://doi.org/10.3390/w15030453