# Individual-Tree Diameter Growth Models for Mixed Nothofagus Second Growth Forests in Southern Chile

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}emp) of 0.56, relative root mean square error of 44.49% and relative bias of −1.96% for annual DBH growth predictions, and R

^{2}emp of 0.98 and 0.97 for DBH projection at 6 and 12 years, respectively. This model constitutes a simple and useful tool to support management plans for these forest ecosystems.

## 1. Introduction

^{3}ha

^{−1}year

^{−1}[5] and are some of the highest rates for Southamerican temperate native forests. Hence, given its large geographical distribution, timber market value and attractive growth rates, these RORACO forests present high economic and social potential value within the Chilean forestry sector [6].

^{2}, including four growth zones). Thus, the aim of this study is to develop an individual-tree growth model to estimate annual increment in DBH for second growth mixed forest stands of N. alpina, N. obliqua, and N. dombeyi, including a set of tree- and stand-level predictors to explain diameter growth. The specific objectives of this study are: (1) to compare different multiple linear fitting procedures to obtain the best individual-tree growth model; (2) to generalize diameter growth models by incorporating growth zone and species; and (3) to evaluate the final diameter growth model using an independent validation dataset. Accordingly, to meet these objectives, advanced statistical model predictor selection methods are used in currently available forest inventory and plot data that spans RORACO’s ecologically and geographically diverse area.

## 2. Materials and Methods

#### 2.1. Data Description

^{2}, but for higher densities (>4800 trees ha

^{−1}) plot area was reduced to 250 m

^{2}. All trees with a DBH of 5 cm or larger, and taller than 2 m in height were measured. Total height was obtained in a subsample of 15 trees per plot, and tree increment cores were extracted from this subsample at DBH to estimate breast height age and obtain past diameter growth. The plots from the TP network were formed by two circular subplots with an area of 125 m

^{2}each. As with the PP network, all trees with DBH > 5 cm and taller than 2 m were measured. In each subplot, two trees were felled for complete stem analysis. Further details of these datasets are presented in Gezan and Moreno [11]. This study used annual average DBH growth based on the last years, which originated from the increment cores (PP network) and tree sections (TP network) at breast height of the penultimate and antepenultimate growth years. The last growth year was discarded as plots were established at different months within a year, without all completing the current growing season.

^{2}ha

^{−1}), sum of basal area of larger trees in relation to the target individual for Nothofagus (BALn, m

^{2}ha

^{−1}), sociological status (SS, defined according to vertical stratification with dominant (1); codominant (2); intermediate (3) or suppressed (4)), and relative basal area of larger trees (BALr, with BALr = BAL/BA). The response variable, as indicated earlier, corresponded to the average annual increment in DBH (AIDBH, mm year

^{−1}). Summary statistics of the above attributes are presented in Table 1; a total of 158 plots are presented here as some were dropped given that they were not dominated by Nothofagus or they lacked information on breast height age.

^{2}ha

^{−1}), trees per hectare (N, trees ha

^{−1}), quadratic mean diameter (QD, cm), dominant height (Hd, m), dominant breast height stand age (Ad, years), dominant species (DS), site index (SI, m), basal area of Nothofagus (BAN, m

^{2}ha

^{−1}), stand density index (SDI, trees ha

^{−1}) and relative spacing (RS). Zone was defined by Gezan and Moreno [11] based mainly on climate, soil and site productivity (Figure 1). Hd and Ad were calculated using the 100 largest trees per hectare in terms of DBH. SI was estimated using available dominant height-site models obtained from the same datasets for an index age of 20 years [7]. BAN was calculated as the basal area sum of N. alpina, N. obliqua, and N. dombeyi. DS corresponded to the Nothofagus species with the largest proportion of BAN. SDI was calculated using the expression SDI = N × (25.4/QD)

^{β}[40], where β = −1.4112, as reported by Gezan et al. [6]. Finally, RS was calculated using the equation RS = [(10,000/N)

^{0.5}]/Hd [40]. Hence, a total of 1108 trees constituted the final dataset originating from 158 plots with a single measurement that have the total of variables for fitting. Note that some plots were dropped as these lacked HD or Ad.

