On the Construction of Growth Models via Symmetric Copulas and Stochastic Differential Equations

Abstract

By nature, growth regulatory networks in biology are dynamic and stochastic, and feedback regulates their growth function at different ages. In this study, we carried out a stochastic modeling of growth networks and demonstrated this method using three mixed effect four-parameter Gompertz-type diffusion processes and a combination thereof using the conditional normal copula function. Using the conditional normal copula, newly derived univariate distributions can be combined into trivariate and bivariate distributions, and their corresponding conditional bivariate and univariate distributions. The link between the predictor variable and the remaining one or two explanatory variables can be formalized using copula-type densities and a numerical integration procedure. In this study, for parameter estimation, we used a semiparametric maximum pseudo-likelihood estimator procedure, which was characterized by a two-step technique, namely, separately estimating the parameters of the marginal distributions and the parameters of the copula. The results were illustrated using two observed longitudinal datasets, the first of which included the age, diameter, and potentially available area of 39,437 trees (48 stands), while the second included the age, diameter, potentially available area, and height of 8604 trees (47 stands) covering uneven mixed-species (pine, spruce, and birch) stands. All results were implemented using the MAPLE symbolic algebra system.

1. Introduction

A copula is a multivariate cumulative distribution function with standard uniform univariate margins. The copula function methodology provides an opportunity to model the structure of dependencies between different outcomes, with certain assumptions about the marginal distributions. It is important to note that the transformation of any marginal univariate distribution into a uniform one does not change the methodology based on the “copula approach” for the analysis of dependence. By limiting ourselves to the trivariate case, we can change the postulation that two random vectors, $\left(X_1,X_2,X_3\right)$ and $\left(Y_1,Y_2,Y_3\right)$, have the same dependence structure to the fact that these vectors have the same copula function. As a general rule, researchers seek to quantify the dependence between random variables, namely, how much particular random variables depend on each other. The classical and most commonly used measure of dependence in practice is the Pearson correlation coefficient. One characteristic of the normal copula is that it describes dependence using the usual linear correlation matrix of the corresponding multivariate normal distribution. Research in recent decades has shown an advantage over other types of copulas for formalizing variable dependence, as it enables much more flexibility.

The copula method is particularly useful because it allows us to predict the multivariate distributions of random vectors by knowing the distributions of the individual components. The application of copula-based multivariate probability distributions is also appropriate when combining unequal numbers of observed datasets in univariate spaces. In this study, we focused on the trivariate normal copula, which is a traditional candidate for dependency modeling. The normal copula belongs to the class of elliptic copulas, which have elliptically contoured distributions. Recently, the use of copulas in modeling complex processes has become popular in the econometric literature [1,2,3,4]. The notion of the time evolution of copulas has also been introduced in risk management [4]. In order to account for the dynamic of the marginal distributions, we considered conditional copulas [5] to formalize situations where the marginal distributions and their copula were affected by the values of a particular covariate [6], for example, time, T. With the restriction that the copula belongs on only one parameter, ρ, it is most appropriate to model the influence of the covariate variable, T, on copulas by selecting the functional relationship between the copula parameter and the covariate variable, $\rho =f\left(t\right)$ [7].

Modeling biological science processes in the mathematical domain is indeed a challenging task, as these processes are open to various random disturbances. In recent decades, stochastic differential equations have been established in many scientific fields to model systems that are influenced by many different random noises. Mathematical models described by stochastic differential equations have been published in the form of continuous stochastic processes intended to evaluate and predict the biological growth of an individual, for example, in forestry (tree diameter, height, crown width, and volume) [8,9,10,11], medicine [12], environment [13], and biology [14]. It should be emphasized that the maximum likelihood parameter estimation procedure for multivariate stochastic differential equations consumes a great deal of computer time and resources in order to achieve the convergence [15,16]. Therefore, it would be desirable to simplify this model by partitioning the parameters to those corresponding to marginal stochastic differential equations using a multistage parameter estimation procedure, taking into account the fact that the measurements of some of the considered random variables in the datasets may be of different lengths. One possible frame for the analysis of nonlinear multivariate stochastic equation models could be the copula method, which combines univariate nonlinear processes through their corresponding copula functions.

This study was conducted in order to investigate the growth trends of a whole stand and individual trees in the stand and use the information provided by the newly developed model to improve the prediction of target variables in forestry, such as tree diameter, height, and the number of trees per hectare [17]. The purpose of this study was to develop an integrated system by deriving dynamic models of individual tree diameter, $X_1\left(t\right)$; potentially available area, $X_2\left(t\right)$; and height, $X_3\left(t\right)$ (t is the age of the tree) in uneven mixed species stands. The marginal individual tree variables, $X_1\left(t\right)$, $X_2\left(t\right)$, and $X_3\left(t\right)$, were modeled using sigmoid-type diffusion processes [18,19]. The use of these three variables highlights the importance of jointly considering compound effects in the dynamics of the response variable. However, precisely fitting the joint distribution of these three random variables and other related conditional distributions is not simple, as their degree of dependence may vary with changes in spatiotemporal scales. Finally, we describe here the statistical methods for the implementation and modeling of the trivariate normal copula function and show that the copula method performs better than the trivariate diffusion process by modeling a real observed dataset from Lithuania. All results were implemented in the MAPLE symbolic algebra system [20].

