# Regularized Regression: A New Tool for Investigating and Predicting Tree Growth

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Tree Growth Data

#### 2.2. Likelihood Model

#### 2.3. Regularized Regression Model

#### 2.4. Comparing Inferential Performance

#### 2.5. Evaluating Predictive Performance

^{2}) of the model when applied to its corresponding test set, which was entirely unused in the fitting of that model (see Section 2.1 Tree growth data for how training and test sets were defined). This metric measures the proportion of variance around the mean value of the dependent variable explained by the model. The maximum possible value for a coefficient of determination is 1 (all variance explained), but negative values can exist when a model is applied to unseen test data if there is more unexplained variation around model predictions than exists around the mean growth value in the test data.

## 3. Results

#### 3.1. Comparing Inferential Performance

#### 3.2. Evaluating Predictive Performance

## 4. Discussion

#### 4.1. Using Regularized Regression for Inference

#### 4.2. Using Neighborhood Models for Prediction

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Which neighbor species are associated with the highest/lowest growth of ABAM focals? For each neighbor species, there are two rows of colored bars. The top row of bars shows the likelihood model results, and the bottom row shows the regularized regression model results. Each row of bars is divided into four sections, to show the results according to the models fitted to each of the four training sets. The color of the bars indicates the growth rate of ABAM in the presence of the neighbor species that row represents (see inset legend). The numbers on the right of the figure indicate the number of neighbors of each species averaged across training sets.

**Figure 2.**Training fit (

**a**) and out-of-sample predictive skill (

**b**) of the likelihood and regularized regression models. Likelihood models using AIC and cross-validation for model selection are shown in green and purple, respectively. Regularized regression models are shown in orange. Points and error bars represent the mean and range of coefficients of determination across the four training sets. Raw values are provided in Table S17.

**Table 1.**Conditions used to draw conclusions regarding inferential questions in the likelihood and regularized regression models.

Conclusion | Condition | |
---|---|---|

Likelihood | Regularized Regression ^{1} | |

Focal growth is influenced by neighboring trees | Best model is: equivalent, conspecific vs. heterospecific, or species-specific | At least one species identity, size, proximity, or density variable retained |

Focal growth is influenced by neighbor species identity | Best model is: conspecific vs. heterospecific or species-specific | At least one species identity or species-specific density variable retained |

Focal growth is higher in the presence of conspecifics | In the best conspecific vs. heterospecific model, λ_{het}–λ_{con} > 0 | Coefficient of: neighbor species = focal species and/or focal species density > 0 ^{2} |

Neighbor species X is associated with: (1) high; (2) medium; (3) low focal growth relative to the average neighbor | In the best species-specific model: (1) λ_{X} > 0.66, (2) 0.33 ≤ λ_{X} ≤ 0.66, (3) λ_{X} < 0.33 | Coefficient of neighbor species = X is: (1) positive, (2) dropped from model, (3) negative ^{3} |

^{1}Conclusions were drawn separately for each of the 100 regularized regression models run for each focal species × training set combination.

^{2}If coefficients of neighbor species = focal species and focal species density had opposite signs, we concluded that focal growth was unaffected by whether neighbors were conspecific or heterospecific in that particular model run.

^{3}To enable comparison with λ

_{X}in the likelihood model, we calculated the number of the 100 regularized regression models where the coefficient of neighbor species = X was positive minus the number where this coefficient was negative to obtain a number between −100 and +100, then rescaled these values to a range of 0–1.

**Table 2.**Is focal growth higher in the presence of conspecific or heterospecific neighbors? Higher growth in the presence of conspecifics represents positive feedback on growth and is indicated with ‘+’. Negative feedback and the absence of feedback are indicated with ‘−’ and ‘0’, respectively. NA values for regularized regression indicate that the model did not find growth to be substantially higher in the presence of conspecifics or heterospecifics. See Table S16 for numerical outputs.

Focal Species | Likelihood | Regularized Regression | ||||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | |

ABAM | + | + | + | + | + | + | + | + |

CANO | − | − | − | − | − | − | − | − |

PSME | + | − | − | − | NA | NA | NA | NA |

THPL | − | − | 0 | − | NA | − | NA | NA |

TSHE | − | − | − | − | − | − | − | − |

TSME | − | − | − | − | − | − | − | − |

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**MDPI and ACS Style**

Graham, S.I.; Rokem, A.; Fortunel, C.; Kraft, N.J.B.; Hille Ris Lambers, J.
Regularized Regression: A New Tool for Investigating and Predicting Tree Growth. *Forests* **2021**, *12*, 1283.
https://doi.org/10.3390/f12091283

**AMA Style**

Graham SI, Rokem A, Fortunel C, Kraft NJB, Hille Ris Lambers J.
Regularized Regression: A New Tool for Investigating and Predicting Tree Growth. *Forests*. 2021; 12(9):1283.
https://doi.org/10.3390/f12091283

**Chicago/Turabian Style**

Graham, Stuart I., Ariel Rokem, Claire Fortunel, Nathan J. B. Kraft, and Janneke Hille Ris Lambers.
2021. "Regularized Regression: A New Tool for Investigating and Predicting Tree Growth" *Forests* 12, no. 9: 1283.
https://doi.org/10.3390/f12091283