Assessing Slope Stability of Open Pit Mines using PCA and Fisher Discriminant Analysis

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1 Assessing Sloe Stabilit of Oen Pit Mines using PCA and Fisher Discriminant Analsis Shiao Xiang College of Geoscience and Surveing Engineering, China Universit of Mining and echnolog (Beijing), Beijing , China Abstract In order to determine the sloe stabilit of oen-it mine effectivel from available statistical data, this aer rooses a discriminant analsis model based on Princial Comonent Analsis (PCA) and Fisher discriminant analsis. he 6 discriminant indices which affect the sloe stabilit of oen-it mine including gravit densit of rocks, cohesion force, internal friction angle, sloe angle, sloe height and ore water ressure were reduced into 3 comrehensive indices including the first, second and third rincial comonents b PCA to eliminate the mutual influence of the discriminant indices and avoid miscalculation due to different index dimensions. hen, a discriminant analsis model for sloe stabilit was formulated b Fisher discriminant analsis which has no secial requirements for data distribution. he back substitution method was used to test the model. he results of training and discrimination for 32 training samles and 7 ending samles show that this method can discriminate accuratel and it can be a new method to discriminate the sloe stabilit of oen-it mine. Comared with others, this method can imrove the accurac of discriminant analsis. Kewords - Oen-it mine; Sloe stabilit; PCA; Fisher discriminant analsis; Dimensions reduction I. INRODUCION With the increase of oen-it mining deth and the continuous exansion of mining boundar, sloe stabilit has become one of the imortant factors affecting the safet of mining roduction. Sloe stabilit directl affects the ersonnel and equiment safet and economic benefits of mine area [1]. Monitoring the oen-it mine sloes and discriminating the stabilit of them have been imortant research contents of mine surveing domain and mine safet domain [2]. At resent, there are man methods to discriminate the sloe stabilit which have been used in ractice, such as quantitative analsis methods which include the limit equilibrium method [3], numerical analsis method [4], [5], [6], block theor method [7] and so on, and qualitative analsis methods which include the analogism [8], grahic method [9] and so on. Aiming at the comlicated and changeable influence factors of sloe stabilit, the methods to redict unknown stabilit of new monitoring sloes b carring out statistical analsis of existing sloe data have been generated, such as distance discriminant analsis method [10], suort vector machine method [11], neural network method [12], [13], random forest rediction method [14] and so on. If there were too man discriminant indexes which affected sloe stabilit, the miscalculation would be caused b information overlaing. Meanwhile, the different index dimensions and the unknown distribution of data would also affect the discriminant accurac. According to the questions mentioned above, this aer transforms the indexes with correlation and overlaing information into indeendent linear combinations b rincial comonent analsis (PCA). hen, the linear combinations are used as the new comrehensive discriminant indexes to achieve the uroses of reducing dimensions and eliminating influences of different dimensions. Finall, the discriminant analsis model for sloe stabilit of oen-it mine is built b Fisher discriminant analsis to discriminate the sloe stabilit b statistical methods. II. HEORY AND ALGORIHMS A. PCA PCA is a method which is used for comressing data and extracting feature information. According to the correlation of the indexes, the are transformed into a set of new and unrelated comrehensive indexes. Fewer comrehensive indexes are selected to reflect the original indexes as much as ossible deending on the actual needs. Finall, the goal of reducing dimensions and simlifing structure of data can be achieved. he basic rincile is shown below: Assume X X1, X2,, X is -dimension vector, and do linear transformation as Z AX b vectors of X: DOI /IJSSS.a ISSN: x online, rint

