Average Linkage Hierarchical Clustering Algorithm for Determining the Relationships between Elements in Coal.

The modes of occurrence of elements in coal are important not only because they can provide insights into the sources of mineral matter in coal but also because they are vital in determining the behavior of their environmental and human health impacts. Besides a number of physical and chemical analyses for determining the modes of occurrence in coal, some statistical methods have been commonly adopted to investigate elements in coal. Among many statistical methods, the hierarchy clustering algorithm is the most common method for deducing modes of occurrence of elements in coal. However, different hierarchical clustering algorithms with a number of similarity measures sometimes result in different modes of occurrence of elements in coal, and subsequently in some cases, such results could be confusing. Therefore, which algorithm is more effective in determining the modes of occurrence in coal deserves to be investigated. In this paper, the data sets of coals from the Adaohai coal mine in Inner Mongolia, China, are used for this performance evaluation. From the analytical results with the average linkage hierarchical clustering algorithm on Adaohai coal samples, many instructive and surprising insights can be concluded. For example, selenium, Be, and Tl do not appear to be in agreement with geochemical principles, that is, substituting for P, associated with rare earth elements, and occurring in Fe-sulfides, respectively. In conclusion, the average linkage hierarchical clustering algorithm with correlation similarity is much better in the analysis of the geological processes than the previous statistical method used in Adaohai coal samples, that is, centroid linkage hierarchical clustering algorithm with Pearson correlation similarity.

Introduction

The modes of occurrence of elements in coal are important because they can provide insights into the sources of mineral matter in coal resulting from different geological processes.[1−3] Also, the modes of occurrence of elements in coal are vital in determining the behavior of their environmental and human health impacts.[4] In addition to a number of physical and chemical procedures for determining the modes of occurrence in coal,[5,6] several statistical methods have been commonly adopted to investigate the modes of occurrence of elements in coal, including correlation analysis, cluster analysis, and principal component analysis. Correlation analysis is the simplest method for analyzing the relationships of element concentrations versus ash yields and has been widely used in such studies[1] as it may provide direct indications of the organic and inorganic associations of elements in coal. Principal component analysis is one of the most popular dimensionality reduction techniques.[7] However, in helping deduce modes of occurrence of elements in coal, the hierarchical clustering algorithm is optimum because it represents the affinity of all the elements in coal.[8−11] However, different hierarchical clustering algorithms, that is, average linkage hierarchical clustering algorithm (ALHCA) and centroid linkage hierarchical clustering algorithm (CLHCA), in many cases, may result in different interpretations on modes of occurrence of elements in coal.[12−14]

Several recent publications have noted potential misinterpretations if the hierarchical clustering algorithm of coal chemistry data is not properly used.[15,16] This could be especially misleading when interpreting the modes of occurrence of the elements in a set of coal samples with great variation of ash yields. Nevertheless, if hierarchical clustering algorithms with some similarity measures are properly used, the results on the modes of occurrence of element in coal may be not only reliable but also informative, providing insights that could not be obtained via chemical and physical methods. Here, we look at data of the Late Paleozoic coals from the Adaohai coal mine in Inner Mongolia, China, from a comprehensive study by Dai et al.,[8] as shown in Table . All the analytical results on the 33 Adaohai samples were presented in a dendrogram (Figure ) produced by cluster analysis, as shown in the paper by Dai et al.[8]

Table 1

Chemical Analysis of the Samples (Ash Yield and Oxides, %; Trace Elements, μg/g)a

