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A hybrid rolling grey framework for short time series modelling

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Abstract

Time series modelling is gaining spectacular popularity in the prediction process of decision making, with applications including real-world management and engineering. However, for short time series, prediction has to face unavoidable limitation for modelling extremely complex systems. It has to apply inadequate and incomplete data from short time to predict unknown observations. With such limited data source, existing techniques, such as statistical modelling or machine learning methods, fail to predict short time series effectively. To address this problem, this paper provides a global framework for short time series modelling predictions, integrating the rolling mechanism, grey model, and meta-heuristic optimization algorithms. In addition, dragonfly algorithm and whale optimization algorithm are investigated and deployed to optimize the framework by enhancing its predicting accuracy with less computational costs. To verify the performance of the proposed framework, three industrial cases are introduced as simulation experiments in this paper. Experimental results confirm that the framework solves corresponding short time series modelling predictions with greater accuracy and speed than the standard GM(1,1) models. The source codes of this framework are available at: https://github.com/zhesencui/HybridRollingGreyFramework.git.

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Acknowledgements

This work was supported in part by the Natural Science Foundation of China [Grant No. 61773002], the Center of Collaborative Innovation for Educational Big Data Analysis and Application in Changzhi University 1331 project [Grant No. 020-200624], the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS) [Grant No. CE140100049], Guangdong Basic and Applied Basic Research Foundation [Grant No. 2020A1515011580], and Provincial key platforms and major scientific research projects of Guangdong universities [Grant No. 2018GKTSCX010].

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Authors and Affiliations

  1. Department of Computer Science, Changzhi University, Changzhi, 046031, China

    Zhesen Cui & Wei Lian

  2. School of Mathematical Sciences, Queensland University of Technology, Brisbane, 4001, Australia

    Jinran Wu & Qibin Duan

  3. School of Computer Science, Queensland University of Technology, Brisbane, 4001, Australia

    Zhe Ding

  4. College of Automation and College of Artificial Intelligence, Nanjing University of Posts and Telecommunication, Nanjing, 210023, China

    Yang Yang

  5. School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou, 510320, China

    Taoyun Cao

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  1. Zhesen Cui
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Correspondence to Jinran Wu or Zhe Ding.

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Appendices

Appendix 1: Whale optimization algorithm

The whale optimization algorithm (WOA) is a swarm intelligence optimization algorithm that simulates the hunting behavior of humpback whales in nature [47]. The algorithm has the characteristics of a simple structure, few parameters, a strong search ability, and easy implementation. The position of each whale represents a feasible solution in WOA. In the course of whale hunting, each whale has two behaviors. One is to surround the prey, and all the whales move toward the other whales; the other is the drum net, where the whale’s annular swimming sprays bubbles to drive away the prey. In the whale’s behavior of encircling its prey, whales choose whether to swim toward the optimal position of the whale or choose a whale as its own target and approach it randomly. Furthermore, three steps are presented to search for the optimization as follows: encircling prey, bubble-net attacking method of exploitation phase, and search for prey section of exploration phase.

Appendix 2: Dragonfly algorithm

The dragonfly algorithm (DA) was developed by [48], and the inspiration for DA comes from static and dynamic swarm behaviors. In static swarm, dragonflies fly in a small area with small groups to hunt other flying prey. In dynamic swarms, huge amounts of dragonflies create groups for migrating in one direction over long distances. Based on the behaviors of dragonflies, three principles of the dragonfly algorithm are proposed: (1) separation, which refers to the static collision avoidance between the individuals and other individuals in adjacent areas; (2) alignment, which indicates velocity matching of individuals to that of other individuals nearby; (3) cohesion, which means the trend of individuals toward the center of the mass of neighborhood; and (4) all individuals should be attracted toward food sources and distract enemies.

Appendix 3: Particle swarm optimization

Particle swarm optimization (PSO) was proposed by [58]. It is a swarm intelligence algorithm designed to simulate bird swarming predation behavior. There are different food sources in the region, and the point of bird swarm is to find the largest food sources (global optimal solution). Birds get to know the location of food source by transmitting the information of their respective positions to each other throughout the search process. Finally, the whole bird group can gather around the food source, which is the optimal solution.

The goal of the particle swarm algorithm is to enable all particles to find the optimal solution in multi-dimensional hyper-volume. First of all, initial random positions and initial random velocities are assigned to all particles in space. Then, the position of each particle is advanced according to the velocity of each particle, the optimal global position in the problem space, and the optimal position of the particle in turn. Particles aggregate around the best value by exploring and exploiting known positions in the search space. The advantage of the algorithm is that it retains the information of the optimal global position and the optimal position is known to the particles, which has a positive effect on both faster convergence speed and avoiding falling prematurely into the local optimal solution.

Appendix 4: Artificial bee colony

Artificial bee colony (ABC) is a swarm intelligence optimization algorithm proposed by [50]. Its intuitive background comes from the honey harvesting behavior of bee colonies. Bees carry out different activities based on their respective behaviors and labor in order to find the optimization results of the problem through sharing and communication of information. It provides a population-based search process in which individuals known as food locations are altered by artificial bees over time. The purpose of the bees is to discover the locations of food sources with the highest nectar amount. ABC combines local search measures performed by employed and bystander bees with global search measures actualized by bystanders and scouts, which contribute to balancing exploration and exploitation.

Appendix 5: Cuckoo search

A cuckoo search (CS) is a meta-heuristic optimization algorithm developed by [51]. It solves the parameter optimum problem through simulating the parasitic brood of some species of cuckoo (Brood Parasitism). Meanwhile, CS also uses the relevant Levy flight search mechanism rather than the more simple isotropic flight in order to obtain an optimal nest, which can achieve an efficient optimization mode. CS has three idealized rules: (1) Each cuckoo lays an egg each time and dumps it randomly into a selected nest; (2) the best nests with premium eggs will be brought to the next generation; (3) the number of available nests is fixed, and the host finds the eggs laid by the cuckoo via probability. Furthermore, the host can destroy the egg or discard the old nest to build a new one. CS has the advantages of having few parameters, simple operation, easy implementation, random search path optimization, and strong optimization ability.

Appendix 6: Grey wolf optimizer

Grey wolf optimizer (GWO) is a group intelligent optimization algorithm proposed by [52]. This algorithm is inspired by the prey activities of grey wolves and has the characteristics of strong convergence performance, few parameters, and easy implementation. It has been successfully applied in parameter optimization, image classification, and so on.

GWO has the following advantages: (1) The search method is unique, and all grey wolves rely on the optimal gray wolf to update its position, which means GWO has better global search performance; (2) GWO uses two parameters to balance exploration and exploitation, so it can obtain faster convergence speed and stronger local search ability; and (3) the fitness values are calculated after all gray wolf positions were updated, and the positions of \(\lambda\), \(\beta,\) and \(\delta\) are updated by greedy algorithm, which means the calculation can be done in parallel in order to achieve a higher speed of operation.

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Cui, Z., Wu, J., Ding, Z. et al. A hybrid rolling grey framework for short time series modelling. Neural Comput & Applic 33, 11339–11353 (2021). https://doi.org/10.1007/s00521-020-05658-0

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