### 1. Introduction

### 2. Interpretation of Black-Box Classifiers

### 2.1 ANN and SVM

##### (2)

$$f(x)=sign\hspace{0.17em}\left(\sum _{s=1}^{sv}{\alpha}_{s}{y}_{s}K({\chi}_{s},\chi )+b\right),$$*lhs*is last hidden layer,

*ω*

*is the weight from last hidden layer to the output layer in equation (1), and where*

^{i}*sv*is the model support vectors,

*α*

*is Lagrange multiplier,*

_{s}*K*(

*χ*

_{s}*; χ*) is a kernel function in equation (2), respectively.

### 2.2 Rule Extraction Using Decision Tree and Fuzzy Logic-Based Classifier Implementation

Step 1: Normalize given learning data.

Step 2: Divide the normalized data into four pieces, and name the divided data as A, B, C, and D, respectively.

Step 3: Implement SVM and ANN using the dataset A.

Step 4: Apply dataset B to implemented SVM and ANN, described as B

_{1}and B_{2}. The results can be considered as representations of each classifier.Step 5: Implement decision tree using B

_{1}and B_{2}.Step 6: Verify performances of decision trees.

Step 7: Derive a series of crisp rules from each decision tree.

Step 8: Convert crisp rules into fuzzy rules for constructing fuzzy logic-based classifiers.

Step 9: Set input and output membership parameters and build fuzzy logic-based classifiers.

_{1}), an average reflectivity data (x

_{2}), a maximum reflectivity data (x

_{3}), an average Doppler velocity (x

_{4}), and a minimum Doppler velocity (x

_{5}). In Section 3, we describe how the input variables are generated. And the decision tree indicates there is only 3 important inputs for separating anomalous propagation echo. Also, we can construct fuzzy rules using the decision tree as follows. The first fuzzy rules are derived from Figure 2, and the second fuzzy rules are derived from Figure 4.

Rule 1: If

**x**_{1}is*small*, then**y**is*NOTAP*.Rule 2: If

**x**_{1}is*large*and**x**_{5}is*small*, then**y**is*NOTAP*.Rule 3: If

**x**_{1}is*large*and**x**_{5}is*large*and**x**_{2}is*small*, then**y**is*AP.*Rule 4: If

**x**_{1}is*large*and**x**_{5}is*large*and**x**_{2}is*large*, then**y**is*NOTAP*.Rule 1: If

**x**_{5}is*large*, then**y**is*NOTAP*.Rule 2: If

**x**_{5}is*small*and**x**_{3}is*small*, then**y**is*NOTAP*.Rule 3: If

**x**_{5}is*small*and**x**_{3}is*large*and**x**_{1}is*small*, then**y**is*NOTAP*.Rule 4: If

**x**_{5}is*small*and**x**_{3}is*large*and**x**_{1}is*large*, then**y**is*AP*.

**x**

_{1}and

**x**

_{5}, and the different input variables are

**x**

_{2}and

**x**

_{3}. Further,

**x**

_{4}seems not significant influence because it is not shown in the trees and rules. Their input and output membership functions are generated as shown in Figures 4 and 5, respectively. The functions are trapezoidal shaped function.

### 3. Anomalous Propagation Echo

_{1}), average reflectivity data (x

_{2}), maximum reflectivity data (x

_{3}), average Doppler velocity (x

_{4}), and minimum Doppler velocity (x

_{5}). The reason why we select the centroid altitude of the cluster is that the anomalous propagation echo appears in low altitude by its own properties.