Abstract: The consumption of natural gas is an important indicator reflecting a country’s energy utilization and demand. If it can be scientifically and reasonably predicted, this will be of great significance for the formulation of natural gas pricing strategies, national economic accounting, and optimization of pipeline network design. However, existing methods for predicting natural gas consumption have insufficient performance in describing causal relationships and modeling dynamic systems. Therefore, this paper proposes to use high-order fuzzy cognitive maps for predicting natural gas consumption. This method combines the advantages of cognitive map models and fuzzy theory, and has many advantages in adaptability and fuzzy reasoning.
Regarding the problem of one-dimensional time series prediction, some scholars have proposed a wavelet high-order fuzzy cognitive map, which uses redundant Hale wavelet transformation to transform one-dimensional time series data into multi-dimensional for prediction. However, the limitation of this algorithm is that it requires prior knowledge of training, validation, and prediction data, which is unreasonable. The high-order fuzzy cognitive map prediction based on biconvex transformation proposed in this article only requires knowledge of the data to be processed, namely training and validation data, and can be directly predicted without the need to know the specific values of the original prediction data. That is to say, wavelet high-order fuzzy cognitive map prediction is more about solving a fitting problem than a prediction problem. The method proposed in this article is more in line with the true meaning of prediction.
This article proposes a high-order fuzzy cognitive map based on biconvex transformation to solve the problem of difficult processing of one-dimensional natural gas consumption time series prediction using high-order fuzzy cognitive maps. Firstly, when inputting data, the high-order fuzzy cognitive graph learning algorithm requires the input data to be a time series larger than one dimension, while the obtained data is only one dimension. To increase the data dimension, it is proposed to perform corresponding mathematical transformation on the obtained one-dimensional time series to meet the input requirements of the high-order fuzzy cognitive graph learning algorithm. There are many functions for transforming data, but considering practicality and the simplicity of subsequent inverse calculations, it is proposed to perform different forms of nonlinear transformation on one-dimensional time series data.
Secondly, the transfer function is an important component of high-order fuzzy cognitive maps, controlling the range of output. This article aims to improve the nonlinear expression ability of the transfer function for numerical transformation between small cells. A new transfer function with concave convex features is obtained by redesigning it using a non-linear quadratic function fitting method. It should be noted that the concept of concavity convex design here is different from the data dimensionality enhancement technique of concavity convex transformation on one-dimensional time series, and only because both have concavity convex characteristics, they are called double concavity convex transformation.
Finally, an empirical analysis is conducted based on natural gas consumption data from 2000 to 2019 across the country and 30 provinces. The empirical process of prediction mainly includes three modules: data preprocessing, model construction, and data prediction. (1)Data preprocessing module: Firstly, we divide the input data into three parts: training set, validation set, and testing set. Next, the obtained one-dimensional time series data with a length of M is subjected to concavity convex transformation to increase its dimensionality, converting the one-dimensional data into three-dimensional data. Finally, the three-dimensional data is normalized. (2)Model construction module: it mainly includes weight matrix solving and cross validation. The solution of weight matrix involves using a newly designed g(x)-transfer function to solve the weight vector w_n of high-order fuzzy cognitive maps, thereby further constructing the weight matrix of K-order fuzzy cognitive maps. Cross validation: it mainly utilizes cross validation of the validation set to optimize the order K and regularization parameter α of the ridge regression problem. (3)Data prediction module: we obtain corresponding values through a high-order fuzzy cognitive map model, and perform normalization and non-linear data concavity convex transformation to obtain the final predicted value.
The empirical results show that, firstly, on the basis of increasing the dimensionality of the data through concavity convex transformation, the prediction results of the high-order fuzzy cognitive map using the newly designed transfer function are better than those based on the traditional transfer function. The applicability rate of the new transfer function is as high as 96.4%. Secondly, comparing the predicted results of both methods with ARIMA and GM(1,1), it is found that the applicability of the proposed method can reach 87.1%, further verifying the effectiveness of the proposed method.
This article mainly focuses on the prediction of one-dimensional time series using high-order fuzzy cognitive maps. Future research directions will consider the modeling and prediction of multi-dimensional time series using high-order fuzzy cognitive maps, as well as the prediction of one-dimensional and multidimensional time series using different fuzzy cognitive maps.

Key words: double concave convex transformation, high order fuzzy cognitive maps, natural gas consumption forecast

摘要： 为解决高阶模糊认知图难以处理一维时间序列预测问题,提出基于双凹凸变换的高阶模糊认知图。为增加数据维度,采用特定函数对一维时间序列作凹凸变换,其次为改善高阶模糊认知图非线性表达能力,设计了具有凹凸特征的新传递函数,并结合全国及30个省份的2000—2019年天然气消费数据进行实证分析。结果表明,在对数据作增维凹凸变换基础上,采用新设计传递函数的高阶模糊认知图和基于传统传递函数的高阶模糊认知图预测结果相比更优,新传递函数适用率高达96.4%;其次将其两者预测结果和ARIMA及GM(1,1)相比,得出所提方法的适用率可达87.1%,进一步验证了提出方法的有效性。

关键词: 双凹凸变换, 高阶模糊认知图, 天然气消费预测