J. Cent. South Univ. Technol. (2008) 15: 117-120 

DOI: 10.1007/s11771-008-0023-x 

Stability classification model of mine-lane surrounding rock based on distance discriminant analysis method

ZHANG Wei(张 伟)¹, LI Xi-bing(李夕兵)^1,2, GONG Feng-qiang(宫凤强)^1,2

(1. School of Resources and Safety Engineering, Central South University, Changsha 410083, China;

2. Hunan Key Laboratory of Resources Exploitation and Hazard Control for Deep Metal Mines,

Changsha 410083, China)

Abstract: Based on the principle of Mahalanobis distance discriminant analysis (DDA) theory, a stability classification model for mine-lane surrounding rock was established, including six indexes of discriminant factors that reflect the engineering quality of surrounding rock: lane depth below surface, span of lane, ratio of directly top layer thickness to coal thickness, uniaxial comprehensive strength of surrounding rock, development degree coefficient of surrounding rock joint and range of broken surrounding rock zone. A DDA model was obtained through training 15 practical measuring samples. The re-substitution method was introduced to verify the stability of DDA model and the ratio of mis-discrimination is zero. The DDA model was used to discriminate 3 new samples and the results are identical with actual rock kind. Compared with the artificial neural network method and support vector mechanic method, the results show that this model has high prediction accuracy and can be used in practical engineering.

Key words: distance discriminant analysis; stability; classification; lane surrounding rock

1 Introduction

Unlike other engineering materials in nature, rock or rock mass is inhomogeneous, leading to high uncertainties in underground rock engineering^[1-4]. The quality of rock or rock mass is the synthetic index reflecting the engineering geological properties and its classification is a basic and complex problem^[5]. The  stability classification of lane surrounding rock is an essential work in mine and rock engineering because the subsequent optimization of types of lane and supporting parameters depend on the result of classification. Many researchers have been involved in the research on this filed and provided many methods^[6-8]. However, these traditional surrounding rock classification systems used for design of underground lane and constructions have some major shortcomings: rock classification parameters are not well defined or sufficient to select adequate parameters due to engineering geological factors; complex properties of surrounding rock cannot be satisfactorily described by a single number; the same rating can be achieved by various combinations of classification parameters, even though the surrounding rock behavior could be different^{[1, 9]}. In recent years, the fuzzy pattern recognition (FPR) theory^[10], fuzzy clustering analysis (FCA) theory^[11] and dynamic engineering classification (DEC) technology have been developed in the classification of lane surrounding rock. The artificial neural network (ANN)^[12-14] and support vector machines (SVM)^[15] have also been introduced in the classification of lane surrounding rock.

In this work, a new approximate solution, i.e. distance discriminant analysis (DDA) method was proposed to estimate the classification of lane surrounding rock. The distance discriminant analysis is a classical statistics approach for classifying samples of unknown classes, based on training samples with known classes. Several indexes of mine-lane surrounding rock were selected as the discriminant factors and a distance discriminant analysis model was obtained through training a large number of expansive samples and used to discriminate the new sample.

2 Distance discriminat analysis theory

The basic principle of distance discriminant analysis theory can be divided into two steps: 1) collect existent collectivities’ information from training samples and construct a corresponding discriminant criterion; 2) according to the discriminant criterion in the first step, discriminate which collectivity that the new given sample belongs to. Here, Mahalanobis distance was introduced as follows.

2.1 Mahalanobis distance

Supposing a collectivity  with m member indexes, a sample can be expressed asand the expected value of X_i denoted by μ_i, is^[16]

 (i=1, 2, …, m).

And then the expected value μ of G can be expressed as

The covariance matrix of G is

               (1)

The Mahalanobis distance between sample X and collectivity G is defined as

              (2)

2.2 Distance discriminant criterion between many collectivities

Supposing that there are many collectivities: (k＞2), and  G_p and G_q are two collectivities taken out from G_k stochastically. The square difference of Mahalanobis distance between sample X and collectivity G_p and G_q is defined as^[17]

     (3)

where 

,.

And then

d²(X, G_q)≥d²(X, G_p)W_p(X)≥W_q(X)         (4a)

d²(X, G_q)＜d²(X, G_p)W_p(X)＜W_q(X)          (4b)

Generally, the expected value vectors μ₁, μ₂, …, μ_k and public covariance matrix Σ are unknown and their estimation values can be obtained from training samples. Supposing that there is one sample from G_q, then the unbiased estimation of μ_q can be defined as

              (5)

When  the unbiased estimation of Σ can be written as

                    (6)

where n_q is the number of training sample from G_q, μ_q and Σ are displaced by and S respectively, then the estimation of W_q(X) (q=1, 2, …, k, t=1, 2, …, n_q) can be obtained as

       (7)

The distance discriminant criterion between many collectivities can be written as follows:

If G_q0 satisfies the condition

                        (8)

then

.

