Incorporate long association into high-order fuzzy logical relationship based time series forecasting

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Abstract

In fuzzy logical relationship (FLR) based forecasting models, FLRs play a vital role. In each FLR of such models, the engaged observations in premise and consequent are consecutive at time. The association (named short association in this paper) implied by the premise and consequent of such an FLR can reflect some of the properties and regularities hidden in a time series, which has been verified by more successful applications. However, there exists other kind of associations that cannot be described by short association. In this paper, the concept of long association to describe these different associations is proposed. And long-association FLRs are then proposed to reflect such long associations. For a long-association FLR, the engaged observations in the premise and consequent are not consecutive at time. This new kind of FLRs exists more and can overcome one deficiency of the existing FLR based forecasting models: no FLR available for forecasting occurred often at some prediction moments. Departure from long-association FLRs, we construct trend long-association FLRs to reflect the trend long associations in time series and design a novel forecasting model. The advantages of the proposed (trend) long-association FLRs and the superiority of the proposed model are verified in experiments with comparisons with other forecasting models.

Introduction

Time series forecasting is one of the important tasks in time series analysis. Many kinds of models work on this task. For example, Support Vector Machine (SVM) (Tian, 2020, Li et al., 2018), Nonlinear Autoregressive Neural Networks (NAR) (Tian, 2019, Wu et al., 2020), Fuzzy Cognitive Map based Model (FCM-Model) (Homenda et al., 2014) and Fuzzy Time Series based Model (Pattanayak et al., 2021, Xian et al., 2020). Among these models, the forecasting model based on fuzzy time series is a widely used one in recent years. This forecasting model transforms time series into fuzzy time series, and analyzes the time series without any assumption usually appearing in the conventional methods (Azahari et al., 2017, Jain et al., 2018). More importantly, this model can handle time series forecasting problems with linguistic values.

Fuzzy time series based forecasting model was firstly proposed by Song and Chissom to solve the forecasting of college enrollment (Song and Chissom, 1991, Song and Chissom, 1993, Song and Chissom, 1994), which achieved good performance. With the needs of reality, this kind of models has been used in many fields, such as stock forecasting (Tahseen and Syed, 2008), environment forecasting (Wang et al., 2018), tourists forecasting (Guney et al., 2018, Aladag and Egrioglu, 2014) and energy forecasting (Severiano et al., 2017, Severiano et al., 2021). In order to advance the forecasting performance of these models, various kinds of improved models have been put forward. The improvements usually concentrate on four aspects. The first one is concerned to the optimal interval partition which takes the distribution of time series into consideration (Wang et al., 2013, Bose and Mali, 2018). The second one is concerned to the fuzzification methods on time series, among which the establishments of triangular fuzzy sets and intuitionistic fuzzy sets are common methods (Kumar and Gangwar, 2015, Egrioglu et al., 2019). The third one is concerned to the methods for constructing FLRs (Lee et al., 2006, Lu et al., 2014). And the last one is concerned to the methods of weights assignment of different FLRs (Cheng and Chen, 2018, Liu et al., 2012, Qiu et al., 2011). From the fuzzy time series based model, the hybrid forecasting models are established by combining it with other intelligent methods (SVM, NAR or Autoregressive Integrated Moving Average (ARIMA)) (Sadaei et al., 2016b, Sadaei et al., 2016a). Among these studies, constructing suitable FLRs in forecasting occupies an important position, which has a significant impact on predictions.

FLRs play an important role in the fuzzy time series based forecasting models. FLRs with different forms imply different associations between the premise observation(s) and the consequent observation, and thus yield different predictions. The original forecasting model based on fuzzy time series considers only one-order FLRs, each of which has only one premise observation at one moment. Every FLR of such kind reflects only the association between the observation at one moment and its left closest moment (Song and Chissom, 1994, Guo et al., 2019). For example, one-order FLR Ft1F(t) describes the association between the premise observation at moment t1 and the consequent observation at moment t. However, this kind of FLRs cannot reflect the trend influence of premise observations on the consequent observation. Chen et al. put forward the concept of high-order FLRs and high-order trend FLRs whose number of the premise moments is larger than one (Bas et al., 2018, Chen, 2002). For example, high-order FLR FthFt2Ft1F(t) describes the trend influence of premise observations at moments (t1,t2,,th) on the consequent observation at moment t. The high-order (trend) FLRs can raise the accuracy of forecasting effectively.

High-order FLRs as well as one-order FLRs have a common characteristic: the premise moment(s) and the consequent moment are consecutive. In other words, they describe the associations between the premise observation(s) and the consequent observation at consecutive moments. We name such FLRs as short-association FLRs, while the association reflected by short-association FLRs are called short associations. The short associations reflect some of the properties and regularities hidden in a time series. This has been verified by more successful applications realized by researchers (Chen and Chen, 2011). However, there also exists such a different kind of associations between the premise observation(s) and the consequent observation where the premise moment(s) and the consequent moment are non-consecutive, such as periodic or periodic-like regularity, seasonal or seasonal-like regularity in time series, and so on. For such time series with the above-mentioned regularities, the associations existing among some observations at non-consecutive moments are more important than the short associations in reflecting the characteristics of them. In order to distinguish this kind of associations from the short association ones, we call them long associations. The FLRs reflect the long associations are named as long-association FLRs.

