Determination of stability of epimetamorphic rock slope using Minimax Probability Machine

Figures & data

Table 1. Summary of the available methods for slope stability.

Reference	Brief methodology
Fellenius (1936)	Fellenius assumed that the resultant of inter-slice forces acts in a direction parallel to the base of each slice.
Taylor (1948)	Taylor (1948) used the friction circle method to obtain the stability numbers (Ns ) for homogeneous soil slopes.
Bishop (1955)	Bishop (1955) used the method of slices in obtaining stability of slopes.
Janbu (1957)	This method also used the method of slices to solve the problem. Janbu solved the problem by assuming the point of application of the inter-slices forces.
Morgenstern and Price (1965)	Morgenstern and Price (1965) attempted to satisfy all the equations of statical equilibrium in obtaining the solution of the stability problem using the method of slices.
Chen (1975)	Chen (1975) used the upper bound theorem of the limit analysis to obtain the critical heights for homogenous soil slopes.
Michalowski (1994)	Michalowski (1994) also used the upper bound theorem of limit analysis in order to obtain the stability numbers for homogenous soil slopes.
Michalowski (1995)	Using the upper bound limit analysis, Michalowski (1995) presented a stability analysis of slopes based on a translational mechanism failure mechanism.
Michalowski (2002)	Michalowski (2002) also used the upper bound theorem of limit analysis in order to obtain the stability numbers for homogeneous slopes in the presence of pore water pressures as well as pseudo-static horizontal earthquake body forces.

Table 2. Statistical parameters of the data-set.

Input variable	Mean	Standard deviation	Skewness	Kurtosis
d (kN/m^3)	23.88	2.60	−0.19	1.33
H (m)	46.45	16.09	0.50	3.82
β (°)	32.67	7.60	0.17	3.90
C (kPa)	33.69	8.77	−1.35	4.70
φ (°)	29.94	5.85	−0.26	1.74

Table 3. Performance of training data-set.

d (kN/m^3)	H (m)	β (°)	C (kPa)	φ (°)	Actual class	Predicted class by LMPM	Predicted class by KMPM
20	10	10	8	20	−1	−1	−1
27.3	30	30	37.3	31	1	1	1
20.6	35	25	26.31	22	−1	−1	−1
21.6	50	40	6.5	19	−1	−1	−1
22.4	35	28	28.9	24	−1	−1	−1
23.2	33	30	31.2	23	−1	1	−1
26.8	26	30	37.5	32	1	1	1
27.4	42	25	38.1	31	1	1	1
21.8	50	50	32.7	27	−1	−1	−1
21.8	60	35	27.6	25	−1	−1	−1
26.5	21	30	35.4	32	1	1	1
26.5	39	35	36.1	31	1	1	1
27	69	30	35.8	32	1	1	1
27	22	25	38.4	33	1	1	1
21.4	52	50	28.8	20	−1	−1	−1
26	55	38	42.4	37	1	1	1
26	30	25	39.4	36	1	1	1
25.6	26	25	38.8	36	1	1	1
20	53	45	30.3	25	−1	−1	−1
25.8	50	30	34.7	33	1	1	1
21.8	99	35	28.8	26	−1	−1	−1
21.8	60	30	31.2	25	−1	−1	−1
24	51	30	41.5	36	1	1	1
24	50	35	40.8	35	1	1	1
20.6	70	35	27.8	27	−1	−1	−1
20.6	55	35	32.4	26	−1	−1	−1
25.8	40	27	38.2	33	1	1	1
25.8	45	25	39.4	33	1	1	1
21.1	31	40	33.5	28	−1	−1	−1
21.1	75	30	34.2	26	−1	−1	−1
26.6	52	25	42.4	37	1	1	1
26.6	42	35	44.1	38	1	1	1
26.6	60	35	40.7	35	1	1	1
25.8	40	30	41.2	35	1	−1	1
25.8	33	30	43.3	37	1	1	1
21.7	60	45	32	27	−1	−1	−1
20.6	65	40	28.5	27	−1	−1	−1
21.5	70	40	29.8	26	−1	−1	−1
26.5	36	34	42.9	38	1	1	1
20.8	45	30	15.6	20	−1	−1	−1
20.8	40	30	14.8	21	−1	−1	−1

Table 4. Performance of testing data-set.

d (kN/m^3)	H (m)	β (°)	C (kPa)	φ (°)	Actual class	Predicted class by LMPM	Predicted class by KMPM
19.6	58	40	29.6	23	−1	−1	−1
25.4	35	20	33	33	1	1	1
22.4	50	50	29.3	26	−1	−1	−1
26.2	30	35	41.5	36	1	1	1
26.2	36	23	42.3	36	1	−1	1
25.6	32	30	39.8	36	1	1	1
25.6	60	35	36.8	34	1	1	1
26.2	37	30	42.8	37	1	1	1
26.2	68	35	43.8	38	1	1	1
20.6	42	30	32.4	26	−1	−1	−1
26.5	54	42	41.8	36	1	1	1
20.8	53	30	15.4	21	−1	−1	−1

Table 5. Performance of the LMPM and KMPM.

Models	Training performance (%)	Testing performance (%)	α
LMPM	95.12	91.67	96%
KMPM	100	100	100%

Figure 1. The effect of σ on training performance (%).

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