2. Copulas, Stochastic Differential Equations, and Measures of Association

2.1. Stochastic Differential Equation Structure

This study focused on a 4-parameter Gompertz-type sigmoidal curve with a single inflection point. The univariate mixed-effects Gompertz-type stochastic differential equation for the tree diameter at breast height (1.3 m), $X_1\left(t\right)$; potentially available area, $X_2\left(t\right)$; and height, $X_3\left(t\right)$ is defined by:

$$d{X}_j^i\left(t\right)=\left(\left({\alpha }_j+{\varphi }_j^i\right)-{\beta }_{j}ln\left(X_j^i\left(t\right)-{ɣ}_j\right)\right)\left(X_j^i\left(t\right)-{ɣ}_j\right)dt+\sqrt{{\sigma }_j}\left(X_j^i\left(t\right)-{ɣ}_j\right)\cdot d{W}_j^i\left(t\right),$$

(1)

where $P\left(X_j^i\left(t_0\right)=x_{j0}\right)=1$; i = 1, …, M (where M is the number of stands); j = 1, 2, 3; the initial condition takes the following forms—if t = t₀, then $X_1^i\left(t_0\right)=x_{10}, X_2^i\left(t_0\right)=x_{20}=\delta ,$ and $X_3^i\left(t_0\right)=x_{30}$; ${\alpha }_j$ and ${\beta }_j$ are the birth and death rate parameters (${\alpha }_j$ > 0, ${\beta }_j$ > 0), respectively; ${\sigma }_j$ is the volatility parameter; ${ɣ}_j$ is the threshold parameter; δ is the fixed effect parameter; the random effects ${\varphi }_j^i$ are independent and normally distributed random variables with zero mean and constant variances, respectively, ${\varphi }_j^i~N\left(0;{\sigma }_{x_j}^2\right)$; and $d{W}_j^i\left(t\right)$ represents the Brownian motion increments considered independent across all stands. The unknown fixed effect parameters $\mathsf{\theta }$ = {${\alpha }_j$,${\beta }_j$,${ɣ}_j$,${\sigma }_j,\delta ;\mathrm{i}=1,2,3$} must be estimated.

Using state transformation $e^{{\beta }_{i}t}ln\left(X_j^i\left(t\right)\right)$ i = 1, …, M, j = 1, 2, 3, and Itô’s lemma [21], the stochastic differential Equation (1) can be transformed into one that confirms the existence and uniqueness conditions of a strong solution (see, e.g., [22]). By integrating the transformed equation from t₀ to t and taking antilogarithms, we obtain that $(X_j^i\left(t\right)|X_j^i\left(t_0\right)=x_{j0})$ has a lognormal distribution $L{N}_1({\mu }_j^i(t);\hspace{0.33em}{v}_j(t))$, with the mean ${\mu }_j^i\left(t\right)$, variance $v_j\left(t\right),$ and probability density function $f_j^i(x_j,t|\mathsf{\theta },{\varphi }_j^i)$ defined as:

$${\mu }_j^i\left(t\right)=ln\left(x_{j0}-{ɣ}_j\right)e^{-{\beta }_j\left(t-t_0\right)}+\frac{{\alpha }_j+{\varphi }_j^i}{{\beta }_j}\left(1-e^{-{\beta }_j\left(t-t_0\right)}\right),$$

(2)

$$v_j\left(t\right)=\frac{1-e^{-2{\beta }_j\left(t-t_0\right)}}{2{\beta }_j}{\sigma }_j,$$

(3)

$$f_j^i(x_j,t\left|\mathsf{\theta },{\varphi }_j^i\right.)=\frac{1}{\sqrt{2\pi v_j\left(t\right)}\left(x-{ɣ}_j\right)}\mathit{exp}\left(-\frac{{(\ln \left(x-{ɣ}_j\right)-{\mu }_j^i\left(t\right))}^2}{2v_j\left(t\right)}\right).$$

(4)

The dynamics of the mean, $m_j^i\left(t\right)$; median, $m{e}_j^i\left(t\right)$; mode, $m{o}_j^i\left(t\right)$; qth quantile (0 < q < 1), $q{m}_j^i\left(t\right)$; and variance, $w_j^i\left(t\right)$, i = 1, …, M, j = 1, 2, 3 of the tree diameter, potentially available area, and height can be presented by analogy to the 3-parameter Gompertz stochastic differential equation model [23], as shown in the following expressions:

$$m_j^i\left(t\right)={ɣ}_j+\mathit{exp}\left({\mu }_j^i\left(t\right)+\frac{1}{2}{v}_j\left(t\right)\right),$$

(5)

$$m{e}_j^i\left(t\right)={ɣ}_j+\mathit{exp}\left({\mu }_j^i\left(t\right)\right),$$

(6)

$$m{o}_j^i\left(t\right)={ɣ}_j+\mathit{exp}\left({\mu }_j^i\left(t\right)-v_j\left(t\right)\right),$$

(7)

$$q{m}_j^i\left(t\right)={ɣ}_j+\mathit{exp}\left({\mu }_j^i\left(t\right)+\sqrt{v_j\left(t\right)}{\Phi }_q^{-1}\left(0;1\right)\right),$$

(8)

$$w_j^i\left(t\right)=\mathit{exp}\left(2{\mu }_j^i\left(t\right)\left(t\right)+v_j\left(t\right)\right)\cdot \left(\mathit{exp}\left(v\left(t\right)\right)-1\right).$$

(9)

2.2. Trivariate Normal Copula

Copula models are based on Sklar’s theorem [24], which states that every multivariate cumulative distribution function can be expressed as a copula function depending on univariate marginal distributions that captures the structure of the dependencies between the marginal components of a random vector. Let $\left(X_1,X_2,X_3\right)$ represents a trivariate random vector; then, the joint and marginal distribution functions conditioned by T = t are denoted as the following forms:

$$\begin{gathered} H_t\left(X_1,X_2,X_3\right)=P\left(X_1\left(t\right)\le x_1, X_2\left(t\right)\le x_2,X_3\left(t\right)\le x_3\right), \\ {F}_1\left(x_1,t\right)=P\left(X_1\left(t\right)\le x_1\right), F_2\left(x_2,t\right)=P\left(X_2\left(t\right)\le x_2\right), F_3\left(x_3,t\right)=P\left(X_3\left(t\right)\le x_3\right). \end{gathered}$$

(10)

As the newly developed lognormal densities defined by Equations (2)–(4) are continuous, according to Sklar’s theorem there exists a unique trivariate conditional copula function C_t, which for all u₁, u₂, and u₃ ∈ [0, 1] takes the following form:

$$C\left(u_1,u_2,u_3;t\right)=H_t\left(F_1^{-1}\left(u_1,t\right),F_2^{-1}\left(u_2,t\right),F_3^{-1}\left(u_3,t\right)\right),$$

(11)

where $F_j^{-1}$, j = 1, 2, 3 is the conditional quantile function of X_j given that T = t.