sample	ash	Al2O3	SiO2	CaO	K2O	TiO2	Fe2O3	MgO	Na2O	MnO	P2O5	Li	Be	F	Sc	V	Cr	Co	Ni	Cu	Zn
A1	33.45	15	17	0.17	0.014	0.78	0.19	0.02	0.03	0.002	0.022	19	1.17	237	9.79	180	23.7	2.81	5.32	44.4	43.7
A2	33.07	14.88	16.84	0.19	0.026	0.6	0.38	0.01	0.034	0.006	0.016	15	2.26	201	11.3	64.5	16.7	2.13	4.2	18	23.1
A3	24.55	10.73	12.1	0.22	0.032	0.88	0.4	0.02	0.022	0.007	0.01	6.6	2.44	130	11.4	49.2	17.8	1.66	3.11	14.4	36.9
A4	23.49	10.51	11.91	0.11	0.021	0.58	0.24	0.02	0.018	0.004	0.009	5.99	1.6	132	5.99	33.6	10.2	2.01	3.5	9.12	8.14
A5	21.89	9.5	10.59	0.38	0.023	0.78	0.38	0.02	0.018	0.006	0.008	6.16	1.88	165	9.71	41.6	13.6	2.26	4.04	11.4	33
A6	11.17	4.81	5.27	0.23	0.01	0.53	0.16	0.02	0.008	0.002	0.004	4.06	2.09	87.6	11.3	39.4	14.9	3.49	3.89	92.2	28.1
A7	45.22	14.54	14.45	5.34	0.047	0.35	6.12	1.57	0.05	0.123	0.015	12.5	1.32	205	5.49	23.5	10.2	2.1	4.21	8.62	14.8
A8	35.63	16.11	18.36	0.08	0.062	0.55	0.11	0.02	0.028	0.0004	0.03	5.01	2.28	114	6.79	23	9.94	2.23	1.48	12.1	12.7
A9	39.05	18.01	20.22	0.1	0.046	0.45	0.09	0.01	0.043	0.005	0.138	7.73	2.89	270	9.22	28.6	13.5	1.3	4.07	10.6	23.5
A10	12.55	5.59	5.83	0.54	0.024	0.23	0.06	0.01	0.012	0.001	0.052	1.37	2.73	79.2	12.3	45.4	10.7	3.41	2.86	14.1	31.2
A11	38.25	17.65	18.73	0.3	0.075	0.82	0.14	0.05	0.037	0.005	0.133	11.5	2.58	371	5.78	28.5	13.4	1.36	2.51	15.1	37.9
A13	5.24	3.28	0.72	0.41	0.005	0.14	0.29	0.05	0.01	0.004	0.077	2.08	0.91	153	1.27	15.2	12.1	2.04	3.72	9.93	76
A14	5.37	3.53	0.1	0.35	0.003	0.25	0.78	0.15	0.004	0.012	0.05	0.92	1.08	112	2.33	19.7	12.6	1.4	2.77	17.3	21.1
A15	12.58	4.54	0.1	3.95	0.001	0.1	1.18	0.91	0.005	0.023	0.107	0.005	1.18	199	2.78	11.7	7.3	1.85	2.43	8.1	15.7
A17	19.98	4.38	2.01	6.49	0.005	0.1	4.13	1.45	0.026	0.103	0.028	3.04	0.87	204	2.32	12.4	12.6	3.02	6.68	7.54	13.2
A18	22.59	9.58	7.64	3.42	0.028	0.31	0.33	0.21	0.028	0.004	0.061	8.15	1.01	555	2.14	15.2	13.2	3.08	4.68	7.7	24.2
