Time series prediction of environmental electric field intensity with generalized correlation entropy loss function-based Transformer model

doi:10.11772/j.issn.1001-9081.2025050560

Journal of Computer Applications ›› 2026, Vol. 46 ›› Issue (6): 1872-1880.DOI: 10.11772/j.issn.1001-9081.2025050560

• Data science and technology • Previous Articles

Time series prediction of environmental electric field intensity with generalized correlation entropy loss function-based Transformer model

Wenjun FENG, Xinwei SONG(), Yuntao YUE

School of Intelligent Science and Technology，Beijing University of Civil Engineering and Architecture，Beijing 100044，China

Received:2025-05-21 Revised:2025-08-03 Accepted:2025-08-18 Online:2025-08-22 Published:2026-06-10
Contact: Xinwei SONG
About author:FENG Wenjun， born in 2001， M. S. candidate. His research interests include electromagnetic environment， machine learning.
YUE Yuntao， born in 1971， Ph. D.， associate professor. His research interests include converter pulse width modulation， motor drive and microprocessor applications.
First author contact:SONG Xinwei， born in 1990， Ph. D.， associate professor. Her research interests include electromagnetic environment， electromagnetic compatibility.
Supported by:
Scientific Research Program of Beijing Municipal Education Commission(KM202210016005)

基于广义相关熵损失函数Transformer模型的环境电场强度时序预测

丰文君, 宋欣蔚(), 岳云涛

北京建筑大学智能科学与技术学院，北京 100044

通讯作者: 宋欣蔚
作者简介:丰文君（2001—），男，安徽池州人，硕士研究生，主要研究方向：电磁环境、机器学习
岳云涛（1971—），男，黑龙江宝清人，副教授，博士，主要研究方向：变换器脉宽调制、电机驱动及微处理器应用。
第一联系人：宋欣蔚（1990—），女，河北衡水人，副教授，博士，主要研究方向：电磁环境、电磁兼容
基金资助:
北京市教育委员会科学研究计划项目(KM202210016005)

Abstract

Abstract:

Predicting the time series of electromagnetic radiation in the environment is of great significance for public health protection and the adaptability of electronic devices to the electromagnetic environment. Aiming at the high volatility of the environmental electric field intensity time series， which leads to more outliers and interferes with model training， a Generalized Correlation entropy Loss function-based Transformer （GCL-Transformer） model was proposed. By applying nonlinear weighting to errors through kernel mapping， this model combined the gradient smoothing of Mean Square Error （MSE） with the outlier robustness of Mean Absolute Error （MAE）， effectively weakening the interference of outliers on the model training. Data were collected at three typical electromagnetic exposure monitoring sites in Beijing， and validation was carried out by multiple sets of cross-time scale prediction experiments. Comparisons were made with the traditional Transformer model， the variant model TOEformer （Temporal-Optimized Enhanced Transformer）， and the Long Short-Term Memory （LSTM） model. Experimental results indicate that GCL-Transformer model significantly outperforms the comparison models in terms of prediction accuracy. In short-term tasks with a prediction interval of one hour， the Root Mean Square Error （RMSE） of GCL-Transformer reaches 0.090 6 V/m， which is 30.6% lower than that of the traditional Transformer model （0.130 7 V/m）. Moreover， as the prediction interval extends to 72 hours， its error growth rate is the slowest （RMSE increases only from 0.090 6 V/m to 0.123 4 V/m）， demonstrating excellent long-term prediction stability.

Key words: electric field intensity, time series prediction, Transformer, generalized correlation entropy, loss function

摘要：

预测环境中电磁辐射的时间序列对公众健康防护、电子设备的电磁环境适应性具有重要意义。针对环境电场强度时间序列的高波动性导致离群点较多、干扰模型训练的问题，提出基于广义相关熵损失函数的Transformer （GCL-Transformer）模型。该模型通过核映射对误差进行非线性加权，兼具均方误差（MSE）的梯度平滑性和平均绝对误差（MAE）的异常值鲁棒性，可以有效削弱离群点对模型训练的干扰。在北京市3个典型的电磁暴露监测点采集数据，通过多组跨时间尺度预测实验进行验证，并与传统Transformer模型、变体TOEformer（Temporal-Optimized Enhanced Transformer）模型和长短时记忆（LSTM）模型进行对比。实验结果表明，GCL-Transformer模型在预测精度上显著优于对比模型。在预测间隔为1 h的短期任务中，GCL-Transformer的均方根误差（RMSE）为0.090 6 V/m，相较于传统Transformer模型（0.130 7 V/m）降低30.6%；且随着预测间隔延长至72 h，它的误差增长最慢（RMSE仅从0.090 6 V/m增至0.123 4 V/m），展现了优异的长期预测稳定性。

关键词: 电场强度, 时间序列预测, Transformer, 广义相关熵, 损失函数

CLC Number:

TP391.9

Wenjun FENG, Xinwei SONG, Yuntao YUE. Time series prediction of environmental electric field intensity with generalized correlation entropy loss function-based Transformer model[J]. Journal of Computer Applications, 2026, 46(6): 1872-1880.