In this study, we focused on a trivariate normal copula that could be defined by a trivariate normal probability density function ${\phi }_3\left(x_1,x,x_3;P\right)$. The probability density function of the trivariate normal distribution $N_3\left(0;P\right)$ with a correlation matrix P is defined as:

$$P=\left(\begin{array}{ccc}1& {\rho }_{12}& {\rho }_{13}\\ {\rho }_{12}& 1& {\rho }_{23}\\ {\rho }_{13}& {\rho }_{23}& 1\end{array}\right),$$

(12)

$${\phi }_3\left(x_1,x_2,x_3;P\right)=\frac{e^{-\frac{w}{2\left({\rho }_{12}^2+{\rho }_{13}^2\pm {\rho }_{23}^2-2{\rho }_{12}{\rho }_{13}{\rho }_{23}\right)}}}{2{\pi }^{\frac{3}{2}}\sqrt{1-\left({\rho }_{12}^2+{\rho }_{13}^2+{\rho }_{23}^2\right)+2{\rho }_{12}{\rho }_{13}{\rho }_{23}}},$$

(13)

$$\begin{array}{c}w=x_1^2\left({\rho }_{23}^2-1\right)+x_2^2\left({\rho }_{13}^2-1\right)+x_3^2\left({\rho }_{12}^2-1\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+2(x_1{x}_2\left({\rho }_{12}-{\rho }_{13}{\rho }_{23}\right)+x_1{x}_3\left({\rho }_{13}-{\rho }_{12}{\rho }_{23}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+x_2{x}_3\left({\rho }_{23}-{\rho }_{12}{\rho }_{13}\right)).\end{array}$$

(14)

In addition, its trivariate normal distribution function is given by:

$${\Phi }_3\left(x_1,x_2,x_3;P\right)={{\displaystyle \int }}_{-\infty }^{x_1}{{\displaystyle \int }}_{-\infty }^{x_2}{{\displaystyle \int }}_{-\infty }^{x_3}{\phi }_3\left(z_1,z_2,z_3;P\right)d{z}_{1}d{z}_{2}d{z}_3.$$

(15)

The univariate probability density and distribution functions of the standard normal distribution are defined, respectively, as:

$$\phi \left(x\right)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}},$$

(16)

$$\Phi \left(x\right)={{\displaystyle \int }}_{-\infty }^x\phi \left(z\right)dz.$$

(17)

The trivariate normal copula distribution function with correlation matrix P is then defined by:

$$C_3\left(u_1,u_2,u_3;P,t\right)={\Phi }_3\left({\Phi }^{-1}\left(u_1\right),{\Phi }^{-1}\left(u_2\right),{\Phi }^{-1}\left(u_3\right);P\right),$$

(18)

where $u_1=F_1\left(x_1,t\right)$, $u_2=F_2\left(x_2,t\right)$, and $u_3=F_3\left(x_3,t\right)$.

The trivariate normal copula probability density function $c_3\left(u_1,u_2,u_3;P,t\right)$ takes the following form:

$$\begin{gathered} c_3\left(u_1,u_2,u_3;P,t\right)=\frac{{\partial }^3}{\partial u_1\partial u_2\partial u_3}{C}_3\left(u_1,u_2,u_3;P,t\right) \\ =\frac{{\phi }_3\left({\Phi }^{-1}\left(u_1\right),{\Phi }^{-1}\left(u_2\right),{\Phi }^{-1}\left(u_3\right);P\right)}{\begin{array}{c}\phi \left({\Phi }^{-1}\left(u_1\right)\right)\phi \left({\Phi }^{-1}\left(u_1\right)\right)\phi \left({\Phi }^{-1}\left(u_1\right)\right)\\ =\frac{1}{\sqrt{1-\left({\rho }_{12}^2+{\rho }_{13}^2+{\rho }_{23}^2\right)+2{\rho }_{12}{\rho }_{13}{\rho }_{23}}}{e}^{-\frac{w}{2\left({\rho }_{12}^2+{\rho }_{13}^2\pm {\rho }_{23}^2-2{\rho }_{12}{\rho }_{13}{\rho }_{23}-1\right)},}\end{array}} \end{gathered}$$

(19)

$$\begin{array}{c}w=x_1^2\left(2{\rho }_{12}{\rho }_{13}{\rho }_{23}-{\rho }_{12}^2-{\rho }_{13}^2\right)+x_2^2\left(2{\rho }_{12}{\rho }_{13}{\rho }_{23}-{\rho }_{12}^2-{\rho }_{23}^2\right)\\ +x_3^2\left(2{\rho }_{12}{\rho }_{13}{\rho }_{23}-{\rho }_{13}^2-{\rho }_{23}^2\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +2(x_1{x}_2({\rho }_{12}-{\rho }_{13}{\rho }_{23})+x_1{x}_3({\rho }_{13}-{\rho }_{12}{\rho }_{23})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+x_2{x}_3({\rho }_{23}-{\rho }_{12}{\rho }_{13})),\hfill \end{array}$$

(20)

where $x_1={\Phi }^{-1}\left(u_1\right)$, $x_2={\Phi }^{-1}\left(u_2\right)$, and $x_3={\Phi }^{-1}\left(u_3\right)$.