A19	18.44	8.79	7.48	1.42	0.016	0.22	0.41	0.2	0.034	0.005	0.03	9.48	0.89	165	2.99	15.8	11.1	3.6	3.7	10.4	20.9
A21	20	8.85	6.57	2.01	0.012	0.3	0.68	0.33	0.036	0.012	0.343	7.56	1.39	300	6.66	25.7	12.6	7.08	4.83	16.4	24
A22	25.33	10.6	9.08	2.53	0.024	0.41	0.7	0.41	0.037	0.008	0.169	9.06	1.45	151	8.76	45.2	16.2	6.58	6.51	16.4	23.1
A23	12.26	8.24	0.92	0.45	0.011	0.37	0.23	0.04	0.01	0.003	0.952	2.33	1.19	429	9.14	43.1	10.1	10.5	2.91	16.6	27
A24	15.55	10.69	2.41	0.39	0.039	0.44	0.29	0.04	0.016	0.004	1.023	3.08	2	824	7.83	30.6	15	8.13	10.5	17.5	44.6
A25	34.7	16.16	17.03	0.19	0.06	0.86	0.25	0.03	0.051	0.001	0.038	13.8	2.46	256	16.9	53.2	23.3	3.96	12.4	25.4	22.5
A28	22.58	12.49	6.01	1.04	0.046	0.57	0.58	0.16	0.026	0.006	0.302	5.19	1.92	248	9.52	41.4	18.7	8.57	20.6	21.4	26.6
A29	23.28	11.07	10.6	0.2	0.028	0.66	0.24	0.04	0.024	0.001	0.088	6.12	2.28	115	14.6	57.9	21.1	5.86	18.2	15.1	29.2
A30	36.85	17.4	18.2	0.21	0.017	0.68	0.29	0.03	0.04	0.002	0.056	8.73	2.52	117	12.3	54.2	18.6	5.96	10.5	19.9	20
A31	46.48	21.39	22.04	0.2	0.029	0.83	0.4	0.04	0.049	0.001	0.093	9.55	2.83	196	10.8	45	10.9	3.93	18.6	14.5	10.9
A32	32.84	7.93	7.1	8.38	0.024	0.48	4.73	2.46	0.025	0.054	0.055	5.01	1.93	124	18.1	81.1	28.7	3.73	9.27	33.2	25.3
A34	38.6	17.19	18.67	0.57	0.034	0.87	0.68	0.12	0.054	0.006	0.074	9.46	3.09	128	20	64.8	30.6	2.31	7.92	54.5	32.1
A40	23.57	9.77	9.63	1.79	0.019	0.53	0.62	0.17	0.036	0.008	0.031	5.82	2.15	124	9.46	60.4	21.7	10.6	17.3	68.4	96.2
A41	23.26	6.62	5.65	7.94	0.009	0.22	2.18	0.5	0.03	0.059	0.01	5.11	1.39	95.6	7.45	63.8	25.7	9.76	13.2	69.6	44.7
A42	18.97	4.49	4.3	3.92	0.026	1.81	2.84	1.21	0.017	0.028	0.029	2.81	2.48	141	27.9	75.2	26.1	3.75	9.92	35.1	21.8
A43	34.16	14.46	16.06	1.06	0.063	0.94	0.39	0.12	0.035	0.004	0.031	7.76	1.65	119	16.6	105	21.4	8.87	6.6	22.5	18.5
A44	14.17	6.04	5.97	1.22	0.007	0.35	0.36	0.14	0.022	0.003	0.017	4.6	1.05	91	7.73	62.7	15.2	10.6	7.71	16.5	16.7

