A Comparative Analysis of Machine Learning and Grey Models
机器学习与灰色模型的比较分析
ABSTRACT Artificial Intelligence (AI) has recently shown its capabilities for almost every field of life. Machine Learning, which is a subset of AI, is a ‘hot’ topic for researchers. Machine Learning outperforms other classical forecasting techniques in almost all-natural applications and it is a crucial part of modern research. Many modern Machine Learning methods require a large amount of training data. Due to the small datasets, the researchers may not prefer to use Machine Learning algorithms that require large training data. To tackle this issue, this survey illustrates, and demonstrates related studies for significance of Grey Machine Learning (GML). Which is capable of handling large datasets as well as small datasets for time series forecasting likely outcomes. This survey presents a comprehensive overview of the existing grey models and machine learning forecasting techniques. To allow an in-depth understanding for the readers, a brief description of Machine Learning, as well as various forms of conventional grey forecasting models are discussed. Moreover, a brief description on the importance of GML framework is presented.
摘要 人工智能 (AI) 最近展示了其在几乎所有生活领域的能力。机器学习作为 AI 的一个子集，是研究者的热门话题。在几乎所有自然应用中，机器学习都优于其他经典预测技术，并且是现代研究的关键部分。许多现代机器学习方法需要大量的训练数据。由于数据集较小，研究者可能不愿意使用需要大量训练数据的机器学习算法。为了解决这个问题，本调查展示并演示了灰色机器学习 (Grey Machine Learning, GML) 在时间序列预测中的重要性相关研究。GML 能够处理大数据集以及小数据集的时间序列预测。本调查全面概述了现有的灰色模型和机器学习预测技术。为了让读者深入理解，简要介绍了机器学习以及各种形式的传统灰色预测模型。此外，还简要介绍了 GML 框架的重要性。
INDEX TERMS Machine Learning, Grey Models, Grey Machine Learning, Forecasting, Small Sample Learning
索引术语 机器学习 (Machine Learning)，灰色模型 (Grey Models)，灰色机器学习 (Grey Machine Learning)，预测 (Forecasting)，小样本学习 (Small Sample Learning)
INTRODUCTION
引言
ACHINE learning techniques plays an essential role especially for forecasting [1, 2]. Every country has a relevant organization that analyzes, and collects the economic facts, and figures to predict future tendency for several economic indicators and assist policy-makers in their decision-making [3, 4]. Data gathered from industries (e.g., demand and sale) remain insufficient. Recently, several types of forecasting methods were proposed and can be divided into two main categories: (i) Qualitative and (ii) Quantitative. Qualitative methods include expert system, trend prediction, Delphi, etc. Meanwhile, the quantitative methods include multi-linear regression analysis, exponential smoothing, time series analysis, and genetic algorithms [5, 6]. These forecasting methods are constrained by the lack of data, complicated input variables, and predicted environmental changes [7]. There are more than 300 studies related to forecasting. However, only a few of such studies are reliable. Despite the well-developed scientific technologies, There are several social and natural factors which are unexplained, uncertain, or incomplete. Besides the availability of an extensive range of technologies and frameworks, which can be used for big datasets [8, 9].
机器学习技术在预测中扮演着至关重要的角色 [1, 2]。每个国家都有一个相关组织，负责分析和收集经济事实和数据，以预测未来若干经济指标的趋势，并协助决策者进行决策 [3, 4]。从行业收集的数据（例如需求和销售）仍然不足。最近，提出了几种类型的预测方法，可以分为两大类：(i) 定性方法和 (ii) 定量方法。定性方法包括专家系统、趋势预测、德尔菲法等。同时，定量方法包括多元线性回归分析、指数平滑、时间序列分析和遗传算法 [5, 6]。这些预测方法受到数据缺乏、复杂的输入变量和预测环境变化的限制 [7]。有超过 300 项与预测相关的研究。然而，只有少数研究是可靠的。尽管科学技术已经相当发达，但仍有一些社会和自然因素无法解释、不确定或不完整。此外，尽管有广泛的技术和框架可用于处理大数据集 [8, 9]。
Recent forecasting review studies offer a systematic overview of current forecasting models and their classification. Hippert et al. [10] presented a review on short-term load forecasting. Mat Daut et al. presented a review on building electrical energy consumption forecasting analysis using conventional and AI methods [11]. Zhao et al. classified and reviewed the existing methods for building energy consumption prediction [12]. A review on energy demand models for forecasting proposed by Suganthi and Samuel [13]. Fumo et al. presented a detailed study on building energy estimation and classification [14]. Furthermore, Martinez-Alvarez et al. presented a survey on data mining techniques for time series forecasting of electricity [15]. Raza and Khosravi presented a review on short-term load forecasting techniques based on AI techniques [16]. Wang et al. proposed a review of AI based building energy prediction with a focus on ensemble prediction models [17].
最近的预测综述研究对当前的预测模型及其分类进行了系统概述。Hippert 等人 [10] 对短期负荷预测进行了综述。Mat Daut 等人对使用传统和 AI 方法进行建筑电力消耗预测分析进行了综述 [11]。Zhao 等人对现有的建筑能耗预测方法进行了分类和综述 [12]。Suganthi 和 Samuel 对用于预测的能源需求模型进行了综述 [13]。Fumo 等人对建筑能源估算和分类进行了详细研究 [14]。此外，Martinez-Alvarez 等人对用于电力时间序列预测的数据挖掘技术进行了调查 [15]。Raza 和 Khosravi 对基于 AI 技术的短期负荷预测技术进行了综述 [16]。Wang 等人对基于 AI 的建筑能源预测进行了综述，重点关注集成预测模型 [17]。
Recently, AI has shown its capabilities for almost every field of life. In 2015, The authors designed a machine to understand Mongolian and passed the Turing test [18]. Such research has shown that the computer can function as humans in handwriting tasks. In 2016, David Silver et al. published AlphaGo’s first paper, stating that the machine could beat the Go game practitioners for the very first time. This article was published after AlphaGo defeated the World Champion Lee Sedol. The success of AlphaGo had also demonstrated that the machine can not only be like humans, but can also be smarter than humans, and the output of this research gives AI enormous confidence [19]. Chirag et al. presented a review on time series forecasting techniques for building energy consumption [20]. The superior performance of the hybrid and ensemble models for time series forecasting was also proposed in recent review studies [21–25]. All the above surveys and studies provided vital information on forecasting models on different scales.
近年来，AI 几乎在生活的各个领域都展示了其能力。2015 年，作者设计了一台能够理解蒙古语的机器，并通过了图灵测试 [18]。此类研究表明，计算机在手写任务中可以像人类一样工作。2016 年，David Silver 等人发表了 AlphaGo 的第一篇论文，指出该机器首次能够击败围棋从业者。这篇文章是在 AlphaGo 击败世界冠军李世石之后发表的。AlphaGo 的成功也表明，机器不仅可以像人类一样，甚至可以比人类更聪明，这项研究的成果为 AI 带来了巨大的信心 [19]。Chirag 等人对建筑能耗的时间序列预测技术进行了综述 [20]。最近的综述研究也提出了混合模型和集成模型在时间序列预测中的优越性能 [21–25]。上述所有调查和研究为不同规模的预测模型提供了重要信息。
FIGURE 1. Illustration the criteria and selection process of this survey paper within the domain of building optimization.
图 1: 本综述论文在建筑优化领域中的标准和选择过程示意图。
Besides this, a question that needs to be addressed is how can researchers use Machine Learning techniques using extremely small datasets to acquire high accuracy and speed?
此外，一个需要解决的问题是，研究人员如何利用极小的数据集使用机器学习技术来获得高准确性和速度？
To answer this, a comprehensive overview of Machine Learning, grey models, and GML framework illustrated in this survey to highlight the present outlooks, and enhancement for the classical proposed methods.
为了回答这个问题，本调查对机器学习、灰色模型和GML框架进行了全面概述，以突出当前的前景，并对经典提出的方法进行了改进。
A. OBJECTIVES OF THE SURVEY
A. 调查目标
A forecasting model can be rely on static data that compares a dependent variable to a collection of independent variables, or it can be rely on composite or simultaneous time series data [20]. The importance of time series analysis has evolved as people more convinced of the significance of real-time data monitoring and storage [26]. The aim of this survey is to understand more about the existing time series forecasting framework known as GML. The key contributions of this article can be summarized as follows:
预测模型可以依赖于将因变量与一组自变量进行比较的静态数据，也可以依赖于复合或同步时间序列数据 [20]。随着人们越来越认识到实时数据监控和存储的重要性，时间序列分析的重要性也在不断提升 [26]。本次调查的目的是更深入地了解现有的时间序列预测框架 GML。本文的主要贡献可以总结如下：
This subsection describes the process of this paper and provides a summary of the articles discussed in Figure 1. The structure of this survey is given as follows: To illustrate the core concept of GML. Firstly, we provide a primer survey of Machine Learning algorithms and their applications in Section II. The general forms of the conventional grey models, including popular models, are illustrated in Section III. The general formulations including the computational details of GML are discussed in Section IV. A brief discussion and future perspectives of GML have been presented in Section V. Finally, the conclusion is presented in Section VI.
本小节描述了本文的过程，并对图1中讨论的文章进行了总结。本调查的结构如下：为了说明GML的核心概念，首先在第二部分提供了机器学习算法及其应用的初步调查。第三部分展示了传统灰色模型的一般形式，包括流行模型。第四部分讨论了GML的一般公式，包括计算细节。第五部分简要讨论了GML的未来展望。最后，第六部分给出了结论。
II. MACHINE LEARNING
II. 机器学习
Machine Learning is a subset of AI specifically developed to simulate human intelligence. It is essentially a data processing approach that automates the construction of an empirical model. In other words, it is focus on the premise that algorithms can learn from data, recognize patterns and make decisions with minimal human intervention. Recently, Machine Learning techniques have been used for Big Data [27] and has been widely implemented in some of the areas, varying from computer vision [28], finance [29], spacecraft engineering [30], entertainment [31], pattern recognition [32], and computational biology to biomedical applications [33]. In this overview, the types of Machine Learning algorithms as well as popular techniques of Machine Learning are presented.
机器学习是人工智能的一个子集，专门用于模拟人类智能。它本质上是一种自动化构建经验模型的数据处理方法。换句话说，它关注的前提是算法可以从数据中学习、识别模式并在最少人为干预的情况下做出决策。最近，机器学习技术已被用于大数据 [27]，并在一些领域得到了广泛应用，包括计算机视觉 [28]、金融 [29]、航天工程 [30]、娱乐 [31]、模式识别 [32] 以及计算生物学和生物医学应用 [33]。在本概述中，将介绍机器学习算法的类型以及流行的机器学习技术。
A. OVERVIEW OF MACHINE LEARNING
A. 机器学习概述
In this subsection, a brief analysis over time to explore the history of Machine Learning as well as the most significant
在本小节中，我们将简要分析机器学习的历史及其最重要的
milestones are presented. Furthermore, we have divided this overview into several categories to make it more understandable.
里程碑被呈现出来。此外，我们将此概述分为几个类别，以便更易于理解。
1) Classical work
1) 经典工作
Although Arthur Samuel et al. was an American pioneer of AI and inventor of the term “Machine Learning” in International Business Machines (IBM) a leading US computer manufacturer in 1959 [34], the year 1950 marks the first time Alan Turing et al. proposed the “Turing Machine” while posing such questions as, “Can machines think?” and “Can machines do what we can do? (as thinking entities)”. For instance, if a machine has true intellect, then the machine device must have the ability to trick a human being into thinking that it is also a human [35]. In Turings’ proposed study, there are many features that could be exhibited by machine intelligence and the different consequences for architecture are revealed, this is the first discovery in the field of Machine Learning. Earlier research works planned to connect the computer with human interaction. For this purpose, the first Machine Learning program was published in 1953, which is written by Arthur Samuel et al. [34]. The software was a game (Checkers). The IBM machine improved the design of the game and its progress, helped to refine the winning tactics, and integrated certain movements into the software. Based on the above studies, it can be analyzed that, in the early stages of Machine Learning developments, scientists and engineers introduced Machine Learning applications to improve computer intelligence.
虽然 Arthur Samuel 等人是 AI 的先驱，并于 1959 年在美国领先的计算机制造商 IBM (International Business Machines) 中提出了“机器学习 (Machine Learning)”这一术语 [34]，但 1950 年标志着 Alan Turing 等人首次提出了“图灵机 (Turing Machine)”，同时提出了诸如“机器能思考吗？”和“机器能做我们能做的事吗？（作为思考实体）”等问题。例如，如果一台机器具有真正的智能，那么该机器设备必须具备欺骗人类使其认为它也是人类的能力 [35]。在 Turing 提出的研究中，机器智能可以展示许多特征，并揭示了架构的不同后果，这是机器学习领域的首次发现。早期的研究工作计划将计算机与人类互动连接起来。为此，1953 年发布了第一个机器学习程序，由 Arthur Samuel 等人编写 [34]。该软件是一个游戏（跳棋）。IBM 机器改进了游戏的设计及其进展，帮助完善了获胜策略，并将某些动作集成到软件中。基于上述研究，可以分析出，在机器学习发展的早期阶段，科学家和工程师引入了机器学习应用来提高计算机的智能。
In 1957, Frank Rosenblatt et al. proposed the first neural network for computers namely called “the perceptron” [36], which simulates the thinking patterns in the human brain. Moreover, T.M. Cover and P.E. Hart proposed the “nearest neighbor” algorithm in 1967 [37]. This kind of algorithm is used for simple pattern recognition. Essentially, It was used to make a path for passengers beginning with a random location, but to make sure that they reach all cities on a short ride. Moreover, Stanford University students developed the “Stanford Cart” in 1979 [38], which can navigate obstacles automatically. The Cart was a moderately remotely controlled television-equipped mobile robot in the Stanford Artificial Intelligence Laboratory (SAIL). Other researchers proposed several explanation-based approaches, which can be linked back to the MACROPS learning strategies used in STRIPS [39]. The key sheet of Machine Learning algorithms have been discussed in Figure 2.