In an analogous way, we can write the distribution and density functions of the bivariate normal copula, respectively, as:

$${\phi }_2\left(x_1,x_2;{\rho }_{12}\right)=\frac{1}{2\pi \sqrt{1-{\rho }_{12}^2}}{e}^{-\frac{x_1^2-2{\rho }_{12}{x}_1{x}_2+x_2^2}{2\left(1-{\rho }_{12}^2\right)}},$$

(21)

$${\Phi }_2\left(x_1,x_2;{\rho }_{12}\right)={{\displaystyle \int }}_{-\infty }^{x_1}{{\displaystyle \int }}_{-\infty }^{x_2}{\phi }_2\left(z_1,z_2;{\rho }_{12}\right)d{z}_{1}d{z}_2,$$

(22)

$$C_2\left(u_1,u_2;{\rho }_{12},t\right)={\Phi }_2\left({\Phi }^{-1}\left(u_1\right),{\Phi }^{-1}\left(u_2\right);{\rho }_{12}\right),$$

(23)

$$c_2\left(u_1,u_2;{\rho }_{12},t\right)=\frac{1}{\sqrt{1-{\rho }_{12}^2}}{e}^{-\frac{{\rho }_{12}^2(x_1^2+x_2^2)-2{\rho }_{12}{x}_1{x}_2}{2\left(1-{\rho }_{12}^2\right)}},$$

(24)

where $x_1={\Phi }^{-1}\left(u_1\right)$ and $x_2={\Phi }^{-1}\left(u_2\right)$. The joint trivariate copula-type probability density function $f_{3c}\left(x_1,x_2,x_3,t\right)$ is given as:

$$f_{3c}\left(x_1,x_2,x_3,t\right)=c_3\left(F_1\left(x_1,t\right),F_2\left(x_2,t\right),F_3\left(x_3,t\right);P,t\right)f_1\left(x_1,t\right)f_2\left(x_2,t\right)f_3\left(x_3,t\right),$$

(25)

where $f_i\left(x_i,t\right)$, j = 1, 2, 3 is the probability density function of X_j given that T = t. The joint bivariate copula-type probability density function $f_{2c}\left(x_1,x_2,t\right)$ is given as:

$$\begin{gathered} f_{2c}\left(x_1,x_2,t\right)=c_2\left(F_1\left(x_1,t\right),F_2\left(x_2,t\right);{\rho }_{12},t\right)f_1\left(x_1,t\right)f_2\left(x_2,t\right), \\ {f}_{2c}\left(x_1,x_3,t\right)=c_2\left(F_1\left(x_1,t\right),F_3\left(x_3,t\right);{\rho }_{13},t\right)f_1\left(x_1,t\right)f_3\left(x_3,t\right), \\ {f}_{2c}\left(x_2,x_3,t\right)=c_2\left(F_2\left(x_2,t\right),F_3\left(x_3,t\right);{\rho }_{23},t\right)f_2\left(x_2,t\right)f_3\left(x_3,t\right). \end{gathered}$$

(26)

The conditional probability density function of $X_i\left(t\right)$, i = 1, 2, 3 at a given $\left(X_j\left(t\right)=x_j,X_k\left(t\right)=x_k\right)$, j, k ≠ i, is defined as:

$$\begin{gathered} f_{1|2,3}\left(x_1,t|x_2,x_3\right)=\frac{f_{3c}\left(x_1,x_2,x_3,t\right)}{f_{2c}\left(x_2,x_3,t\right)}, \\ {f}_{2|1,3}\left(x_2,t|x_1,x_3\right)=\frac{f_{3c}\left(x_1,x_2,x_3,t\right)}{f_{2c}\left(x_1,x_3,t\right)}, \\ {f}_{3|1,2}\left(x_2,t|x_1,x_2\right)=\frac{f_{3c}\left(x_1,x_2,x_3,t\right)}{f_{2c}\left(x_1,x_2,t\right)}. \end{gathered}$$

(27)

The conditional probability density function of $X_i\left(t\right)$, i = 1, 2, 3 at a given $\left(X_j\left(t\right)=x_j\right)$, j ≠ i, is defined as:

$$\begin{gathered} f_{1|2}\left(x_1,t|x_2\right)=\frac{f_{2c}\left(x_1,x_2,t\right)}{f_2\left(x_2,t\right)}, \\ {f}_{1|3}\left(x_1,t|x_3\right)=\frac{f_{2c}\left(x_1,x_2,t\right)}{f_3\left(x_3,t\right)}, \\ {f}_{2|1}\left(x_2,t|x_1\right)=\frac{f_{2c}\left(x_1,x_2,t\right)}{f_1\left(x_1,t\right)}, \\ {f}_{2|3}\left(x_2,t|x_3\right)=\frac{f_{2c}\left(x_1,x_2,t\right)}{f_3\left(x_3,t\right)}, \\ {f}_{3|1}\left(x_3,t|x_1\right)=\frac{f_{2c}\left(x_1,x_2,t\right)}{f_1\left(x_1,t\right)}, \\ {f}_{3|2}\left(x_3,t|x_2\right)=\frac{f_{2c}\left(x_1,x_2,t\right)}{f_2\left(x_2,t\right)}. \end{gathered}$$

(28)

