3.5. Models on Branch Occlusion and Knot Attributes
Generalized linear mixed models were used to explore the relationship between branch occlusion and knot attributes. As to the time of branch occlusion (OT), a log link function with a negative binomial distribution was applied to estimate it:
where a
0, a
1, a
2 and a
3 are parameter estimates (
Table 1). The time of branch occlusion was correlated with the natural logarithm of occluded branch diameter (OBD), the radius of dead portion of knot (RDP) and the stem diameter growth rate during branch occlusion (SDGR). Other variables and random effects were omitted in the model due to their non-significance (
Table 6). There was no significant trend in the raw residuals when plotted against independent variables OBD and SDGR (
Figure 2A,C), while a remarkable decreasing trend was found for RDP larger than 55 mm (
Figure 2B) and for predicted OT longer than 9 years (
Figure 2D). Nonetheless, as RDP and OT exceeded the above values only in very few cases, the model was acceptable. The error statistics also showed that there was no obvious bias (
Table 6), and the model explained 79.61% of total variance.
Table 6.
Parameter estimates of the model for the time of branch occlusion, radius of dead knot, radius of knot and insertion angle.
Table 6.
Parameter estimates of the model for the time of branch occlusion, radius of dead knot, radius of knot and insertion angle.
Parameter | Estimate | Standard Error | T-value | Cl-Lower | Cl-Upper | Significance |
---|
Time of branch occlusion (Equation (1)) |
Fixed parameters |
a0 | 0.616 | 0.177 | 3.478 | 0.268 | 0.963 | <0.001 |
a1 | 0.198 | 0.044 | 4.453 | 0.111 | 0.285 | <0.001 |
a2 | 0.033 | 0.002 | 19.071 | 0.030 | 0.037 | <0.001 |
a3 | −0.050 | 0.004 | −13.398 | −0.057 | −0.043 | <0.001 |
Φ | 0.608 | | | | | |
Error statistics | E = 0.0000 | |E| = 0.3623 | E² = 0.4996 |
Radius of dead portion of knot (Equation (2)) |
Fixed parameters |
b0 | 2.010 | 0.139 | 14.474 | 1.738 | 2.282 | <0.001 |
b1 | 0.535 | 0.032 | 16.530 | 0.472 | 0.599 | <0.001 |
Random parameters |
βptb | 53.110 | 2.071 | 25.644 | 49.202 | 57.328 | <0.001 |
Error statistics | E = 0.0142 | |E| = 4.7864 | E² = 106.4202 |
Total radius of knot (Equation (3)) |
Fixed parameters |
c0 | 2.003 | 0.081 | 24.765 | 1.844 | 2.162 | <0.001 |
c1 | 0.678 | 0.017 | 40.059 | 0.645 | 0.711 | <0.001 |
Random parameters |
γptb | 122.748 | 4.787 | 25.643 | 113.715 | 132.498 | <0.001 |
Error statistics | E = −0.0124 | |E| = 8.1513 | E² = 123.8880 |
Insertion angle of the occluded branch (Equation (4)) |
Fixed parameters |
d0 | 4.151 | 0.031 | 134.398 | 4.090 | 4.211 | <0.001 |
d1 | −0.047 | 0.010 | −4.776 | −0.066 | −0.028 | <0.001 |
Random parameters |
γptb | 77.414 | 3.018 | 25.652 | 71.720 | 83.561 | <0.001 |
Error statistics | E = −0.01254 | |E| = 6.5097 | E² = 139.9267 |
Figure 2.
Raw residuals of the model for time of branch occlusion plotted against the independent variables (A, B and C) and predicted variables(D). The edges of the box are 25th and 75th percentiles; the vertical lines are drawn from the box to the most extreme point within 1.5 interquartile ranges; the transverse lines in boxes connect the medians.
Figure 2.
Raw residuals of the model for time of branch occlusion plotted against the independent variables (A, B and C) and predicted variables(D). The edges of the box are 25th and 75th percentiles; the vertical lines are drawn from the box to the most extreme point within 1.5 interquartile ranges; the transverse lines in boxes connect the medians.
The radius of the dead portion of the knot (RDP) was estimated using a log linear model with a normal distribution of response:
where b
0 and b
1 are the parameter estimates, β
ptb is the random effect at branch level (
Table 1). The RDP was only correlated with the natural logarithm of OBD (
Table 6). There was no significant trend in the raw residuals when plotted against OBD and predicted RDP (
Figure 3). The error statistics inferred that some other factors were not included in the model (
Table 6), and the model only explained 31.27% of total variance.