#### 2.2. Model Fitting

^{2}adj), Mallows’ Cp, Schwartz’s information criterion (BIC), residual sum of squares (RSS), and the mean squared error averaged over the K folds (MSE

_{K}) [32,42]. The second procedure used for model selection and model fit was LASSO regression [32]. The λ tuning parameter for this approach was determined by cross-validation using the fitting database with the mean square error as criteria to obtain the final candidate model.

_{0}* = $\widehat{\mathsf{\beta}}$

_{0}× exp(MSE/2) the adjusted intercept, where MSE is the mean square error for a given model obtained based on the training dataset. To evaluate the predictions performance of these models, the following goodness-of-fit statistics were used on the test data: empirical coefficient of correlation (R

^{2}emp), root mean square error (RMSE), relative root mean square error (RMSE%), bias (BIAS), relative bias (BIAS%) and, Theil’s inequality coefficient (U2) [44,45]. Their formulae are:

## 3. Results

#### 3.1. Model Fitting

^{−1}for N. alpina, N. obliqua and N. dombeyi, respectively. These rates vary greatly according to growth zone, where zone 3 had the highest annual DBH growth average with 3.8 mm year

^{−1}and zone 4 had the lowest growth with 2.7 mm year

^{−1}.

^{2}adj, Cp, BIC, RSS, and MSE

_{K}(Figure 2). All model parameter estimates were significant with p < 0.001, and estimated parameters are presented in Table 2. The resulting expression of the model is:

^{2}emp of 0.56, with a RMSE% of 44.16% and a BIAS% of −1.96% (see Table 3). Thus, the model predictions have moderate accuracy (in terms of RMSE%), but have negligible bias (Figure 3).

^{−4}. In contrast to the CV regression model, LASSO also incorporated BALr and Ad as predictors, so the final expression of this model was:

^{2}emp of 0.57, RMSE% of 44.16 and BIAS% of −2.94.

_{ij}is the intercept parameter for the ith species in the jth zone, and ${\mathsf{\beta}}_{1}$ to ${\mathsf{\beta}}_{4}$ are slope parameters to estimate. SpZone

_{ij}are the dummy variables associated with the ij

^{th}group specie and zone. The species were coded as: 1 (N. alpina), 2 (N. obliqua) and 3 (N. dombeyi). The other model terms were described previously. Note that with the incorporation of SpZone

_{ij}, the predictor SDI was removed from the model as it resulted not significant (p = 0.297). The goodness-of-fit statistics for this extended model were R

^{2}emp = 0.56, RMSE% = 44.49%, and BIAS% = −2.29%. This model had similar performance than the CV model (Table 4); however, from the analysis of variance, the factor SpZone resulted highly significant (p < 0.001).

^{−3}). Here, the final model obtained was:

^{2}emp = 0.54, RMSE% = 45% and BIAS% = −4.31%, values that are worst in performance with respect to other three previous models evaluated (Table 4).

_{0}* based on Baskerville [43] correction of CV, LASSO, CV + SpZone, and LASSO + SpZone models were 2.546, −0.160, 2.829, and −1.239, respectively.

#### 3.2. Model Projection

^{2}emp > 0.97 (Table 5). The CV and LASSO regression had better goodness-of-fit statistics, while LASSO + SpZone had the worst performance. A six year projection for DBH showed that LASSO regression had a better R

^{2}emp of 0.99 and RMSE% of 7.26 but with marginally larger BIAS% of 0.36 than the CV model, although all models had excellent statistics. CV and LASSO regression for a simulation period of 12 years continued with good prediction performance. CV regression presented R

^{2}emp of 0.97, RMSE% of 9.66 and BIAS% of 1.75, at the same projection time, with similar results for LASSO regression with R

^{2}emp of 0.97, RMSE% of 9.50 and BIAS% of 1.96. Surprisingly, the extended models with species and growth zone factors were not clearly better than those without them.

#### 3.3. Selected Model

## 4. Discussion

^{−1}against 4.05 mm year

^{−1}). Other studies also have found contrasting differences in growth between Nothofagus species and growth zones [5,9,10,59,60,61,62,63,64]. The incorporation of species and/or zone factors to improve growth models has been previously reported [24,25]. In the present study, the combined specie-zone factor (SpZone, with 11 levels), explained more variability than the individual two factors (with seven levels for both species and zone), indicating that a greater disaggregation of the data is required to model growth rate accurately. Interestingly, the LASSO + SpZone model selected as predictors for the combined factor SpZone the extreme groups (Table 6), by combining the central five levels into a single group class.

