Applications of machine learning algorithms in forest growth and yield prediction
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摘要: 森林生长收获预估是森林经理学的一个重要方向，采用模型技术进行森林生长收获估计是森林经营决策的重要前提。传统的统计模型如线性及非线性回归模型、混合效应模型、分位数回归、度量误差模型等统计方法已被广泛应用于研究林木生长，但这些统计方法在应用时常常需满足一定的统计假设前提，诸如数据独立、正态分布和等方差等。由于森林生长数据的连续观测和层次性，上述假设通常难以满足。近年来随着人工智能技术的发展，机器学习算法为森林生长收获预估提供了一种新的手段，它具有对输入数据的分布形式没有假设前提、能够揭示数据中的隐含结构、预测结果好等优点，但在森林生长收获预估中的应用仍十分有限。文章对分类和回归树、多元自适应样条、bagging回归、增强回归树、随机森林、人工神经网络、支持向量机、K最近邻等方法在森林生长收获预估中的应用、软件及调参等进行了综述，讨论了机器学习方法的优势和挑战，认为机器学习方法在森林生长收获预估方面有很大的潜力，必将得到广泛应用，并和传统统计模型相结合成为生长收获模型发展的一种趋势。Abstract: Forest growth and yield prediction is an important field of forest management science, and modelling forest growth and yield is key to forest management decision-making. The traditional statistical growth models such as linear and nonlinear regression model, mixed-effect model, quantile regression, variable-in-error model are often applied under certain statistical assumptions, such as the data are independent, normally distributed and homoscedastic. The above requirements are usually difficult to be met for forest data with repeated observation and hierarchy. With the development of AI techniques, machine learning provides a new way for forest growth modeling, with the advantages of no requirements on data distribution, extracting deep knowledge from the data, and high accuracy. The applications in forest growth and yield are still less than other domains. We reviewed the main machine learning algorithms including classification and regression tree (CART), multivariate adaptive regression splines (MARS), bagging regression, boosted regression tree (BRT), random forest (RF), artificial neural networks (ANN), k-nearest neighbors (k-NN), and support vector machine (SVM), parameter tuning, software, advantages and challenge. We conclude that machine learning would be widely applied with great potential and its combination with traditional statistical methods would become a trend in forest growth and yield prediction.
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表 1 R软件中机器学习包及调参方法
Table 1. R packages and parameter tuning of machine learning algorithms
机器学习算法
ML algorithmR 程序包
R package可调超参数
Adjustable hyper-parameter调参方法
Method of parameter tuning默认值（回归）
Default value (regression)CART rpart 复杂度参数
Complexity parameter (cp)cp取介于0 ~ 1的实数。可基于交叉验证（如10折交叉验证），建立调参网格对cp进行调优（如cp可设置为0.000 1，0.001，0.01，0.1等不同值）
The value of cp is between 0 and 1. The optimal one could be obtained by grid search (with different cp values of 0.000 1, 0.001, 0.01, 0.1 for example）cp = 0.01 MARS earth ①自变量间交互的阶数
The degree of interaction of input variables (degree)
②模型中最大的项数
Maximum number of terms (including intercept) in the model (nprune)degree为大于等于1的整数，Hastie等[93]建议为交互项degree设置一个上限（如degree ≤ 3）。2 ≤ nprune ≤ nk，nk的计算公式为：nk = min(200, max(20, 2 × ncol(x)))，式中，min()与max()分别表示取最小值和取最大值，ncol(x)表示自变量x的总数。可基于交叉验证，建立调参网格对degree和prune进行调优
The value of degree is the integer ≥ 1. Hastie et al.[93] suggested degree should be set an upper limit (degree ≤ 3, for example). 2 ≤ nprune ≤ nk，nk = min(200, max(20, 2 × ncol(x)))，where ncol(x)is the number of input variables. The values of degree and prune could be attained by grid search based on cross validationdegree = 1
nprune无默认值
No default provided for npruneBagging
回归树
Bagging regression treeipred 决策树的数量