The total radius of knot (TRK) was modeled using a log-link function with a normal distribution of dependent variable:
where c
0 and c
1 are the parameter estimates of the model and γ
ptb is the random effect at branch level (
Table 1). The TRK was also only correlated with natural log-transformed OBD (
Table 6). The error statistics showed there was some unexplained variation left (
Table 6). No trend was found in the raw residuals when they were plotted against the predictor and predicted variables (
Figure 4). The model explained 62.62% of the total variance.
The insertion angle (IA) of occluded branch was modelled in a log link function with a normal distribution of response:
where d
0 and d
1 were the parameter estimates, δ
ptb was the random effect at branch level (
Table 1). The branch insertion angle was only correlated with the natural logarithm of OBD (
Table 6). No trend in the raw residuals was found when plotted against independent and predicted variables (
Figure 5). The error statistics inferred that there was still a considerable unexplained residual variation (
Table 6). The model only explained 19.09% of total variance.
Figure 3.
Raw residuals of the model on radius of dead portion of knot plotted against the independent variables (A) and predicted variables (B). The edges of the box are 25th and 75th percentiles; the vertical lines are drawn from the box to the most extreme point within 1.5 interquartile ranges; the transverse lines in boxes connect the medians.
Figure 3.
Raw residuals of the model on radius of dead portion of knot plotted against the independent variables (A) and predicted variables (B). The edges of the box are 25th and 75th percentiles; the vertical lines are drawn from the box to the most extreme point within 1.5 interquartile ranges; the transverse lines in boxes connect the medians.
Figure 4.
Raw residuals of the model on total radius of knot plotted against the independent (A) and predicted (B) variables. The edges of the box are 25th and 75th percentiles; the vertical lines are drawn from the box to the most extreme point within 1.5 interquartile ranges; the transverse lines in boxes connect the medians.
Figure 4.
Raw residuals of the model on total radius of knot plotted against the independent (A) and predicted (B) variables. The edges of the box are 25th and 75th percentiles; the vertical lines are drawn from the box to the most extreme point within 1.5 interquartile ranges; the transverse lines in boxes connect the medians.
Figure 5.
Raw residuals of the model on insertion angel of occluded branch plotted against independent variables (A) and predicted (B) variables. The edges of the box are 25th and 75th percentiles; the vertical lines are drawn from the box to the most extreme point within 1.5 interquartile ranges; the transverse lines in boxes connect the medians.
Figure 5.
Raw residuals of the model on insertion angel of occluded branch plotted against independent variables (A) and predicted (B) variables. The edges of the box are 25th and 75th percentiles; the vertical lines are drawn from the box to the most extreme point within 1.5 interquartile ranges; the transverse lines in boxes connect the medians.
Figure 6.
Predicted time of branch occlusion. (A) From three levels of stem diameter growth rates during branch occlusion (SDGR) and a radius of dead portion of knot (RDP) of 20mm plotted against occluded branch diameter (ODB) (Equation (1)); (B) from three levels of OBD and RDP of 20mm plotted against SDGR; and (C) from three levels of OBD and SDGR of 20mm year−1 plotted against RDP.
Figure 6.
Predicted time of branch occlusion. (A) From three levels of stem diameter growth rates during branch occlusion (SDGR) and a radius of dead portion of knot (RDP) of 20mm plotted against occluded branch diameter (ODB) (Equation (1)); (B) from three levels of OBD and RDP of 20mm plotted against SDGR; and (C) from three levels of OBD and SDGR of 20mm year−1 plotted against RDP.
3.6. Simulations
Time of branch occlusion (OT), radius of dead portion of knot (RDP), total radius of knot (TRK) and insertion angle of occluded branch (IA) were simulated according to above equations (
Figure 6 and
Figure 7). It was shown that the predicted OT increased with the increment of occluded branch diameter and RDP (
Figure 6A,C), and trees with higher stem diameter growth rate (SDGR) occluded faster (
Figure 6B). The occluded branch diameter influenced the simulated values of OT less than RDP and SDGR during the whole range of values from the sampled dataset. In addition, when the SDGR was larger than 20 mm year
−1, OT tended to be less than 3 years. Both RDP and TRK were significantly positively correlated with the occluded branch diameter (
Figure 7A,B). The predicted IA decreased slightly with an increase in occluded branch diameter (
Figure 7C). When the occluded branch diameter increased over the whole range of sampled values, the simulated RDP and TRK still increased markedly. However, IA did not decrease significantly when the occluded branches were larger than 15 mm in diameter.
Figure 7.
Predicted radius of dead portion of knot ((A), Equation (2)), total radius of knot ((B), Equation (3)) and insertion angle of occluded branch ((C), Equation (4)) plotted against occluded branch diameter.
Figure 7.
Predicted radius of dead portion of knot ((A), Equation (2)), total radius of knot ((B), Equation (3)) and insertion angle of occluded branch ((C), Equation (4)) plotted against occluded branch diameter.