^{2}emp ranging from 0.54 to 0.57, RMSE% of ~44%, and a BIAS% smaller than 3%. Similar goodness-of-fit statistics have been reported in other species for AIDBH models with R

^{2}emp ranging from 0.26 to 0.68 [24,26,51,65]. AIDBH projections for 6 and 12 years resulted in R

^{2}emp ranges of 0.23–0.28 and 0.15–0.24, respectively. This drop in performance is likely to have been affected by the quality of the projection database, that included, for example, 6 years with 20% of the trees with null or negative DBH increments, probably due to field measurement errors and low growth rates for these species. In contrast, for both Nothofagus and companion species, DBH projections for 6 and 12 years resulted in overall R

^{2}emp values greater than 0.97. Small diameter trees (DBH < 15 cm) presented lower correlations, with values of 0.88 and 0.58 for projections of 6 and 12 years, respectively. In addition, as expected, goodness-of-fit statistics were better for the Nothofagus cohort than the companion species cohort.

_{K}) when compared to other stepwise methods. LASSO is an alternative procedure that regularizes the coefficient estimates, forcing some of them to be exactly equal to zero, and hence, performing variable selection. Both procedures, CV and LASSO, have been used broadly due their advantages to identify those variables that help to better predict a given response [32]. Similarly, the importance of incorporating dummy variables (or factors) in a linear model helps to expand model specificity, where additional factors can modify intercepts or slopes from the original regression model to make obtain more generic, and therefore robust, models that apply for wider conditions.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Geographical localization of permanent plots (PP) and temporary plots (TP) networks. Growth zones defined by Gezan and Moreno [11] are also shown.

**Figure 2.**Goodness-of-fit statistics of adjusted r-squared (R

^{2}adj), Statistics Mallows’ C

_{p}, Schwartz’s information criterion (BIC), residual sum of squares (RSS), and the mean squared error averaged over the K folds (MSE

_{K}) for different number of predictors with cross-validation (CV) regression selection model using the fitting database.

**Figure 3.**Fit and residuals plots for the natural logarithm of annual increment in diameter at breast height (AIDBH, mm year

^{−1}) of selected models using test dataset. Cross-validation (CV, (

**a**,

**b**)); LASSO (

**c**,

**d**); cross-validation with specie-zone factor (CV + SpZone, (

**e**,

**f**)); and LASSO with specie-zone factor (LASSO + SpZone, (

**g**,

**h**)).

**Figure 4.**Relative root mean square error (RMSE%, (

**a**)) and relative bias (BIAS%, (

**b**)) calculated by DBH classes for selected models using the test dataset.

**Figure 5.**Projection for diameter at breast height (cm) using a period of 6 years (2000 to 2006) (

**a**) and 12 years (2000 to 2012) (

**b**), using the projection database for the CV + SpZone (cross validation with species-zone factor) model. For further details and goodness-of-fit statistics see Table 5.

**Table 1.**Summary statistics of Nothofagus spp. individual and stand attributes for fitting database based on data from the permanent plots (PP) and temporary plots (TP) network.

Tree Species | Nothofagus alpina | Nothofagus obliqua | Nothofagus dombeyi | ||||||||||||

Tree Variables | n | Mean | SD | Min | Max | n | Mean | SD | Min | Max | n | Mean | SD | Min | Max |

DBH | 202 | 16.7 | 9.8 | 5.0 | 42.7 | 627 | 17.0 | 10.0 | 5.0 | 62.1 | 279 | 19.3 | 11.3 | 5.0 | 60.2 |

H | 202 | 15.2 | 6.6 | 4.2 | 34.5 | 627 | 16.1 | 7.3 | 3.5 | 45.0 | 279 | 16.2 | 6.1 | 4.2 | 34.5 |

A | 202 | 36.7 | 14.2 | 11.0 | 104.0 | 627 | 30.6 | 14.6 | 8.0 | 95.0 | 279 | 32.5 | 13.2 | 9.0 | 80.0 |

BAL | 202 | 26.0 | 16.8 | 0.0 | 83.4 | 627 | 19.0 | 14.0 | 0.0 | 65.1 | 279 | 28.0 | 20.6 | 0.0 | 92.4 |

BALn | 202 | 20.9 | 15.0 | 0.0 | 82.3 | 627 | 15.7 | 12.3 | 0.0 | 54.9 | 279 | 20.6 | 16.6 | 0.0 | 82.2 |

