# A Nonlinear Mixed-Effects Height-to-Diameter Ratio Model for Several Tree Species Based on Czech National Forest Inventory Data

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Materials

#### 2.2. Training Dataset

#### 2.3. Validation Dataset

^{2}to 4900 m

^{2}. Sample plots are laid out in different parts of the Czech Republic (Figure 1b). Several stand characteristics (stand structure, natural regeneration, dead wood stock, and stand development stage and site quality) were considered as main criteria while laying out sample plots. Sample plot locations vary from 240 m to 1,370 m a.s.l., where mean climate characteristics (i.e., annual temperature (4–9.5° C), mean annual precipitation (500–1450 mm), and growing season length (45–180 days) also largely vary. In-depth descriptions of this dataset are also available in the literature [43,54,59,69,70,71,72,73,74]. This dataset comprises 24% monospecific sample plots and 76% mixed species sample plots. The definition of the monospecific sample plots considered the inclusion of all individuals other than a species of interest if they had over-bark DBH <4 cm [43]. Measurements were carried out between April 2007 and March 2016. However, no repeated measurement was involved, meaning that there was no temporal variation in this dataset. Total height and DBH were measured with precisions of 0.1 m and 1 mm, respectively. The number of trees by species in the validation data is given in Table S2. Graphs of HDR plotted against DBH in both training and validation datasets are presented in Figure 2.

#### 2.4. Data Analysis

#### 2.4.1. Deriving Stand-Level Variables

^{−1}), stand basal area (BA, m

^{2}ha

^{−1}), arithmetic mean DBH (AMD, cm), and quadratic mean DBH (QMD, cm) per sample plot. We also evaluated two tree-centered competition measures: DBH-to-QMD ratio (dq) and basal area of the trees larger in diameters than a subject tree (BAL, m

^{2}ha

^{−1}) for their potential contributions to the HDR model. We evaluated the relative spacing index (RS), which is defined by $\mathrm{RS}=\sqrt{10000/N}/\mathrm{HDOM}$ [55,76,77] for its potential contribution to the HDR model. The RS is assumed as a more effective density measure than others, because it incorporates number of stems per hectare, site quality and stand development through HDOM. All measures were calculated using the dendrometric measurements of all trees regardless of species per sample plot. Summary statistics of important variables in both training and validation datasets are presented in Table 1. The number of HDR sample trees per sample plot in the validation dataset is higher compared to that in the training dataset, because of larger sample plot size (2500–4900 m

^{2}) in the former dataset compared to the latter plot size (500 m

^{2}). However, the number of stems per hectare in the training dataset is higher compared to that in the validation dataset. It is because that training dataset includes all age and DBH classes of the forest stands, whereas the majority of the validation data came from middle-aged and mature forest stands.

#### 2.4.2. Model Construction

_{ij}and DBH

_{ij}are the height-to-diameter ratio (m cm

^{−1}) and diameter at breast height of the j

^{th}tree (cm) on the i

^{th}sample plot, respectively; b

_{3}= 0.5; b

_{1}and b

_{2}are the parameters to be estimated; and ${\epsilon}_{ij}$ is an error term.

_{1}, b

_{2}) plotted against each variable and its transformation and interactions with other predictors were also examined, and those showing stronger correlations with b

_{1}or b

_{2}were identified. Because of its biological logic and interpretation, the two-stage approach has frequently been used for modelling tree characteristics [53,70,80,81]. Among various predictors evaluated using the two-stage approach, HDOM, DDOM, RS, and dq showed stronger relationships only with b

_{1}of Equation (1). Following the principles of modelling categorical variables [62,82], we formed two dummy variables (CC

_{1}, CC

_{2}) to account for HDR variations caused by three canopy layers denoted by CHC, as below:

Canopy Height Class | CC_{1} | CC_{2} |

CHC1 | 0 | 0 |

CHC2 | 1 | 0 |

CHC3 | 0 | 1 |

_{1}, TS

_{2}, TS

_{3}, TS

_{4}, TS

_{5}, TS

_{6}) for seven species to describe species-specific or group-specic HDR variaitons, as below:

Tree Species | TS_{1} | TS_{2} | TS_{3} | TS_{4} | TS_{5} | TS_{6} |

Norway spruce | 0 | 0 | 0 | 0 | 0 | 0 |

Scots pine | 1 | 0 | 0 | 0 | 0 | 0 |

European larch | 0 | 1 | 0 | 0 | 0 | 0 |

Fir species | 0 | 0 | 1 | 0 | 0 | 0 |

European beech | 0 | 0 | 0 | 1 | 0 | 0 |

Oak species | 0 | 0 | 0 | 0 | 1 | 0 |

Birch and alder speices | 0 | 0 | 0 | 0 | 0 | 1 |

_{1}, CC

_{2}, TS

_{1}, TS

_{2}, TS

_{3}, TS

_{4}, TS

_{5}, TS

_{6}) were included into the base function (Equation (1)) through expression of b

_{1}as a function of all these variables as in Equation (2). This expression was chosen as this provided the convergence with the smallest residual variations (smallest sum of squared errors) in fitting the model.

_{i}) added to b

_{1}. Details of the mixed-effects model formulation are found in the statistical textbooks [48,83]. We present only a final form of a nonlinear mixed-effects model (Equation (3)) that describes HDR variations with the smallest residual variations in our data.

_{k}is a dummy variable (k = 1,2); TS

_{k}is dummy variable (k = 1, 2, …, 6); b

_{3}= 0.5; b

_{2}and α

_{k}(k = 1,2, …, 13) are parameters to be estimated; and ${\epsilon}_{ij}$ is an error term; the other symbols and abbreviations are the same as in Equations (1) and (2). When CC belongs to k, then CC

_{k}= 1, and otherwise is 0; similarly, when TS belongs to k, then TS

_{k}= 1 and otherwise is 0 (see above two tables of dummy variable formation). When b

_{3}in Equations (1) and (3) was tried to be estimated along with other parameters by optimization, the convergence with a global minimum was not achieved. We then compared the sum of squared errors (SSE) produced from the fitting with several alternative values of b

_{3}(0.1 to 2 by 0.1 increment) iteratively, and b

_{3}= 0.5 resulted in the smallest SSE. This is the reason that b

_{3}= 0.5 was chosen in both Equations (1) and (3).

_{i}) are defined by ${\mathit{\epsilon}}_{\mathit{i}}~\mathit{N}\left(0,\text{}\mathit{R}\right)$ and ${\mathit{u}}_{\mathit{i}}~\mathit{N}\left(0,\mathit{D}\right)$ respectively. It was assumed that error vector ${\mathit{\epsilon}}_{\mathit{i}}$ would have a normal distribution with zero mean and within-plot variance–covariance matrix ${\mathit{R}}_{\mathit{i}}$ defined by:

^{2}is a scaling factor and is equivalent to a residual variance of the estimated HDR model and common to all plots [50]. A matrix ${\mathsf{\Gamma}}_{\mathit{i}}$ accounts for within-plot residual autocorrelations. However, we assumed this matrix as an identity matrix, as our data lacked significant autocorrelations. The matrix

**G**is a diagonal matrix that accounted for a within-sample plot heteroscedasticity. The random effect vector u

_{i}_{i}was assumed to have a normal distribution with zero mean and plot variance–covariance matrix

**D**defined by:

#### 2.4.3. Estimating Model Parameters and Evaluation

^{2}

_{adj}), and Akaike information criterion (AIC). The AIC is based on minimizing the Kull-back-Lieber distance and it imposes a penalty for the number of parameters in the model [87,88]. Formulae of all these statistical measures are found in the statistical textbooks [89]. The graphs of residuals, prediction errors and simulated HDR curves produced with fitted HDR model were also examined. Unless otherwise specified, we used 1% level of significance in our analyses. Realizing the fact that validation with an external independent dataset increases the credibility of the model, we tested our HDR model using validation data. Sample plot-specific HDR curves generated using the calibrated mixed-effects HDR model after adjustment of the random effects predicted with the empirical best linear unbiased prediction (EBLUP) theory [48] were examined for all sample plots in the validation data.