The conditional probability density functions defined by Equations (27) and (28) allow us to formalize copula-type regression relationships. One-dimensional numerical integration is needed to compute the conditional mean of the continuous random variables $\left(X_1\left(t\right),X_2\left(t\right),X_3\left(t\right)\right)$. Let $X_i\left(t\right)$, i = 1, 2, 3, be a tentatively assumed response variable and $X_j\left(t\right)=x_j$, j ≠ i, be a tentative predictor variable; then, the conditional mean regression of $X_i\left(t\right)$ on $X_j\left(t\right)$, denoted by $m_{i|j}\left(x_j,t\right)$, using Equation (28) takes the following form:

$$m_{i|j}\left(x_j,t\right)={{\displaystyle \int }}_{{ɣ}_i}^{+\infty }{x}_i{f}_{i|j}\left(x_i,t|x_j\right)d{x}_i.$$

(29)

In addition, if $X_i\left(t\right)$, i = 1, 2, 3 is a tentatively assumed response variable and $\left(X_j\left(t\right)=x_j,X_k\left(t\right)=x_k\right)$, j, k ≠ i, are tentative predictor variables, then the conditional mean regression of $X_i\left(t\right)$ on $\left(X_j\left(t\right),X_k\left(t\right)\right)$, denoted by $m_{i|j,k}\left(x_j,x_k,t\right)$, using Equation (27) takes the form:

$$m_{i|j,k}\left(x_j,x_k,t\right)={{\displaystyle \int }}_{{ɣ}_i}^{+\infty }{x}_i{f}_{i|j,k}\left(x_i,t|x_j,x_k\right)d{x}_i.$$

(30)

It can be seen that we derived three different regression equations to define the dynamics of the response variable: the first (Equation (5)) defines the dynamics of the response variable only over time (age); the second (Equation (29)) describes the dynamics of the response variable over time and with one additional explanatory variable; and, finally, the third (Equation (30)) describes the dynamics of the response variable as a function of time and the other two explanatory variables.

2.3. Semiparametric Maximum Pseudo-Likelihood Procedure for Copulas

There are many methods for estimating the copula parameters of a given data sample. In this study, we used a semiparametric maximum pseudo-likelihood estimator for the copula parameters. Semi-parametric parameter estimation, which was introduced by Genest et al. [25], is characterized by a two-step technique, namely, separately estimating the parameters of the marginal distributions and the parameters of the copula. For the trivariate normal copula density function, defined by Equations (19) and (20), with a parameter matrix P, which consists of $\left({\rho }_{12},{\rho }_{13},{\rho }_{23}\right)$, the pseudo-log-likelihood function is then represented as:

$$LL\left({\rho }_{12},{\rho }_{13},{\rho }_{23}\right)={\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=1}^{n_i}ln\left(c_3\left({\Phi }^{-1}\left(F_1\left(x_{1j}^i,t_j^i\right)\right),{\Phi }^{-1}\left(F_2\left(x_{2j}^i,t_j^i\right)\right),{\Phi }^{-1}\left(F_3\left(x_{3j}^i,t_j^i\right)\right)\left|\left({\rho }_{12},{\rho }_{13},{\rho }_{23}\right),t_j^i\right.\right)\right),$$

(31)

for a given set of discrete trivariate process measurements of tree diameter (x₁), potentially available area (x₂), and height (x₃) $\left\{\left(x_{11}^i,x_{21}^i,x_{31}^i\right),\left(x_{12}^i,x_{22}^i,x_{32}^i\right),\dots ,\left(x_{1n_i}^i,x_{2n_i}^i,x_{3n_i}^i\right)\right\}$ at discrete times (ages) $\left\{t_1^i,t_2^i,\dots ,t_{n_i}^i\right\}$ (n_i is the number of observed trees of the ith stand, i = 1, …, M). The normal copula parameter estimates obtained through the semiparametric maximum pseudo-likelihood procedure provide the asymptotic normality result for estimations [26]. The estimates of standard errors can be obtained as the inverse of the observed Fisher information matrix (see, for instance, [19]). The approximate asymptotic standard errors of the normal copula parameters are defined by the diagonal elements of the observed Fisher information matrix ${\left[\tilde{I}\left(\widehat{\mathsf{\theta }}\right)\right]}^{-1}$, $\mathsf{\theta }=\left\{{\rho }_{12},{\rho }_{13},{\rho }_{23}\right\}:$

$$SE\left(\widehat{\mathsf{\theta }}\right)=\sqrt{{\left[\tilde{I}\left(\widehat{\mathsf{\theta }}\right)\right]}^{-1}},$$

(32)

where the matrix is defined as:

$$\tilde{I}\left(\widehat{\mathsf{\theta }}\right)={\left[-\frac{{\partial }^{2}LL\left(\mathsf{\theta }\right)}{\partial {\mathsf{\theta }}_i\partial {\mathsf{\theta }}_j}\right]}^T{|}_{\mathsf{\theta }=\widehat{\mathsf{\theta }}}.$$

(33)

3. Results and Discussion

3.1. Parameter Estimates

The model defined by Equation (1) was fitted to analyze the discrete measurements of tree diameter (x₁), potentially available area (x₂), and height (x₃) $\left\{\left(x_{11}^i,x_{21}^i,x_{31}^i\right),\left(x_{12}^i,x_{22}^i,x_{32}^i\right),\dots ,\left(x_{1n_i}^i,x_{2n_i}^i,x_{3n_i}^i\right)\right\}$ at discrete times (ages) $\left\{t_1^i,t_2^i,\dots ,t_{n_i}^i\right\}$ [16]. During the experiment, the following characteristics were recorded for each tree in the considered plots: tree species, age, location (position of plane coordinates x and y), and diameter at breast height. The height of approximately every 5th tree was measured. To account for this, two different datasets were used to obtain parameter estimates: the first (47 plots; 39,437 mixed-species trees) was used to estimate the parameters of tree diameter and potentially available area for the fixed effects Equation (1), while the second (48 plots; 8604 mixed-species trees) was used to estimate the fixed effect parameters of tree height in Equation (1), calibrate random effects using Equation (34), and estimate the parameters of the trivariate normal copula probability density function defined by (19).