SS | 202 | 2.3 | 1.1 | 1.0 | 4.0 | 627 | 2.3 | 1.0 | 1.0 | 4.0 | 279 | 2.2 | 1.0 | 1.0 | 4.0 |

BALr | 202 | 0.6 | 0.3 | 0.0 | 1.0 | 627 | 0.6 | 0.3 | 0.0 | 1.0 | 279 | 0.6 | 0.3 | 0.0 | 1.0 |

AIDBH | 202 | 2.6 | 1.6 | 0.2 | 7.7 | 627 | 3.0 | 2.1 | 0.2 | 12.1 | 279 | 3.6 | 2.2 | 0.1 | 10.2 |

Dominant Specie | Nothofagus alpina | Nothofagus obliqua | Nothofagus dombeyi | ||||||||||||

Stand Variables | n | Mean | SD | Min | Max | n | Mean | SD | Min | Max | n | Mean | SD | Min | Max |

BA | 24 | 46.7 | 13.8 | 10.8 | 72.3 | 87 | 35.7 | 13.0 | 9.5 | 71.5 | 47 | 55.2 | 17.8 | 23.4 | 98.4 |

N | 24 | 2564 | 1184 | 320 | 5000 | 87 | 2378 | 1119 | 320 | 4640 | 47 | 2900 | 1357 | 880 | 5600 |

QD | 24 | 16.4 | 4.8 | 8.5 | 30.5 | 87 | 15.4 | 6.2 | 6.8 | 33.3 | 47 | 16.9 | 5.4 | 8.4 | 30.4 |

Hd | 24 | 21.2 | 5.7 | 10.2 | 35.1 | 87 | 20.8 | 7.0 | 7.8 | 42.4 | 47 | 20.8 | 5.9 | 9.9 | 34.1 |

Ad | 24 | 47.5 | 10.9 | 23.0 | 77.0 | 87 | 36.6 | 14.7 | 12.7 | 86.8 | 47 | 40.2 | 13.1 | 21.3 | 85.1 |

SI | 24 | 10.3 | 3.9 | 4.1 | 24.3 | 87 | 12.5 | 3.9 | 2.0 | 22.9 | 47 | 11.4 | 3.4 | 3.9 | 18.4 |

BAN | 24 | 37.6 | 12.4 | 10.8 | 59.7 | 87 | 30.4 | 11.5 | 8.8 | 57.8 | 47 | 46.5 | 18.7 | 8.0 | 89.6 |

SDI | 24 | 1212 | 336 | 406 | 1944 | 87 | 956 | 258 | 410 | 1674 | 47 | 1400 | 361 | 640 | 2305 |

RS | 24 | 0.1 | 0.0 | 0.1 | 0.2 | 87 | 0.1 | 0.0 | 0.1 | 0.3 | 47 | 0.1 | 0.0 | 0.1 | 0.2 |

^{2}ha

^{−1}); BALn, basal area of larger trees for Nothofagus (m

^{2}ha

^{−1}); SS, sociological status; BALr, relative BAL; AIDBH, annual increment in DBH (mm year

^{−1}); BA, stand basal area (m

^{2}ha

^{−1}); N, number of trees (trees ha

^{−1}); QD, quadratic diameter (cm); Hd, dominant height (m); Ad, dominant breast height stand age (years); SI, site index (m); BAN, basal area of Nothofagus (m

^{2}ha

^{−1}); SDI, stand density index (trees ha

^{−1}); RS, relative spacing.

**Table 2.**Parameter estimates of selected models for estimation of the logarithm of annual increment in diameter at breast height (AIDBH, mm year

^{−1}) for cross-validation (CV) and least absolute shrinkage and selection operator (LASSO) procedures.