丰文君, 宋欣蔚, 岳云涛. 基于广义相关熵损失函数Transformer模型的环境电场强度时序预测[J]. 《计算机应用》唯一官方网站, 2026, 46(6): 1872-1880.

Figures/Tables 17

Fig. 1 Model structure

Fig. 2 Input sequence feature transformation

Fig. 3 Training process of model

Fig. 4 Sensitivity analysis of Gaussian kernel function on prediction error

Fig.5 Joint influence surface of generalized correlation parameter （σ,α） on error sensitivity

Fig.6 Schematic diagram of electric field intensity data acquisition device

Tab. 1 Electromagnetic field time series statistics for location dataset

样本	样本总数	平均值	标准差	最小值	最大值
样本一	43 920	1.50	0.28	0.71	2.22
样本二	43 920	2.28	0.66	1.20	4.28
样本三	43 920	0.76	0.10	0.44	1.10

Fig. 7 Time series of electric field intensity

Fig. 8 Multi-step prediction strategy of model

Tab. 2 Settings of input and output steps of electric field intensity time series prediction model

组号	I_step	P_step
1	240（1d）	10（1h）
2	240（1d）	30（3h）
3	1 680（7d）	240（1d）
4	1 680（7d）	720（3d）

Tab. 3 Model parameter setting

模型	类型	数量	参数
LSTM	LSTM层	1	神经元数量=50，激活函数=‘tanh’，内核初始化=HeNormal（）
LSTM	全连接层	1	—
Transformer/GCL-Transformer	嵌入层	1	神经元数量=16，激活函数=‘ReLU’，权重正则化=L2正则化
	归一化层	1	—
	多头注意力层	2	注意力头数=2，键向量维度=32
	Dropout层	2	速率=0.1
	前馈网络	2	神经元数量=16，激活函数=‘ReLU’
	编码器层	1	—
	解码器层	1	—
TOEformer	嵌入层	1	神经元数=32，无激活函数（线性变换）
	归一化层	2	—
	多头注意力层	2	注意力头数=2，键向量维度=64，Dropout率=0.1
	前馈网络	1	卷积滤波器数=64，激活函数=‘ReLU’

Tab. 4 Parameter setting of scaling parameter γ and nonlinear adjustment parameter α

编号	样本一	样本二	样本三
1	$α$ =2.5， $γ$ =1.0	$α$ =2.5， $γ$ =1.5	$α$ =2.0， $γ$ =1.0
2	$α$ =2.5， $γ$ =1.0	$α$ =2.5， $γ$ =1.5	$α$ =2.0， $γ$ =1.0
3	$α$ =2.5， $γ$ =1.5	$α$ =3.0， $γ$ =2.0	$α$ =2.0， $γ$ =1.0
4	$α$ =2.5， $γ$ =1.5	$α$ =3.0， $γ$ =2.0	$α$ =2.0， $γ$ =1.0

Tab. 4 Parameter setting of scaling parameter γ and nonlinear adjustment parameter α

编号	样本一	样本二	样本三
1	$α$ =2.5， $γ$ =1.0	$α$ =2.5， $γ$ =1.5	$α$ =2.0， $γ$ =1.0
2	$α$ =2.5， $γ$ =1.0	$α$ =2.5， $γ$ =1.5	$α$ =2.0， $γ$ =1.0
3	$α$ =2.5， $γ$ =1.5	$α$ =3.0， $γ$ =2.0	$α$ =2.0， $γ$ =1.0
4	$α$ =2.5， $γ$ =1.5	$α$ =3.0， $γ$ =2.0	$α$ =2.0， $γ$ =1.0

Tab.5 Results of ablation experiments of self-attention layers

预测间隔/h	N	样本一				样本二				样本三
预测间隔/h	N	RMSE/ （V·m^-1）	MAE/ （V·m^-1）	PCC	时间/min	RMSE/ （V·m^-1）	MAE/ （V·m^-1）	PCC	时间/min	RMSE/ （V·m^-1）	MAE/ （V·m^-1）	PCC	时间/min
1	1	0.09	0.07	0.96	0.25	0.15	0.12	0.97	0.25	0.03	0.02	0.95	0.25
	2	0.35	0.26	0.18	0.50	0.24	0.17	0.94	0.50	0.05	0.04	0.90	0.50
	3	0.36	0.26	0.17	0.80	0.70	0.49	0.39	0.80	0.10	0.08	0.35	0.80
72	1	0.12	0.10	0.94	8.00	0.26	0.17	0.94	8.00	0.06	0.04	0.82	8.00
	2	0.32	0.25	0.33	20.00	0.34	0.27	0.66	20.00	0.08	0.06	0.62	20.00
	3	0.35	0.27	0.31	130.00	0.73	0.60	0.54	130.00	0.11	0.08	0.32	130.00

