# Stochastic Models to Qualify Stem Tapers
## 2 Materials and Methods
### Regression Stem Paper Models  
Forest statisticians have developed multiple advanced regression models for describing tree stem tapers using more and more complex nonlinear functions. The newly developed SDE models for stem tapers will be compared with a well-known and frequently used stem taper model [14; 15; 16]. The segmented polynomial model published by [PERSON] [14]. **Model 9**: \(\frac{d^{2}}{D^{2}}=\beta_{1}(z-1)\beta_{1}(z-1)+\beta_{2}\Big{(}z^{2}-1\Big{)} +\beta_{3}(\beta_{5}-z)I_{1}(\beta_{5}-z)+\beta_{4}(\beta_{6}-z)^{2}I_{2}( \beta_{6}-z)\)  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Species**} & \multirow{2}{*}{**Data**} & \multicolumn{2}{c}{**Number**} & \multirow{2}{*}{**Min. **} & \multirow{2}{*}{**Max. **} & \multirow{2}{*}{**Mean**} & \multirow{2}{*}{**SD**} & \multicolumn{2}{c}{**Number**} & \multirow{2}{*}{**Min. **} & \multirow{2}{*}{**Max. **} & \multirow{2}{*}{**Mean**} & \multirow{2}{*}{**SD**} \\  & & & & & & & & & & & & \\ \hline \multicolumn{11}{c}{**Estimation**} & \multicolumn{5}{c}{**Validation**} \\ \hline \multirow{3}{*}{**Oak**} & **d (cm)** & \(101\) & 7.0 & 49.6 & 21.33 & 8.98 & 45 & 6.0 & 43.0 & 24.40 & 8.05 \\ \cline{2-11}  & **h (m)** & \(101\) & 9.6 & 29.9 & 21.60 & 4.24 & 45 & 11.3 & 29.0 & 22.68 & 3.82 \\ \cline{2-11}  & **v (m\({}^{3}\))** & \(101\) & 0.026 & 2.801 & 0.490 & 0.481 & 45 & 0.020 & 2.047 & 0.591 & 0.424 \\ \hline \multirow{3}{*}{**Ash**} & **d (cm)** & 27 & 8.0 & 29.0 & 18.33 & 4.60 & 11 & 10 & 28 & 19.21 & 5.96 \\ \cline{2-11}  & **h (m)** & 27 & 11.2 & 25.7 & 19.29 & 2.98 & 11 & 13 & 25.1 & 19.17 & 3.12 \\ \cline{2-11}  & **v (m\({}^{3}\))** & 27 & 0.029 & 0.831 & 0.273 & 0.175 & 11 & 0.052 & 0.622 & 0.295 & 0.183 \\ \hline \multirow{3}{*}{**Birch**} & **d (cm)** & 230 & 7.6 & 51.2 & 23.26 & 9.04 & 103 & 8.0 & 49.3 & 22.78 & 8.65 \\ \cline{2-11}  & **h (m)** & 230 & 10.2 & 31.9 & 21.39 & 4.58 & 103 & 8.4 & 31.3 & 21.37 & 4.26 \\ \cline{2-11}  & **v (m\({}^{3}\))** & 230 & 0.027 & 2.583 & 0.504 & 0.439 & 103 & 0.027 & 2.190 & 0.470 & 0.391 \\ \hline \multirow{3}{*}{**Black alder**} & **d (cm)** & 136 & 8.0 & 39.9 & 22.74 & 5.65 & 61 & 12.0 & 33.5 & 22.02 & 5.03 \\ \cline{2-11}  & **h (m)** & 136 & 8.5 & 27.6 & 20.59 & 3.57 & 61 & 13.2 & 27.4 & 20.66 & 3.42 \\ \cline{2-11}  & **v (m\({}^{3}\))** & 136 & 0.019 & 1.806 & 0.467 & 0.282 & 61 & 0.077 & 0.883 & 0.426 & 0.215 \\ \hline \multirow{3}{*}{**White alder**} & **d (cm)** & 16 & 7.7 & 8.0 & 17.64 & 5.55 & 6 & 12.0 & 24.2 & 17.68 & 4.26 \\ \cline{2-11}  & **h (m)** & 16 & 11.3 & 23.4 & 17.26 & 3.10 & 6 & 15.4 & 20.2 & 17.22 & 1.72 \\ \cline{2-11}  & **v (m\({}^{3}\))** & 16 & 0.029 & 0.693 & 0.240 & 0.166 & 6 & 0.097 & 0.403 & 0.221 & 0.112 \\ \hline \multirow{3}{*}{**Aspen**} & **d (cm)** & 102 & 7.0 & 49.6 & 21.34 & 8.94 & 44 & 6.0 & 43.0 & 24.45 & 8.13 \\ \cline{2-11}  & **h (m)** & 102 & 9.6 & 29.9 & 21.65 & 4.24 & 44 & 11.3 & 29.0 & 22.61 & 3.83 \\ \cline{2-11}  & **v (m\({}^{3}\))** & 102 & 0.026 & 2.801 & 0.489 & 0.479 & 44 & 0.020 & 2.049 & 0.595 & 0.429 \\ \hline \multirow{3}{*}{**Pine**} & **d (cm)** & 1344 & 5.0 & 58.4 & 25.25 & 9.86 & 567 & 3.8 & 58.5 & 23.95 & 9.87 \\ \cline{2-11}  & **h (m)** & 1344 & 4.5 & 35.2 & 20.85 & 5.23 & 567 & 3.8 & 33.5 & 20.14 & 5.50 \\ \cline{2-11}  & **v (m\({}^{3}\))** & 1344 & 0.006 & 3.129 & 0.626 & 0.562 & 567 & 0.003 & 3.398 & 0.570 & 0.563 \\ \hline \multirow{3}{*}{**Spruce**} & **d (cm)** & 661 & 7.9 & 52.4 & 23.14 & 8.78 & 249 & 8.0 & 49.8 & 23.34 & 8.31 \\ \cline{2-11}  & **h (m)** & 661 & 7.0 & 32.7 & 20.92 & 5.54 & 249 & 7.5 & 33.8 & 21.12 & 5.23 \\ \cline{2-11}  & **v (m\({}^{3}\))** & 661 & 0.0 where \(d\) is the diameter at any particular height \(h\), \(D\) is the diameter at breast height, \(H\) is the total tree height, \(z\) is the relative height (\(z=\frac{h}{H}\)), \(\beta_{5}\) and \(\beta_{6}\) are the joining points of three segments, and \(\beta_{1}\)-\(\beta_{6}\) are the unknown parameters to be estimated. Additionally,  
\[I_{i}(\beta_{i}-z)=\left\{