Algorithm 1 BRF learning

1. T: the maximum number of decision trees to grow for BRF

D: the maximum depth of trees to extend

M: number of classes

Sn: Training set, including positive (river and lake) and negative (land, mountain and building) samples with their labels and weight, {x1, y1, w1},…,{xN, yN, wN}; xi ∈ X, y ∈ M

Initialize sample weight $w_i^{\mathrm{(1)}}$ = 1/N

2. For t = 1 to T do

Select subset s from training set Sn

Grow an unpruned tree using the s subset samples with their corresponding weights.

For d = 1 to D do

Each internal node randomly selects p variables and determines the best split function using only these variables.

Loop: Using different p-th variables, the split function f(vp) iteratively splits the training data into left (Il) and right (Ir) subsets using Equation (6).

$$\begin{array}{l}Il=\{p\in In|f(vp)<t\},\\ Ir=In\backslash Il\end{array}$$ (6)

The threshold t is randomly chosen by the split function f(vp) in the range $t\in (\underset{p}{\min }f(v_p),\underset{p}{\max }f(v_p))$.

Compute information gain ΔG function f(vp)

If (ΔG= max) then Determine the best split function f(vp) for the node d

Else goto Loop.

End For

Store the probability distribution P (C | lt) to leaf node

Output: A weak decision tree

Estimate class label ${\widehat{y}}_i$ of the training data with the trained decision trees: $${\widehat{y}}_i=\arg \text{}\underset{c}{\max }P(c|l_t)$$ (7)

Calculate the error of decision tree εt: $${\mathrm{\epsilon }}_t={\displaystyle {\sum }_{i:y_i\ne {\widehat{y}}_i}^N{w}_i^{(t)}/{\displaystyle {\sum }_i^N{w}_i^{(t)}}}$$ (8)

Compute weight of the t-th decision tree αt: $${\mathrm{\alpha }}_t=\frac{1}{2}\log \frac{(M-1)(1-{\mathrm{\epsilon }}_t)}{{\mathrm{\epsilon }}_t}$$ (9)

If α > 0, then

Update weight of training sample $w_i^{(t+1)}$: $$w_i^{(t+1)}=\{\begin{array}{c}{w}_i^{(t)}\exp ({\mathrm{\alpha }}_t)if{y}_i\ne {\widehat{y}}_i\\ w_i^{(t)}\exp (-{\mathrm{\alpha }}_t)otherwise\end{array}$$ (10)

else

Reject the decision tree

End For

3. Final output: A BRF consists of N decision trees (N ≤ T)