2 Z1 a11x1 a21x2 a 1X Z2 a12x1 a22x2 a2x Z a X a X a X where Z 1 is the first rincial comonent, Z 2 is the second rincial comonent,, and Z is the th rincial comonent. Eq.(1) satisfies the following conditions: i i i (1) a1 a2 a 1, i 1,, ; Z i and Z i j;, i j 1,, are unrelated; he variance of Z 1 is j the biggest, the variance of Z 2 is the second biggest,, and the variance of Z is the smallest. he general rocess of calculating rincial comonents is as follows: Standardize the original data to eliminate the influence of different dimensions and calculate the correlation matrix R. Calculate the characteristic roots which include 1, 2 b correlation matrix R and order them in decreasing order, such as 1 2 and then, calculate the corresonding unit characteristic vectors which include a 1, a 2 a. he values of unit characteristic vectors are the coefficients of Eq.(1). In this wa, Z1, Z2, Z can be got. Meanwhile, the data in Eq.(1) is the standardized data. Calculate the cumulative contribution rate of the first m characteristic roots as m k i. If the k1 i1 cumulative contribution rate is 80%~90%, the first m rincial comonents can basicall contain most information of original indexes and the can be used for relacing the original indexes [15]. B. Fisher discriminant analsis Fisher discriminant analsis was ut forward in his method has no secial requirements for oulation distribution, so it is suitable for the oulation with unknown distribution [16]. he basic rincile is shown below: ake two oulations for examle. rain the samles with indexes from oulations G 1 and G 2, and construct discriminant function c1x1c2x2 c x. he values of c1, c2 c are determined in order to maximize the disersion between two oulations and minimize the disersion within each oulation. Calculate the values of ending samles b discriminant function and comare them with the critical value 0 to determine the samles belonging to which oulation. he calculation rocess is as follows: Construct function: Q I F n n1 2 (1) (1) 2 (2) (2) 2 ( i ) ( i ) i1 i1 2 (2) where Q is the disersion between two oulations and F is the disersion within each oulation. Calculate c1, c2c to maximize I through the necessar conditions for extreme value of calculus. hen, Eq.(3) can be got: where C c1, c1,, c ; n i 2 () i () i () i () i xj x xj x S ; i1 j1 D d1, d2,, d 1 C S D (3) x1 x1, x2 x2,, x x. In this wa, the discriminant function can be obtained. DOI /IJSSS.a ISSN: x online, rint

3 Calculate the critical value 0. he critical value can be got b Eq.(4): n n 0 n n Discriminant criterion. Calculate the value of each ending samle b discriminant function and discriminate the samles b the following criterion: If and 0, then X G1 and 0, then X G2 result could not be determined. If and 0, then X G2 and 0, then X G1 result could not be determined. C. Evaluation Criterion (4) and 0, the and 0, the his aer uses back substitution method based on training samles to evaluate the discriminant analsis model. Assume that G 1 and G 2 are two oulations and the caacities of them are n 1 and n 2. Calculate the results of n 1 +n 2 training samles b discriminant function. he misjudgment rate is calculated b Eq.(5), as shown below: * * n12 n 21 n1 n 2 (5) G 1 but are wrongl discriminated into G 2 ; * n is the number of samles which belong to G 2 but are wrongl discriminated into G 1. D. Algorithm Flow he algorithm flow of establishing discriminant analsis model and discriminating ending samles is shown as Fig. 1. Firstl, Calculate the samles with -dimension vector b PCA, then select the first m rincial comonents whose characteristic roots could m meet i 80% to reduce the dimensions and k k1 i1 obtain the comrehensive indexes. Secondl, comute the samles with m-dimension vector b Fisher discriminant analsis to establish the discriminant analsis model and evaluate the model b evaluation criterion. If the model satisfies the evaluation criterion, it can be ut into use. If the model does not satisf the evaluation criterion, the training samles require to be increased to reeat the whole flow. hirdl, standardize the ending samles and calculate standardized samles b Eq.(6), Eq.(7) and Eq.(8) to reduce the dimensions and get the ending samles with m-dimension vector. Finall, discriminate the ending samles b discriminant analsis model for sloe stabilit of oen-it mine. In this wa, the whole discrimination flow based on PCA and Fisher discriminant analsis can be finished. 21 Where * 12 n is the number of samles which belong to DOI /IJSSS.a ISSN: x online, rint