1957年，Frank Rosenblatt等人提出了第一个用于计算机的神经网络，称为“感知器 (perceptron)” [36]，它模拟了人脑的思维模式。此外，T.M. Cover和P.E. Hart在1967年提出了“最近邻 (nearest neighbor)”算法 [37]。这种算法用于简单的模式识别。本质上，它用于为乘客从随机位置开始规划路径，但确保他们能在短途旅行中到达所有城市。此外，斯坦福大学的学生在1979年开发了“斯坦福车 (Stanford Cart)” [38]，它可以自动避开障碍物。该车是斯坦福人工智能实验室 (SAIL) 中一种中等远程控制的配备电视的移动机器人。其他研究人员提出了几种基于解释的方法，这些方法可以追溯到STRIPS中使用的MACROPS学习策略 [39]。图2讨论了机器学习算法的关键内容。
According to the expansion and enhancement, it must be credited to Silver, Mitchell, and DeJong. At the same period, each of these researchers built very broad Explanation Based Learning (EBL) computer systems-LP (Silver), ESA (DeJong), and LEX2 (Mitchell). This kind of systems analyzes training data and generates a basic law that can be enforced by discarding the redundant data. It presents a historical account of the development of EBL and discusses some of the important outstanding research tasks [40]. Due to the increasing size of data, in 1998, the Scientists attracted a large number of data programs and applications (Data-Driven Approach using Machine Learning) [41, 42].
根据扩展和增强的内容，必须归功于 Silver、Mitchell 和 DeJong。在同一时期，这些研究人员各自构建了非常广泛的基于解释的学习 (Explanation Based Learning, EBL) 计算机系统——LP (Silver)、ESA (DeJong) 和 LEX2 (Mitchell)。这类系统通过分析训练数据并生成一个基本法则，该法则可以通过丢弃冗余数据来强制执行。它提供了 EBL 发展的历史记录，并讨论了一些重要的未解决的研究任务 [40]。由于数据量的不断增加，1998 年，科学家们吸引了大量的数据程序和应用程序（使用机器学习的基于数据驱动的方法）[41, 42]。
FIGURE 2. Popular Machine Learning Algorithms.
图 2: 流行的机器学习算法。
The above history shows that many researchers have been doing their research from time to time and played a key role in the field of Machine Learning. In the 21st century, the new millennium introduced an abundance of integrated technology. There is Machine Learning where adaptive programs are expected. These algorithms are capable of detecting patterns, extracting new knowledge from inputs, learning from practice, and optimizing the efficiency and accuracy of their analysis and output. Essentially, the researchers have taken the main ideas from the earlier proposed works and have enhanced their work in the form of new inventions to develop new algorithms, software, games, as well as probabilistic reasoning, particularly in the field of automated medical diagnosis [43].
上述历史表明，许多研究人员一直在不断进行研究，并在机器学习领域发挥了关键作用。21世纪，新千年带来了大量集成技术。在需要自适应程序的地方，机器学习应运而生。这些算法能够检测模式，从输入中提取新知识，从实践中学习，并优化其分析和输出的效率和准确性。本质上，研究人员从早期提出的工作中汲取了主要思想，并以新发明的形式增强了他们的工作，开发了新的算法、软件、游戏，以及概率推理，特别是在自动化医疗诊断领域 [43]。
In 2000, Thomas G. Dietterich et al. [44] proposed the ensemble methods in Machine Learning. As we know the original set approach is Bayesian averaging, but several techniques offer error-correcting output coding, boosting, and bagging. The authors analyzed these approaches and clarified why the ensembles can often do better than any single classifier [45]. In 2002, the authors designed a new technique for gene selection using Support Vector Machine (SVM) approaches focused on Recursive Feature Elimination (RFE) for cancer classification [46]. Experimentally, they proved that the genes identified by their strategies yield better detection efficiency and are biologically important to cancer.
2000年，Thomas G. Dietterich等人[44]提出了机器学习中的集成方法。众所周知，最初的集成方法是贝叶斯平均法，但还有几种技术提供了纠错输出编码、提升法和装袋法。作者分析了这些方法，并阐明了为什么集成方法通常比任何单一分类器表现更好[45]。2002年，作者设计了一种新的基因选择技术，使用支持向量机（SVM）方法，专注于递归特征消除（RFE）用于癌症分类[46]。实验证明，通过他们的策略识别出的基因具有更好的检测效率，并且对癌症具有重要的生物学意义。
This research shows that Machine Learning models have already been used to diagnose human diseases. In other research, the authors suggested that Machine Learning should also be a faster and better approach to corner-detection [47].
研究表明，机器学习模型已被用于诊断人类疾病。在其他研究中，作者提出机器学习也应成为角点检测更快、更好的方法 [47]。
2) State-of-the-art
2) 最新技术
Recently, several researchers focusing on Machine Learning technologies and algorithms merge diverse areas to increase production efficiencies, such as cloud computing, biomedical engineering, network security, image processing, forecasting, Internet of Things (IoT), and Big Data technology [48– 53]. In 2019, Wenrui Yang et al. introduced a system for sports image detection using Machine Learning. The purpose of this study was to develop the identification of athletes, the judgment on sport behavior, the perception of motion, and the development of a test framework to validate the effectiveness of the research process [54]. In another study, the authors suggested a combination of big data processing using cloud computing and Machine Learning [55]. In this study, they proposed that, Machine Learning seems to be an ideal solution for exploring the possibilities concealed for big data. Recently, the authors published a performance analysis of the new Machine Learning algorithm and the Logistic Prediction Method called MLIA [56]. In this review, the authors demonstrated that Machine Learning has a significant predictive effect of MLIA on the measurement of financial credit risk and can provide a theoretical basis for subsequent relevant studies.
最近，一些专注于机器学习技术和算法的研究人员将云计算、生物医学工程、网络安全、图像处理、预测、物联网 (IoT) 和大数据技术等不同领域结合起来，以提高生产效率 [48–53]。2019 年，Wenrui Yang 等人介绍了一种使用机器学习进行体育图像检测的系统。该研究的目的是开发运动员识别、运动行为判断、运动感知以及开发测试框架以验证研究过程的有效性 [54]。在另一项研究中，作者提出了结合云计算和机器学习的大数据处理方法 [55]。在这项研究中，他们认为机器学习似乎是探索大数据隐藏可能性的理想解决方案。最近，作者发表了一种新的机器学习算法和称为 MLIA 的逻辑预测方法的性能分析 [56]。在这篇综述中，作者证明了机器学习对金融信用风险测量具有显著的预测效果，并可以为后续相关研究提供理论基础。
FIGURE 3. Key sheet of Machine Learning Algorithms.
图 3: 机器学习算法关键表
In addition to the role of Machine Learning in wireless communication, Machine Learning techniques are anticipated to play a significant part in the implementation of the fifth-generation (5G). Researchers have recently presented an overview of wireless communication channel modeling based on Machine Learning [57]. In this overview, the writers addressed 5G with massive Multiple-Input / Multiple-Output (MIMO), quick handover, higher data rate, and channel simulation becoming more complicated than other conventional stochastic or deterministic models. To this purpose, scholars and academics are looking forward to more effective methods that are less complicated and more reliable. For example, Emerging Machine Learning Methods can offer a new direction for the analysis of big data and traffic data. To diagnosing human diseases by using Deep learning techniques [58] such as, the Convolutional Neural Networks (CNNs) have provided significant performance to boost the fields related to diagnosing human diseases. CNN techniques have been applied successfully for several tasks, like computer-aided diagnosis, image enhancement/generation, classification, and segmentation [59–64]. In comparison of Machine Learning with IoT and Big Data, Jangam J.S Mani and Sandhya Rani Kasireddy propounded a framework that classifies the population into four classes based on diet efficiency. After 30 days of dietary retrieval, they devour as normal, unbalanced, almost balanced, and almost unbalanced by using logistic regression, linear discriminant analysis (LDA), and random forest algorithm [65]. In Figure 3, the popular Machine Learning algorithms have been demonstrated.
除了机器学习在无线通信中的作用外，机器学习技术预计还将在第五代（5G）的实现中发挥重要作用。研究人员最近提出了基于机器学习的无线通信信道建模概述 [57]。在这篇概述中，作者讨论了5G的大规模多输入/多输出（MIMO）、快速切换、更高的数据速率以及信道模拟比其他传统的随机或确定性模型更加复杂的问题。为此，学者们期待更有效、更简单且更可靠的方法。例如，新兴的机器学习方法可以为大数据和流量数据的分析提供新的方向。通过使用深度学习技术 [58] 如卷积神经网络（CNNs），在诊断人类疾病的相关领域中提供了显著的性能提升。CNN技术已成功应用于多个任务，如计算机辅助诊断、图像增强/生成、分类和分割 [59–64]。在机器学习与物联网（IoT）和大数据的比较中，Jangam J.S Mani 和 Sandhya Rani Kasireddy 提出了一个框架，该框架根据饮食效率将人群分为四类。在30天的饮食数据检索后，他们使用逻辑回归、线性判别分析（LDA）和随机森林算法将人群分为正常、不平衡、几乎平衡和几乎不平衡 [65]。图3展示了流行的机器学习算法。
To discuss the data processing architecture, Machine Learning approaches are used to allow precise tuning to train a classifier for large-scale datasets. To serve IoT applications, these infrastructures use Machine Learning or AIbased techniques which evaluate entity or system data to produce valuable knowledge that can be used for service or decision-making. Machine Learning methods make it possible for machines to communicate with people, drive cars automatically, forecasting, writing and publishing sport match reports, and identifying criminal suspects as well. Furthermore, Machine Learning has a serious impact on most businesses and employees inside them, that is why a professional will at least have a context about what Machine Learning is, and how it is evolving [2, 66, 67].
为了讨论数据处理架构，机器学习方法被用于精确调整以训练大规模数据集的分类器。为了服务于物联网应用，这些基础设施使用机器学习或基于人工智能的技术来评估实体或系统数据，从而产生可用于服务或决策的有价值知识。机器学习方法使机器能够与人交流、自动驾驶汽车、进行预测、撰写和发布体育比赛报告，以及识别犯罪嫌疑人。此外，机器学习对大多数企业和内部员工产生了重大影响，这就是为什么专业人士至少需要了解机器学习是什么以及它是如何发展的 [2, 66, 67]。
B. POPULAR ALGORITHMS OF MACHINE LEARNING In this subsection, popular algorithms of Machine Learning are presented. Machine Learning algorithms are specifically programmed to create predictive models dependent on the underlying algorithm and dataset. Input data for Machine Learning algorithms usually consist of “label” and “features” over a range of samples. Labels are what the purpose of a Machine Learning algorithm is to determine, which is the output of the model, whereas the features are the quantities of all tests, either raw or mathematically transformed [68]. The most common Machine Learning algorithms can be divided into two key categories-Supervised learning and Unsupervised learning [69, 70]. Apart from these, some other methodologies of Machine Learning are also discussed in Figure 4.
B. 机器学习中的流行算法
在本小节中，介绍了机器学习中的流行算法。机器学习算法经过专门编程，以创建依赖于基础算法和数据集的预测模型。机器学习算法的输入数据通常由一系列样本中的“标签”和“特征”组成。标签是机器学习算法旨在确定的目标，即模型的输出，而特征是所有测试的量，无论是原始的还是经过数学变换的 [68]。最常见的机器学习算法可以分为两个主要类别——监督学习和无监督学习 [69, 70]。除此之外，图 4 中还讨论了其他一些机器学习方法。
1) Supervised Machine Learning Algorithms
1) 监督式机器学习算法
Supervised Machine Learning is the cognitive activity of discovering relationships between parameters in annotated data (training set). Using this knowledge, making a forecasting model capable of inferring annotations for new data in which annotations are unknown. This kind of algorithm uses the characteristics and annotations of the training set to induce the model to predict the annotations of instances in the test set [71, 72].
监督式机器学习是一种在标注数据（训练集）中发现参数之间关系的认知活动。利用这些知识，建立一个能够推断新数据（其中标注未知）的预测模型。这类算法利用训练集的特征和标注来引导模型预测测试集中实例的标注 [71, 72]。
FIGURE 4. Classification of Machine Learning Algorithms.
图 4: 机器学习算法的分类
2) Unsupervised Machine Learning Algorithms
2) 无监督机器学习算法
Compared with Supervised learning algorithms, Unsupervised learning algorithms work without the desired output label. For instance, an unsupervised learning algorithm analyzes the $a$ without needing the $b$ , whereas a Supervised Machine Learning algorithm usually learns from a method that maps an input $a$ into the $b$ output. Unsupervised learning strategies may be motivated by theoretical and Bayesian concepts of intelligence. Unsupervised learning algorithms usually using to expand the data and train a model for finding suitable internal representation, such as sorting data into clusters [73, 74].
与监督学习算法相比，无监督学习算法在没有期望输出标签的情况下工作。例如，无监督学习算法分析 $a$ 时不需要 $b$，而监督机器学习算法通常通过将输入 $a$ 映射到输出 $b$ 的方法进行学习。无监督学习策略可能受到理论和贝叶斯智能概念的启发。无监督学习算法通常用于扩展数据并训练模型以找到合适的内部表示，例如将数据分类为簇 [73, 74]。
3) Semi-supervised Machine Learning Algorithms
3) 半监督机器学习算法
Traditional class if i ers need to train labeled data (features/label pairs). However, labeled instances are sometimes challenging to procure, time taking and costly, since they involve the efforts of the skilled human annotator. In the meantime, unlabeled data can be relatively easy to get but difficult to use. Semi-supervised learning solves this issue by creating stronger class if i ers utilizing vast volumes of unlabeled data, coupled with the labeled data needing fewer human intervention and higher accuracy [75, 76]. Semisupervised learning algorithms are used in such cases where the labels are missing. For instance, only a limited amount of training data is labeled and the goal is to improve the output of the model that can be done either by avoiding the labels and performing unsupervised learning or by ignoring unlabeled data and performing supervised learning. It is the great interest in both practical and theoretical aspects [68, 77, 78].