The data are from Dai et al.[8]

Figure 3

CLHCA on Adaohai samples.

The Adaohai mine is an important coal-hosted critical metal deposit, that is, Al-Ga-REE deposit in coal [rare earth elements (REEs)].[8] The modes of occurrence of elements in the Adaohai coals from the CP2 seam (CP2 is the ID of the coal seam) have been comprehensively investigated by integrated methods, including optical microscopy, X-ray diffraction analysis (XRD), scanning electron microscopy–energy-dispersive X-ray spectroscopy (SEM–EDS), low-temperature ashing (LTA), X-ray fluorescence spectrometry, and inductively coupled plasma mass spectrometry. Also, the sources of the elements in coal (i.e., sediment source region and igneous intrusion) have been identified based on strong evidence of geochemistry, petrology, and mineralogy. Therefore, these samples provide us a good opportunity to compare the validity of the ALHCA and CLHCA methods for determining the modes of occurrence of elements.

Background Information of the Coal from Adaohai Mine

The Adaohai coal mine is located in the southeast of the Daqingshan Coalfield in Inner Mongolia, northern China. There are 16 mines in the Daqinshan Coalfield (Figure B). The coal-bearing strata in the Daqingshan Coalfield include the Pennsylvanian Shuanmazhuang Formation and the Early Permian Zahuaigou Formation.[11,17−19] The CP2 coal, with a thickness varying from 4.72 to 42.79 m and averaging 22.58 m, is the major minable coalbed in the coalfield. The roof of the coalbed mainly consists of mudstones and sandy mudstones with variable thickness. The floor strata of the CP2 coal seam consist of medium-coarse and fine sandstones. The Cambrian-Ordovician strata underlying the coal-bearing sequence mainly consist of limestone thick layers, interlayered with silty mudstone. Due to the Yanshan movement in the Late Jurassic and Early Cretaceous Epochs, the zonal distribution of coal ranks is dominantly related to the intrusion of igneous rocks.[20] In the eastern coalfield, igneous rocks intruded into the coal-bearing strata. The coal ranks range from northwest to southeast, from high volatile bitumen, through medium volatile bitumen, to low volatile bitumen (lvb). The CP2 coal comprises 3 to 42 parting layers which are from 0.02 to 3.4 m thick.

Figure 1

Location of the Daqingshan Coalfield (A) and coal-rank distribution in the different mines of the coalfield (B).aaThe data are from Dai et al.[8]

Samples involved in this study include a total of 33 bench samples[8] collected from the face of the mined coal at the Adaohai Mine, Daqingshan Coalfield. Proximate analyses, total sulfur, vitrinite reflectance, and mineral compositions in low-temperature ashes of coal are given in Table . Table lists the concentrations of major-element oxides and trace elements. The average vitrinite reflectance and average volatile matter (on a dry and ash-free basis) of the coals are 1.58 and 21.65%, respectively, indicating an lvb coal according to the ASTM classification (ASTM D 388–99).[8] Compared with averages of element concentrations in Chinese and world coals reported by Dai et al.[21] and Ketris and Yudovich,[22] the coals from the Adaohai coal mine have a lower SiO2/Al2O3 ratio (0.93) and are higher in CaO (1.69%), MgO (0.32%), P2O5 (0.124%), F (207 μg/g), Ga (16.3 μg/g), Zr (446 μg/g), Ba (276 μg/g), Hg (0.33 μg/g), Pb (25.6 μg/g), and Th (12.4 μg/g).[8] The light REEs are enriched in the coal, and the light and heavy rare earths elements have been highly fractionated.

