# Stochastic Models to Qualify Stem Tapers
## 2 Materials and Methods
### Parameters Estimation
\right.,\] (32) \[\theta_{4}=\{\alpha_{B},\beta_{B},\sigma_{B},\alpha_{M},\beta_{M}, \sigma_{M},\sigma_{T},\beta_{T},\sigma_{T}\}\] \[p_{5}(y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\theta_{5},y^{i}_{j+1},x^{ i}_{j+1})=\left\{ \begin{array}{l}p_{C}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j} \Big{|}\alpha_{B},\beta_{B},\sigma_{B},y^{i}_{0}\Big{)},0\leq x^{i}_{j}\leq a_{ 0}\\ p_{V}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\alpha_{M},\beta_{M},\sigma_{M},y ^{i}_{j+1},x^{i}_{j+1}\Big{)}\,a_{0}<x^{i}_{j}\leq a_{1}\\ p_{V}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\alpha_{T},\beta_{T},\sigma_{T}, 0,1\Big{)}\,a_{1}<x^{i}_{j}\leq 1\end{array} \right.,\] (33) \[\theta_{5}=\{\alpha_{B},\beta_{B},\sigma_{B},\alpha_{M},\beta_{M}, \sigma_{M},\sigma_{T},\beta_{T},\sigma_{T}\}\] \[p_{6}(y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\theta_{6},y^{i}_{j-1}, x^{i}_{j-1})=\left\{ \begin{array}{l}p_{C}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j} \Big{|}\alpha_{B},\beta_{B},\sigma_{B},y^{i}_{0}\Big{)},0\leq x^{i}_{j}<\frac{1.3}{h^{i}}\\ p_{V}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\alpha_{M},\beta_{M},\sigma_{M},y ^{i}_{j+1},x^{i}_{j+1}\Big{)}\,\frac{1.3}{h^{i}}\leq x^{i}_{j}<a_{1}\\ p_{V}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\alpha_{T},\beta_{T},\sigma_{T}, 0,1\Big{)}\,a_{1}\leq x^{i}_{j}\leq 1\end{array} \right.,\] (34) \[\theta_{6}=\{\mu_{0},\sigma_{0},\alpha_{B},\beta_{B},\sigma_{B}, \alpha_{M},\beta_{M},\sigma_{M},\sigma_{T},\beta_{T},\sigma_{T},\gamma\}\] \[p_{7}(y^{i}_{j},x^{i}_{j}\Big{|}\theta_{7})=\left\{ \begin{array}[] {l}p_{C}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\alpha_{B},\beta_{B},\sigma_{ B},y^{i}_{0}\Big{)},0\leq x^{i}_{j}\leq a_{0}\\ p_{V}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\alpha_{T},\beta_{T},\sigma_{T}, 0,1\Big{)},a_{0}<x^{i}_{j}\leq 1\end{array} \right.,\ \theta_{7}=\{\alpha_{B},\beta_{B},\sigma_{B},\alpha_{T},\beta_{T}, \sigma_{T}\},\] (35) \[p_{8}(y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\theta_{8})=\left\{ \begin{array}{l}p_{C}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\alpha_{B}, \beta_{B},\sigma_{B},\gamma\Big{)}\,0\leq x^{i}_{j}\leq a_{0}\\ p_{V}\Big{(}y^{i}_{j^{\prime}},x^{i}_{j}\Big{|}\alpha_{T},\beta_{T},\sigma_{T},0,1 \Big{)}\,a_{0}<x^{i}_{j}\leq 1\end{array} \right.,\ \theta_{8}=\{\mu_{0},\sigma_{0},\alpha_{B},\beta_{B}, \sigma_{B},\sigma_{T},\beta_{T},\sigma_{T},\gamma\}. \tag{36}\]  
In this paper, we apply the theory of a maximum likelihood procedure of the parameter estimates for stem taper in Models 1-8, as all models have closed form transition probability density functions defined by Equations (29)-(36). The fixed and mixed parameters estimation for discretely observed SDE is a complex problem and during the past decades it has attracted an attention of numerous researchers [30; 31; 32]. The log-likelihood functions are given, respectively, as follows:  
\[L_{k}(\theta_{k})=\sum_{i=1}^{M}\sum_{j=1}^{m_{i}}\ln(p_{k}(y^{i}_{j^{\prime}},x^{i}_ {j}\Big{|}\theta_{k},y^{i}_{j-1},x^{i}_{j-1})),\;k=1,\;3,\;4,\;6, \tag{37}\]  
\[L_{k}(\theta_{k})=\sum_{i=1}^{M}\sum_{j=1}^{m_{i}}\ln(p_{k}(y^{i}_{j^{\prime}},x^{i}_ {j}\Big{|}\theta_{k},y^{i}_{j+1},x^{i}_{j+1})),\;k=2,\;5, \tag{38}\]\[L_{k}(\theta_{k})=\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{n_{i}}\ln(p_{k}(y^{i}_ {j^{\prime}},x^{i}_{j}|\theta_{8})),\ k=7,\ 8. \tag{39}\]  
As the transition density functions are known, the maximum log-likelihood calculation and its maximization with respect to \(\theta_{k}\), where \(k\) = 1-8, for a given set of discretely observed datasets is straightforward. However, if \(Y^{i}(0)=\gamma\sim LN_{1}\big{(}\mu_{0};\sigma_{0}^{2}\big{)}\), then there are two additional parameters that must be estimated. The estimation of parameters \(\mu_{0}\) and \(\sigma_{0}^{2}\) depends only on the measurement of each tree diameter at the height \(h\) = 0 and does not influence the other parameters.