传统的分类器需要训练带标签的数据（特征/标签对）。然而，带标签的实例有时难以获取，耗时且成本高，因为它们需要熟练的人工标注者的努力。与此同时，未标记的数据相对容易获取，但难以使用。半监督学习通过利用大量未标记数据和少量带标签数据来创建更强的分类器，从而解决了这个问题，减少了人工干预并提高了准确性 [75, 76]。半监督学习算法用于标签缺失的情况。例如，只有少量训练数据被标记，目标是通过避免标签并执行无监督学习，或忽略未标记数据并执行监督学习来改进模型的输出。这在实践和理论方面都引起了极大的兴趣 [68, 77, 78]。
4) Reinforcement Machine Learning Algorithms
4) 强化机器学习算法
Reinforcement learning is a branch of Machine Learning algorithms in which the learner or software entity tries to perform a sequence of acts that will optimize accumulated incentives, such as winning a checker or chess game. It is an area of research that has been able to overcome a broad variety of complicated decision-making problems which were historically been out of control for the machine. It also opens up a range of new opportunities in areas such as infrastructure, automation, smart grids, banking, and much more [79– 81].
强化学习是机器学习算法的一个分支，其中学习者或软件实体尝试执行一系列行为，以优化累积的激励，例如赢得跳棋或国际象棋比赛。这是一个能够克服历史上机器无法控制的多种复杂决策问题的研究领域。它还在基础设施、自动化、智能电网、银行等领域开辟了一系列新的机会 [79–81]。
These methods led to impressive advances in AI, going beyond human performance in domains ranging from Atari to Go to no-limit poker [82]. These signs of progress attracted the attention of cognitive scientists interested in understanding human learning. Over the last few years, because of its performance in solving the complexities of sequential decision-making, it has become increasingly popular. Some of the achievements were attributed to the combination of rein for cement learning and deep learning methodologies [83– 86].
这些方法在人工智能领域取得了令人瞩目的进展，在从Atari到围棋再到无限注德州扑克等多个领域超越了人类表现 [82]。这些进步迹象吸引了关注人类学习理解的认知科学家的注意。过去几年中，由于其在解决序列决策复杂性方面的表现，该方法变得越来越受欢迎。部分成就归功于强化学习与深度学习方法的结合 [83–86]。
5) Recommend er Systems
5) 推荐系统
Recommend er systems have been built in coexistence with the internet. Initially, this kind of systems were focused on statistical, content-based and shared filtering. Such systems currently integrate social knowledge. Recommend er systems can also be described as learning techniques through which online customers can design their websites to match the customer’s tastes such as, an internet customer may get a product and/or associated products ranking while looking for things based on an established recommendation system. There are mainly two methods, namely content-based recommendation, and collective recommendation. This type of system allows users to get access to it and collect info, principles, intelligent and novel suggestions. Several e-commerce pages use this program [87–90].
推荐系统与互联网共存而生。最初，这类系统主要关注统计、基于内容和共享过滤的方法。如今，这些系统已整合了社交知识。推荐系统也可被视作一种学习技术，通过它，在线客户可以设计自己的网站以匹配客户的喜好。例如，互联网客户在搜索商品时，可能会根据已建立的推荐系统获得产品及/或相关产品的排名。主要有两种方法，即基于内容的推荐和集体推荐。这类系统允许用户访问并收集信息、原则、智能和新颖的建议。多个电子商务页面都采用了这一程序 [87–90]。
FIGURE 5. Popular Applications of Machine Learning.
图 5: 机器学习的流行应用
C. POPULAR APPLICATIONS OF MACHINE LEARNING Many companies and industries dealing with data packages have recognized the importance of Machine Learning technology. By using Machine Learning approaches, businesses can work more reliably and effectively as well as gain an advantage over competitors [91, 92].
C. 机器学习的流行应用
许多处理数据包的公司和行业已经认识到机器学习技术的重要性。通过使用机器学习方法，企业可以更可靠、更有效地工作，并在竞争中占据优势 [91, 92]。
Furthermore, with the aid of compelling articles for clarity to the reader, several applications of Machine Learning are also discussed and divided into different sections in Figure 5, which are given below:
此外，为了帮助读者更好地理解，本文还通过引人入胜的文章讨论了机器学习的几种应用，并在图 5 中将其分为不同的部分，具体如下：
1) Precision Agriculture
1) 精准农业
The most common concepts of Machine Learning in the field of agriculture were proposed by A. Kukuta et al. [93]. Precision agriculture, satellite farming, or site-specific crop management are agricultural management terms focused on observation, estimation, and the reaction of inter-and intrafield crop variability. The key goal of precision agriculture analysis is to set up a decision support system for agricultural management to optimize the return on inputs while maintaining energy [94–96].
农业领域中最常见的机器学习概念由 A. Kukuta 等人 [93] 提出。精准农业 (Precision Agriculture)、卫星农业 (Satellite Farming) 或特定地点作物管理 (Site-Specific Crop Management) 是农业管理术语，侧重于观察、估计和应对田间和田间作物变异。精准农业分析的关键目标是建立一个农业管理决策支持系统，以优化投入回报，同时保持能源 [94–96]。
2) Health care
2) 医疗保健
Machine Learning is an emerging field and fast-growing phenomenon in the field of health care [97, 98]. The advent of smart applications and devices that can use data to assess the health of patients in real-time. The medical professionals analyze data for the detection of patterns or warning flags that can contribute to better diagnosis and care [99–102].
机器学习是医疗保健领域中的一个新兴且快速发展的现象 [97, 98]。智能应用和设备的出现使得能够利用数据实时评估患者的健康状况。医疗专业人员通过分析数据来检测模式或警示标志，从而有助于更好的诊断和护理 [99–102]。
3) Retail
3) 零售
Websites recommend items that you would buy based on prior purchases using a Machine Learning method called a ‘recommendation system’ to evaluate your experience about
网站会根据你之前的购买记录，使用一种称为“推荐系统 (recommendation system)”的机器学习方法，评估你的体验，从而推荐你可能购买的商品。
purchasing the item. Retailers depend on Machine Learning technology to record, interpret, and customize their shopping experience [103, 104].
购买商品。零售商依赖机器学习技术来记录、解释并定制他们的购物体验 [103, 104]。
4) Government
4) 政府
State departments, such as infrastructure and public health have a strong need for artificial intelligence because they offer several data points that can be exploited for information. For instance, analyzing the data of the system to find opportunities to boost efficiency and save money. Machine Learning can also help spot fraud and prevent data theft [105, 106].
基础设施和公共卫生等政府部门对人工智能有着强烈的需求，因为它们提供了多个可被利用的数据点。例如，通过分析系统数据来寻找提高效率和节省成本的机会。机器学习还能帮助发现欺诈行为并防止数据盗窃 [105, 106]。
5) Computational Finance
5) 计算金融
Banks and certain organizations in the business industry are utilizing Machine Learning technologies for two key purposes: the first is to identify valuable insights from data and the second is to prevent fraud. Insights can recognize investment opportunities and allow shareholders to know when to sell. Data mining can also recognize high-risk clients or use cyber monitoring to detect warning signals of fraud [107, 108].
银行和商业领域的某些组织正在利用机器学习技术实现两个关键目标：一是从数据中识别有价值的洞察，二是预防欺诈。洞察能够识别投资机会，并让股东知晓何时出售。数据挖掘还能识别高风险客户，或通过网络监控发现欺诈的预警信号 [107, 108]。
6) Transportation
6) 交通
Analyzing data to detect patterns and developments is crucial for the transport sector, which focuses on keeping roads more effective and finding possible challenges to improve productivity. Information modeling and simulation elements of Machine Learning are useful tools for logistics firms, urban transportation’s, and other transit organizations [109, 110].
分析数据以检测模式和发展对于交通部门至关重要，该部门专注于提高道路效率并发现可能的挑战以提高生产力。机器学习的信息建模和仿真元素是物流公司、城市交通和其他运输组织的有用工具 [109, 110]。
7) Oil and gas
7) 石油和天然气
Use cases include finding new sources of energy, analyzing elements in rocks, predicting malfunction of refinery sensor, streamlining the production of oil to make it more reliable
用例包括寻找新能源、分析岩石中的元素、预测炼油厂传感器故障、简化石油生产以提高可靠性
and cost-effective. The amount of use cases of Machine Learning in this sector is overwhelming and continues to increase [111–114].
机器学习在该领域的应用案例数量庞大且持续增长 [111–114]。
8) Computational linguistics
8) 计算语言学
Computational linguistics has historically been conducted by computer scientists who have specialized in the use of computers for the analysis of natural languages. Nowadays, computer linguists often work as part of interdisciplinary teams, which can include computer scientists, target language specialists, and professional linguists [115, 116].
历史上，计算语言学一直由专门使用计算机分析自然语言的计算机科学家进行。如今，计算机语言学家通常作为跨学科团队的一部分工作，这些团队可能包括计算机科学家、目标语言专家和专业语言学家 [115, 116]。
III. CONVENTIONAL GREY MODELS
III. 传统灰色模型
Basic Grey Model Theory is an interdisciplinary research discipline that was proposed by [117] in 1980. He provided a classic continuous GM(1,1) model in which procedures begin with a differential equation namely ‘whitening equation’. As long as knowledge is concerned, systems that suffer from a lack of information, such as operating mechanism, structure message, and actions log, are referred to as Grey Systems. Throughout the background, the human body, livestock, climate, etc., are the Grey Systems, where “grey” implies incomplete, unknown, poor, etc. The goal of the Grey Model and its applications is to bridge the distance between natural science and social science [118]. This concept has been very popular in terms of its potential to work with systems that have partly unknown parameters. As an improvement over traditional predictive models, grey forecasting models only need small datasets to determine the actions of unknown processes [119]. Some of the existing grey models with a continuous whitening function follow the same linear formula as follows:
基础灰色模型理论是由 [117] 在 1980 年提出的一个跨学科研究领域。他提供了一个经典的连续 GM(1,1) 模型，该模型的流程从一个微分方程开始，即“白化方程”。就知识而言，缺乏信息的系统，如操作机制、结构信息和行为日志，被称为灰色系统。在背景中，人体、牲畜、气候等都是灰色系统，其中“灰色”意味着不完整、未知、贫乏等。灰色模型及其应用的目标是弥合自然科学与社会科学之间的距离 [118]。这一概念在应对部分参数未知的系统方面非常受欢迎。作为对传统预测模型的改进，灰色预测模型只需要少量数据集即可确定未知过程的行为 [119]。一些现有的具有连续白化函数的灰色模型遵循以下相同的线性公式：
\frac{d X^{(1)}(t)}{d t}+a x_{1}^{(1)}(t)=f(\pmb{\theta};t),```
while the series $X_{1}^{(1)}$ is usually referred to as an input series. Sometimes, the function $f(\pmb\theta;t)$ differs by time $t$ or dependency sequence (or input series) $X_{i}^{(1)},i=\dot{2},3,\dots,n$ , with unknown parameters $\theta$ . In case of discrete grey models, the general linear equation can also be written as,
而序列 $X_{1}^{(1)}$ 通常被称为输入序列。有时，函数 $f(\pmb\theta;t)$ 会随时间 $t$ 或依赖序列（或输入序列） $X_{i}^{(1)},i=\dot{2},3,\dots,n$ 的不同而变化，且参数 $\theta$ 未知。在离散灰色模型的情况下，一般线性方程也可以写成：
x_{1}^{(1)}(k+1)=\alpha x_{1}^{(1)}(k)+f(\pmb\theta;k).```
It is clear that the solutions of grey models in the abovementioned formulations also contain similar formulations and the same equations. Several grey models also use the initial condition \$\dot{X}_{1}^{(1)}(1)\,=\,X_{1}^{(0)}\check{(1)}\$ . The general formulation of these models can thus easily be accessed.
显然，上述公式中的灰色模型解也包含类似的公式和相同的方程。一些灰色模型还使用初始条件 \$\dot{X}_{1}^{(1)}(1)\,=\,X_{1}^{(0)}\check{(1)}\$。因此，这些模型的一般公式可以很容易地获得。
For continuous models, the answer can always be described as the following convolution equation:
对于连续模型，答案总是可以描述为以下卷积方程：
\hat{X}{1}^{(1)}(t)=X{1}^{(0)}(1)\cdot e^{-a(t-1)}+\int_{1}^{t}e^{-a(t-\tau)}f(\pmb\theta;\tau)d\tau.```
Whereas, in the case of discrete models, the solution can always be described as the following discrete convolution equation:
在离散模型的情况下，解总是可以描述为以下离散卷积方程：
\hat{X}_{1}^{(1)}(k+1)=X_{1}^{(0)}(1)\cdot\alpha^{k}+\sum_{\tau=2}^{k+1}\alpha^{(k+1-\tau)}f(\pmb\theta;\tau).```
Grey forecasting models are essentially divided into two categories: Univariate and Multivariate. Single-variable models are called univariate while multivariable models are called multivariate [120]. For descriptive purposes, the whitening equation, time response function, and the restored values of grey models are provided in this section and also summarized in Figure 6.
灰色预测模型主要分为两类：单变量和多变量。单变量模型被称为单变量模型，而多变量模型被称为多变量模型 [120]。为了便于描述，本节提供了灰色模型的白化方程、时间响应函数和还原值，并在图 6 中进行了总结。
A. EVOLUTION OF GREY FORECASTING MODELS Since the study of Lin and Liu, the propagation of this Grey System theory took place as follows: Scholarly periodical for the presentation of research results followed by ‘Journal of Grey System’ that started to be published in England in 1989. More than 300 various scientific journals recognize and publish papers relevant to the grey system in the world. Moreover, In the early 1990s, several universities based throughout China, Taiwan, Australia, United States, and Japan began offering grey system theory courses. The Chinese Grey System Association (CGSA) was founded in 1996. Every year, CGSA conducts a conference on Grey System Theory and its application [121]. In the last four decades, the Grey System Theory has developed quickly and drawn the interest of several researchers. It has been widely and effectively extended to many applications such as commercial, manufacturing, transport, medical, military, mechanical, meteorological, civil, political, financial, science and technology, agricultural, hydrological, geological, etc. Furthermore, the conventional grey model called GM(1,1) has been widely adopted, and its forecasting efficiency could also be improved. To date, several researchers have proposed new approaches for improving the performance of the model as Deng et al. [122] have proposed the modifiable residual sequence method. Whereas, Mu et al. [123] obtained the formula of optimum grey derivative whitening values. He proposed an unbiased GM(1,1) model to develop a framework for estimating the parameters.