The fixed effect parameters of the fixed effect scenario stochastic differential Equation (1) (random effects ${\varphi }_j^i=E{\varphi }_j^i=0$, i = 1, …, M, j = 1, 2, 3) for all tree size components (diameter, potentially available area, and height) and the dependence parameters ρ_ij, 1 ≤ i, j ≤ 3, of the copula function (19) were estimated using the maximum likelihood procedure. The fixed effect parameters of the mixed-effect scenario stochastic differential Equation (1) for all tree size components were estimated using the approximated maximum likelihood procedure [18]. The parameter estimation results for both scenario models, defined by Equation (1), are presented in

Table 1. Parameter estimates (standard errors) for both scenarios of the stochastic differential Equation (1).

Attribute	α	β	ɣ	σ	δ	σx
Fixed effect scenario models (1)
Diameter	0.0860 (0.0014)	0.0226 (0.0005)	−11.9338 (0.4604)	0.0037 (0.0002)	-	-
Area	0.0641 (0.0011)	0.0186 (0.0004)	−1.3008 (0.0514)	0.0139 (0.0003)	1.7570 (0.0277)	-
Height	0.0827 (0.0026)	0.0214 (0.0001)	−14.1101 (1.6421)	0.0017 (0.0003)	-	-
Mixed effect scenario models (1)
Diameter	0.0850 (0.0006)	0.0226 (0.0002)	−7.1108 (0.0953)	0.0042 (0.0001)	-	0.0069 (0.0010)
Area	0.0617 (0.0006)	0.0186 (0.0002)	−1.3260 (0.0358)	0.0102 (0.0001)	1.6151 (0.0237)	0.0094 (0.0014)
Height	0.0827 (0.0011)	0.0213 (0.0003)	−13.3459 (0.3489)	0.0013 (0.00005)	-	0.0042 (0.0006)

. All parameters were statistically significant (p < 0.05). The volatility parameter σ was lower for mixed effect scenario models compared to fixed effect scenario models. The parameter estimates and their standard errors for the copula probability density function are presented in

Table 2. Parameter estimates (standard errors) for copula probability density Equation (19).

ρ12	ρ13	ρ23
Fixed effect scenario
0.2390 (0.0103)	0.8619 (0.0021)	0.1720 (0.0100)
Mixed effect scenario
0.1513 (0.0101)	0.8528 (0.0022)	0.0877 (0.0100)

. The dependence parameters ρ₁₂, ρ₁₃, ρ₂₃ for fixed effect scenario models acquire higher values compared to mixed effect scenario models.

In the following discussion of the results, we will take the fixed effect parameters for both scenario models from Table 1 and Table 2. However, when discussing the mixed effect scenario model, we will calibrate the random effects for each variable (diameter, potentially available area, and height) with respect to the stand observed dataset $\left\{\left(x_{11},x_{21},x_{31}\right), \left(x_{12},x_{22},x_{32}\right), \dots , \left(x_{1m},x_{2m},x_{3m}\right)\right\}$ at discrete times $\left(t_1,t_2, \dots ,t_m\right)$ in the following form:

$$\widehat{{\varphi }_i}=\underset{\left({\varphi }_i\right)}{argmax}\left({\displaystyle \sum }_{j=1}^{m}ln\left(f_i\left(x_{ij},t|\widehat{\mathsf{\theta }},{\varphi }_i\right)\right)+ln\left(p({\varphi }_i|\widehat{{\sigma }_{x_i}^2})\right)\right),$$

(34)

where m is the number of measured trees and $p({\varphi }_i|\widehat{{\sigma }_i^2})$, i = 1, 2, 3, is a normal probability density function with zero mean and constant variance.

3.2. Visualization of Trivariate, Bivariate, and Univariate Conditional Copula Densities

The visualization of the joint trivariate copula-type density function faces challenges in adequately representing the five dependent variables ($x_1,x_2,x_3,t,f_{3c}\left(x_1,x_2,x_3,t\right)$). The main challenge and goal of visualizing the joint trivariate copula-type density function is to understand this complex dependence more simply. In the interest of clarity, we will simplify the representation of the function $f_{3c}\left(x_1,x_2,x_3,t\right)$ by taking a fixed time t at 60 years and changing the trivariate density to the corresponding three bivariate conditional densities, defined as:

$$\begin{gathered} f_{3c|3}\left(x_1,x_2,t|x_3\right)=\frac{f_{3c}\left(x_1,x_2,x_3,t\right)}{f_3\left(x_3,t\right)}, \\ {f}_{3c|2}\left(x_1,x_3,t|x_2\right)=\frac{f_{3c}\left(x_1,x_2,x_3,t\right)}{f_2\left(x_2,t\right)}, \\ {f}_{3c|1}\left(x_2,x_3,t|x_1\right)=\frac{f_{3c}\left(x_1,x_2,x_3,t\right)}{f_1\left(x_1,t\right)}. \end{gathered}$$

(35)

Figure 1 and Figure 2 show the conditional bivariate densities (see Equation (33)) under the condition that the remaining third variable takes some fixed value and a fixed time of 60 years for the fixed and mixed effect scenarios, respectively. Comparing Figure 1 and Figure 2, we notice that the conditional densities of the fixed effect scenario had a higher dispersion than the conditional densities with an additional random effect. This feature can be explained by the fact that the conditional densities of the fixed effect scenario covered a very wide region of stands, while the density of the mixed effect scenario reflected only a single specific stand. It can be added that the conditional densities became steeper for a smaller value of the conditional variable, which means that the variation was smaller.

