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🔥 Content Introduction
1. Research Background and Significance
In multivariate time series scenarios such as energy load forecasting, economic indicator forecasting, and environmental parameter forecasting, traditional regression prediction models (e.g., ARIMA, single neural networks) often only output “point prediction” results, failing to quantify prediction uncertainty. In practical applications, “interval probability forecasting” (i.e., providing the range of future target values at different confidence levels and the corresponding probabilities) is crucial for decision-making: for example, in power dispatch, it is necessary to know the load fluctuation range for the next hour at a 95% confidence level to avoid supply-demand imbalance; in economic decision-making, risk response strategies need to be formulated based on the probability range of GDP growth rate.
Current interval probability forecasting methods face two major bottlenecks: first, insufficient feature extraction, where the long-term and short-term dependencies and key feature weight allocation issues in multivariate data remain unresolved, leading to larger point prediction errors and subsequently affecting interval accuracy; second, weak adaptability in probability estimation, where traditional kernel density estimation (KDE) requires manual setting of kernel functions and bandwidth, which can easily result in “over-smoothing” or “underfitting” in dynamically changing data distribution scenarios, failing to accurately characterize the probability distribution of prediction errors.
To address this, this paper proposes the BiLSTM-Multihead-Att-ABKDE fusion model: by using the “Bidirectional Long Short-Term Memory Network (BiLSTM)” to capture the bidirectional long-term and short-term dependencies of multivariate time series data, the “Multihead Attention Mechanism (Multihead-Att)” to enhance the weight allocation of key features, and combining the “Improved Adaptive Kernel Density Estimation (ABKDE)” to dynamically optimize kernel function parameters, achieving the dual objectives of “accurate point prediction + reliable interval probability prediction,” providing a solution for multivariate regression prediction that combines accuracy with uncertainty quantification capabilities.
2. Overall Model Architecture and Core Components
2.1 Overall Model Framework
The BiLSTM-Multihead-Att-ABKDE model consists of three layers: “Multivariate Feature Extraction Layer,” “Point Prediction Layer,” and “Error Probability Modeling Layer,” with the process as follows (Figure 1 is a schematic diagram of the framework):
- Multivariate Feature Extraction Layer: Input multivariate time series data, learn bidirectional temporal dependencies through BiLSTM, then focus on key features through Multihead-Att, outputting high-dimensional fused features;
- Point Prediction Layer: Based on the fused features, output the point prediction value of the target variable through a fully connected network, while calculating the point prediction error (the difference between the true value and the predicted value);
- Error Probability Modeling Layer: Using the improved ABKDE to perform probability density estimation on historical point prediction errors, dynamically generating interval prediction results at different confidence levels (e.g., 90%, 95%, 99%) and outputting the corresponding probability distribution for the intervals.
Figure 1 Schematic Diagram of the BiLSTM-Multihead-Att-ABKDE Model Framework
(Note: This is a textual description; the actual diagram should include the process flow from input layer → BiLSTM → Multihead-Att → fully connected layer (point prediction) → ABKDE (error probability estimation) → output layer (point prediction + interval probability prediction), marking the dimensional changes of each layer’s data.)
⛳️ Running Results
📣 Sample Code
🔗 References
🎈 Some theoretical references are from online literature; please contact the author for removal if there is any infringement.
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