Tab. 6 Error parameters of five models on samples at different prediction intervals

模型	预测间隔/h	样本一				样本二				样本三
模型	预测间隔/h	RMSE/ （V·m^-1）	MAE/（V·m^-1）	MAPE/ %	PCC	RMSE/ （V·m^-1）	MAE/（V·m^-1）	MAPE/ %	PCC	RMSE/ （V·m^-1）	MAE/（V·m^-1）	MAPE/ %	PCC
GCL- Transformer	1	0.090 6	0.070 8	4.865 2	0.963 1	0.155 1	0.116 3	4.915 4	0.975 8	0.032 4	0.023 0	3.092 6	0.953 8
	3	0.102 3	0.080 6	5.602 0	0.952 7	0.196 9	0.147 5	6.300 7	0.959 1	0.040 0	0.028 9	3.868 6	0.929 6
	24	0.112 5	0.090 2	6.466 1	0.948 6	0.247 9	0.169 5	6.499 2	0.948 3	0.057 8	0.042 6	5.910 0	0.846 4
	72	0.123 4	0.098 7	7.064 8	0.943 0	0.257 5	0.174 5	6.842 8	0.937 7	0.061 2	0.043 1	6.197 8	0.824 7
Transformer1	1	0.130 7	0.103 2	7.260 5	0.932 6	0.271 5	0.190 4	7.691 9	0.925 0	0.036 9	0.026 5	3.576 7	0.940 9
	3	0.130 3	0.102 7	7.115 2	0.939 3	0.240 7	0.174 1	7.260 4	0.939 4	0.042 6	0.030 9	4.165 1	0.918 9
	24	0.125 0	0.100 0	6.683 1	0.940 9	0.298 7	0.238 3	10.890 2	0.933 3	0.058 7	0.042 9	5.960 1	0.852 9
	72	0.130 1	0.105 0	7.213 9	0.930 8	0.292 1	0.215 7	8.355 7	0.928 5	0.061 7	0.045 7	6.307 5	0.822 7
Transformer2	1	0.119 4	0.094 5	6.719 8	0.941 8	0.214 9	0.160 4	6.902 3	0.951 4	0.033 2	0.024 1	3.243 9	0.949 9
	3	0.111 5	0.088 9	6.409 5	0.949 4	0.213 8	0.159 1	6.931 0	0.951 9	0.042 3	0.031 2	4.226 3	0.922 6
	24	0.115 1	0.090 6	6.499 1	0.944 3	0.250 6	0.184 2	7.337 5	0.943 2	0.059 0	0.043 9	6.039 3	0.843 5
	72	0.124 0	0.100 4	7.132 7	0.942 1	0.277 5	0.198 2	7.714 3	0.932 6	0.061 9	0.044 9	6.228 5	0.823 4
LSTM	1	0.096 7	0.074 9	5.340 2	0.961 2	0.164 5	0.121 0	5.023 8	0.972 2	0.032 5	0.023 1	3.109 1	0.953 0
	3	0.121 1	0.093 0	6.729 7	0.937 5	0.198 2	0.147 7	6.314 9	0.960 5	0.046 6	0.033 3	4.582 0	0.900 4
	24	0.137 4	0.112 1	7.889 7	0.931 2	0.328 3	0.243 6	9.368 8	0.901 1	0.075 4	0.055 2	7.483 2	0.708 9
	72	0.145 6	0.118 5	8.338 0	0.928 7	0.455 4	0.354 8	13.114 5	0.822 1	0.074 3	0.052 5	7.209 1	0.718 5
TOEformer	1	0.092 2	0.072 3	5.089 2	0.962 3	0.160 8	0.120 9	5.033 8	0.968 2	0.032 6	0.023 1	3.100 2	0.952 3
	3	0.107 3	0.084 2	6.131 1	0.948 5	0.205 8	0.153 6	6.872 8	0.957 6	0.040 1	0.029 8	4.029 8	0.929 2
	24	0.124 4	0.101 1	6.838 3	0.941 7	0.256 1	0.190 3	7.438 5	0.940 0	0.058 0	0.043 4	5.944 4	0.845 4
	72	0.132 0	0.103 0	7.253 1	0.937 7	0.282 0	0.190 4	7.222 0	0.933 6	0.062 5	0.043 9	6.488 7	0.815 1