As well known, when doing predictions with the existing high-order short-association FLR based models, we often meet with such situation: there is no available FLR for forecasting, or in other words, none of the premise of the constructed short-association FLRs is successfully matched. With the help of the additional long-association FLRs, the possibility of occurrence of such situation will be decreased. In fact, we may well find available long-association FLRs when no available short-association FLR exists. It will be true especially for the forecasting made by high-order long-association FLRs. This is verified in the experimental studies of this paper.

Regarding high-order (trend) short-association FLR based forecasting, we list three of their problems as follows:

Problem 1

The existing high-order short-association FLRs cannot reflect the associations between the premise observations and the consequent observation at the non-consecutive moments.

Problem 2

When doing forecasting with high-order short-association FLRs based models, it occurred often that no available short-association FLR can be found for forecasting.

Problem 3

When doing forecasting with high-order trend short-association FLRs based models, it occurred often that no available trend short-association FLR can be found for forecasting.

From the above analyses, we can find that it is necessary to construct the FLRs by using the observations at non-consecutive moments. In order to describe the associations among observations at non-consecutive moments, we raise the high-order long-association FLRs in this paper. Forecasting from such kind of FLRs (constructed by some observations at non-consecutive moments) can make up for the deficiency of the existing forecasting models.

In this paper, we first construct high-order long-association FLRs which can reflect the long associations in time series. Different from the existing high-order short-association FLRs, the high-order long-association FLRs are constructed by some observations at non-consecutive moments. After that, we construct another kind of FLRs, namely high-order trend long-association FLRs, which can reflect the trend long associations in time series. After the high-order trend long-association FLRs being constructed from the high-order long-association FLRs, a novel forecasting model is built up. In this model, the high-order trend long-association FLRs as well as the high-order trend short-association FLRs are integrated together to calculate the predictions.

The main contributions of this paper are outlined as follows:

(1) Two kinds of long-association FLRs are proposed.

One kind of the proposed FLRs is high-order long-association FLRs which describe the association among observations at non-consecutive moments. They can make up for the deficiencies of the existing forecasting models: neglect the long associations in time series and often cause no available FLR for forecasting.

The other kind of the proposed FLRs is high-order trend long-association FLRs corresponding to high-order long-association FLRs, which used to reflect the trend influence on the consequent observation from the premise observations at the left non-closest non-consecutive moments. They can raise the possibility of finding available FLRs for forecasting.

(2) One (trend) long-association FLR based forecasting model is put forward.

Based on the high-order (trend) long-association FLRs, a novel forecasting model is proposed. In this model, a new reasoning model is given, where the predictions are calculated by the high-order trend long-association FLRs and the high-order trend short-association FLRs.

This paper is organized as follows. In Section 2, some necessary definitions and a typical fuzzy logical relationship based forecasting model are introduced. Two kinds of FLRs and one FLR based forecasting model are proposed in Section 3. Experiments are presented in Section 4 to show the advantages of the proposed FLRs and model. The last section concludes this study.

Section snippets

Preliminaries

Some basic definitions of fuzzy time series and a typical fuzzy logical relationship based forecasting model are briefly reviewed in this section. More details can be found in literatures (Zadeh, 1965, Xian et al., 2018, Zhang et al., 2018, Bisht and Kumar, 2016, Yu and Huarng, 2008).

High-order long-association fuzzy logical relationship based forecasting

Aiming at dealing with the problems pointed out in Introduction, high-order long-association fuzzy logical relationships and high-order trend long-association fuzzy logical relationships are proposed in this section. Based on these FLRs, a novel forecasting model is designed here.

Experimental study

In order to demonstrate the benefits of the new proposed high-order (trend) long-association FLRs (introduced in Section 3.1) and the superiority of the proposed forecasting model (described in Section 3.2), four experiments and their comparisons with other three high-order FLR based forecasting models and seven classical computing intelligent forecasting models are presented in this section. The comparative models are:

  • Three high-order FLR based forecasting models: Chen’s model (FLR-1, Chen,

Conclusion

In this paper, two new concepts of high-order long-association fuzzy logical relationship and high-order trend long-association fuzzy logical relationship are proposed. These two kinds of FLRs emphasize the influence and the trend influence on the consequent observation from the premise observations at the left non-closest non-consecutive moments. In other words, these new kinds of FLRs can describe the associations among some observations at non-consecutive moments as well as the trend

CRediT authorship contribution statement

Fang Li: Propose the high-order (trend) long-association fuzzy logical relationship and the high-order (trend) long-association fuzzy logical relationship based forecasting model, Writing – original draft & review. Chen Liu: Writing – review & editing. Xiyang Yang: Verify the above method and idea through the four experiments on time series, Experimental verification.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is supported by the Fujian Natural Science Foundation Project (No. 2021J01001).

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