Parameter | Estimate | SE | p-Value | VIF* | Parameter | Estimate | VIF* |
---|---|---|---|---|---|---|---|

CV (Equation (8)) | LASSO (Equation (9)) | ||||||

${\mathsf{\beta}}_{0}$ | 2.410 × 10^{0} | 2.173 × 10^{−1} | <0.001 | - | ${\mathsf{\beta}}_{0}$ | −3.098 × 10^{−1} | - |

${\mathsf{\beta}}_{1}$ | −7.062 × 10^{−3} | 2.064 × 10^{−3} | <0.001 | 1.74 | ${\mathsf{\beta}}_{1}$ | −7.255 × 10^{−3} | 3.20 |

${\mathsf{\beta}}_{2}$ | 2.745 × 10^{−4} | 6.787 × 10^{−5} | <0.001 | 1.22 | ${\mathsf{\beta}}_{2}$ | 2.215 × 10^{−4} | 1.40 |

${\mathsf{\beta}}_{3}$ | 9.046 × 10^{−1} | 6.798 × 10^{−2} | <0.001 | 3.29 | ${\mathsf{\beta}}_{3}$ | 7.892 × 10^{−1} | 5.28 |

${\mathsf{\beta}}_{4}$ | −1.138 × 10^{0} | 7.488 × 10^{−2} | <0.001 | 2.26 | ${\mathsf{\beta}}_{4}$ | −1.073 × 10^{0} | 6.73 |

${\mathsf{\beta}}_{5}$ | −1.336 × 10^{−1} | 3.149 × 10^{−2} | <0.001 | 2.22 | ${\mathsf{\beta}}_{5}$ | −1.325 × 10^{−1} | 2.44 |

${\mathsf{\beta}}_{6}$ | - | - | - | - | ${\mathsf{\beta}}_{6}$ | −4.318 × 10^{−2} | 5.36 |

${\mathsf{\beta}}_{7}$ | - | - | - | - | ${\mathsf{\beta}}_{7}$ | 9.792 × 10^{0} | 6.59 |

CV + SpZone (Equation (10)) | LASSO + SpZone (Equation (11)) | ||||||

${\mathsf{\alpha}}_{11}$ | 2.702 × 10^{0} | 2.562 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{0}$ | −1.387 × 10^{0} | - |

${\mathsf{\alpha}}_{12}$ | 2.908 × 10^{0} | 2.345 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{1}$ | 6.790 × 10^{−2} | 2.28 |

${\mathsf{\alpha}}_{14}$ | 3.065 × 10^{0} | 2.397 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{2}$ | −2.507 × 10^{−1} | 2.28 |

${\mathsf{\alpha}}_{21}$ | 2.538 × 10^{0} | 2.120 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{3}$ | −1.702 × 10^{−1} | 2.28 |

${\mathsf{\alpha}}_{22}$ | 2.587 × 10^{0} | 2.219 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{4}$ | −1.955 × 10^{−1} | 2.28 |

${\mathsf{\alpha}}_{23}$ | 2.841 × 10^{0} | 2.272 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{5}$ | 3.455 × 10^{−2} | 2.28 |

${\mathsf{\alpha}}_{24}$ | 2.678 × 10^{0} | 2.329 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{6}$ | 5.922 × 10^{−2} | 2.28 |

${\mathsf{\alpha}}_{31}$ | 2.946 × 10^{0} | 2.298 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{7}$ | −7.117 × 10^{−3} | 3.30 |

${\mathsf{\alpha}}_{32}$ | 2.948 × 10^{0} | 2.680 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{8}$ | 8.700 × 10^{−5} | 2.07 |

${\mathsf{\alpha}}_{33}$ | 2.941 × 10^{0} | 2.353 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{9}$ | 7.699 × 10^{−1} | 6.21 |

${\mathsf{\alpha}}_{34}$ | 2.902 × 10^{0} | 2.358 × 10^{−1} | <0.001 | 2.09 | ${\mathsf{\beta}}_{10}$ | −1.096 × 10^{0} | 7.30 |

${\mathsf{\beta}}_{1}$ | −6.517 × 10^{−3} | 2.002 × 10^{−3} | 0.0012 | 1.82 | ${\mathsf{\beta}}_{11}$ | −1.293 × 10^{−1} | 2.53 |

${\mathsf{\beta}}_{2}$ | 9.307 × 10^{−1} | 6.970 × 10^{−2} | <0.001 | 3.82 | ${\mathsf{\beta}}_{12}$ | −2.250 × 10^{−2} | 5.56 |

${\mathsf{\beta}}_{3}$ | −1.175 × 10^{0} | 7.797 × 10^{−2} | <0.001 | 2.61 | ${\mathsf{\beta}}_{13}$ | 1.419 × 10^{−1} | 6.91 |

${\mathsf{\beta}}_{4}$ | −1.401 × 10^{−1} | 3.092 × 10^{−2} | <0.001 | 2.29 | ${\mathsf{\beta}}_{14}$ | - | - |