Table 2

Coal Characteristics of the CP2 Coal in the Adaohai Minea,b

sample	proximate analysis			St,d	Ro,ran	mineral compositions of parting and LTA samples of coal
	Mad	Ad	Vdaf			LTA yield	quartz	kaolinite	illite	diaspore	boehmite	calcite	dolomite	siderite	gorceixite	anatase	pyrite
A44	0.28	14.17	19.37	1.09	1.57	21.68		70.2	2.8	1.6	1.2	6.8	17.5
A43	0.45	34.16	22.84	0.71	1.55	nd		66.9	26.5	2.4		2	2.1
A42	0.42	18.97	18.98	0.79	1.61	nd		33.4	9.3	0.9		0.1	43.1	9.1		4.1
A41	0.45	23.26	23.28	0.71	1.59	35.56		34.4	2.4	2.8	0.9	32.1	22.2	5.2
A40	0.19	23.57	21.74	0.90	1.66	30.52		71.3	2.5	2.9	3.3	8.8	10.7			0.6
A34	0.61	38.6	23.79	0.54	1.62	45.49		92	2.3	1.4	1.7		2.5
A32	0.48	32.84	27.99	0.58	1.61	45.72		31.2	5.6	2.3	1.1	5.2	50.4	4.2
A31	0.58	46.48	26.99	0.47	1.59	53.4		89	4.5	4.1	1.7			0.8
A30	0.49	36.85	23.47	0.62	1.63	43.18		86.3	2.8	6.8	3.2	0.4		0.5
A29	0.53	23.28	19.18	0.85	1.63	27.91		78.8	6.9	7.1	6.1				0.6	0.5
A28	0.51	22.58	21.08	0.91	1.46	27.29		52.3	6.7	9.4	17.9	3	8.7	0.6	0.9	0.4
A25	0.64	34.7	22.1	0.71	1.61	40.99		76.8	17	3	1.9	0.5		0.5		0.3
A24	0.51	15.55	19.09	1.01	1.55	nd		15.2	8.8	23.4	38.2	0.2		5.6	8.6
A23	0.06	12.26	18.86	1.08	1.59	15.01		8.6		35.3	34.2			6.6	15.3
A22	0.24	25.33	22.62	0.86	1.57	30.02		50.1	12.3	11.8	2.3	7.6	15.3	0.2	0.3
A21	0.11	20	22.25	0.91	1.54	23.82		40.4	4.5	21.4	4.5	5.5	21	0.5	2.1
A19	0.22	18.44	20.95	0.94	1.66	22.37		42.1	21.3	10.9	3	8.3	14.5
A18	0.31	22.59	19.74	0.67	1.49	26.72		26.4	24	16.1	2.9	20.5	10.1
A17	0.27	19.98	24.21	0.82	1.61	nd		9.5		9.1	2.3	14.4	49.5	15.3
A15	0.43	12.58	16.4	0.84	1.56	nd				28.5	4.4	18.8	47	0.9		0.5
A14	0.31	5.37	16.4	0.93	1.62	6.13				62.2	3.5	1.7	12.6	18.9		1.1
A13	0.26	5.24	16.26	0.98	1.6	6.44		5.5	11.2	54.3	2.2	11.1	5.6	7.2	1.6	1.3
A11	0.43	38.25	23.63	0.56	1.57	68.41		80.4	12.4	0.6		2.2	3	1.1
A10	0.15	12.55	18.29	0.86	1.53	15.01	0.3	79.5	4	1.9	5	8.1		0.5	0.9
A9	0.52	39.05	23.44	0.53	1.56	45.95	0.2	95.6	2.6	0.5	0.6				0.5
A8	0.35	35.63	22.65	0.58	1.47	40.81		77.8	21.5		0.7
A7	0.34	45.22	38.03	1.08	1.57	nd		55	9.6		0.3	3.5	16.7	12.6			2.3
A6	0.26	11.17	17.18	0.86	1.63	12.37		86.6	3.9		3.5	2.9		1.8		1.3
A5	0.34	21.89	19.57	0.72	1.53	25.52		86.3	3.5		0.6	4.4		4.3		0.9
A4	0.36	23.49	19.74	0.70	1.55	27.58		95.2	2.7		0.4	0.4		1		0.4
A3	0.44	24.55	19.58	0.67	1.6	nd	0.2	89.9	5.6			1.2		1.9		1.2
A2	0.5	33.07	22.07	0.58	1.69	39		93.3	4.2			0.9		1.6
A1	0.46	33.45	22.75	0.67	1.5	39.51		95.7	2.7			0.5		0.6		0.5
Av.	0.38	25.1	21.65	0.78	1.58

Proximate analyses of the CP2 coal in the Adaohai Mine (%); St, total sulfur (%); Ro, ran, random vitrinite reflectance of the CP2 coal in the Adaohai Mine (%); mineral compositions of parting and LTA samples of coal by XRD analysis and Siroquant (%; on organic matter-free basis).

The data are from Dai et al.[8]

Methods

The hierarchical clustering algorithm is a method which splits the data sets recursively into smaller clusters. It divides unlabeled data into different clusters according to their similarity. The data similarity is measured based on the distance and the correlation coefficient.[12] Distance-based similarity states that two data points with a small distance have a large similarity, involving mainly Euclidean distance, Mahalanobis distance, Minkowski distance,[12,14,23] whereas correlation-based similarity state that two data points with a large correlation have a large similarity, involving mainly the Pearson correlation coefficient and cosine similarity. Hierarchy cluster algorithms involve single linkage, complete linkage, average linkage, centroid linkage, and Ward’s method.[23] Usually, the similarity for modes of elements in coal is calculated based on the Pearson correlation coefficient. Only if the similarity is same for different hierarchical clustering algorithms, the performance of different hierarchical clustering algorithms can be evaluated effectively.

