# Random Forests and Cubist Algorithms for Predicting Shear Strengths of Rockfill Materials
## 2 Materials and Methods
### Indicator Analysis  
The shear strength behavior of RFM is influenced by many aspects, such as the compressive strength of the original rock, saturation percent, stress condition, grain shape and grain size distribution [6]. These influence factors can be approximately subdivided into five categories [20]: material gradation, fineness and gradation moduli, material hardness, material density, and normal stress. The interpretation of the selection of the input parameters in this paper is discussed as follows:  
Some researchers [37; 38; 39] have observed that the RFM gradation makes a difference in the rockfill shear strength. With the development of the study of the rockfill shear strength, several scholars [40] have suggested that the larger particle size reduces the effect of the shear strength. To prove this effect, scaling of the RFM size tends to be implemented instead of the sophisticated in situ experiments because of the excessively large in situ particle sizes. Based on the past experience and research [20], the percent passing sieve sizes that correspond to 10%, 30%, 60%, and 90% (i.e., D\({}_{10}\), D\({}_{30}\), D\({}_{60}\), and D\({}_{90}\), respectively) were adopted in reference to the scaled-down gradation sizes. Additionally, the coefficients of uniformity (Cu) and curvature (Cc) (Equations (1) and (2) related to the material gradation [20] were also employed as parameters in the random forests and Cubist models:  
\[\mathrm{Cu}=\mathrm{D}_{60}/\mathrm{D}_{10} \tag{1}\]  
\[\mathrm{Cc}=(\mathrm{D}_{30})^{2}/(\mathrm{D}_{10}\times\mathrm{D}_{60}). \tag{2}\]  
(2) Fineness and gradation moduli  
The fineness modulus (FM) and gradation modulus (GM) were also key indicators in the characterization of RFM. A detailed description of the FM and GM is provided in the literature [41; 42; 43; 44]. (3) Material hardness  
Hardness is defined as the resistance of materials to the local invasion of external objects. Affected by the confinement of rockfill, the material hardness also influences the dilatancy of individual rock particles [40]. Many standards for hardness and methods for measuring the compressive strength index exist. In this paper, a standard suggested by the International Society of Rock Mechanics (ISRM) [45] was referenced to characterize the material hardness. The minimum uniaxial compressive strength (UCS) and the maximum UCS of the testing data were employed for the separate input parameter. (4) Material density \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Inputs and outputs** & **Variable form in model** & **Description** & **Range** \\ \hline \multirow{8}{*}{Inputs variables} & D\({}_{10}\) (mm) & 10\% percent passing sieve sizes & 0.01–33.90 \\  & D\({}_{30}\) (mm) & 30\% percent passing sieve sizes & 0.56–42.40 \\  & D\({}_{60}\) (mm) & 60\% percent passing sieve sizes & 1.2–80.1 \\  & D\({}_{30}\) (mm) & 90\% percent passing sieve sizes & 2.6–100.0 \\  & C\({}_{\mathrm{C}}\) & Coefficients of uniformity & 0.12–22.72 \\  & Cu & Curvature & 1.36–1040.00 \\  & G\({}_{\mathrm{m}}\) & Gradation modulus & 0.2–6.0 \\  & F\({}_{\mathrm{m}}\) & Fineness modulus & 3.0–8.8 \\  & R & ISRM hardness rating & 1–6 \\  & UCS\({}_{\mathrm{min}}\) (Mpa) & Minimum uniaxial compression strength of group & 1–250 \\  & UCS\({}_{\mathrm{max}}\) (Mpa) & Maximum uniaxial compression strength of group & 5–400 \\  & \(\gamma\) (kN/m\({}^{3}\)) & Dry unit weight & 9.3195–38.900 \\  & N\({}_{\mathrm{s}}\) (Mpa) & Normal stress & 0.01–33.90 \\ \hline Output variable & Shear strength (Mpa) & Shear strength of RFM & 0.56–42.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Explanation of input and output variables. Several researchers [6; 43] have confirmed that the shear strength of RFM typically increases with an increase in the material density. The higher material density, the greater is the interaction among the material particles; thus, the shear strength is enhanced.