4 raining Samles X 1,X 2,,X PCA m k k1 i1 80%? i Yes Z 1,Z 2,,Z m Fisher Discriminant Analsis Increase raining Samles No Evaluation Criterion Yes Pending Samles X 1,X 2,,X Standardization Eq.(6),(7),(8) Z 1,Z 2,,Z m Discriminant Analsis Model G 1? G 2? Figure 1. Algorithm flow. III. ESABLISHMEN OF HE DISCRIMINAN ANALYSIS MODEL FOR SLOPE SABILIY OF OPEN-PI MINE A. Selection of discriminant indexes he sloe of oen-it mine is a comlex sstem and its stabilit is affected b a variet of factors, such as engineering geological conditions, hdrologic conditions, litholog, geometr, and so on [17]. According to the theoretical research, field investigation and related research results [11], [14], the 6 discriminant indexes which affect the sloe stabilit of oen-it mine kn including gravit densit of rocks X 1 ( ), cohesion 3 m force X 2 (kp a ), internal friction angle X 3 ( ), sloe angle X 4 ( ), sloe height X 5 (m) and ore water ressure X 6 are selected, considering the geometric factors, geotechnical mechanics index. he tes of sloes are divided into stable and destructive. B. Establishment of the discriminant analsis model his aer selects 32 sets of samles with clear conclusions in reference [11] as training samles. he samles have comlete indexes which include gravit densit of rocks, cohesion force, internal friction angle and so on. he can be divided into two oulations. he first oulation includes 14 sets of samles and their te is destructive. he second oulation includes 18 sets of samles and their te is stable. he distribution of the oulations is unknown. he data are shown in able I. ABLE I. DAA OF SLOPE SABILIY DISCRIMINAN NO. Discriminant Indexes X1 X2 X3 X4 X5 X6 Actual es D D D D D D DOI /IJSSS.a ISSN: x online, rint

5 S S S S S S Note: D - destructive; S - stable. 1) Data analsis of PCA for training samles In engineering ractice, the data often have different dimensions. he influence of different dimensions can be eliminated through standardizing the data in able II. hen, the correlation matrix of samles and the corresonding eigenvalues, contribution rates and cumulative contribution rates of each rincial comonent can be obtained. We order rincial comonents in decreasing order, as shown in able II. ABLE II. EIGENVALUES, CONRIBUION RAES AND CUMULAIVE Princial Comonent CONRIBUION RAES OF EACH PRINCIPAL COMPONEN Eigenvalue Contribution Rate/% Cumulative Contribution Rate/% Z Z Z able II shows that the bigger the rincial comonent is, the greater the contribution rate is. Generall, when the cumulative contribution rate reaches 80%~90%, the rincial comonents can basicall contain most information of original indexes. he cumulative contribution rate of the first three rincial comonents has reached %, so the are able to reflect the basic information of original indexes well. he unit characteristic vectors can be calculated b corresonding eigenvalues. hen, we can get the functions of the first, the second and the third rincial comonent as Eq.(6), Eq.(7) and Eq.(8). Z X X X X4 (6) 0.394X X6 Z X X2 0.5X X4 (7) X X6 Z X X X X4 (8) X X6 Eq.(6) shows that the first rincial comonent has strong ositive correlation with X 1 and X 4, which can be regarded as a comrehensive index of gravit densit of rocks and sloe angle. Eq.(7) shows that the second rincial comonent has strong ositive correlation with X 3 and X 6, which can be regarded as a comrehensive index of internal friction angle and ore water ressure. Eq.(8) shows that the third rincial comonent has strong negative correlation with X 2 and strong ositive correlation with X 5, which can be regarded as a comrehensive index of cohesion force and sloe height. Calculate the standardized original data through Eq.(6), Eq.(7) and Eq.(8). he results are shown in able III. ABLE III. HE RESULS CALCULAED BY PCA NO. Comrehensive Indexes Actual Z1 Z2 Z3 es D D D D D D S S S S S S Note: D - destructive; S - stable. In this wa, Z 1, Z 2 and Z 3 can be new discriminant indexes insteading of X 1, X 2, X 3, X 4, X 5 and X 6 to achieve the urose of dimensions reduction. 2) Data analsis of Fisher discriminant analsis for training samles For the data in able III, we regard Z 1, Z 2 and Z 3 as the discriminant indexes. he samles belonging to destructive te have been classified into G 1 and the ones belonging to stable te have been classified into G 2. Finall, the discriminant function can be got b Fisher discriminant analsis. he function is shown as Eq.(9) Z Z Z3 (9) Eq.(9) is the discriminant analsis model for sloe DOI /IJSSS.a ISSN: x online, rint