The elements in coal are assumed as vector = [x,x,...,x],i = 1...n, m and n are the sample size and element number, respectively. The similarity between element and element is assumed as D(,). The similarity for the elements in coal is expressed as . The cluster whose elements and are assigned is assumed as C = {,}. The average linkage clustering algorithm uses the average similarity between the clusters, as shown in Table . It is expressed as . The centroid linkage clustering algorithm uses the minimum similarity between the centroids in clusters, as shown in Table . It is expressed as . Tables and 4 clearly describe the difference between the average and centroid linkage clustering algorithm.

Table 3

Average-Linkage Algorithm for the Hierarchical Clustering

Input: The element xi, i = 1...n; elements number n; the method for calculating similarity D(xi, xj).

Output: Print hierarchical clustering records.

Begin:

Initialize:

Elements clusters C = {C1,C2, ..., Cn} = {{x1},{x2}, ..., {xn}},

Minimum distance dmin, cluster index a,b.

While length(C) > 1 do

dmin = Infinity.

For i = 1 → (length(C) – 1) do

For j = (i+1) → length(C) do

Calculate the distance between Ci and Cj:

If d < dmin then dmin = d, a = i, b = j.

End for

End for

Merge cluster Ca and Cb: Ctmp = Ca ∪ Cb. Delete Ca and Cb from C. Append Ctmp to C.

Print elements {V(xi)|xi ∈ Ctmp}.

End While

End

Table 4

Centroid-Linkage Algorithm for the Hierarchical Clustering

Input: The element xi, i = 1...n; elements number n; the method for calculating similarity D(xi, xj).

Output: Print hierarchical clustering records.

Begin:

Initialize:

Elements clusters C = {C1,C2, ..., Cn} = {{x1},{x2}, ..., {xn}},

Minimum distance dmin, Cluster index a,b.

While length(C) > 1 do

dmin = Infinity.

For i = 1 → (length(C) – 1) do

For j = (i+1) → length(C) do

Calculate the distance between Ci and Cj:

If d < dmin then dmin = d, a = i, b = j.

End for

End for

Merge cluster Ca and Cb: Ctmp = Ca ∪ Cb. Delete Ca and Cb from C. Append Ctmp to C.

Print elements {V(xi)|xi ∈ Ctmp}.

End While

End

Results and Discussion

Comparison of Two Different Hierarchical Clustering Algorithms

The data set from the Adaohai mine is used for a different hierarchical clustering algorithm than used by Dai et al.[8] The result is shown in Figure with the ALHCA, while Figure is with the CLHCA.

Figure 2

ALHCA on Adaohai samples.

The differences between Figure and the cluster results of Figure are due to the different hierarchical clustering algorithms, that is, the difference between the centroid and the average algorithms. For both the centroid and average algorithms, Sr versus Ba, Sn versus Hg, Cd versus Zn, Zr versus Hf, and Nb versus Ta are clustered in the first stage. Additionally, the major elements including Ca, Mg, Mn, and Fe are clustered in the first stage. Also, Al and Si are clustered in the first stage. However, the ALHCA method is much better in terms of the similarity of Sn versus Hg and Sr versus Ba. For the ALHCA method, Sn versus Hg and Sr versus Ba are clustered in the lower stage than the centroid linkage algorithm. Note that the REEs and Y (REY) are clustered in the lower stage than that of Dai et al. Although Figure is much better than Figure in terms of the similarity of Zr versus Hf in the ALHCA method being clustered in the higher stage than that of Dai et al., ALHCA is generally better than that of CLHCA. The result shows that Figure is much better for determining the modes of occurrence of the elements based on the performance evaluation of the two methods because the clustering of the elements is totally consistent with geochemical principles. The cophenetic correlation coefficient for ALHCA and CLHCA is 0.8782 and 0.8786, respectively. Although the centroid linkage is only a little higher than the ALHCA, clustering of the elements is totally consistent with geochemical principles with the ALHCA.

The data from the Datanhao mine and Hailiushu mine are also analyzed using the two different methods. In terms of the modes of occurrence of the elements, results in Figures and 6 derived from the ALHCA are better than those in Figures and 7 derived from the CLHCA, consistent with those in the Adaohai mine.