Tab. 6 Error parameters of five models on samples at different prediction intervals

模型	预测间隔/h	样本一				样本二				样本三
模型	预测间隔/h	RMSE/ （V·m^-1）	MAE/（V·m^-1）	MAPE/ %	PCC	RMSE/ （V·m^-1）	MAE/（V·m^-1）	MAPE/ %	PCC	RMSE/ （V·m^-1）	MAE/（V·m^-1）	MAPE/ %	PCC
GCL- Transformer	1	0.090 6	0.070 8	4.865 2	0.963 1	0.155 1	0.116 3	4.915 4	0.975 8	0.032 4	0.023 0	3.092 6	0.953 8
	3	0.102 3	0.080 6	5.602 0	0.952 7	0.196 9	0.147 5	6.300 7	0.959 1	0.040 0	0.028 9	3.868 6	0.929 6
	24	0.112 5	0.090 2	6.466 1	0.948 6	0.247 9	0.169 5	6.499 2	0.948 3	0.057 8	0.042 6	5.910 0	0.846 4
	72	0.123 4	0.098 7	7.064 8	0.943 0	0.257 5	0.174 5	6.842 8	0.937 7	0.061 2	0.043 1	6.197 8	0.824 7
Transformer1	1	0.130 7	0.103 2	7.260 5	0.932 6	0.271 5	0.190 4	7.691 9	0.925 0	0.036 9	0.026 5	3.576 7	0.940 9
	3	0.130 3	0.102 7	7.115 2	0.939 3	0.240 7	0.174 1	7.260 4	0.939 4	0.042 6	0.030 9	4.165 1	0.918 9
	24	0.125 0	0.100 0	6.683 1	0.940 9	0.298 7	0.238 3	10.890 2	0.933 3	0.058 7	0.042 9	5.960 1	0.852 9
	72	0.130 1	0.105 0	7.213 9	0.930 8	0.292 1	0.215 7	8.355 7	0.928 5	0.061 7	0.045 7	6.307 5	0.822 7
Transformer2	1	0.119 4	0.094 5	6.719 8	0.941 8	0.214 9	0.160 4	6.902 3	0.951 4	0.033 2	0.024 1	3.243 9	0.949 9
	3	0.111 5	0.088 9	6.409 5	0.949 4	0.213 8	0.159 1	6.931 0	0.951 9	0.042 3	0.031 2	4.226 3	0.922 6
	24	0.115 1	0.090 6	6.499 1	0.944 3	0.250 6	0.184 2	7.337 5	0.943 2	0.059 0	0.043 9	6.039 3	0.843 5
	72	0.124 0	0.100 4	7.132 7	0.942 1	0.277 5	0.198 2	7.714 3	0.932 6	0.061 9	0.044 9	6.228 5	0.823 4
LSTM	1	0.096 7	0.074 9	5.340 2	0.961 2	0.164 5	0.121 0	5.023 8	0.972 2	0.032 5	0.023 1	3.109 1	0.953 0
	3	0.121 1	0.093 0	6.729 7	0.937 5	0.198 2	0.147 7	6.314 9	0.960 5	0.046 6	0.033 3	4.582 0	0.900 4
	24	0.137 4	0.112 1	7.889 7	0.931 2	0.328 3	0.243 6	9.368 8	0.901 1	0.075 4	0.055 2	7.483 2	0.708 9
	72	0.145 6	0.118 5	8.338 0	0.928 7	0.455 4	0.354 8	13.114 5	0.822 1	0.074 3	0.052 5	7.209 1	0.718 5
TOEformer	1	0.092 2	0.072 3	5.089 2	0.962 3	0.160 8	0.120 9	5.033 8	0.968 2	0.032 6	0.023 1	3.100 2	0.952 3
	3	0.107 3	0.084 2	6.131 1	0.948 5	0.205 8	0.153 6	6.872 8	0.957 6	0.040 1	0.029 8	4.029 8	0.929 2
	24	0.124 4	0.101 1	6.838 3	0.941 7	0.256 1	0.190 3	7.438 5	0.940 0	0.058 0	0.043 4	5.944 4	0.845 4
	72	0.132 0	0.103 0	7.253 1	0.937 7	0.282 0	0.190 4	7.222 0	0.933 6	0.062 5	0.043 9	6.488 7	0.815 1

Fig. 9 Prediction results for prediction interval of one hour

Fig. 10 Prediction results for prediction interval of three days

Fig. 11 RMSE of predicted results

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Time series prediction of environmental electric field intensity with generalized correlation entropy loss function-based Transformer model

基于广义相关熵损失函数Transformer模型的环境电场强度时序预测

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