A. 灰色预测模型的演变
自 Lin 和 Liu 的研究以来，灰色系统理论的传播如下：研究成果的学术期刊发布，随后《灰色系统杂志》于 1989 年在英国开始出版。全球有超过 300 种科学期刊认可并发表与灰色系统相关的论文。此外，在 20 世纪 90 年代初，中国、台湾、澳大利亚、美国和日本的几所大学开始开设灰色系统理论课程。中国灰色系统协会 (CGSA) 于 1996 年成立。每年，CGSA 都会举办关于灰色系统理论及其应用的会议 [121]。在过去的四十年中，灰色系统理论发展迅速，并引起了众多研究人员的兴趣。它已被广泛且有效地扩展到许多应用领域，如商业、制造、交通、医疗、军事、机械、气象、土木、政治、金融、科技、农业、水文、地质等。此外，传统的灰色模型 GM(1,1) 已被广泛采用，其预测效率也得到了提升。迄今为止，许多研究人员提出了改进模型性能的新方法，如 Deng 等人 [122] 提出了可修改残差序列方法。而 Mu 等人 [123] 获得了最优灰色导数白化值的公式。他提出了一个无偏 GM(1,1) 模型，用于开发参数估计的框架。
Furthermore, Song and Wang developed the center approach of the alteration of grey model GM(1,1). They designed an adjusting grey model [124, 125]. Tan et al. supported the structure method of the background values in the GM(1,1) model and a basic approximation of the background value function was re-established, which had strong adaptability [126]. In another studies, the authors presented the optimal time-response sequence formula and used the least square approach to measure the constant number in the time-response series of the basic GM(1,1) model [126– 128]. Several researchers examined the appropriate scope and simulation accuracy of the GM(1,1) model [129, 130]. Moreover, some key approaches shall include center approach method [124], discrete models [131–133], correcting the residues [122], constructing background values [126], and optimization of the grey derivative [125]. A part of these, some other grey forecasting theory methods proposed by scholars [134–145].
此外，Song 和 Wang 开发了灰色模型 GM(1,1) 的改进中心方法。他们设计了一种调整灰色模型 [124, 125]。Tan 等人支持 GM(1,1) 模型中背景值的结构方法，并重新建立了背景值函数的基本近似，具有较强的适应性 [126]。在其他研究中，作者提出了最优时间响应序列公式，并使用最小二乘法来测量基本 GM(1,1) 模型时间响应序列中的常数 [126–128]。一些研究人员研究了 GM(1,1) 模型的适用范围和模拟精度 [129, 130]。此外，一些关键方法包括中心方法 [124]、离散模型 [131–133]、修正残差 [122]、构建背景值 [126] 和优化灰色导数 [125]。除此之外，学者们还提出了其他一些灰色预测理论方法 [134–145]。
FIGURE 6. Popular Grey Forecasting Models.
图 6: 常见的灰色预测模型
# 1) Univariate Grey Model
# 1) 单变量灰色模型
The single parameter grey prediction model is GM(1,1) with one vector and one first-order equation, and a simulation unit is a single time sequence. It forecasts the future of the program by defining machine operating rules contained in a series focused on grey generation approaches. The singlevariable grey model might not take into account the effect of related factors on the system and thus, it has the benefit of the basic modeling process [146].
单参数灰色预测模型是GM(1,1)，它包含一个向量和一个一阶方程，模拟单元是一个单一的时间序列。它通过定义一系列专注于灰色生成方法中包含的机器操作规则来预测程序的未来。单变量灰色模型可能没有考虑相关因素对系统的影响，因此它具有基本建模过程的优势 [146]。
Classic GM(1,1) model is an effective method for precise predictions for small samples. Therefore, it is not surprising that GM(1,1) is commonly used as a predictive tool [120]. In order to define the GM(1,1) model, the first ‘1’ represents for the ‘first order’, while the second ‘1’ stands the ‘univariate’ [147]. The system parameters are calculated by de-escalating the whitening equation and using the least square method. For instance, The definition of the Grey Theory is the ‘Grey Box’ where information is established and knowledge is uncertain. Grey system theory is an important tool for determining unknown issues with limited samples and incomplete information [117]. The Whitening equations of the univariate grey forecasting models have been addressed in Table 1.
经典 GM(1,1) 模型是一种针对小样本进行精确预测的有效方法。因此，GM(1,1) 常被用作预测工具并不令人意外 [120]。为了定义 GM(1,1) 模型，第一个“1”代表“一阶”，而第二个“1”代表“单变量” [147]。系统参数通过降阶白化方程并使用最小二乘法计算得出。例如，灰色理论的定义是“灰箱”，其中信息是已知的，而知识是不确定的。灰色系统理论是在样本有限且信息不完整的情况下确定未知问题的重要工具 [117]。单变量灰色预测模型的白化方程已在表 1 中列出。
# 1. GM(1,1)
# 1. GM(1,1)
Among the families of grey models, the GM(1,1) model is the most widely used, due to its simplicity and high accuracy for limited datasets [148]. Based on this essential function, it has been successfully implemented in several fields. Widely used applications of GM (1,1) include energy production [149–152], the prediction of stock price [137], the oil production in China [153, 154], the consumption of energy [155, 156], detection [157], and the electricity consumption [149, 158]. Let \$X^{(0)}\,=\,\left\{x^{(0)}(1),x^{(0)}(2),\dots,x^{(0)}(n)\bar{\}\right.\$ denote original data, \$X^{(1)}\,=\,\left\{x^{(1)}(1),x^{(1)}(2),\dots,x_{\mathrm{~}}^{(1)}(n)\right\}\$ is the first order accumulation generator, \$\begin{array}{r l r l}{Z^{(1)}}&{{}}&{=}\end{array}\$ \$\left\{z^{(1)}(1),z^{(1)}(2),\ldots,z^{(1)}(n)\right\}\$ is the background of \$X^{(0)}\$ where \$z^{(1)}(k)=0.5\left(x^{(1)}(k)\bar{+}\,x^{(1)}(k-1)\right)\!.\$ . Hence,
在灰色模型家族中，GM(1,1) 模型因其简单性和对有限数据集的高准确性而得到广泛应用 [148]。基于这一基本功能，它已成功应用于多个领域。GM(1,1) 的广泛应用包括能源生产 [149–152]、股票价格预测 [137]、中国石油生产 [153, 154]、能源消耗 [155, 156]、检测 [157] 以及电力消耗 [149, 158]。设 \$X^{(0)}\,=\,\left\{x^{(0)}(1),x^{(0)}(2),\dots,x^{(0)}(n)\bar{\}\right.\$ 表示原始数据，\$X^{(1)}\,=\,\left\{x^{(1)}(1),x^{(1)}(2),\dots,x_{\mathrm{~}}^{(1)}(n)\right\}\$ 为一阶累加生成序列，\$\begin{array}{r l r l}{Z^{(1)}}&{{}}&{=}\end{array}\$ \$\left\{z^{(1)}(1),z^{(1)}(2),\ldots,z^{(1)}(n)\right\}\$ 是 \$X^{(0)}\$ 的背景值，其中 \$z^{(1)}(k)=0.5\left(x^{(1)}(k)\bar{+}\,x^{(1)}(k-1)\right)\!.\$。因此，
x^{(0)}(k)+a z^{(1)}(k)=b```
is known as Grey model GM(1,1). The restrictions of $-a$ and $b$ in the grey basic form of GM(1,1) model are referred to as development coefficient and grey action quantity, respectively. Whereas the time response signal of GM(1,1) is,
被称为灰色模型 GM(1,1)。GM(1,1) 模型的灰色基本形式中 $-a$ 和 $b$ 的限制分别被称为发展系数和灰色作用量。而 GM(1,1) 的时间响应信号为，
\hat{x}^{(1)}(k+1)=\left(x^{(0)}(1)-\frac{b}{a}\right)\mathrm{e}^{-a k}+\frac{b}{a},\quad k=1,2,\dots,n```
while the predictive value of GM(1,1) is obtained as,
GM(1,1) 的预测值如下：
\begin{array}{c}{{\hat{x}^{(0)}(k+1)=\hat{x}^{(1)}(k+1)-\hat{x}^{(1)}(k)}}\ {{{}}}\ {{{}=(1-\mathrm{e}^{a})\left(x^{(0)}(1)-{\frac{b}{a}}\right)\mathrm{e}^{-a k},}}\ {{{}}}\ {{{\displaystyle k=1,2,\ldots,n}}}\end{array}```
Let $\begin{array}{c c l}{{\widehat{X}^{(0)}}}&{{=}}&{{\left{\hat{x}^{(0)}(1),\hat{x}^{(0)}(2),\ldots,\hat{x}^{(0)}(n)\right}}}\end{array}$ . Therefore, $X^{(0)}$ is the simulation sequence and $X^{(0)},\cdot,\hat{x}^{(1)}(k+1)$ is the simulation data of $x^{(\bar{1})}(k+1)$ . According to the (3), it is simple to show that the simulation data sequence is the geometric series [121], and thus, the growth rate of the simulation series is constant:
设 $\begin{array}{c c l}{{\widehat{X}^{(0)}}}&{{=}}&{{\left{\hat{x}^{(0)}(1),\hat{x}^{(0)}(2),\ldots,\hat{x}^{(0)}(n)\right}}}\end{array}$ 。因此，$X^{(0)}$ 是模拟序列，$X^{(0)},\cdot,\hat{x}^{(1)}(k+1)$ 是 $x^{(\bar{1})}(k+1)$ 的模拟数据。根据 (3)，可以简单证明模拟数据序列是几何级数 [121]，因此模拟序列的增长率为常数：
\hat{u}(k)=\frac{\hat{x}^{(0)}(k+1)-\hat{x}^{(0)}(k)}{\hat{x}^{(0)}(k)}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}\\ {=\displaystyle\frac{\hat{x}^{(0)}(k+1)}{\hat{x}^{(0)}(k)}-1={\mathrm{e}}^{-a}-1.\qquad\qquad\qquad}```
# \$2.\mathsf {N G M}(1,1,k,c)\$
# \$2.\mathsf {N G M}(1,1,k,c)\$
Chen and Yu designed a parameter optimization approach to boost the \$\mathbf{NGM}(1,\,1,\,k,\,c)\$ model combine with grey action quantity \$b t+c\$ [134]. Based on the specified form of the latest NGM \$(1,1,K,c)\$ model, the differential equation for the \$\mathbf{NGM}(1,1,k,c)\$ is obtained as,
Chen 和 Yu 设计了一种参数优化方法，结合灰色作用量 \$b t+c\$ 来提升 \$\mathbf{NGM}(1,\,1,\,k,\,c)\$ 模型 [134]。基于最新的 NGM \$(1,1,K,c)\$ 模型的特定形式，\$\mathbf{NGM}(1,1,k,c)\$ 的微分方程如下：
{\frac{\mathrm{d}x^{(1)}(t)}{\mathrm{d}t}}+a x^{(1)}(t)=b t+c,```
while $a$ is the developing coefficient, and $b t+c$ is a grey action quantity. Furthermore, the time response function for the $\mathbf{NGM}(1,1,k,c)$ is,
其中 $a$ 是发展系数，$b t+c$ 是灰色作用量。此外，$\mathbf{NGM}(1,1,k,c)$ 的时间响应函数为，
\begin{array}{r}{\hat{x}^{(1)}(k)=\bigg(x^{(0)}(1)+\frac{b}{a^{2}}-\frac{b}{a}-\frac{c}{a}\bigg)\,e^{-a(k-1)}}\\ {+\frac{b}{a}k-\frac{b}{a^{2}}+\frac{c}{a},k=2,3,\dots,n,}\end{array}```
Whereas the restored function of \$\mathbf{NGM}(1,1,k,c)\$ model can be written as,
而 \$\mathbf{NGM}(1,1,k,c)\$ 模型的恢复函数可以表示为,
\begin{array}{r}{\hat{x}^{(0)}(k)=\bigg(x^{(0)}(1)+\cfrac{b}{a^{2}}-\cfrac{b}{a}-\cfrac{c}{a}\bigg)}\ {(1-e^{a}),e^{-a(k-1)}+\cfrac{b}{a},}\ {k=2,3,,.,.,,n.}\end{array}```
3. $},}(1,1,t^)$
3. $},}(1,1,t^)$
In 2012, Qian et al. developed a new forecasting grey model called $\mathrm{GM}(1,1,\mathit{t}^{\alpha})$ with grey action quantity of $b t^{\alpha}+c$ , and used it to forecast the settlement of the foundation [135]. The whitening differential equation of $\mathrm{GM}(1,,1,!t^{\alpha})$ model based on [159] is,
2012年，Qian等人开发了一种新的预测灰色模型，称为$\mathrm{GM}(1,1,\mathit{t}^{\alpha})$，其灰色作用量为$b t^{\alpha}+c$，并用于预测地基沉降[135]。基于[159]的$\mathrm{GM}(1,,1,!t^{\alpha})$模型的白化微分方程为，
\frac{d x^{(1)}(t)}{d t}+a x^{(1)}(t)=b t^{\alpha}+c,r>0,\alpha>0,```
Whereas the time response function of \$\mathrm{GM}(1,1,\!t^{\alpha})\$ model is,
\$\mathrm{GM}(1,1,\!t^{\alpha})\$ 模型的时间响应函数为
\begin{array}{r}{x^{(r)}(k)=\left(x^{(0)}(1)-\displaystyle\frac{c}{a}\right)e^{-a(k-1)}+\displaystyle\frac{c}{a}+\frac{b}{2}e^{-a(k-1)}}\ {\displaystyle\sum_{-1}^{k-1}\left(\tau^{\alpha}e^{a(\tau-1)}+(\tau+1)^{\alpha}e^{a\tau}\right),}\ {k=2,3,\dots,n,}\end{array}```
and restored value of $\hat{x}^{(0)}(k)k=2,3,\ldots,n$ is given by,
$\hat{x}^{(0)}(k)k=2,3,\ldots,n$ 的还原值由下式给出，
x^{(0)}(k)=x^{(1)}(k)-x^{(1)}(k-1).```
# \$4.\; {\sf G M P}(1,1, {\cal N})\$
# \$4.\; {\sf G M P}(1,1, {\cal N})\$
In 2017, Luo and Wei [160] proposed a grey model with a polynomial term called \$\mathrm{GMP}(1,1,\!N)\$ where the grey action term is a time polynomial function. The GM(1,1) model, the \$\mathrm{NGM}(1,1,\boldsymbol{k})\$ model, and the \$\mathrm{GM}(1,1,\ t^{\alpha})\$ model have been shown to be special cases of the \$\mathrm{GMP}(1,1,\!N)\$ model. The differential form of the GMP \$(1,1,\!N)\$ model obtained as,
2017年，Luo和Wei [160] 提出了一种带有多项式项的灰色模型，称为 \$\mathrm{GMP}(1,1,\!N)\$，其中灰色作用项是时间多项式函数。GM(1,1) 模型、\$\mathrm{NGM}(1,1,\boldsymbol{k})\$ 模型和 \$\mathrm{GM}(1,1,\ t^{\alpha})\$ 模型已被证明是 \$\mathrm{GMP}(1,1,\!N)\$ 模型的特殊情况。GMP \$(1,1,\!N)\$ 模型的微分形式如下：
\frac{d x^{(1)}(t)}{d t}+a x^{(1)}(t)=\beta_{0}+\beta_{1}t+\beta_{2}t^{2}+\cdot\cdot\cdot+\beta_{\mu}t^{n},```
Whereas the time response function for the GMP $(1,1,!N)$ is,
而 GMP $(1,1,!N)$ 的时间响应函数为，
x^{(1)}(k)=\left(x^{(0)}(1)-\sum_{r=0}^{k}r_{i}\right)e^{-a(k-1)}+\sum_{i=0}^{N}\left(r^{i}k^{i}\right),```
Finally, the restored value for the GMP \$\left(1,1,\!N\right)\$ is,
最后，GMP \$\left(1,1,\!N\right)\$ 的恢复值为，
x^{(0)}(k)=x^{(1)}(k)-x^{(1)}(k-1).```
$5.;\mathsf(1,1)$
$5.;\mathsf(1,1)$
The Grey Verhulst model GVM(1,1) is suitable for forecasting the frequency of sequences that have a single apex or whose development delayed [164]. Suppose that, there is a positive sequence of data $\begin{array}{r l}{X^{(0)}}&{{}=}\end{array}$ $\left{x^{(0)}(1),x^{(0)}(2),\ldots,\stackrel{\cdot}{x}^{(0)}(n)\right}$ , and $X^{(1)}$ is accumulated generating operation (AGO) of $X^{(0)}$ , written as X(1) $X^{(1)}\ \ =\ \ \left{x^{(1)}(1),x^{(1)}(2),\ldots,x^{(1)}(n)\right}$ , where, $\begin{array}{r l r}{x^{(1)}(k)}&{{}=}&{\sum_{i=1}^{K}\overset{\setminus}{x^{(0)}}(\overset{.}{i})}\end{array}$ , k = 1, 2, . . . , n. $Z^{(1)}\quad=$ $\left{z^{(1)}(1),z^{(1)}({\overline{{2)}}},\ldots,z^{(1)}(n)\right}$ is the mean sequence of $\bar{x}^{(1)}(k)$ , while, $z^{(1)}(k);=;\textstyle{\frac{1}{2}}:\bigl(x^{(1)}(k)+x^{(1)}(k-1)\bigr)$ , $k,=$ $2,3,\ldots,n$ [165].