**Table 3.**Goodness-of-fit statistics for the natural logarithm of annual increment in diameter breast height (AIDBH, mm year

^{−1}) for the CV regression model using test data.

n | R^{2}emp | RMSE | RMSE% | BIAS | BIAS% | U2 | |
---|---|---|---|---|---|---|---|

Zone | 551 | 0.55 | 1.37 | 44.82 | −0.07 | −2.29 | 0.37 |

Sp | 551 | 0.56 | 1.36 | 44.49 | −0.07 | −2.29 | 0.37 |

Zone + Sp | 551 | 0.55 | 1.38 | 45.14 | −0.07 | −2.29 | 0.37 |

SpZone | 551 | 0.56 | 1.36 | 44.49 | −0.06 | −1.96 | 0.37 |

^{2}emp, empirical coefficient of correlation; RMSE, root mean square error; RMSE%, relative root mean square error; BIAS%, relative bias; U2, Theil’s inequality coefficient.

**Table 4.**Goodness-of-fit statistics for the natural logarithm of annual increment in diameter at breast height (AIDBH, mm year

^{−1}) for the selected models using test data.

Model | n | R^{2}emp | RMSE | RMSE% | BIAS | BIAS% | U2 |
---|---|---|---|---|---|---|---|

CV | 551 | 0.56 | 1.35 | 44.16 | −0.06 | −1.96 | 0.37 |

LASSO | 551 | 0.57 | 1.35 | 44.16 | −0.09 | −2.94 | 0.37 |

CV + SpZone | 551 | 0.56 | 1.36 | 44.49 | −0.07 | −2.29 | 0.37 |

LASSO + SpZone | 551 | 0.54 | 1.36 | 45.13 | −0.13 | −4.31 | 0.38 |

^{2}emp, empirical coefficient of correlation; RMSE, root mean square error; RMSE%, relative root mean square error; BIAS%, relative bias.

**Table 5.**Projection goodness-of-fit statistics for diameter at breast height (cm) using a period of 6 and 12 years based on the validation database. Values in bold correspond to best models.

Projection = 6 Years | ||||||||||

CV | CV + SpZone | |||||||||

n | R^{2}emp | RMSE% | BIAS% | U2 | n | R^{2}emp | RMSE% | BIAS% | U2 | |

Total | 1455 | 0.98 | 7.32 | 0.18 | 0.06 | 1455 | 0.98 | 7.44 | 0.54 | 0.06 |

DBH (5–15) | 777 | 0.88 | 10.06 | −0.42 | 0.10 | 777 | 0.89 | 9.75 | 0.00 | 0.09 |

DBH (15–30) | 510 | 0.88 | 6.94 | 0.94 | 0.07 | 510 | 0.87 | 7.27 | 1.46 | 0.07 |

DBH (>30) | 168 | 0.97 | 4.11 | 0.46 | 0.04 | 168 | 0.97 | 4.19 | 0.32 | 0.04 |

Nothofagus | 943 | 0.99 | 6.33 | −0.26 | 0.06 | 943 | 0.99 | 6.48 | 0.31 | 0.06 |

Companion | 512 | 0.96 | 10.47 | 1.46 | 0.09 | 512 | 0.96 | 10.47 | 1.20 | 0.09 |

LASSO | LASSO + SpZone | |||||||||

n | R^{2}emp | RMSE% | BIAS% | U2 | n | R^{2}emp | RMSE% | BIAS% | U2 | |

Total | 1455 | 0.99 | 7.26 | 0.36 | 0.06 | 1455 | 0.98 | 7.32 | 0.60 | 0.06 |

DBH (5–15) | 777 | 0.88 | 9.96 | −0.52 | 0.10 | 777 | 0.89 | 9.85 | −0.31 | 0.09 |

DBH (15–30) | 510 | 0.88 | 6.94 | 1.13 | 0.07 | 510 | 0.88 | 7.12 | 1.51 | 0.07 |

DBH (>30) | 168 | 0.97 | 3.96 | 0.13 | 0.04 | 168 | 0.97 | 4.02 | −0.03 | 0.04 |

Nothofagus | 943 | 0.99 | 6.28 | −0.05 | 0.06 | 943 | 0.99 | 6.38 | 0.36 | 0.06 |

