Canonical Forest

Abstract

We propose a new classification ensemble method named Canonical Forest. The new method uses canonical linear discriminant analysis (CLDA) and bootstrapping to obtain accurate and diverse classifiers that constitute an ensemble. We note CLDA serves as a linear transformation tool rather than a dimension reduction tool. Since CLDA will find the transformed space that separates the classes farther in distribution, classifiers built on this space will be more accurate than those on the original space. To further facilitate the diversity of the classifiers in an ensemble, CLDA is applied only on a partial feature space for each bootstrapped data. To compare the performance of Canonical Forest and other widely used ensemble methods, we tested them on 29 real or artificial data sets. Canonical Forest performed significantly better in accuracy than other ensemble methods in most data sets. According to the investigation on the bias and variance decomposition, the success of Canonical Forest can be attributed to the variance reduction.

Appendices

Appendix 1: Pseudocode of CLDA

Given:

• $X$: the objects in the training data set (an $n \times p$ matrix)

• $C$: number of classes

• $p$: number of variables

• $S_i$: covariance of class $i$

Procedure:

• 1. Compute the class centroid matrix $M_{C \mathbf{x} p}$, where the $(i,j)$ entry is the mean of class $i$ for variable $j$

• 2. Compute the common covariance matrix W:

  $$\begin{aligned} W = \sum _{i=1}^C \left( n_i - 1\right) S_i \end{aligned}$$

• 3. Compute $M^*$ = $MW^{-1/2}$ by using eigen-decomposition of $W$

• 4. Obtain the between covariance matrix $B^*$ by computing the covariance matrix of $M^*$

• 5. Do the eigenvalue-decomposition of $B^*$ such that $B^* = VDV^T$

• 6. The columns $v_i$ of $V$ define the coordinates of the optimal subspaces

• 7. Convert $X$ to the coordinates in the new subspace:

  $$\begin{aligned} Z_i = v_i^TW^{-1/2}X \end{aligned}$$

• 8. $Z_i$ is the $i^{th}$ canonical coordinate

Appendix 2: Pseudocode of Canonical Forest

Input:

• $X$: training data composed of $n$ instances (an $n \times p$ matrix)

• $Y$: the labels of the training data (an $n \times 1$ vector)

• $B$: number of classifiers in an ensemble

• $K$: number of subsets

• $w = (1,\ldots , C)$: set of class labels

Training Phase: For $i = 1,\ldots , B$

• 1. Randomly split $F$ (the feature set) into $K$ subsets: $F_{i,j}$ (for $j = 1,\ldots , K$)

• 2. For $j = 1,\ldots , K$

  • $\diamond$ Let $X_{i,j}$ be the data matrix that corresponds to the features in $F_{i,j}$

  • $\diamond$ Draw a bootstrap sample $X'_{i,j}$ (with sample size 75 % of the number of instances in $X_{i,j}$) from $X_{i,j}$

  • $\diamond$ Apply CLDA on $X'_{i,j}$ to obtain a coefficient matrix $A_{i,j}$

• 3. Arrange $A_{i,j}$ ($j = 1 ,\ldots , K$) into a block diagonal matrix $R_i$

• 4. Construct the rotation matrix $R^a_i$ by rearranging the rows of $R_i$ so that they correspond to the original order of features in $F$

• 5. Use ($XR^a_i, Y$) as the training data to build a classifier $L_i$

Test Phase: For a given instance $x$, the predicted class label from classifier $L$ is:

$$\begin{aligned} L(x) = \mathop {\arg \max }\limits _{y\in w} \sum _{i=1}^B I(L_i(xR^a_i) = y) \end{aligned}$$