Figure 4

ALHCA on Datanhao samples.

Figure 6

ALHCA on Hailiushu samples.

Figure 5

CLHCA on Datanhao samples.

Figure 7

CLHCA on Hailiushu samples.

Modes of Occurrence

On the basis of trace-elements’ geochemical nature and on the investigations by Zhao et al.[24] and Dai et al.[8] using directed analysis (such as SEM–EDS and XRD), the geochemical nature of each pair of the following elements, that is, Sr versus Ba, Sn versus Hg, Cd versus Zn, Zr versus Hf, and Nb versus Ta, is similar. Additionally, the major elements including Ca, Mg, Mn, and Fe would be expected to be clustered together as they generally coexist in carbonate minerals. Also, Al and Si are both largely associated with silicate minerals and should be expected to be clustered. In representing the similarities among all the major elements including Ca, Mg, Mn, and Fe, the two methods are similar. However, the ALHCA method is much better in terms of the similarity of Sn versus Hg and Sr versus Ba. It would be expected that the REEs and Y (REY) should generally be clustered together if a method is effective. Note that the REY is not clustered in Figure .[8] However, the result of REY clustering with the average linkage method is much better than that of Dai et al. Although Figure is much better than Figure in terms of the similarity of Zr versus Hf, the average linkage method is generally more consistent with geochemical principles than that of the CLHCA.

For the Datanhao mine and Hailiushu mine, the results of REY clustering with the average linkage method are much better than those of the centroid linkage method. Although in representing, Sr versus Ba, Sn versus Hg, Cd versus Zn, Zr versus Hf, Nb versus Ta, and the major elements including Ca, Mg, Mn, and Fe are similar, ALHCA is generally more consistent with geochemical principles than that of the CLHCA.

Co, Mo, Ni, and Re

Starting on the left-hand side, it can be seen that Co, Mo, Ni, and Re are closely linked. This is to be expected for these siderophile elements.[25] This relationship is exceedingly interesting as there is very little information on the modes of occurrence of Re. Only a few articles discussing rhenium in coal could be located.[26−28]Figure does not inform us of the modes of occurrence of these elements; rather, it indicates that they came from the same source and were not separated during deposition or during diagenesis of the coal. From the proximity to ash yields, it can be inferred for these trace components (none more than 37 ppm) that they are detrital rather than of epigenetic in origin. Their proximity to the lithophile elements indicates that they are, in part, associated with the silicate minerals and a lesser portion is associated with the nearby chalophile elements in sulfide minerals.

There are two possibilities emerging. These elements, as occurring in sulfides, were weathered out of the source rocks and were transported into the peat swamp where they are largely associated with secondary silicate minerals, but a portion survived as sulfides. In other words, none of these elements has been gained or lost during transportation and diagenesis.

Cs, K, Rb, Na, Al, Si, Zr, and Ash

Cesium, K, Rb, Li, and Si are all group I alkali metals commonly found in alumino-silicates. Therefore, a close correlation with silica and alumina is expected. As the alumino-silicates constitute the bulk of the ash, so linking with the ash also is consistent for these lithophile elements, likely primarily in clay minerals. Zirconium is commonly suspected to occur as detrital zircons, although some may be in clays. Finkelman et al.[29] estimated that 25% of the zirconium in a suite of 20 coals was in the clays in both bituminous and low-rank coals. Dai et al.,[8] based on strong correlations with silicon and aluminum, suggest that Zr occurs in association with the clay minerals. It is certainly possible that this close relationship could indicate that zircons and detrital illite were both washed into the peat swamp from a single source.

Sn, Zn, Cd, Hg, and As

These are all chalcophile elements except for Sn which are associated with each other in Figure , while Figure is not. Tin in coal is dominantly associated with various minerals in coal, for example, aluminosilicate.[30−33]

Although some studies, for example, Vassilev et al.[34] and Tian et al.,[35] showed that Sn in coal dominantly has a strong association with sulfides, Finkelman et al.[6] do not indicate that a tin sulfide mineral was reported to occur in coal. Finkelman[36] did report on finding three unidentified tin sulfides and several other tin-bearing minerals by SEM/EDS. The relationship of tin with the chalcophile elements provides a strong indication that in this coal, tin occurs primarily in sulfide minerals.