Companion | 512 | 0.96 | 10.29 | 1.54 | 0.09 | 512 | 0.96 | 10.38 | 1.29 | 0.09 |

Projection = 12 Years | ||||||||||

CV | CV + SpZone | |||||||||

n | R^{2}emp | RMSE% | BIAS% | U2 | n | R^{2}emp | RMSE% | BIAS% | U2 | |

Total | 389 | 0.97 | 9.66 | 1.75 | 0.08 | 389 | 0.97 | 9.82 | 2.34 | 0.09 |

DBH (5–15) | 177 | 0.58 | 17.07 | 5.03 | 0.16 | 177 | 0.59 | 16.87 | 6.02 | 0.16 |

DBH (15–30) | 151 | 0.83 | 8.39 | 2.10 | 0.08 | 151 | 0.81 | 8.86 | 2.70 | 0.09 |

DBH (>30) | 61 | 0.89 | 5.55 | 1.36 | 0.05 | 61 | 0.89 | 5.60 | 1.00 | 0.05 |

Nothofagus | 295 | 0.98 | 7.93 | −0.29 | 0.07 | 295 | 0.97 | 8.17 | 0.53 | 0.07 |

Companion | 94 | 0.83 | 18.45 | 12.49 | 0.17 | 94 | 0.83 | 18.20 | 11.92 | 0.17 |

LASSO | LASSO + SpZone | |||||||||

n | R^{2}emp | RMSE% | BIAS% | U2 | n | R^{2}emp | RMSE% | BIAS% | U2 | |

Total | 389 | 0.97 | 9.50 | 1.96 | 0.08 | 389 | 0.97 | 9.50 | 2.39 | 0.08 |

DBH (5–15) | 177 | 0.58 | 17.07 | 4.74 | 0.16 | 177 | 0.59 | 16.87 | 5.13 | 0.16 |

DBH(15–30) | 151 | 0.83 | 8.30 | 2.42 | 0.08 | 151 | 0.82 | 8.49 | 2.80 | 0.08 |

DBH (>30) | 61 | 0.90 | 5.30 | 0.79 | 0.05 | 61 | 0.90 | 5.27 | 0.37 | 0.05 |

Nothofagus | 295 | 0.98 | 7.74 | 0.05 | 0.07 | 295 | 0.98 | 7.83 | 0.62 | 0.07 |

Companion | 94 | 0.83 | 18.45 | 12.41 | 0.17 | 94 | 0.83 | 18.04 | 11.76 | 0.16 |

^{2}emp, empirical coefficient of correlation; RMSE%, relative root mean square error; BIAS%, relative bias.

**Table 6.**Least square means (standard errors in parentheses) statistics of pairwise Tukey comparisons between all 11 groups in the CV + SpZone model (Equation (9)).

N. alpina | N. obliqua | N. dombeyi | |
---|---|---|---|

Zone 1 | 0.794 (0.125) abcd | 0.652 (0.057) a | 1.020 (0.079) cd |

Zone 2 | 0.996 (0.072) cd | 0.697 (0.053) ab | 1.027 (0.154) abcd |

Zone 3 | - | 0.948 (0.063) bcd | 1.034 (0.090) cd |

Zone 4 | 1.153 (0.086) d | 0.784 (0.069) abc | 0.960 (0.087) abcd |

© 2017 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**

Moreno, P.C.; Palmas, S.; Escobedo, F.J.; Cropper, W.P.; Gezan, S.A.
Individual-Tree Diameter Growth Models for Mixed *Nothofagus* Second Growth Forests in Southern Chile. *Forests* **2017**, *8*, 506.
https://doi.org/10.3390/f8120506

**AMA Style**

Moreno PC, Palmas S, Escobedo FJ, Cropper WP, Gezan SA.
Individual-Tree Diameter Growth Models for Mixed *Nothofagus* Second Growth Forests in Southern Chile. *Forests*. 2017; 8(12):506.
https://doi.org/10.3390/f8120506

**Chicago/Turabian Style**

Moreno, Paulo C., Sebastian Palmas, Francisco J. Escobedo, Wendell P. Cropper, and Salvador A. Gezan.
2017. "Individual-Tree Diameter Growth Models for Mixed *Nothofagus* Second Growth Forests in Southern Chile" *Forests* 8, no. 12: 506.
https://doi.org/10.3390/f8120506