An improvement of the traditional gray system model, GM(1,1), to enhance forecast accuracy, has been realized using the particle swarm optimization (PSO) algorithm. Unlike the GM(1,1) which uses a fixed adjacent neighbor weight for all data sets, the proposed PSO-improved model, PSO-GM(1,1), determines an optimal adjacent neighbor weight, based on the presented data set. This optimal adjacent neighbor weight so determined is the principal factor that enhances forecast accuracy. The performance of the proposed model was evaluated using generated monotonic increasing and decreasing data sets as well as measured energy consumption data for a laptop computer, desktop computer, printer, and photocopier. The performance of PSO-GM(1,1) was compared with that of GM(1,1), and two other models in literature that sought to improve the performance of GM(1,1). The PSO-GM(1,1) outperformed the traditional model and the two other models. For the monotonic increasing data, the mean absolute percentage error (MAPE) for the proposed model was 0.007% as against a MAPE value of 20.383% for the GM(1,1). For the monotonic decreasing data, the PSO-GM(1,1) again outperformed GM(1,1), yielding a MAPE of 0.057% compared to a value of 13.407% for the traditional model. For the measured laptop computer energy data, the obtained MAPE for the PSO-GM(1,1) was 0.675% while the values for the two models were 4.052% and 2.991%. For the measured desktop computer energy data, the obtained MAPE for the PSO-GM(1,1) was 0.0018% while the values for the two models were 0.0018% and 1.163%. For the data associated with the printer, the MAPEs were 8.414% for the PSO-GM(1,1), 20.957% for the first model and 9.080% for the second model. For the measured photocopier energy data, the obtained MAPE for the PSO-GM(1,1) was 0.901% while the values for the two models were 3.799% and 0.943%. Thus, the proposed PSO-GM(1,1) greatly improves forecast accuracy and is recommended for adoption, for forecasting.

中文翻译：

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基于粒子群算法的灰色系统模型改进

使用粒子群优化（PSO）算法已经实现了对传统灰色系统模型GM（1,1）的改进，以提高预测精度。与GM（1,1）对所有数据集使用固定的相邻邻居权重不同，建议的PSO改进模型PSO-GM（1,1）根据给出的数据集确定最佳的相邻邻居权重。这样确定的最佳相邻邻权重是提高预测准确性的主要因素。使用生成的单调递增和递减数据集以及便携式计算机，台式计算机，打印机和影印机的测量能耗数据来评估所提出模型的性能。将PSO-GM（1,1）的性能与GM（1,1）的性能进行了比较，并且在文献中试图提高GM（1,1）的性能的其他两个模型也进行了比较。PSO-GM（1,1）优于传统模型和其他两个模型。对于单调递增数据，建议模型的平均绝对百分比误差（MAPE）为0.007％，而GM（1,1）的MAPE值为20.383％。对于单调递减数据，PSO-GM（1,1）再次优于GM（1,1），MAPE为0.057％，而传统模型的MAPE为13.407％。对于所测量的膝上型计算机能量数据，获得的PSO-GM（1,1）的MAPE为0.675％，而两个模型的值为4.052％和2.991％。对于所测量的台式计算机能量数据，获得的PSO-GM（1,1）的MAPE为0.0018％，而两个模型的值分别为0.0018％和1.163％。对于与打印机相关的数据，PSO-GM（1,1）的MAPE为8.414％，第一个型号的MAPE为20.957％，而9为9。对于第二个型号，为080％。对于测得的复印机能量数据，获得的PSO-GM（1,1）的MAPE为0.901％，而两个模型的值为3.799％和0.943％。因此，提出的PSO-GM（1,1）大大提高了预测准确性，并建议采用该方法进行预测。