The number of decision trees (nbagg)nbagg取值为大于等于1的整数，该参数取值仍需依具体数据而定，为保证预估结果的可靠性且不影响计算效率，可将nbagg设置为大于25的值（如50等）
The value of nbagg is the integer ≥ 1. It could be set as the integer ≥25 (50 for example) for prediction reliability and computation efficiencynbagg = 25 RF randomForest ①随机森林中决策树的数目
The number of decision trees (ntree)
②树节点随机抽选的变量个数
The number of input variables randomly sampled as candidates at each node (mtry)ntree和mtry取值均为大于1的整数，当ntree在500以后整体误差便趋于稳定，但仍需依据具体数据而定，为保证预估结果的可靠性且不会影响计算效率，ntree可以取大于500的值（如1 000等）；1 ≤ mtry ≤ P, P为全部自变量数目，可基于交叉验证，建立调参网格对mtry进行调优
The values of ntree and mtry are the integers ≥ 1. The bias could be stable when ntree is larger than 500. It could be set as the integer ≥ 500 (1 000 for example) for prediction reliability and computation efficiency. mtry is less than the number of all input variables P, and the optimal value could be attained by grid search based on cross validationntree = 500；mtry为全部自变量数目的三分之一（取整）
The value of mtry is the one-third of all input variables
(integer)BRT gbm ①损失函数的形式
The name of the distribution (distribution)
②决策树的数目（或称迭代次数）
Integer specifying the total number of trees to fit (n.trees)
③学习速率（或称收缩参数）
The learning rate or step-size reduction (shrinkage)
④再抽样比率
The fraction of the training set observations randomly selected to propose the next tree in the expansion (bag.fraction)
⑤变量交互的深度
The maximum depth of variable interactions (interaction.depth)对于回归问题distribution设置为gaussian；bag.fraction (0 < bag.fraction ≤ 1)，Friedman[96]推荐将bag.fraction设置在0.5左右；shrinkage (0 < shrinkage ≤ 1)，n.trees为大于1的整数，shrinkage 影响超参数n.trees的取值，Ridgeway[95]建议将shrinkage 的取值范围设置在0.01至0.001之间，同时n.trees取值介于3 000至10 000之间；interaction.depth取值为大于等于1的整数，为了平衡计算开销和模型性能，该超参数可尝试若干值进行调优（如1，3，5，7，9）。可基于交叉验证，建立调参网格对上述超参数进行调优
gaussian is assumed for regression; bag.fraction is suggested to be set as 0.5 by Friedman[96], shrinkage is suggested to be between 0.01 and 0.001 and n.trees between 3 000 and 10 000 by Ridgeway[95]. Shrinkage affects the value of n.trees. The value of interaction.depth is a integer ≥ 1, and several specific values could be tested (1, 3, 5, 7, 9 for example) for the balance of model performance and computation efficiency. All optimal values of these hyper-parameters could be attained by grid search based on cross validationDistribution = gaussian；
n.trees = 100；
shrinkage = 0.1；
bag.fraction = 0.5；
interaction.dept = 1SVM kernlab ①核函数
Kernel function (kernel)
②代价参数
Cost of constraints violation (C)对于回归问题，线性核函数以及径向基核函数是两类常用的kernel；C为大于0的实数（如0.01，0.1，1，10，100，…）。可基于交叉验证，建立调参网格对上述超参数进行调优
Linear or radical basis functions are commonly used for regressions. The optimal values of C (0.01, 0.1, 1, 10, 100, … for example) could be attained by grid search based on cross validationkernel为径向
基核函数；
Radical basis function; C = 1ANN nnet ①隐藏节点个数
The number of hidden node (size)
②权重衰减
Weight decay (decay)size为大于等于0的整数，一般使用的size确定方法为，$\scriptstyle{\rm{size}} = \sqrt {P + O} + m$，P表示输入层自变量的个数，O表示输出层因变量的个数，m的取值为0 ~ 10之间的整数；decay为0至0.1的实数。可基于交叉验证（如10折交叉验证），建立调参网格对上述超参数进行调优$\scriptstyle{\rm{size}} = \sqrt {P + O} + m$
Size is the integar ≥ 0 and determined as $\scriptstyle{\rm{size}} =\sqrt {P + O} + m $, where P is the number of input variables, O is the number of the output variables, m is a integer between 0 and 10. Decay is a real between 0 and 0.1. The optimal values of these parameters could be attained by grid search based on cross validationdecay = 0
