# Concept Drift Detection Based on Deep Neural Networks and Autoencoders
## 4 Experiment
### Evaluation Metrics  
For concept drift detection, commonly used evaluation metrics are employed to assess the performance of drift detection algorithms. Specifically, we utilize the first two metrics from [34; 39] to evaluate the algorithm's effectiveness in terms of detection delay and accuracy. In the case of a single drift, the First Effective Drift Point (\(FEDP\)) metric is used to quantify the effectiveness of the drift detection algorithm. The calculation is as follows:  
\[FEDP=\sum_{i=1}^{n}(DP-pos)/n \tag{4}\]  
\(DP\) represents the first detected drift point that is greater than or equal to the actual drift onset, while \(pos\) denotes the true drift start point. This metric primarily evaluates the algorithm's ability to promptly detect drift and measure the corresponding detection delay. If no drift point is recorded in the drift list during detection, \(DP\) is set to 10,000 or 100,000. For instance, if the dataset in use is the abrupt SEA, \(pos\) is 5000, and the first detected drift point exceeding this threshold is 5012, then \(FEDP\) is calculated as \(\text{FEDP}=5012-5000=12\). This represents a single \(FEDP\) result. To obtain the final evaluation, the average \(FEDP\) value is computed over \(n\) experiments. For multiple drifts, the Effective Drift Detection Rate (\(EDDR\)) metric [34; 39] is used to measure the number of effective detections made by the algorithm. Initially, the Effective Drift Range (\(EDR\)) is defined to limit the range for effective detection. For synthetic data with gradual and incremental drifts, parameters are set at the data generation stage to determine their values. Thus, the \(EDR\) for these datasets is tied to the drift formation width, as outlined in Table 1. However, for abrupt synthetic datasets, where the drift width is 1, we extend the effective drift span by 100 to maintain consistency with other datasets. This results in an \(EDR\) of (5000, 5100). For real-world datasets, whether involving single or multiple drifts, the drift's effective range is set to a span of 100. In cases of multiple drifts, the effective range encompasses multiple points.The Effective Detected Drift Rate (\(EDDR\)) is used as an evaluation metric for multiple drift points. Assume DP = {DP\({}_{i}\), DP\({}_{i+1}\),..., DP\({}_{i+t}\)}, where \(DP\) represents all drift points detected within the effective range:  
\[valid\ detected=\left\{ \begin{array}{ll}1,&\text{if }\exists DP\&\forall DP \in EDR\\ |DP|/|N|,&\text{if }\exists DP\&\exists DP\in EDR\\ 0,&\text{others}\end{array} \right. \tag{5}\]\[EDDR=\frac{1}{n}\sum_{i=1}^{n}\mathit{valid\ detected} \tag{6}\]  
Let \(N\) represent the number of effective ranges. A valid detection occurs only when the detected drift points are non-empty and all detected points lie within the effective range. When both conditions are satisfied, it is counted as \(1\). If only one or a few of the effective ranges contain valid detected drift points, the value is recorded as \(|DP|/|N|\), which represents the number of detected drift points within the effective range (with each range counted only once) divided by the total number of effective ranges. When no drift points in the \(DP\) are within the effective range, the value is recorded as \(0\). This metric evaluates whether the detection algorithm can identify all drift points when multiple drifts are present, thereby assessing the sensitivity of the detection algorithm. To measure the significant differences between algorithms, the statistical tests proposed in [40; 41]--the Friedman test and its subsequent Bonferroni-Dunn test--are employed to compare the overall performance of the algorithms. First, the Friedman test is applied to verify whether all algorithms perform similarly.