灰色 Verhulst 模型 GVM(1,1) 适用于预测具有单一顶点或发展延迟的序列频率 [164]。假设存在一个正数据序列 $\begin{array}{r l}{X^{(0)}}&{{}=}\end{array}$ $\left{x^{(0)}(1),x^{(0)}(2),\ldots,\stackrel{\cdot}{x}^{(0)}(n)\right}$，且 $X^{(1)}$ 是 $X^{(0)}$ 的累加生成操作 (AGO)，记为 $X^{(1)}\ \ =\ \ \left{x^{(1)}(1),x^{(1)}(2),\ldots,x^{(1)}(n)\right}$，其中 $\begin{array}{r l r}{x^{(1)}(k)}&{{}=}&{\sum_{i=1}^{K}\overset{\setminus}{x^{(0)}}(\overset{.}{i})}\end{array}$，k = 1, 2, . . . , n。$Z^{(1)}\quad=$ $\left{z^{(1)}(1),z^{(1)}({\overline{{2)}}},\ldots,z^{(1)}(n)\right}$ 是 $\bar{x}^{(1)}(k)$ 的均值序列，其中 $z^{(1)}(k);=;\textstyle{\frac{1}{2}}:\bigl(x^{(1)}(k)+x^{(1)}(k-1)\bigr)$，$k,=$ $2,3,\ldots,n$ [165]。
y^{(1)}(k+1)=\beta_{0}+\beta_{1}k+\beta_{2}y^{(1)}(k),```
Equation (18) is an optimized discrete Verhulst model. Whereas the differential equation of the conventional grey verhulst model [161] is given below:
方程 (18) 是一个优化的离散 Verhulst 模型。而传统的灰色 Verhulst 模型 [161] 的微分方程如下：
\frac{d\left(x^{(1)}\right)}{d t}+a x^{(1)}=b\left(x^{(1)}\right)^{2},```
Whereas the time response function for Grey Verhulst model is,
而灰色 Verhulst 模型的时间响应函数为，
x^{(1)}(k+1)=\frac{a x^{(1)}(0)}{b x^{(1)}(0)+\left(a-b x^{(1)}(0)\right)E X P(a k)},```
while the restored value of grey verhulst model is,
灰色Verhulst模型的恢复值为，
\hat{x}^{(0)}(k)=\hat{x}^{(1)}(k)-\hat{x}^{(1)}(k-1),k=2,3,\cdot\cdot\cdot\cdot,n.```
- NGBM(1,1)
- NGBM(1,1)
The nonlinear grey Bernoulli model NGBM(1,1) is a branch of the Grey Verhulst model and GM(1,1) [8, 166]. The key advantage of NGBM(1,1) is the power exponents of the model will effectively represent the non-linear features of the true structure and evaluate the form of the model in a versatile manner. Therefore, the framework can not only forecast sequences that increase or decrease monotonically but can also respond favorably to nonlinear processes with small sample sizes [162, 167]. The parameters of the NGBM(1,1) are also calculated by the least square estimation approach known as the Levenberge Marquardt (LM) optimization theory, and then use the modified model to allow analytical comparisons with the regression models, the traditional GM(1,1) and the Grey Verhulst models, which suggests that NGBM(1,1) shows the desirable prediction accuracy [168, 169]. Therefore, it is empirically proved that the performance of the NGBM(1,1) is higher than that of the GM(1,1) and Grey Verhulst model [170]. The differential equation of the NGBM(1,1) model is given by [162],
非线性灰色伯努利模型 NGBM(1,1) 是灰色 Verhulst 模型和 GM(1,1) 模型的一个分支 [8, 166]。NGBM(1,1) 的关键优势在于其幂指数能够有效表示真实结构的非线性特征，并以一种多功能的方式评估模型的形式。因此，该框架不仅能够预测单调递增或递减的序列，还能在小样本情况下对非线性过程做出良好的响应 [162, 167]。NGBM(1,1) 的参数也是通过最小二乘估计方法（称为 Levenberge Marquardt (LM) 优化理论）计算的，然后使用修改后的模型进行与回归模型、传统 GM(1,1) 模型和灰色 Verhulst 模型的对比分析，结果表明 NGBM(1,1) 具有理想的预测精度 [168, 169]。因此，经验证明 NGBM(1,1) 的性能高于 GM(1,1) 和灰色 Verhulst 模型 [170]。NGBM(1,1) 模型的微分方程由 [162] 给出，
TABLE 1. The definition of the comparative Univariate grey models used.
表 1. 使用的比较单变量灰色模型定义
| No. | 模型名称 | 微分方程 | 参考文献 |
|---|---|---|---|
| 1 | GM(1,1) | dx(1)(t) +ax(1)(t) =b p | [121] |
| 2 | NGM(1,1,k,c) | x(0)(k) =ax(0)(k - 1) + b | [134] |
| 3 | GM(1,1,tα) | [135] | |
| 4 | GMP(1,1,N) | [160] | |
| 5 | GVM(1,1) | q(I+-u)+(-)(0)∞==(?)(0)= | [161] |
| 6 | NGBM(1,1) | d(l(+ax(1)(t)=b(x(1)(t)² 7P | [162] |
| 7 | GBM(1,1) | dx(1)(t) =a(M- x(1)(t) + bx(1)(t) dt | x(1)(t) [163] M |
\frac{\mathrm{d}x^{(1)}(t)}{\mathrm{d}t}+a x^{(1)}(t)=b\left(x^{(1)}(t)\right)^{\gamma},```
Furthermore, the time response function of the NGBM(1,1) model is obtained as,
此外，NGBM(1,1) 模型的时间响应函数为，
x^{(r)}(t)=\left[\left(\left(x^{(r)}(1)\right)^{1-\gamma}-\frac{b}{a}\right)e^{-a(1-\gamma)(t-1)}+\frac{b}{a}\right]^{\frac{1}{1-\gamma}},```
while the prediction value of NGBM(1,1) is,
NGBM(1,1) 的预测值为,
\begin{array}{r}{x^{(r)}(k)=\left[\left(\left(x^{(r)}(1)\right)^{1-\gamma}-\frac{b}{a}\right)e^{-a(1-\gamma)(k-1)}+\frac{b}{a}\right]^{\frac{1}{1-\gamma}}}\\ {k=2,3,\ldots,n.}\end{array}```
# 7. GBM(1,1)
# 7. GBM(1,1)
In 1969, Bass et al. proposed once the purchasing model of durable goods by market research on the prevalence of 11 types of durable goods, abbreviated as a Grey Bass model GBM(1,1) [163]. Due to the simple form and the clear economic sense of the parameters, the Bass model is commonly used in new product forecasts [171, 172], technological diffusion [173], and business model diffusion [174–176]. Moreover, the mathematical form of the grey bass model is,
1969年，Bass等人通过对11种耐用品的市场调研，提出了耐用品的购买模型，简称为灰色Bass模型GBM(1,1) [163]。由于模型形式简单且参数具有明确的经济意义，Bass模型常被用于新产品预测 [171, 172]、技术扩散 [173] 以及商业模式扩散 [174–176]。此外，灰色Bass模型的数学形式为，
\frac{d x^{(1)}(t)}{d t}=a\left(M-x^{(1)}(t)\right)+b x^{(1)}(t)\left(1-\frac{x^{(1)}(t)}{M}\right),```
while the time response function of the grey bass model is,
灰色Bass模型的时间响应函数为，
\hat{x}^{(1)}(k)=M\left[\frac{1-e^{-(a+b)k}}{1+\frac{b}{a}e^{-(a+b)k}}\right],k=1,2,\cdot\cdot\cdot\cdot,n.```
At last, the restored value of the grey bass model is,
最后，灰色贝斯模型的恢复值为，
\hat{x}^{(0)}(k+1)=\hat{x}^{(1)}(k+1)-\hat{x}^{(1)}(k),k=1,2,\cdot\cdot\cdot\cdot,n.```
2) Multivariate Grey Models
2) 多元灰色模型
The multivariate grey forecasting model is represented by ${\mathrm{GM}}(1{,}n)$ . This model consists of system-specific sequences (or dependent variable sequences) and $(n{-}1)$ related sequences of variables (or independent variable sequences). The modeling method takes complete account of the effect of relevant variables on system transition and is a standard causal forecasting model. The ${\mathrm{GM}}(1{,}n)$ model has some similarities to the multi-regression model, yet it is fundamentally distinct. The former is focused on grey theory, while the latter is centered on probability statistics. The multivariate gray forecasting model is the limitation of the single-fold framework and the limited simulation potential of single-variable models. This model is mainly used as a tool for determining the similarities between system features sequences and associated factor sequences [140–144]. The detailed differential equation of multivariate grey forecasting models have been summarized in Table 2
多元灰色预测模型由 ${\mathrm{GM}}(1{,}n)$ 表示。该模型由系统特定序列（或因变量序列）和 $(n{-}1)$ 个相关变量序列（或自变量序列）组成。建模方法充分考虑了相关变量对系统转换的影响，是一种标准的因果预测模型。${\mathrm{GM}}(1{,}n)$ 模型与多元回归模型有一些相似之处，但本质上有所不同。前者基于灰色理论，而后者基于概率统计。多元灰色预测模型是单变量模型的局限性和有限模拟潜力的扩展。该模型主要用于确定系统特征序列与相关因子序列之间的相似性 [140–144]。多元灰色预测模型的详细微分方程已在表 2 中总结。
1. $\mathsf(1,!n)$
1. $\mathsf(1,!n)$
${\mathrm{GM}}(1{,}n)$ demonstrated that the approximate whitening time response function of ${\mathrm{GM}}(1{\mathrm{,}}n)$ could often contribute to an inappropriate experimental error [177, 178]. Furthermore, $\textstyle\sum_{i=2}^{N}b_{i}x_{i}^{(1)}$ ios fd ethfien edw hbiyt,ening equation $\begin{array}{r l}{\frac{d x_{1}^{(1)}}{d t};+;a x_{1}^{(1)}}&{=}\end{array}$
${\mathrm{GM}}(1{,}n)$ 表明，${\mathrm{GM}}(1{\mathrm{,}}n)$ 的近似白化时间响应函数常常会导致不适当的实验误差 [177, 178]。此外，$\textstyle\sum_{i=2}^{N}b_{i}x_{i}^{(1)}$ ios fd ethfien edw hbiyt,ening 方程 $\begin{array}{r l}{\frac{d x_{1}^{(1)}}{d t};+;a x_{1}^{(1)}}&{=}\end{array}$
