Determining the transition from juvenile to mature wood microfibril angle in lodgepole pine: a comparison of six different two-segment models

Table 2 The six two-segment regression models considered in this study

Model	f ₁(x)	f ₂(x)	Constraints
LIN_C	\( y = a + bx \)	\( y = c \)	\( c = a + b{x_0} \)
LIN_LIN	\( y = a + bx \)	\( y = {a_2} + {b_2}x \)	\( {a_2} + {b_2}{x_0} = a + b{x_0} \)
EXP_C	\( y = a{e^{{ - bx}}} \)	\( y = c \)	\( c = a{e^{{ - b{x_0}}}} \)
EXP_LIN	\( y = a{e^{{ - bx}}} \)	\( y = {a_2} + {b_2}x \)	\( \matrix{ {{a_2} + {b_2}{x_0} = a{e^{{ - b{x_0}}}}} \hfill \\ }<!end array> \)
Q_C	\( y = a + bx + c{x^2} \)	\( y = c\prime \)	\( {c\prime = a + b{x_0} + cx_0^2} \ {x_0} = - \frac{b}{{2c}}\hfill \\ \)
Q_LIN	\( y = a + bx + c{x^2} \)	\( y = {a_2} + {b_2}x \)	\( \matrix{ {{a_2} + {b_2}{x_0} = a + b{x_0} + cx_0^2} \hfill \\ {{x_0} = - \frac{b}{{2c}}} \hfill \\ }<!end array> \)

Note: f ₁(x) applies for \( x \leqslant {x_0} \) and f ₂(x) for \( x \geqslant {x_0}. \). The transition point of the two quadratic models can be obtained from the minimum point of the first quadratic segment, and for Q_C, it coincides with the smooth requirement (the first derivative of the quadratic segment being zero at the transition point)

ISSN: 1297-966X