#### 2.4.4. Calibrating Mixed-Effects Model and Predicting Sample Plot-Specific HDR

_{i}is a random effect vector that describes plot-level HDR variations for the i

^{th}plot. Equations (4) and (5) provide the elements of matrices ${\mathit{R}}_{\mathit{i}}$ and $\mathit{D}$ respectively. Vector ${\mathit{\epsilon}}_{\mathit{i}}$ of the prediction errors was derived from the mean model (only fixed part of the mixed-effects HDR model). The element (z

_{ij}) of the designed matrix

**Z**for each tree and sample plot was determined from partial derivatives of a nonlinear mixed-effects model (Equation (3)) with respect to its fixed-effect parameter (b

_{i}_{1}) to which random effect (u

_{i}) has been added [48,53,58,91]. The following equation was obtained for this purpose.

## 3. Results

^{2}

_{adj}= 0.7863) in the training data without significant trends in the residuals.

_{i}= 0) (Figure 4, Figure 5, Figure 6 and Figure 7). The same scale in the y-axis for all these figures was used for better comparison of the effects of predictors on the HDR. A wide range of the observed data of two predictors HDOM and DDOM in each of the canopy height classes in the training data resulted in several HDR curves for each species or group of species (Figure 4 and Figure 5). However, a relatively narrower range of the observed data of two predictors RS and dq in the suppressed canopy heights (CHC = 3) produced fewer curves for some individual species or groups of species (Figure 6 and Figure 7). The combined effects of site quality and stand development stage described by HDOM emerged as the biggest effect, followed by the effect of tree crowding or competition described by DDOM and dq. For a given condition of stand density and inter-tree spacing, HDR increased with increasing site quality and stand development stage (increased HDOM), and increasing competition (decreased DDOM and dq) and increased RS). The magnitudes of the effects by each predictor on HDR for each canopy height class significantly differed.

^{2}= 0.9443; RMSE = 0.06745) and more than 85% of HDR variations for individual species or groups of species in this dataset, with a description of the largest part for European beech and the smallest part for fir species (Table 3). The mixed-effects HDR model was calibrated with the random effect predicted from measured HDR of one to four randomly selected trees of a particular species or group of species depending on the availability of their numbers on each sample plot. The model test confirmed that a single HDR model developed with the pooled dataset from several species in the NFI sample plots was precise enough for predicting the HDR of the corresponding tree species in the research sample plot data. Graphs of the prediction errors of the calibrated mixed-effects HDR model for each species or group of species showed a promising appearance, i.e., absence of significant trends in the prediction errors for each species and group of species (Figure 8). The prediction errors were found falling within the ±0.25 range for almost all sample plots, indicating that the calibrated mixed-effects HDR model was precise for the validation data.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of sample plots [59]: National forest inventory sample plots in the training data (

**a**) and research sample plots in the validation data (

**b**).

**Figure 2.**Height-to-diameter ratio (HDR) plotted against diameter at breast height (DBH) for each canopy height class (CHC); classification was adapted from Sharma et al. [43]; CHC1: height >66% height of the tallest tree, CHC2: 33% height of the tallest tree < height < 66% height of the tallest tree, and CHC3: height <33% height of the tallest tree per sample plot.

**Figure 3.**Box plots of the standardized residuals of the height-to-diameter ratio (HDR) model. The larger box length represents interquartile range (IQR), whisker length represents class minimum and maximum values in the IQR, smaller boxes represent the observations 1.5 times beyond the IQR (outlier observations lying far away from the median), and horizontal lines and plus signs in a larger box represent class median and mean values, respectively.

**Figure 4.**Effects of dominant height (HDOM) on height-to-diameter ratio (HDR). Curves were generated with the fixed-part of the mixed-effects HDR model using the mean of each of the three measurable covariate predictors and allowing HDOM and diameter at breast height (DBH) to vary from approximately minimum to maximum for each canopy height class (CHC) in the training data. Dummy codes were applied in the same way as in Equation (3) for generating curves.