Figure 3 shows the joint bivariate copula-type density functions for the fixed effect scenario defined by Equation (26), with fixed ages of 30 years and 80 years, and the corresponding contour lines (level ‘c’) of these bivariate distributions with a bivariate sample dataset, i.e., realizations ($x_i$, $x_j$) of two random variables $X_i$ and $X_j$, 1 ≤ i, j ≤ 3. It can be clearly seen that the joint bivariate fixed effect scenario densities at 80 years old were significantly steeper (up to six times) than the densities at 30 years old, which means that the variance of the analyzed variables increased strongly with increasing age. Considering the fact that the trees in the stands were not the same age, the contour lines shown in Figure 3 show a relatively high dependence of tree height on the diameter and a small influence of the potentially available area on the diameter and height of the trees.

The joint bivariate copula-type densities of the mixed effect scenario for two different stands, defined by Equation (26) and shown in Figure 4a–c, shows some influence of the stands. The representation of the contour lines and the corresponding measurement data in one drawing allowed us to conclude that the newly derived copula-based bivariate densities for the mixed effect scenario expressed the interdependencies of the examined quantities and matched the observed datasets well. Comparing the joint bivariate copula-type densities of the fixed effect scenario with the densities of the mixed effect scenario, we concluded that the latter had lower variances.

Additionally, it should be noted that all trivariate, bivariate, and conditional densities derived with the help of the normal copula are characterized by the fact that we can write stationary means, variances, and, of course, copula distributions for them.

Copula-based conditional densities for both the fixed and mixed effect scenarios with respect to one variable are presented in Figure 5 and Figure 6. Graphs (a2) and (c1) in Figure 5 show a very strong interdependence of tree diameter and height. Graphs (a1) and (c2) in Figure 5 show that the area potentially occupied by the tree had a very slight influence on the changes in diameter and height. Graphs (b1) and (b2) in Figure 5 show that tree diameter and height had a small influence on the potentially occupied area.

The graphs of the univariate conditional distributions presented in Figure 6 confirm the structure of the dependencies of tree diameter, height, and area. Additionally, it should be noted that the univariate conditional densities of the mixed effect scenario presented in Figure 6 were significantly higher than the corresponding densities of the fixed effect scenario shown in Figure 5. Consequently, the univariate conditional densities of the mixed effect scenario were controlled by significantly lower variances reflecting only a particular stand and not a certain region of stands.

3.3. Tree Diameter, Potentially Available Area, and Height Dynamic

The basic structure of the model system used in forestry can be classified as a regression, disaggregated, deterministic, and static stand model [27,28]. In general, the growth functions describing the change in the size of an individual tree must correspond to the principles of biological growth, namely, the starting point is fixed and has an asymptote, and the growth pattern is sigmoidal with one inflection point. The solution of the stochastic differential Equation (1) defining the transition probability density functions meets the requirements formulated earlier. Using the normal copula method, these univariate distributions can be combined into trivariate and bivariate distributions and their corresponding conditional bivariate and univariate distributions. Growth in the diameter, potentially available area, and height can be expressed as exact relationships by means or other moments, defined by Equations (5)–(9), using marginal distributions. The link between the variable of tree growth and the remaining one or two tree size variables can be formalized using copula-type densities and the numerical integration procedure, respectively, with Equations (29) and (30). For the mixed effect scenario, the statistical goodness-of-fit measures for the growth models expressed in Equations (5), (29) and (30) for the individual tree and whole-stand cases are presented in

Table 3. Statistical measures and p-values of the Student’s t-test for the prediction of the individual tree and whole-stand cases.

Predictors (Equation)	Individual Tree Case					Whole-Stand Case
	B * (%)	AB (%)	RMSE (%)	R2	T p-Value	B (%)	AB (%)	RMSE (%)	R2	T p-Value
Tree diameter
t: (5)	−0.1180 (−34.83)	4.8240 (55.80)	6.5364 (40.52)	0.6348	0.0938	−0.4769(−2.13)	1.5016 (7.51)	2.0481 (10.05)	0.9102	0.0172
t, A: (29)	0.0909 (−33.57)	4.7460 (55.12)	6.4493 (39.98)	0.6445	0.1906	−0.3283 (−1.98)	1.3798 (7.13)	1.9083 (9.36)	0.9238	0.0766
t, H: (29)	−0.0920 (−8.45)	2.4136 (22.08)	3.4921 (21.65)	0.8958	0.0145	0.1634 (0.94)	0.8813 (4.67)	1.3568 (6.66)	0.9621	0.2134
T, A, H: (30)	−0.1093 (−8.49)	2.3725 (21.92)	3.4242 (21.23)	0.8998	0.0044	0.0387 (0.31)	0.8684 (4.61)	1.3495 (6.62)	0.9630	0.7661
Tree area
t: (5)	−0.1092 (−37.26)	3.9093 (60.74)	6.0131 (63.98)	0.4390	0.0919	−0.2376 (−2.24)	1.0706 (9.16)	1.5444 (12.37)	0.9412	0.1127
t, D: (29)	−0.0840 (−35.98)	3.8417 (59.31)	5.9327 (63.12)	0.4541	0.1886	−0.2128 (−2.48)	0.9978 (8.86)	1.3948 (11.17)	0.9521	0.1157
t, H: (29)	−0.0864 (−36.59)	3.8832 (60.09)	5.9846 (63.67)	0.4445	0.1801	−0.1691 (−2.06)	0.9867 (8.74)	1.4184 (11.36)	0.9509	0.2178
T, D, H: (30)	−0.0775 (−35.92)	3.8191(59.12)	5.9044 (62.82)	0.4593	0.2229	−0.2495 (−2.80)	1.0414 (9.22)	1.4316 (11.46)	0.9491	0.0728
Tree height
t: (5)	−0.0035 (−18.37)	3.3658 (35.56)	4.5945 (29.41)	0.7487	0.9436	−0.2657 (−1.44)	0.9825 (5.41)	1.2777 (6.61)	0.9524	0.0329
t, D: (29)	0.0229 (−6.22)	1.6828 (15.84)	2.3341 (14.94)	0.9351	0.3609	−0.3530 (−2.20)	0.6799 (3.90)	0.9609 (4.46)	0.9758	0.0001
t, A: (29)	0.0292 (−18.30)	3.3495 (35.43)	4.5647 (29.22)	0.7518	0.5516	−0.3061 (−1.76)	0.9416 (5.28)	1.2044 (6.23)	0.9568	0.0095
T, D, A: (30)	0.0235 (−6.12)	1.6721 (15.75)	2.3232 (14.87)	0.9357	0.3462	−0.3062 (−1.95)	0.6572 (3.77)	0.8661 (14.03)	0.9764	0.0004

