Understanding the Characteristics of Non-industrial Private Forest Landowners Who Harvest Trees

Abstract

Achieving regional and national goals of renewable energy production will depend on sufficient supply of biomass from private forests, the majority of which are controlled by non-industrial private forest landowners (NIPF). Considering the diversity in management objectives and changing demographic dynamics in this ownership group, it is important to understand the characteristics of landowners that may supply woody biomass. This study developed linear discriminant analysis (LDA) and classification tree (CT) models to examine the characteristics and motivation of such NIPF landowners. Thirteen combinations of CT variable selection and split-point selection models were used in conjunction with LDA. The “importance of income” from a woody biomass harvest was the most important factor influencing NIPF landowners’ decisions in supplying woody biomass. Another significant interrelated variable was “farmer or non-farmer” forestland ownership, which was also related with “years of residency”, “availability of a multi-management plan,” and “ownership of multi-tracts of land.” CT models provided higher-level explanatory information when compared with LDA models. Study findings provide useful insight for land managers, wood procurement managers, and policy-makers in identifying the landowner groups with interest in biomass supply, and in understanding the factors influencing their decisions.

Appendix A: Linear Discriminant Analysis (LDA)

LDA attempts to find linear combinations of predictor (independent) variables that best separate the groups of observations. These combinations are called discriminant functions (Mika et al. 1999). There are K functions of the form:

$$d_{k} \left( x \right) \, = \, x^{T} \sum ^{ - 1} \mu_{k} {-} \, 1/2\mu_{k}^{T} \sum ^{ - 1} \mu_{k} + \, \log \pi_{k}$$

(1)

for the K groups, each assumed to have a multivariate normal distribution with mean vectors μ _k (k = 1,…, K) and common covariance matrix ∑, ∑⁻¹ is inverse of covariance matrix, where π_k = n _k /n is the proportion of group k, such that π ₁  + π ₂  + ··· + π _k  = 1. LDA classifies observations x _i to the group k, which minimize the within-group variance,

$${\text{k }} = {\text{ argmin}}_{\text{k}} ({\text{x}}_{\text{i}} -\upmu_{\text{k}} )^{\text{T}} \sum^{ - 1} ({\text{x}}_{\text{i}} -\upmu_{\text{k}} )$$

(2)

where argmin stands for “argument of the minimum”, i.e. the set of points of the given argument for which the value of the given expression attains its minimum. LDA has two potential limitations: (a) there must be a statistical significant difference in the mean vectors between groups, and (b) the number of observations in each group must be greater than the number of predictors or independent variables (Eisenbeis 1977). Linear discriminant functions were estimated using SPSS v16.0 (SPSS Inc. 2007).

Two methods were used for validation of the LDA model results. Re-substitution was used which applies the discriminant model to the original training data set to observe the frequency of correctly classified observations. The other method used was q-fold cross-validation (Geisser 1974). For q-fold cross-validation, the original sample is partitioned into q subsamples. A single subsample is retained for validation of the model built from the other q-1 subsamples. The process is repeated q times, with each of the q subsamples used exactly once for validation. The results of the q time iterations are combined to produce a single classification rate estimate.

Classification Trees (CT)

CT methods are elements of decision tree (DT) methodology (Quinlan and Rivest 1989). A DT is a decision tool that uses a graph of a model of decisions and possible outcomes. CTs are used for modeling and predicting categorical response variables from a set of continuous predictors, categorical predictors or both. CTs are promising classification methods because of their simple interpretation, high classification accuracy, and ability to characterize complex interactions among variables (Brieman et al. 1984).

In general, CTs build decision rules by recursive binary or multi-way partitioning of the data space into subspaces that are increasingly homogeneous with respect to the class variable. The homogeneous regions are called nodes. At each step in fitting a CT, an optimization is carried out to select a node, a predictor variable, and a split-point for numeric variables or categorical variables that result in the most homogeneous subgroups for the data (Brieman et al. 1984).

In CTs, selection bias is one of the most important issues. Two sources of bias are: (a) when variables differ significantly in their numbers of splits, and (b) when variables differ in their proportions of missing values. CRUISE (Classification Rule with Unbiased Interaction Selection and Estimation) was used in this analysis (Kim and Loh 2001; Loh 2002; Kim and Loh 2003). CRUISE methods minimize selection bias. CRUISE uses two techniques to improve the interpretability of its trees (Kim et al. 2011). First, it splits each node into multiple subnodes, with one for each class; this reduces CT depth. Second, it selects variables based on one-factor and two-factor effects, therefore immediately identifying variables with significant interactions. V-fold cross-validation was used to validate the CT models. In v-fold cross-validation, a tree of the specified size is computed v times, each time leaving out one of the sub-samples from the computations, and using that sub-sample as a test sample for v-fold cross-validation. Each sub-sample is used (v − 1) times in the training sample and just once as the validation test sample (Brieman et al. 1984).