Investigating adoption patterns of residential low impact development (LID) using classification trees

Abstract

Local governments are under pressure to improve storm water management and often times must comply with consent decrees with the Federal Government. Decentralizing a portion of the storm water management by integrating private landowners into localized retention and infiltration efforts, that is, low impact development (LID) or green infrastructure projects, is becoming increasingly popular. Some wastewater systems have considered incentivizing private land owners to make improvements aimed at retaining storm water or slowing the conveyance to grey infrastructure. This study examines potential opportunities for incentivizing private residential land owners in Washington DC to install LID projects. This study maps LID configurations to a set of adoption strategies and categories. The C4.5 algorithm is then applied to identify a high performance decision tree for classifying parcels by adoption strategy or adoption categories based on property-level attributes.

Appendix A

Decision trees are methods developed for classifying observations based on both continuous and discrete attributes. Decision trees partition the data into classes based on a series of tests. Each test results in a branching of the data into either a terminal classification node, or additional test sequence. While numerous decision tree algorithms exist, the authors applied the C4.5 algorithm, developed by J.R. Quinlan (1993). The C4.5 algorithm was implemented in R using the Rweka package. Equation (1) through Eq. (4) are adopted from Quinlan (1993).

The C4.5 algorithm creates a decision tree by developing rules based on “information gain”. This information gain is calculated by first calculating the entropy of the test set using the following Eq. (1):

$${\text{Entrop}}{{\text{y}}_S}= - \mathop \sum \limits_{{j=1}}^{n} \frac{{{\text{freq}}({C_j},S)}}{{\left| S \right|}} \times {\log _2}\left( {\frac{{{\text{freq}}({C_j},S)}}{{\left| S \right|}}} \right)$$

(1)

Next the portion of the entropy attributable to each variable is calculated as in Eq. (2):

$$~{\text{Entrop}}{{\text{y}}_X}= - \mathop \sum \limits_{{i=1}}^{n} \frac{{\left| {{T_i}} \right|}}{{\left| T \right|}} \times {\text{Entrop}}{{\text{y}}_T}$$

(2)

The gain from each variable, X, is calculated as the difference between the result for Eq. 1 for the whole training set and Eq. (2) for each variable.

$${\text{Gai}}{{\text{n}}_X}={\text{Entrop}}{{\text{y}}_T} - ~{\text{Entrop}}{{\text{y}}_X}$$

(3)

The “gain ratio” is calculated by dividing the entropy of the training set (Eq. 1), by the gain associated with each variable as in Eq. (4). It is this criterion that the j48 implementation uses to determine the splitting rules.

$${\text{Gain~Rati}}{{\text{o}}_X}=\frac{{{\text{Gai}}{{\text{n}}_X}}}{{~{\text{Entrop}}{{\text{y}}_X}}}$$

(3)

where X is the variable being assessed and T is the multi-class data previously partitioned in Eq. (3). The gain for each variable is measured as the difference between Eq. (1) applied to a test set and the weighted entropy for each dependent variable. For example, a model with five dependent variables would entail calculating five weighted entropies.