2.3 Estimation of discriminant criterion

The re-substitution method can be used to estimate the reliability of the discriminant criterion^[17].

For example, supposing there are two collectivities, the sample from G_i(i=1, 2) can be defined as

          (9)

where n_i is the number of training sample from G_i (i=1, 2), and n₁ and n₂ are the numbers of training samples from G₁ and G₂, respectively. All training samples are regarded as the new ones and substituted into the established discriminant criterion to discriminate which collectivity every sample belongs to. Defining  as the number of sample that belongs to collectivity G₁ and is discriminated to collectivity G₂; and defining n₂₁ as the number of sample belonging to collectivity G₂ to be discriminated to collectivity G₁, the ratio of mis-discrimination ability η can be written as

                               (10)

3 DDA model on classification of mine-lane surrounding rock and its application

A DDA model on stability classification of mine- lane surrounding rock was proposed in this section. Classification of mine-lane surrounding rock in Pingding Mountain Coal Mining was modeled subsequently. 18 case histories were collected from Pingding Mountain Coal Mines^[12]. 15 case histories were used as training samples and the other 3 case histories were used as testing samples to verify the accuracy of the DDA model. The DDA model structure adopted is shown in Fig.1.

The inputs of DDA model for the stability classification prediction of mine-lane surrounding rock are as follows: X₁ is the depth of lane below surface, m; X₂ is the span of lane, m; X₃ is the ratio of directly top layer thickness to coal thickness; X₄ is the uniaxial comprehensive strength of rock, MPa; X₅ is the development degree coefficient of rock joint; X₆ is the range of broken rock zone, m.

Fig.1 Model of distance discriminant analysis

The outputs of DDA model for the classification prediction of mine-lane surrounding rock are as follows: G₁ is the typeⅠsurrounding rock; G₂ is the type Ⅱsurrounding rock; G₃ is the type Ⅲ surrounding rock; G₄ is the type Ⅳ surrounding rock; G₅ is the typeⅤ surrounding rock.

Discriminant layer of DDA model for the stability classification prediction of lane surrounding rock is W_i
(i=1, 2, … 5).

By training with the samples, the distance discriminant analysis model can be obtained. To test the reliability and applicability of the distance discriminant analysis model, 15 training samples were re-substituted to this model. The discriminated results are shown in Table 1. It can be seen that all results are identical with actual rock kind and the ratio of mis-discrimination η equals 0.

Three testing samples were used as discriminating classifications by the DDA model. The results are shown in Table 2. The results are identical with actual rock kind and the accuracy of this surrounding rock classification model is 100%. For comparison, the artificial neural network (ANN) method^[12] and support vector mechanic (SVM) method^[15] were also be used to discriminate the three samples and their results are shown in Table 2. It can be seen that one of the results by using ANN method is amphibious between Ⅳ and Ⅴ, whereas one of the results by using SVM method is wrong. It can be concluded that the DDA model can be applied to lane surrounding rock stability classification with high accuracy.

Table 1 Data of cases used as inputs of DDA model  (Training samples)

Table 2 Data of cases used as testing samples of DDA model

4 Conclusions

1) Based on the principle of Mahalanobis distance discriminant analysis (DDA) theory, a forecasting model for stability classification of mine-lane surrounding rock is presented. The factors influencing the stability classification of mine-lane surrounding rock of underground opening are taken into account, including lane depth below surface, span of lane, ratio of directly top layer thickness to coal thickness, uniaxis comprehensive strength of rock, development degree coefficient of rock joint and range of broken rock zone.

2) A DDA model is obtained through training a large number of practical measuring samples and used to discriminate the new samples.

3) The re-substitution method is introduced to verify the stability of DDA model and the ratio of mis-discrimination equals zero. Compared with the artificial neural network method and support vector mechanic method, the results show that the DDA classification model has high prediction accuracy and can be used in practical engineering.

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(Edited by CHEN Wei-ping)

Foundation item: Project(50490274) supported by the National Natural Science Foundation of China

Received date: 2007-05-21; Accepted date: 2007-08-06

Corresponding author: ZHANG Wei, PhD; Tel: +86-15885648899; E-mail: cinf_zw@126.com