REEs, Y, Be, and Se

The tight clustering of the REE with Y is as would be expected. The close linkage of Se and Be to the REE in Figure is interesting because there has been no such relation reported before, as shown in Figure . There are at least two dozen minerals with Be and REE (https://www.mindat.org/element/Beryllium) but no known Se phosphates.[36] Thus, the Be could be associated with the REE, but the Se remains somewhat problematical. The selenium atomic radius is very close to that of P, so perhaps, Se is substituting for P in the REE phosphates.[37]

U, Hf, Th, Bi, Pb, Sc, In, W, Ti, Nb, and Ta

This cluster of elements is common as all of these elements are generally found in the heavy mineral suite. The close linkage of these elements to the REE elements is consistent with a detrital source for all of these elements. Uranium and thorium are commonly associated in a variety of minerals. Niobium and Ta are almost geochemically inseparable. Titanium is most commonly found in coal as rutile or anatase. Bismuth, lead, scandium, indium, and tungsten are usually found in various heavy minerals, although bismuth and lead are often found in sulfide minerals in coal. However, the Adaohai coal appears to have little in the way of pyritic sulfur.

It is unusual for Zr and Hf not to be more closely associated as they both commonly occur in zircons. The coefficients of determination for Zr and Hf are relatively low. If the Zr is, in part, in the clay minerals as Dai et al.[8] suggest, it may explain the lack of clustering with Hf mentioned above.

Cu, V, and Cr

This clustering of elements is similar to the previous set. These elements are also commonly found in the heavy minerals’ suite. Copper does occur in coal as a sulfide, but, as noted earlier, this coal appears to have little in the way of pyritic sulfur.[8]

S, F, Sr, P, and Ba

Dai et al.[8] report that the coal contains fluorapatite and the barium phosphate gorceixite; this clearly accounts for the fluorine, strontium, barium, and phosphorous relationships.

Tl, Fe, Mn, Ca, and Mg

Dai et al.[8] found the carbonates calcite, dolomite, and siderite in most of the 33 samples (Table ) explain the linkage of calcium, iron, magnesium, and manganese. The association of Tl with these elements is a mystery as Tl is predominantly a chalcophile element Mindat (https://www.mindat.org/element/Thallium) which indicates that more than 82% of known Tl minerals are sulfides and there are no known Tl-carbonates. A likely explanation is that some of the Fe occurs in sulfides that host the Tl.

Concluding Thoughts

Based on the ALHCA which is different from the algorithm in Dai et al.,[8] new insights into the affinity of elements in coal can be obtained. For determining the affinity of elements in coal, we recommend the following:

All hierarchical clustering algorithms with correlations based in the literature can be used for the analysis of element affinities.

For coal geochemistry compositional data pertaining to the non-Euclidean space, coal compositional data need to be transformed from non-Euclidean space to Euclidean space with log-ratio transformations, for example, additive log ratio, isometric log ratio, and centered log ratio. The similarity to the Euclidean distance based on hierarchical clustering algorithms can be used for the analysis of the element’s affinity.

Using the ALHCA, the associations of only three of the 63 elements considered do not appear to be in agreement with geochemical principles, that is, Se, Be, and Tl. The analytical error may be an explanation, but there is no supporting evidence. The Tl is likely in Fe-sulfides, but the relationship is masked by the Fe predominantly being associated with carbonates. The Be is known to be associated with the REE, but the Se remains a mystery, although it may be substituting for P. Nevertheless, the statistical approach illustrated here may reveal modes of occurrence of elements in coal heretofore not considered.

The information gleaned from Figure is instructive but not conclusive. To determine the specific modes of occurrence of the elements in these and other coal samples, diagnostic information from SEM/EDS and from XRD must be obtained.

In the future, we plan to use the fuzzy clustering algorithm to provide some insights into the apparently anomalous behavior of certain elements such as Se, Be, and Tl.