**Figure 5.**Effects of dominant diameter (DDOM) on height-to-diameter ratio (HDR). Curves were generated with the fixed-part of the mixed-effects HDR model using the mean of each of the three measurable covariate predictors and allowing DDOM and diameter at breast height (DBH) to vary from approximately minimum to maximum for each canopy height class (CHC) in the training data. Dummy codes were applied in the same way as in Equation (3) for generating curves.

**Figure 6.**Effects of the relative spacing index (RS) on height-to-diameter ratio (HDR). Curves were generated with the fixed-part of the mixed-effects HDR model using the mean of each of the three measurable covariate predictors and allowing RS and diameter at breast height (DBH) to vary from approximately minimum to maximum for each canopy height class (CHC) in the training data. Dummy codes were applied in the same way as in Equation (3) for generating curves.

**Figure 7.**Effects of the ratio of DBH to quadratic mean DBH (dq) on height-to-diameter ratio (HDR). Curves were generated with the fixed-part of the mixed-effects HDR model using the mean of each of the three measurable covariate predictors and allowing dq and diameter at breast height (DBH) to vary from approximately minimum to maximum for each canopy height class (CHC) in the training data. Dummy codes were applied in the same way as in Equation (3) for generating curves.

**Figure 8.**Standardized prediction errors in the validation data. The mixed-effects HDR model was calibrated with the random effects predicted using HDR measurements from one to four randomly selected trees of a particular species or group of species depending on the availability of their numbers per sample plot; DBH: diameter at breast height; CHC: canopy height class, which is the same as defined in Figure 2.

**Figure 9.**Sample plot-specific height-to-diameter ratio (HDR) curves overlaid on the validation data. Curves were produced with the calibrated mixed-effects HDR model using the sample plot-level mean of each of the three measurable covariate variables and allowing individual tree diameter at breast height (DBH)-to-quadratic mean DBH (dq) to vary from approximately minimum to maximum by one for each sample plot. The mixed-effects HDR model was calibrated with the random effect predicted using HDR measurements from one to four randomly selected trees of a particular species or group of species depending on the availability of their numbers per sample plot in the validation data. Dummy codes were applied in the same way as in Equation (3) for generating curves.

**Figure 10.**Relationship between crown ratio (ratio of crown depth to total height) and height-to diameter ratio (HDR) for a particular species and group of species in the validation dataset; mean crown ratio for each canopy height class (CHC) was calculated by HDR class of 0.25 interval. Spec 1: Norway spruce; spec 2: Scots pine; spec 3: European larch; spec 4: Fir species; spec 5: European beech; spec 6: Oak species; spec 7: Birch and alder species.

Variables | Statistics (Mean ± Standard Deviation (Range)) | |
---|---|---|

Training Data | Validation Data | |

Number of sample plots | 13,875 | 220 |

Number of HDR sample trees | 348,980 | 25,146 |

Number of HDR sample trees per sample plot | 36.5 ± 15.9 (4–90) | 232.3 ± 167.8 (8–664) |

Number of stems (N ha^{−1}) | 1514 ± 671 (40–5460) | 940 ± 711 (32–4700) |

Stand basal area (BA, m^{2} ha^{−1}) | 38.3 ± 13.5 (0.07–85.4) | 41.3 ± 14.4 (0.1–81.1) |

BA of trees lager than a subject tree (BAL, m^{2} ha^{−1}) | 28.5 ± 14.4 (0–83.3) | 32.2 ± 17.6 (0–79.5) |

Quadratic mean DBH per sample plot (QMD, cm) | 27.3 ± 6.9 (8.7–84.2) | 28.6 ± 10.3 (9.6–87.3) |

DBH-to-QMD ratio (dq) | 0.95 ± 0.31 (0.13–4.5) | 0.86 ± 0.52 (0.04–7) |

Arithmetic mean DBH per sample plot (AMD, cm) | 25.8 ± 6.6 (8.5–78.4) | 24.9 ± 10.6 (9.1–84.4) |

Dominant height per sample plot (HDOM, m) | 23.2 ± 6.2 (4.4–42.4) | 27.5 ± 6.9 (8–42.8) |