\begin{array}{l}{{x_{1}^{(1)}(t)=e^{-a t}}}\\ {{\displaystyle\left[x_{1}^{(1)}(0)-t\sum_{i=2}^{N}b_{i}x_{i}^{(1)}(0)+\sum_{i=2}^{N}\int b_{i}x_{i}^{(1)}(t)e^{a t}d t\right],}}\end{array}```
When \$\begin{array}{r l r}{X_{i}^{(1)}(i}&{{}=}&{1,2,\dots,N)}\end{array}\$ changes marginally, \$\sum_{i=2}^{N}b_{i}x_{i}^{(1)}(k)\$ is shown as a grey constant. While the estimated time-response sequence of the \${\mathrm{GM}}(1{,}n)\$ model is obtained as,
当 \$\begin{array}{r l r}{X_{i}^{(1)}(i}&{{}=}&{1,2,\dots,N)}\end{array}\$ 发生微小变化时，\$\sum_{i=2}^{N}b_{i}x_{i}^{(1)}(k)\$ 表现为一个灰色常数。而 \${\mathrm{GM}}(1{,}n)\$ 模型的估计时间响应序列为，
\begin{array}{r}{\hat{x}{1}^{(1)}(k+1)=\Bigg[x{1}^{(1)}(0)-\displaystyle\frac{1}{a}\sum_{i=2}^{N}b_{i}x_{i}^{(1)}(k+1)\Bigg],e^{-a k}}\ {+\displaystyle\frac{1}{a}\sum_{i=2}^{N}b_{i}x_{i}^{(1)}(k+1),}\end{array}```
where $x_{1}^{(1)}(0)$ is assumed to be $x_{1}^{(0)}(1)$ , which is the initial value of the ${\mathrm{GM}}(1{\mathrm{,}}n)$ model. Restoration of the inverse accumulation of ${\mathrm{GM}}(1{,}n)$ is obtained by,
其中 $x_{1}^{(1)}(0)$ 被假设为 $x_{1}^{(0)}(1)$ ，这是 ${\mathrm{GM}}(1{\mathrm{,}}n)$ 模型的初始值。通过以下方式获得 ${\mathrm{GM}}(1{,}n)$ 的逆累加恢复：
\hat{x}_{1}^{(0)}(k+1)=\alpha^{(1)}\hat{x}_{1}^{(1)}(k+1)=\hat{x}_{1}^{(1)}(k+1)-\hat{x}_{1}^{(1)}(k).```
# 2. \$ {\mathsf{G M C}}(1,n)\$
# 2. \$ {\mathsf{G M C}}(1,n)\$
To compare \${\mathrm{GMC}}(1,\!n)\$ with other prediction models, the \${\mathrm{GMC}}(1{\mathrm{,}}n)\$ model requires a greater proportion of the forecast core data and is far more significant than the other forecast source data. The accuracy of the indirect estimation and forecast of the \${\mathrm{GMC}}(1,\!n)\$ model can then be presumed. This framework is used to test indirect measurements which cannot be sufficiently accurate for the needs of the analyzer. Since the analyzer can find it appropriate to search for external guidance factors which can improve the accuracy to a certain level [179]. The authors suggested that the \${\mathrm{GMC}}(1{\mathrm{,}}n)\$ model has the following linear differential equation,
为了比较 \${\mathrm{GMC}}(1,\!n)\$ 与其他预测模型，\${\mathrm{GMC}}(1{\mathrm{,}}n)\$ 模型需要更大比例的预测核心数据，并且比其他预测源数据更为重要。因此，可以推测 \${\mathrm{GMC}}(1,\!n)\$ 模型的间接估计和预测的准确性。该框架用于测试无法满足分析者需求的间接测量，因为分析者可能会寻找外部指导因素，以提高准确性至一定水平 [179]。作者建议 \${\mathrm{GMC}}(1{\mathrm{,}}n)\$ 模型具有以下线性微分方程，
x_{1}^{(0)}(r p+t)+b_{1}z_{1}^{(1)}(r p+t)=\sum_{i=2}^{n}b_{i}z_{i}^{(1)}(t)+u,```
while the time response function of ${\mathrm{GMC}}(1{\mathrm{,}}n)$ can be obtained by,
${\mathrm{GMC}}(1{\mathrm{,}}n)$ 的时间响应函数可以通过以下方式获得：
\begin{array}{l}{{\hat{x}_{1}^{(1)}(r p+t)=x_{1}^{(0)}(r p+1)e^{-b_{1}(t-1)}}}\\ {{\qquad\qquad\qquad\qquad+\displaystyle\int_{1}^{t}e^{-b_{1}(t-\tau)}f(\tau)d t,}}\end{array}```
and the predicted value of \${\mathrm{GMC}}(1,\!n)\$ is given as,
\${\mathrm{GMC}}(1,\!n)\$ 的预测值由下式给出，
\hat{x}{1}^{(0)}(r p+t)=\hat{x}{1}^{(1)}(r p+t)-\hat{x}_{1}^{(1)}(r p+t-1).```
$3.\}(1,n)$
$3.\}(1,n)$
The $\mathbf{NGMC}(1,!n)$ model effectively describes and achieves satisfactory predictability as compared to the traditional ${\mathrm{GMC}}(1,!n)$ , and $n{-}1$ power exponents of the expected variables. The $\beta_{2},\beta_{3},\ldots,\beta_{n}$ was introduced in $\mathbf{NGMC}(1,!n)$ model to represent the impact of these upon nonlinear system behaviors and interactions. Unknown parameters are calculated by a computer program that calculates the minimum average relative percentage error of the forecasting model. This enhances the adaptability of the $\mathbf{NGMC}(1,!n)$ model to the initial data and consequently strengthens the accuracy of the prediction [180]. Nonetheless, the current fractional multivariate grey model with convolution integral is known as the following linear differential equation,
$\mathbf{NGMC}(1,!n)$ 模型相比传统的 ${\mathrm{GMC}}(1,!n)$ 模型和 $n{-}1$ 个预期变量的幂指数，能够有效描述并实现令人满意的预测能力。$\mathbf{NGMC}(1,!n)$ 模型中引入了 $\beta_{2},\beta_{3},\ldots,\beta_{n}$ 来表示这些变量对非线性系统行为和交互的影响。未知参数通过计算机程序计算，以最小化预测模型的平均相对百分比误差。这增强了 $\mathbf{NGMC}(1,!n)$ 模型对初始数据的适应性，从而提高了预测的准确性 [180]。然而，当前带有卷积积分的分数阶多元灰色模型被称为以下线性微分方程，
\frac{d x_{1}^{(r)}(t)}{d t}+b_{1}x_{1}^{(r)}(t)=\sum_{i=2}^{n}b_{i}x_{i}^{(r)}(t)+u,```
whereas the continuous time response function can be obtained as,
而连续时间响应函数可以表示为，
\hat{x}{1}^{(1)}(t)=x{1}^{(0)}(1)e^{-b_{1}(t-1)}+\int_{1}^{t}e^{-b_{1}(t-\tau)}f(\tau)d t,```
Finally, the predicted value of $\mathbf{NGMC}(1,!n)$ can be obtained using the inverse fractional-order accumulation as,
最后，$\mathbf{NGMC}(1,!n)$ 的预测值可以通过分数阶累加的逆运算得到，
\begin{array}{r l r}{\lefteqn{\hat{x}_{1}^{(0)}(k)=\Big(\hat{x}^{(r)}(k)\Big)^{(-r)}}}\\ &{}&{=\displaystyle\sum_{i=1}^{k}\left(\begin{array}{c}{k-i-r-1}\\ {k-i}\end{array}\right)\hat{X}^{(r)}(i).}\end{array}```
# 4. RDGM \$(\uparrow,n)\$
# 4. RDGM \$(\uparrow,n)\$
In 2016, Xin Ma et al. [181] presented a novel multivariate grey prediction model represented by \${\mathrm{RDGM}}(1,\!n)\$ . In which he proposed the modeling procedures in detail. The statistical analysis was introduced to demonstrate the interaction and gap between the traditional \${\mathrm{GMC}}(1{\mathrm{,}}n)\$ and \${\mathrm{RDGM}}(1,\!n)\$ as well as the numerical examples were used to validate the output of \${\mathrm{RDGM}}(1,\!n)\$ compared with \${\mathrm{GMC}}(1{\mathrm{,}}n)\$ . The shortcomings of \$\mathsf{R D G M}(1,\!n)\$ were highlighted by the author. The \${\mathrm{RDGM}}(1,\!n)\$ was basically a linear model. Therefore, it can not be feasible to explain the cause and effect of nonlinear processes for which inputs and outputs are nonlinearly linked. The basic form of \${\mathrm{RDGM}}(1,\!n)\$ is obtained as,
2016年，Xin Ma等人[181]提出了一种新型的多元灰色预测模型，表示为\${\mathrm{RDGM}}(1,\!n)\$。其中，他详细提出了建模过程。通过统计分析展示了传统\${\mathrm{GMC}}(1{\mathrm{,}}n)\$与\${\mathrm{RDGM}}(1,\!n)\$之间的交互作用和差距，并通过数值例子验证了\${\mathrm{RDGM}}(1,\!n)\$的输出与\${\mathrm{GMC}}(1{\mathrm{,}}n)\$的对比。作者还指出了\$\mathsf{R D G M}(1,\!n)\$的不足之处。\${\mathrm{RDGM}}(1,\!n)\$基本上是一个线性模型，因此无法解释输入和输出非线性关联的非线性过程的因果关系。\${\mathrm{RDGM}}(1,\!n)\$的基本形式如下：
\begin{array}{l}{x_{1}^{(1)}(r p+k+1)=\beta_{1}x_{1}^{(1)}(r p+k)}\ {\displaystyle+\sum_{i=2}^{n}\beta_{i}z_{i}^{(1)}(k+1)+\mu,}\end{array}```
where the time response function of ${\mathrm{RDGM}}(1,!n)$ is then specified as,
其中 ${\mathrm{RDGM}}(1,!n)$ 的时间响应函数被指定为，
\begin{array}{l}{{x_{1}^{(1)}(r p+k+1)=\displaystyle\frac{1-0.5b_{1}}{1+0.5b_{1}}x_{1}^{(1)}(r p+k)}}\\ {{\displaystyle\ \ \ \ \ \ \ \ \ \ +\sum_{i=2}^{n}\displaystyle\frac{b_{i}}{1+0.5b_{1}}z_{i}^{(1)}(k+1)+\displaystyle\frac{u}{1+0.5b_{1}},}}\end{array}```
\begin{array}{l}{{x_{1}^{(1)}(r p+k+1)=\displaystyle\frac{1-0.5b_{1}}{1+0.5b_{1}}x_{1}^{(1)}(r p+k)}}\ {{\displaystyle\ \ \ \ \ \ \ \ \ \ +\sum_{i=2}^{n}\displaystyle\frac{b_{i}}{1+0.5b_{1}}z_{i}^{(1)}(k+1)+\displaystyle\frac{u}{1+0.5b_{1}},}}\end{array}```
TABLE 2. The definition of the comparative Multivariate grey models used.
