The slope at a knot can be estimated from the lengths and the first divided differences of two adjacent intervals [17, 19]. This interpolation method has been used in several fields of soil and agricultural sciences [20�C22].The log-cubic interpolation selleck MEK162 function defined in (1) passes through all measured points and is both smooth and monotone, which meets essential requirements of a PSD curve.For comparison, commonly used log-linear [15, 16] and log-spline [8] methods were also used. The log-linear interpolation function for PSD can be described by i=1,2,��,n?1,(3)where?di��d��di+1,??i=1,2,��,n?1,Pli(d)=Pih?sh+Pi+1sh,?(3):Pl(d)=Pli(d), Pli(d) is the segment of log-linear interpolation function, Pl(d), for di �� d �� di+1.

The log-spline method for PSD is based on the cubic spline interpolation of P versus ln d, which can be described by i=1,2,��,n?1,(4)where?di��d��di+1,??i=1,2,��,n?1,Psi(d)=Mi(h?s)36h+Mi+1s36h+(Pih?Mih6)(h?s)+(Pi+1h?Mi+1h6)s,?(4):Ps(d)=Psi(d), Psi(d) is the segment of log-spline interpolation function, Ps(d), for di �� d �� di+1; Mi is second derivative at knot xi, i = 1, 2, ��, n, which can be determined from the continuous condition of the first derivative for the cubic spline [19].The interpolation procedure was accomplished by the ��interp1�� function of the Matlab package [19], as described by (5):Pc=interp1(ln?d,P,lndc,��method��),(5)where ln d and P are n-dimensional arrays of the logarithm of measured particle size and cumulative percentage, respectively; lndc is an array representing a desired classification of the logarithm of particle size; Pc is the interpolated cumulative percentage corresponding to lndc; and ��method�� specifies interpolation methods, where ��linear,�� ��spline,�� and ��cubic�� refer to piecewise linear interpolation, cubic spline interpolation, and monotone piecewise cubic interpolation, respectively.

2.2. Evaluation of the Log-Cubic, Log-Linear, and Log-Spline MethodsA leave-one-out cross-validation method was performed to assess the performance of the log-cubic, log-linear, and log-spline methods, using particle size data extracted from UNSODA database [18].In the cross-validation procedure, data of one particle size were left out and interpolated with remaining data. Considering the boundary effect of interpolation, data of the first two and last two particle sizes were not left out in the validation process.

Interpolated values were compared with omitted measured values to calculate the mean error (ME) and root mean square error (RMSE) with (6), which were then Drug_discovery used to assess the performance of interpolation methods:ME=1n?4��i=3n?2(Pi?Pi,i),RMSE=[1n?4��i=3n?2(Pi?Pi,i)2]1/2,(6)where n is the number of particle size grade and Pi,i is the interpolated values of cumulative particle percentage.