Dominant diameter (DDOM, cm) | 31.5 ± 8.1 (8.6–78.4) | 49.7 ± 14.2 (13.3–84.4) |

Total height (H, m) | 21.1 ± 7.1 (1.5–54.3) | 17.2 ± 9.3 (1.4–50.6) |

Diameter at breast height (DBH, cm) | 26.4 ± 11.6 (7–117.3) | 25.2 ± 17.4 (2–118.1) |

Height-to-DBH ratio (HDR, m cm^{−1}) | 0.85 ± 0.19 (0.08–2.15) | 0.81 ± 0.28 (0.11–2.3) |

**Table 2.**Parameter estimates, variance components and fit statistics of the mixed-effects HDR model (Equation (3)).

Estimate | Standard Error | t-Value | Pr > |t| | |
---|---|---|---|---|

Fixed | ||||

α_{1} | −0.4005 | 0.001765 | −226.98 | <0.0001 |

α_{2} | −1.0526 | 0.004407 | −238.81 | <0.0001 |

α_{3} | 0.7998 | 0.004939 | 161.95 | <0.0001 |

α_{4} | 0.4568 | 0.002249 | 203.06 | <0.0001 |

α_{5} | −0.02188 | 0.000223 | −97.98 | <0.0001 |

α_{6} | 0.1236 | 0.01565 | 7.90 | <0.0001 |

α_{7} | −0.4599 | 0.002771 | −165.97 | <0.0001 |

α_{8} | 0.01049 | 0.001457 | 7.20 | <0.0001 |

α_{9} | 0.07762 | 0.002047 | 37.91 | <0.0001 |

α_{10} | −0.04291 | 0.003544 | −12.11 | <0.0001 |

α_{11} | 0.01252 | 0.002087 | 6.00 | <0.0001 |

α_{12} | −0.04845 | 0.002019 | −23.99 | <0.0001 |

α_{13} | 0.04806 | 0.001804 | 26.65 | <0.0001 |

b_{2} | 0.1823 | 0.000747 | 244.09 | <0.0001 |

Variance | ||||

σ^{2}_{ui} | 0.01109 | |||

σ^{2} | 0.006923 | |||

Fit statistics | ||||

RMSE | 0.0923 | |||

R^{2}_{adj} | 0.7863 | |||

AIC | −729778 |

^{2}

_{adj}: adjusted coefficient of determination; RMSE: root mean squared errors; AIC: Akaike’s information criterion; α

_{1}, …, α

_{13}, b

_{2}: fixed-effect parameters; u

_{i}= random effect for i

^{th}sample plot; σ

^{2}: residual variance; other symbols are the same as in Equations (1)–(3).

Species | RMSE | R^{2} |
---|---|---|

Norway spruce | 0.0673 | 0.8922 |

Scots pine | 0.0657 | 0.9039 |

European larch | 0.0667 | 0.8821 |

Fir species | 0.0654 | 0.8574 |

European beech | 0.0676 | 0.9605 |

Oak species | 0.0681 | 0.9176 |

Birch and alder species | 0.0668 | 0.9328 |

^{2}: coefficient of determination.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sharma, R.P.; Vacek, Z.; Vacek, S.; Kučera, M. A Nonlinear Mixed-Effects Height-to-Diameter Ratio Model for Several Tree Species Based on Czech National Forest Inventory Data. *Forests* **2019**, *10*, 70.
https://doi.org/10.3390/f10010070

**AMA Style**

Sharma RP, Vacek Z, Vacek S, Kučera M. A Nonlinear Mixed-Effects Height-to-Diameter Ratio Model for Several Tree Species Based on Czech National Forest Inventory Data. *Forests*. 2019; 10(1):70.
https://doi.org/10.3390/f10010070

**Chicago/Turabian Style**

Sharma, Ram P., Zdeněk Vacek, Stanislav Vacek, and Miloš Kučera. 2019. "A Nonlinear Mixed-Effects Height-to-Diameter Ratio Model for Several Tree Species Based on Czech National Forest Inventory Data" *Forests* 10, no. 1: 70.
https://doi.org/10.3390/f10010070