表 2. 使用的比较多元灰色模型定义
| No. | 模型名称 | 微分方程 | 参考文献 |
|---|---|---|---|
| 1 | GM(1,n) | [m(1(0) -t∑biz(1) (1)(t)eat dt | [177] |
| 2 | GMC(1,n) | ((rp+t)+b12(1)(rp+t)=∑=2bi2(l) (1)(t)+u | [179] |
| 3 | NGMC(1,n) | dx(r)(t) +bx(r)(t)=∑=bix()(t)+u | [180] |
| 4 | RDGM(1,n) | ci | [181] |
| 5 | OGM(1,n) | (k)=∑=2bi(1)(k)-a2(1)(k)+h1(k-1)+h2 | [146] |
| 6 | NGM(1,n) | dx(1)(t) r(1)(t)=∑2bi +ac p | [178] |
| 7 | KGM(1,n) | n+ (4)Φ=(4)(t) C1 | [154] |
while the predicted series of ${\mathrm{RDGM}}(1,!n)$ can be obtained as,
预测的 ${\mathrm{RDGM}}(1,!n)$ 序列可以表示为,
\hat{x}_{1}^{(0)}(r p+k+1)=\hat{x}_{1}^{(1)}(r p+k+1)-\hat{x}_{1}^{(1)}(r p+k).```
# 5 \$.\mathsf {O G M}(1,\!n)\$
# 5 \$.\mathsf {O G M}(1,\!n)\$
In 2016, Zeng et al. [146] were presented a new multivariable grey optimizing model namely called \$\mathrm{OGM}(1,\!n)\$ . The main purpose of this model is to solve the mechanism defect, parameter, and structural defect found in the conventional \${\mathrm{GM}}(1{,}n)\$ model. Theoretically, it has been shown that the \${\mathrm{GM}}(0,n)\$ , DGM(1,1) and NDGM(1,1) models are all similar variants of the \$\mathrm{OGM}(1,\!n)\$ model with varying parameter values. The authors proposed that the differential equation of \$\mathrm{OGM}(1,\!n)\$ can be obtained as,
2016年，Zeng等人[146]提出了一种新的多变量灰色优化模型，称为\$\mathrm{OGM}(1,\!n)\$。该模型的主要目的是解决传统\${\mathrm{GM}}(1{,}n)\$模型中存在的机制缺陷、参数和结构缺陷。理论上已经证明，\${\mathrm{GM}}(0,n)\$、DGM(1,1)和NDGM(1,1)模型都是\$\mathrm{OGM}(1,\!n)\$模型在不同参数值下的变体。作者提出，\$\mathrm{OGM}(1,\!n)\$的微分方程可以表示为：
\hat{x}{1}^{(0)}(k)=\sum{i=2}^{N}b_{i}\hat{x}{i}^{(1)}(k)-a z{1}^{(1)}(k)+h_{1}(k-1)+h_{2},```
where the time-response expression is given by,
其中时间响应表达式为，
\begin{array}{c}{{\displaystyle\hat{x}_{1}^{(1)}(k)=\sum_{t=1}^{k-1}\left[\mu_{1}\sum_{i=2}^{N}\mu_{2}^{t-1}b_{i}x_{i}^{(1)}(k-t+1)\right]}}\\ {{\displaystyle+\,\mu_{2}^{k-1}\hat{x}_{1}^{(1)}(1)+\sum_{j=0}^{k-2}\mu_{2}^{j}\left[(k-j)\mu_{3}+\mu_{4}\right],}}\\ {{\displaystyle\qquad\qquad\qquad\qquad\qquad k=2,3,\cdots\quad(4)}}\end{array}```
and, its restoration expression of inverse accumulation is obtained as,
其逆累积的恢复表达式为，
\hat{x}{1}^{(0)}(k)=\sum{i=2}^{N}b_{i}\hat{x}{i}^{(1)}(k)-a z{1}^{(1)}(k)+h_{1}(k-1)+h_{2}.```
6. $\mathsf(1,!n)$
6. $\mathsf(1,!n)$
To characterize the $\mathrm{{NGM}}(1,!n)$ and its transformed model, a more precise explanation is important about nonlinear interaction between the variables relative to ${\mathrm{GM}}(1{,}n)$ and to enhance the accuracy of the forecasting of the individual data sequences [178]. Furthermore, the authors presented the solution of the whitening equation as,
为了表征 $\mathrm{{NGM}}(1,!n)$ 及其转换模型，关于相对于 ${\mathrm{GM}}(1{,}n)$ 的变量之间的非线性交互作用以及提高个体数据序列预测的准确性，更精确的解释是重要的 [178]。此外，作者提出了白化方程的解为，
\frac{d x_{1}^{(1)}(t)}{d t}+a x_{1}^{(1)}(t)=\sum_{i=2}^{N}b_{i}\left(x_{i}^{(1)}(t)\right)^{\gamma_{i}},```
whereas the time response sequence of the \$\mathrm{{NGM}}(1,\!n)\$ model is then specified as,
而 \$\mathrm{{NGM}}(1,\!n)\$ 模型的时间响应序列则定义为,
\begin{array}{r}{\widehat{x}{1}^{(1)}(k+1)=\left[x{1}^{(1)}(1)-\displaystyle\frac{1}{a}\sum_{i=2}^{N}b_{i}\left(x_{i}^{(1)}(k+1)\right)^{\gamma_{i}}\right]e^{-a k}}\ {+\displaystyle\frac{1}{a}\sum_{i=2}^{N}b_{i}\left(x_{i}^{(1)}(k+1)\right)^{\gamma_{i}},}\end{array}```
The restoration of inverse accumulation is obtained as,
逆累积的恢复结果为，
\widehat{x}_{1}^{(0)}(k+1)=\alpha^{(1)}\widehat{x}_{1}^{(1)}(k+1)=\widehat{x}_{1}^{(1)}(k+1)-\widehat{x}_{1}^{(1)}(k).```
# \$7.\mathsf {K G M}(1,\!n)\$
# \$7.\mathsf {K G M}(1,\!n)\$
In 2018, Xin Ma et al. [154] introduced a novel nonlinear multivariate grey model which is based on the kernel method, namely kernel-based \${\mathrm{GM}}(1{,}n)\$ . Which is also known as \${\mathrm{KGM}}(1,\!n)\$ . The key feature of this model is to include an unknown component of the input series, which can be calculated using the kernel function, and then the \${\mathrm{KGM}}(1,\!n)\$ model is used to explain the nonlinear relationship between the input and output sequence. The differential equation of \${\mathrm{KGM}}(1,\!n)\$ is,
2018年，Xin Ma等人[154]提出了一种基于核方法的新型非线性多元灰色模型，即基于核的\${\mathrm{GM}}(1{,}n)\$，也称为\${\mathrm{KGM}}(1,\!n)\$。该模型的关键特点是包含输入序列的未知分量，可以通过核函数计算得出，然后使用\${\mathrm{KGM}}(1,\!n)\$模型来解释输入和输出序列之间的非线性关系。\${\mathrm{KGM}}(1,\!n)\$的微分方程为，
x_{1}^{(0)}(k)+a z_{1}^{(1)}(k)=\phi(k)+u,```
while the time response function of ${\mathrm{KGM}}(1,!n)$ can be written as,
${\mathrm{KGM}}(1,!n)$ 的时间响应函数可以表示为,
\hat{x}_{1}^{(1)}(k)=\alpha^{k-1}x_{1}^{(0)}(1)+\sum_{\tau=2}^{k}(\psi(\tau)+\mu)\cdot\alpha^{k-\tau},```
and the restored values can be obtained as,
恢复值可以表示为，
\hat{x}{1}^{(0)}(k)=\hat{x}{1}^{(1)}(k)-\hat{x}_{1}^{(1)}(k-1).```
The main structure is also described as shown in the grey model. Besides the free parameters $\alpha$ and $\pmb{\theta}$ , the grey model components are not specified. Essentially, the arrangement of $f(\pmb\theta;t)$ or $f(\pmb\theta;k)$ also plays a key role in enhancing grey models. It is well-known that the non-homogeneous grey NGM model often performs marginally better than the GM(1,1) and related results can be widely found in current studies. Furthermore, the optimum formulation of the $f(\cdot)$ function cannot be always consider in real-world applications. This question contributes to give inspiration to the researchers to apply Machine Learning over Grey Models.
主要结构也如灰色模型所示进行描述。除了自由参数 $\alpha$ 和 $\pmb{\theta}$ 外，灰色模型的组件并未具体指定。本质上，$f(\pmb\theta;t)$ 或 $f(\pmb\theta;k)$ 的排列在增强灰色模型中也起着关键作用。众所周知，非齐次灰色 NGM 模型通常比 GM(1,1) 模型表现略好，相关结果在当前研究中广泛存在。此外，$f(\cdot)$ 函数的最优公式在实际应用中并不总是被考虑。这个问题为研究人员提供了将机器学习应用于灰色模型的灵感。
IV. GREY MACHINE LEARNING (GML)
IV. 灰色机器学习 (GML)
Grey Machine Learning (GML) is a multivariate model based on kernel for time series forecasting, suggested by Xin Ma et al. in 2018 [182]. The main purpose is to investigate and highlight the capability of GML on time series forecasting. GML framework is a kernel-based multivariate model hence it should be used to compensate between the models and data. The detailed timeline of the GML framework have been discussed in Figure 7.
灰色机器学习 (Grey Machine Learning, GML) 是由 Xin Ma 等人在 2018 年提出的一种基于核的多变量时间序列预测模型 [182]。其主要目的是研究和突出 GML 在时间序列预测中的能力。GML 框架是一种基于核的多变量模型，因此它应该用于模型与数据之间的补偿。GML 框架的详细时间线已在图 7 中讨论。
A. OVERVIEW OF GML
A. GML 概述
This brief overview is based on the question of, why the researchers need a reliable framework to train machines for time series datasets. In 2015, Brenden Lake et al. presented a machine to understand Mongolian and passed the Turing test [18]. Such research has shown that the computer can function as humans in handwriting tasks. Another breakthrough is Google DeepMind’s AlphaGo. In 2016, David Silver et al. published AlphaGo’s first paper, stating that the machine could beat the Go game practitioners for the very first time, and this article was published after AlphaGo defeated the World Champion Lee Sedol. The success of AlphaGo has also demonstrated that the machine can not only be like humans, but can also be smarter than humans, and the output of this research gives AI enormous confidence [19, 183].
本简要概述基于研究人员为何需要一个可靠的框架来训练机器处理时间序列数据集的问题。2015年，Brenden Lake等人展示了一台能够理解蒙古语并通过图灵测试的机器 [18]。此类研究表明，计算机在手写任务中可以像人类一样运作。另一个突破是Google DeepMind的AlphaGo。2016年，David Silver等人发表了AlphaGo的第一篇论文，指出该机器首次能够击败围棋从业者，这篇文章是在AlphaGo击败世界冠军李世石之后发表的。AlphaGo的成功也表明，机器不仅可以像人类一样，还可以比人类更聪明，这项研究的成果为AI注入了巨大的信心 [19, 183]。
Defining the role of AI is about creating machines with intelligence, like humans or animals. Mathematical models and algorithms are important for achieving that goal. In the research mentioned above, Brenden Lake et al. used the Bayesian System for handwriting activities, and AlphaGo is built on the neural networks and Monte Carto tree search. Nevertheless, such studies also have other challenges, such as they required a large amount of training data to achieve the best models. It is also really interesting to note that Google has built another Go game machine named AlphaGo Zero [184]. This revolutionary machine trained without any chess manual, only need to teach by using the rules of the Go game. Thus, AlphaGo Zero won the championship successfully and beat AlphaGo [19].
定义 AI 的角色是创造具有智能的机器，类似于人类或动物。数学模型和算法对于实现这一目标非常重要。在上述研究中，Brenden Lake 等人使用贝叶斯系统进行手写活动，而 AlphaGo 则是基于神经网络和蒙特卡洛树搜索构建的。然而，这些研究也面临其他挑战，例如它们需要大量的训练数据才能达到最佳模型。值得注意的是，Google 还构建了另一款名为 AlphaGo Zero 的围棋机器 [184]。这款革命性的机器无需任何棋谱进行训练，只需通过围棋规则进行教学。因此，AlphaGo Zero 成功赢得了冠军并击败了 AlphaGo [19]。
The above studies demonstrated that, it is attainable to train machines utilizing very small time series datasets. This means that Big Data may not be the only way to make AI efficient. Also, researchers have often observed very small data cases in the process of human learning. For starters, a little kid learns the basic techniques of the Go game with a certain lessons. Scientifically, the main methodology used to improve AlphaGo is Deep Learning and Improving training, both of these are focused on the concept of getting the best of human intelligence. Moreover, the variants in the Go game are computationally limitless for computers, although contrary to all instances, the chess guides provided for AlphaGo are restricted. It is thus difficult to train a smart AlphaGo without effective learning methods focused on small samples. Grey system theory is primarily designed for limited datasets and also appears to have a deterministic structure with free parameters to be determined by samples [185].
上述研究表明，利用非常小的时间序列数据集训练机器是可行的。这意味着大数据可能不是使AI高效运作的唯一途径。此外，研究人员在人类学习过程中经常观察到非常小的数据案例。例如，一个小孩子通过几节课就能学会围棋的基本技巧。从科学角度来看，提升AlphaGo的主要方法是深度学习和改进训练，这两种方法都集中在获取人类智能最佳表现的概念上。此外，尽管围棋的变化对计算机来说是计算上无限的，但与所有情况相反，为AlphaGo提供的棋谱是有限的。因此，如果没有专注于小样本的有效学习方法，训练一个聪明的AlphaGo是困难的。灰色系统理论主要是为有限数据集设计的，并且似乎具有确定性结构，其自由参数由样本确定 [185]。
This concept is derived from system science, the deterministic configuration of grey models is an established part of the system, with free parameters as an undefined component. Such kind of phenomenon is quite limiting, as it is really hard to discover the complexity of systems in real-world applications. In most situations, only a small part of the devices can be identified in a limited time or at the least cost. It is obvious to ask the question-how can structures be modelled given that they are partly understood?
这一概念源自系统科学，灰色模型的确定性配置是系统中已确立的部分，而自由参数则是未定义的部分。这种现象具有很大的局限性，因为在现实世界的应用中，很难发现系统的复杂性。在大多数情况下，只能在有限的时间内或以最低的成本识别出设备的一小部分。这就引出了一个显而易见的问题——在部分理解的情况下，如何对结构进行建模？
Based on this question, Xin Ma carried out a range of experiments to merge the principles of grey modeling with the techniques of Machine Learning. The main issues discussed in these studies are structures with a defined complex configuration and an undefined nonlinear interaction between the device state or the output and input series. For instance, previous results in real-world applications also suggested that models built on this concept are significantly more efficient than traditional grey models [6, 182]. A brief flowchart of GML framework have been discussed in Figure 8.
基于这一问题，Xin Ma 进行了一系列实验，将灰色建模原理与机器学习技术相结合。这些研究主要讨论了具有明确复杂配置结构以及设备状态或输出与输入序列之间未定义非线性交互的问题。例如，实际应用中的先前结果也表明，基于这一概念构建的模型比传统灰色模型显著更高效 [6, 182]。图 8 讨论了 GML 框架的简要流程图。
B. GENERAL FORMULATION OF GML
B. GML 的通用公式
The general structure of the GML model is classified into the following equations proposed by [182]:
GML 模型的一般结构根据 [182] 提出的以下方程进行分类：
• The continuous Structure:
• 连续结构：
\frac{d x_{1}^{(1)}(t)}{d t}+a x_{1}^{(1)}(t)=\phi(t)+u```
\frac{d x_{1}^{(1)}(t)}{d t}+a x_{1}^{(1)}(t)=\phi(t)+u```
• The discrete structure:
• 离散结构：
x_{1}^{(1)}(k+1)=\alpha x_{1}^{(1)}(k)+\phi(k)+\mu```
While the element \$\phi(\cdot)\$ is completely unknown, and this is the main difference between GML and conventional grey models.
虽然元素 \$\phi(\cdot)\$ 完全未知，这是 GML 与传统灰色模型之间的主要区别。
FIGURE 7. Timeline of the Grey Machine Learning.