6 stabilit of oen-it mine based on PCA and Fisher IV. MODEL APPLICAION discriminant analsis. We can get that and Discriminate the 7 ending samles in reference [11] b discriminant analsis model for sloe stabilit of oen-it b Eq.(9) and Eq.(4). However, the mine and discriminant criterion. his model is comared discriminant analsis model needs to be evaluated b the with model of ELM(Extreme Learning Machine) and model evaluation criterion. If the model satisfies the evaluation of LSSVM(Least Squares Suort Vector Machine). he criterion, the model can be ut into use. data and results are shown in able IV. B comaring the C. Evaluation for the model calculated results with actual tes, the correct rates of model of PCA and Fisher discriminant analsis, model of Calculate the data in able III b Eq.(9), and comare ELM and model of LSSVM are 100%, 71.43% and 85.71% the results with 0. hen, the misjudgment rate of this resectivel. It is roved that this model can imrove the model can be obtained as % b Eq.(5) and the accurac of discriminant analsis and it can be used to correct rate can be obtained as % accordingl. When determine the sloe stabilit of oen-it mine. the correct rate is close to 80%, the established model can be ut into use [18]. ABLE IV. HE DISCRIMINAN RESULS OF PENDING SAMPLES NO. Discriminant Indexes Actual es PCA+ Fisher X1 X2 X3 X4 X5 X D D D D D D D D D D S * S * S S D * S S S S S S S S S S S S S Note: D - destructive; S stable; * - the samle of miscalculation. ELM LSSVM V. CONCLUSIONS his aer discriminates the sloe stabilit of oen-it mine b statistical method and reduces the dimensions of discriminant indexes b PCA to extract the main information and eliminate the mutual influence of the discriminant indexes. In this wa, we can use fewer indexes to reresent influence factors which affect the sloe stabilit of oen-it mine to effectivel solve the inconvenience of too man indexes. Finall, the discriminant analsis model for sloe stabilit of oen-it mine is built b Fisher discriminant analsis. he more original discriminant indexes are, the greater the advantage of this method is. he standardization for original data b PCA can eliminate the influence of different dimensions. In most cases, the distribution of data is usuall unknown. Since Fisher discriminant analsis has no secial requirements for data distribution, it is suitable for actual situation. he correct rate of discriminating ending samles through the model in this aer has reached 100%, which can rove the feasibilit of the method. Comared with other models, the model can imrove the accurac of discriminant analsis. he samle data is directl related to the accurac and reliabilit of the model, therefore, we should collect data full and widel to extend the samle database in the further research. In this wa, the accurac of discriminant analsis model can be further imroved. REFERENCES [1] F. H. Gu, Failure tes and damage scale analsis of bench sloe based on geometrical characteristics, Metal Mine, No. 7, , [2] W. J. Yu, and Q. Gao, Alication of a snthetic monitoring technique to high and stee sloes, Chinese Journal of Engineering, vol. 32, No. 1, , [3] L. D. Wu, A. J. Su, X. Huo, and Z. Y. Qi, Comarison and analsis of sloe safet factor b different limit equilibrium methods, Water Resources and Power, vol. 31, No. 12, , [4] M. M. Farias, and D. J. Nalor, Safet analsis using finite elements, Comuter and Geotechnics, vol. 22, No. 2, , [5] J. M. Duncan, State of the art: limit equilibrium and finite-element analsis of sloes, Journal of Geotechnical Engineering, vol. 122, No. 7, , [6] B. F. Kong, H. N. Ruan, Z. D. Zhu, W. J. Yuan, and Z. Z. Chen, DOI /IJSSS.a ISSN: x online, rint

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