size无默认值
no default provided
for sizek-NN caret 近邻点的个数
The number of nearest
neighbors (k)k为大于等于1的整数，通常k的取值在3 ~ 10范围内[94]。可基于交叉验证, 建立调参网格对k进行调优
k is the integer ≥ 1 and often set as between 3 and 10 according to Lantz[94]. The optimal value could be attained by grid search based on cross validationk = 5 
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表 2 主要机器学习算法优缺点
Table 2. Advantages and disadvantages of main machine learning algorithms
机器学习算法
ML algorithm优点
Advantage缺点
DisadvantageCART 与分布无关；受共线性和异常值影响小；简单，容易解释，能自动处理交互作用；能处理连续和分类变量
Distribution independent; less affected by collinearity and outliers; easy explained; deal with interactions, continuous and category variables单棵树的结构不稳定，容易出现过拟合，大的决策树不易解释
Unstable structure for single tree; over-fitting; difficulty in explaining big decision treesbagging回归
Bagging regression与分布无关；调节参数少；受共线性和异常值影响小；较CART泛化能力强；能处理连续和分类变量
Distribution independent; less affected by collinearity and outliers; less tuning parameters; better generalization capacity than CART; deal with interactions, continuous and category variables不如单棵决策树容易解释，对过多变量会敏感
Not so easy to be explained as single decision tree; sensitive to too many input variablesRF 与分布无关；调节参数少；受共线性和异常值影响小；能产生变量重要性和偏依赖图；与bagging相比，训练出的模型的方差更小，泛化能力强；能处理连续和分类变量
Distribution independent; less affected by collinearity and outliers; less tuning parameters; better generalization capacity than CART; deal with interactions, continuous and category variables不如单棵决策树容易解释，对过多变量会敏感
Not so easy to be explained as single decision tree; sensitive to too many input variablesBRT 与分布无关；受共线性和异常值影响小，可处理复杂非线性关系；能产生变量偏依赖性和相对重要性图；能处理连续和分类变量
Distribution independent; less affected by collinearity and outliers; deal with complex nonlinear relations, and continuous and category variables; generate variable importance and partial dependence plots超参数过多，调参复杂
Many hyperparameters, complex parameter tuningSVM 与分布无关；可以解决高维问题，能处理非线性特征的相互作用，泛化能力强；能处理连续和分类变量
Distribution independent; workable for high-dimensional variables; deal with continuous and category variables; good generalization capacity黑箱；样本量多时，效率不高；非线性问题较难找到核函数；有太多的定性变量或定性变量水平太多时很难实现，难以解释；易受共线性影响
Black box; low efficiency for large sample size; difficulty in searching kernel function for nonlinear issues and implementing owing to too many category variables or levels; easily affected by collinearityANN 与分布无关；可处理非线性数据；能处理连续和分类变量
Distribution independent; workable for nonlinear relations, continuous and category variables黑箱；存在过拟合风险，自变量过多时预测结果不好；易受共线性影响
Black box; overfitting risk; large prediction errors for many input variables; easily affected by collinearityKNN 与分布无关；建模简单；可处理复杂非线性关系；能处理连续和分类变量
Distribution independent; easy; workable for nonlinear relations, continuous and category variables黑箱；K值需要调试，样本不平衡时，预测误差较大；易受共线性影响
Black box; testing for k values; large prediction errors for imbalanced samples; affected by collinearityMARS 与分布无关；调节参数少；受共线性和异常值影响小；可处理非线性和变量交互问题；计算效率高；能处理连续和分类变量
Distribution independent; less tuning parameters; unaffected by collinearity and outliers; workable for nonlinear relations and interactions; high computation efficiency易受局部数据特征的影响
Easily being affected by local data features
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