* Statistical measures: mean error $B=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\left(y_i-\stackrel{\wedge }{y_i}\right)$; percentage mean error $\%B=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\frac{y_i-\stackrel{\wedge }{y_i}}{y_i}\times 100$; absolute mean error $AB=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\left|y_i-\stackrel{\wedge }{y_i}\right|$; percentage absolute mean error $\%B=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\left|\frac{y_i-\stackrel{\wedge }{y_i}}{y_i}\right|\times 100$; root mean square error $RMSE=\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(y_i-\stackrel{\wedge }{y_i}\right)}^2}$; percentage root mean square error $\%RMSE=\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(\frac{y_i-\stackrel{\wedge }{y_i}}{y_i}\right)}^2}\times 100$; and the coefficient of determination $R^2=1-\frac{{{\displaystyle \sum }}_{i=1}^n{\left(y_i-\stackrel{\wedge }{y_i}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(y_i-\overline{y}\right)}^2}$, where $n={{\displaystyle \sum }}_{i=1}^K{n}_i$ is the total number of observations used to validate the model, K is the number of stands, n_i is the number of measured trees in the ith plot, and $y_i$, $\stackrel{\wedge }{y_i}$, and $\overline{y}$ are the measured, estimated, and average values of the dependent variable, respectively.

.

When starting to examine the fitness of the newly presented tree and stand’s diameter, potentially available area, and height growth models, one should keep in mind that the observed dataset considered uneven stands, most of which were non-pure and consisted of pine, spruce, birch, and other tree species. Tree height as an explanatory variable had a foremost influence on the growth predictions of individual tree and whole-stand diameter. When the explanatory variable was represented by the potentially available area acting alone or in combination with tree height, no significant improvement in tree diameter predictions was shown. For the most part, the individual tree and whole-stand diameter growth models accounted for up to 90% and 96% of the variation, respectively. For both cases, the root mean square error (percentage root mean square error) was reduced to 3.42 cm (21%) and 1.34 cm (7%), respectively. A comparison of the mean prediction error, the mean absolute error, and the Student’s test p-values did not show a significant bias in tree diameter predictions.

For tree height predictions, both the individual tree and whole-stand models showed even higher goodness-of-fit statistical measures than those for diameter, covering variations of 94% and 98%, respectively, and the root mean square error (percentage root mean square error) dropped to 2.32 m (15%) and 0.86 m (4%), respectively. Individual tree height predictions did not show any significant bias; however, the whole-stand height predictions already showed a slight bias. The bias could be explained by the fact that the distribution of the species composition of the trees in both datasets used to fit the parameter estimates was unfortunately different.

The predictions of potentially available area for the individual tree and whole-stand cases showed very high statistical measures. Consequently, the growth model of the tree’s potentially available area expressed by Equation (5) can be successfully used for modeling the number of trees per ha.

Using the dynamic Equations (5)–(8) for tree diameter, potentially available area, and height, Figure 7 shows the mean, median, mode, and 5th and 95th quantiles of two randomly selected stands along with the variables measured in these stands.

The 95% and 5% quantile curves define the area within which we can be 90% confident that similarly constructed areas will contain the curve for the sample mean. The sample mean curve in the same stand varied within the observed dataset due to the natural variability of the sample and, of course, the calibrated random effects represented by Equation (34). The left and right sides of Figure 7, representing two different randomly selected stands, show that the mean curve and the curves of the other attributes could be significantly different in each stand. Additionally, it can be seen from Figure 7 that the distributions of all three examined variables, namely tree diameter, potentially available area, and height, had a positive asymmetry, which was insignificant in young trees and became more pronounced in mature age. In recent years, theoretical studies of tree size growth have confirmed that tree size distributions are positively skewed and sigmoidal in shape, with many small trees and few larger trees due to asymmetric competition, and a number of different non-normal distribution functions have been used to model tree size distributions [29,30]. Despite the fact that two databases of different sizes (the first included the age, diameter, and potential available area of 39,437 trees, while the second included the age, diameter, potential available area, and height of 8604 trees) covering different tree species (pine, spruce, birch, and others) and uneven stands were used to fit the parameters of Equation (1) and the copula density (19), the statistical measures presented in Table 3 show a high prediction accuracy.

4. Conclusions

The combination of stochastic differential equations with the copula method allowed us to successfully analyze the dependencies and dynamics of multivariate variables. Additionally, random effects introduced into the stochastic differential equation enabled a more accurate determination of the variation between individuals (stands). The newly derived multivariate and univariate densities enabled accurate predictions of the values of the modeled variables depending on time (age) and other explanatory variables. As was mentioned earlier, two databases of different sizes (the first included only the diameter and potentially available area of 39,437 trees, while the second included the diameter, potentially available area, and height of 8604 trees) covering different tree species (pine, spruce, birch, and others) and uneven stands were used to present the models, the statistical measures presented in Table 3 show a high prediction accuracy.

In the future, it would be appropriate to expand the combination of the three stochastic differential equations presented in this work to a larger number of variables with the help of the copula function (for example, additionally including the tree crown base height and crown width) using vine copulas and other techniques [4,31].