图 7: 灰色机器学习时间线
# C. AN ESTIMATE OF UNCERTAIN FUNCTION
# C. 不确定函数的估计
There are several phases to approximate the uncertain function \$\phi(\cdot)\$ in the general formulation of the GML model[181, 182], which are given below:
在GML模型的一般公式中，近似不确定函数 \$\phi(\cdot)\$ 有几个阶段 [181, 182]，如下所示：
# 1) Linear representation in the higher dimensional space function
# 1) 高维空间函数中的线性表示
Based on the Theorem of Weierstrass [186], every continuous function at a closed and bound time can be uniformly approximated by polynomials to some degree of accuracy. Furthermore, these are the following forms of the function as given below:
基于魏尔斯特拉斯定理 [186]，每个在闭且有界区间上的连续函数都可以用多项式以某种精度均匀逼近。此外，这些函数的形式如下所示：
\phi(t)=w_{n}x^{n}(t)+w_{n-1}x^{n-1}(t)+\ldots+w_{1}x^{1}(t)+w_{0}```
Lets the mapping be $\varphi;;=;;R;;\rightarrow;;{\mathcal F}$ , where $\mathcal F\quad=$ $\left{\left[x^{n}(t),x^{n-1}\overline{{(t)}},\ldots,\dot{x^{1}}(t),1\right]^{T}\right}$ , and the weights are $\omega=$ $[w_{n},w_{n-1},\ldots,w_{1},w_{0}]^{I^{\prime}}$ . The nonlinear equation can then be written in linear form in the higher dimensional space $\mathcal{F}$ as,
设映射为 $\varphi;;=;;R;;\rightarrow;;{\mathcal F}$，其中 $\mathcal F\quad=$ $\left{\left[x^{n}(t),x^{n-1}\overline{{(t)}},\ldots,\dot{x^{1}}(t),1\right]^{T}\right}$，权重为 $\omega=$ $[w_{n},w_{n-1},\ldots,w_{1},w_{0}]^{I^{\prime}}$。然后，非线性方程可以在高维空间 $\mathcal{F}$ 中以线性形式表示为，
\phi(t)=\omega^{T}\varphi(x(t)).```
Remember that this definition still holds only \$x(t)\$ as a function, and the range of the \$\mathcal{F}\$ element can be infinite.
记住，这个定义仍然只将 \$x(t)\$ 视为一个函数，且 \$\mathcal{F}\$ 元素的范围可以是无限的。
# 2) Non parametric Estimation
# 2) 非参数估计
In the above example, the estimation of arbitrary nonlinear equations can be transformed into a linear problem. With the specified samples \$\{(x(k),y(k))\mid k=1,2,3,\ldots,N\}\$ , which can be estimation from the following formulation [182],
在上述示例中，任意非线性方程的估计可以转化为线性问题。通过指定的样本 \$\{(x(k),y(k))\mid k=1,2,3,\ldots,N\}\$，可以从以下公式 [182] 中进行估计：
y(t)=\boldsymbol{\omega}^{T}\boldsymbol{\varphi}(\boldsymbol{x}(t))+\boldsymbol{b}.```
The issue of regular iz ation is also in the shape of ridge regression is defined as,
正则化问题也以岭回归的形式定义为，
\begin{array}{l}{\displaystyle\operatorname*{min}J(\omega,e)=\frac{||\omega||^{2}}{2}+\frac{C}{2}\sum_{k=1}^{m}e_{k}^{2}}\\ {\mathrm{~s.t.~}e_{k}=y(k)-\omega^{T}\varphi(\boldsymbol{x}(k))-b}\end{array}```
Where the value \$\parallel\cdot\parallel\$ is the 2-norm. The key distinction between the problem of regular iz ation (54) and the commonly adopted least-square solution is that the \$\lVert\omega\rVert^{2}\$ should always be small. The nonlinear mapping \$\varphi\$ is not unique. For example, when the nonlinear equation \$\phi(t)\$ becomes discontinuous for a higher value, it can be generalized for the Taylor Power Series. Therefore, there are various variants where \$\phi(t)\$ becomes applied at specific stages. Mathematically, a brief analysis is required to guarantee that the response is special. Whereas, with the non-deterministic mathematical functions, the problem of regular iz ation is often classified in non parametric estimation formulations. On the other hand, researchers also want the forecast to be smooth enough to make generalization easier or more accurate. The term \$C\$ should be used for this reason. The question of regular iz ation (54) is restricted a quadric programming. Therefore, the Lagrangian should need to describe as a first step.
其中值 \$\parallel\cdot\parallel\$ 是2-范数。正则化问题 (54) 与常用的最小二乘解法的关键区别在于 \$\lVert\omega\rVert^{2}\$ 应该始终很小。非线性映射 \$\varphi\$ 不是唯一的。例如，当非线性方程 \$\phi(t)\$ 在较高值时变得不连续时，可以将其推广为泰勒幂级数。因此，存在多种变体，其中 \$\phi(t)\$ 在特定阶段被应用。数学上，需要简要分析以确保响应是特殊的。然而，对于非确定性数学函数，正则化问题通常被归类为非参数估计公式。另一方面，研究人员也希望预测足够平滑，以便更容易或更准确地进行泛化。因此，应使用术语 \$C\$。正则化问题 (54) 被限制为二次规划。因此，首先需要描述拉格朗日量。
\begin{array}{l}{{\displaystyle{\cal L}:=\frac{|\omega||^{2}}{2}+\frac{C}{2}\sum_{k=1}^{r}e_{k}^{2}}}\ {{\displaystyle\qquad\quad+\sum_{k=1}^{m}\lambda_{k}\left[y(k)-\omega^{T}\phi(x(k))-b-e_{k}\right].}}\end{array}```
The solution to the problem of regular iz ation (54) can be easily achieved by implementing the Karush-Kuhn-Tucker (KKT) conditions as follows:
正则化问题 (54) 的解决方案可以通过实现 Karush-Kuhn-Tucker (KKT) 条件轻松实现：
\left\{\begin{array}{l}{\frac{\partial L}{\partial w}=0\Rightarrow\omega=\sum_{k=1}^{m}\lambda_{k}\varphi(x(k))}\\ {\frac{\partial L}{\partial b}=0\Rightarrow\sum_{k=1}^{m}\lambda_{k}=0}\\ {\frac{\partial L}{\partial\lambda_{j}}=0\Rightarrow e_{k}=\lambda_{k}/C}\\ {\frac{\partial L}{\partial\lambda_{j}}=0\Rightarrow y(k)-\omega^{T}\varphi(\chi(k))-b=e_{k}.}\end{array}\right.```
Equation (56) indicates that \$\omega\$ can be determined using the values of the Lagrangian multipliers \$\lambda_{k}\$ and \$\varphi(\boldsymbol{x}(\boldsymbol{k}))\$ . As a consequence, the computational representation of the nonlinear function can be obtained as,
方程 (56) 表明，\$\omega\$ 可以通过拉格朗日乘数 \$\lambda_{k}\$ 和 \$\varphi(\boldsymbol{x}(\boldsymbol{k}))\$ 的值来确定。因此，非线性函数的计算表示可以表示为，
\omega^{T}\varphi(x(t))=\sum_{k=1}^{m}\lambda_{k}\varphi^{T}(x(k))\varphi(x(t)).```
Although, the author [182] has not yet provided the deterministic form of the nonlinear mapping of $\varphi(\cdot)$ . Although, it is only need to calculate the inner product $<$ $\varphi(x(k)),\varphi(x(t))>=\varphi^{T}(x(k))\varphi(x(t))$ in the $\mathcal{F}$ function. Therefore, if the Mercer conditions under which realize that the author has the following expansion for any symmetric positive specified function $K(\cdot,\cdot)$ , respectively.
尽管作者 [182] 尚未提供非线性映射 $\varphi(\cdot)$ 的确定性形式，但只需在 $\mathcal{F}$ 函数中计算内积 $<$ $\varphi(x(k)),\varphi(x(t))>=\varphi^{T}(x(k))\varphi(x(t))$ 。因此，如果满足 Mercer 条件，作者可以对任何对称正定函数 $K(\cdot,\cdot)$ 分别进行以下展开。
K(x(k),x(t))=\sum_{n=1}^{\infty}\alpha_{n}\psi_{n}(x(k))\psi_{n}(x(t)),```
while \$\{\psi_{n}(x(t))\}_{\begin{array}{l}{n=\infty}\\ {n=1}\end{array}}^{n}\$ is the orthogonal basis, and \$\alpha_{n}\$ is the positive eigenvalues. Such features are also referred to as core functions or kernels. Therefore, the nonlinear mapping corresponding to the given kernel can be easily defined as,
其中 \$\{\psi_{n}(x(t))\}_{\begin{array}{l}{n=\infty}\\ {n=1}\end{array}}^{n}\$ 是正交基，\$\alpha_{n}\$ 是正特征值。这些特征也被称为核心函数或核函数。因此，给定核对应的非线性映射可以轻松定义为，
\varphi(\cdot)=\left[\sqrt{\alpha_{1}}\psi_{1}(\cdot),\sqrt{\alpha_{2}}\psi_{2}(\cdot),\ldots,\sqrt{\alpha_{n}}\psi_{n}(\cdot),\ldots\right]^{T}.```
Clearly, the inner product of nonlinear projection can be described as a kernel,
显然，非线性投影的内积可以描述为一个核函数，
\begin{array}{r l r}{\lefteqn{K(x(k),x(t))=\sum_{n=1}^{\infty}\alpha_{n}\psi_{n}(x(k))\psi_{n}(x(t))}}\\ &{}&{=<\left[\sqrt{\alpha_{1}}\psi_{1}(k),\sqrt{\alpha_{2}}\psi_{2}(k),\ldots,\sqrt{\alpha_{n}}\psi_{n}(k),\ldots\right]^{T},}\\ &{}&{\stackrel{\mathrm{[}}{\sqrt{\alpha_{1}}\psi_{1}(t)},\sqrt{\alpha_{2}}\psi_{2}(t),\ldots,\sqrt{\alpha_{n}}\psi_{n}(t),\ldots\right]^{T}>\quad}\\ &{}&{=\varphi^{T}(x(k))\varphi(x(t)).\ \ \ \ \ \ (60)}\end{array}```
Therefore, through substituting (57), (62) into (56) with the exclusion of \$e_{k}\$ , the solution of the regularized problem is expressed by the following linear framework,
因此，通过将 (57) 和 (62) 代入 (56) 并排除 \$e_{k}\$，正则化问题的解由以下线性框架表示，
\left(\begin{array}{c c c}{{0}}&{{\vdots}}&{{1_{m}^{T}}}\ {{\cdots}}&{{\cdots}}&{{\cdots}}\ {{1_{m}}}&{{\vdots}}&{{\Omega+I_{m}/C}}\end{array}\right)\left(\begin{array}{c}{{b}}\ {{\cdots}}\ {{\lambda}}\end{array}\right)=\left(\begin{array}{c}{{0}}\ {{\cdots}}\ {{Y}}\end{array}\right)```
where,
其中，
\begin{array}{r l r}&{}&{\mathbf{1}_{m}=[1,1,\ldots,1]_{m}^{T},}\\ &{}&{\Omega=[K(x(i),x(j))]_{m\times m},}\\ &{}&{\lambda=[\lambda_{1},\lambda_{2},\lambda_{3},\ldots,\lambda_{m}]^{T}\,,}\\ &{}&{Y=[y(1),y(2),\ldots,y(m)]^{T},}\end{array}```
and \$I_{m}\$ is an identity matrix.
并且 \$I_{m}\$ 是一个单位矩阵。
It can be shown that the solution of the regularized problem (54) can be transformed to a linear model (61), which can be quite easy to solve with a given kernel, and the coefficients are \$\lambda\$ and \$b\$ .
可以证明，正则化问题 (54) 的解可以转化为线性模型 (61)，在给定核函数的情况下，该模型可以很容易地求解，其系数为 \$\lambda\$ 和 \$b\$。
At last point, the statistical formulation of the estimation needs to be provided (53). Under (57) and (62), the estimation of (53) can be obtained as,
最后，需要提供估计的统计公式 (53)。在 (57) 和 (62) 的条件下，可以得到 (53) 的估计为，
\begin{array}{l}{\displaystyle\hat{y}(t)=\omega^{T}\varphi(x(t))+b}\ {\displaystyle\qquad=\sum_{k=1}^{m}\lambda_{k}\varphi^{T}(x(k))\varphi(x(t))+b}\ {\displaystyle\qquad=\sum_{k=1}^{m}\lambda_{k}K(x(k),x(t))+b}\end{array}```
- Semi-parametric Estimation
- 半参数估计
With a non parametric estimation, it is easy to deduce the semi-parametric variant of which the goal is to make an approximation that uses the following terminology,
通过非参数估计，可以轻松推导出半参数变体，其目标是使用以下术语进行近似：
y(t)=\omega_{1}^{T}z(t)+\omega_{2}^{T}\varphi(x(t))+b,```
while \$z(t)\$ is the variable in \$R\$ or the parameter in linear space \$R^{N_{1}}\$ . This form can be converted to a non parametric formulation by representing a new nonlinear mapping of \$\varphi^{\prime}:R^{N_{1}+N_{2}}\stackrel{.}{\rightarrow}\bar{\mathcal{F}}^{\prime}\$ as,
当 \$z(t)\$ 是 \$R\$ 中的变量或线性空间 \$R^{N_{1}}\$ 中的参数时，这种形式可以通过表示一个新的非线性映射 \$\varphi^{\prime}:R^{N_{1}+N_{2}}\stackrel{.}{\rightarrow}\bar{\mathcal{F}}^{\prime}\$ 转换为非参数化形式。
\varphi^{\prime}(\cdot)=\left[\begin{array}{c}{z(\cdot)}\ {\varphi(\cdot)}\end{array}\right]```
and set $\omega^{\prime}=\left[\begin{array}{l}{\omega_{1}}\ {\omega_{2}}\end{array}\right]$
并设 $\omega^{\prime}=\left[\begin{array}{l}{\omega_{1}}\ {\omega_{2}}\end{array}\right]$
Equation (64) can be described as,
式 (64) 可以描述为，
y(t)=\omega^{\prime T}\varphi^{\prime}(x(t))+b.```
In consideration of the fact that this concept is mathematically identical to the non-parametric form (53), all the numerical formulations are the same and thus the primary formulations for the semi-parametric approximation can easily be obtained with some minor adjustments.
考虑到这一概念在数学上与非参数形式 (53) 完全相同，所有数值公式都相同，因此通过一些小的调整，可以轻松获得半参数近似的主要公式。
First of all, \$\Omega\$ in (56) can be rewritten as,
首先，\$\Omega\$ 在 (56) 中可以重写为，
\begin{array}{r l}&{\Omega_{i j}=<\varphi^{\prime}(x(i)),\varphi^{\prime}(x(j))>}\ &{\quad\quad=\left[z^{T}(i),\varphi^{T}(x(i))\right]\left[\begin{array}{c}{z(j)}\ {\varphi(x(j))}\end{array}\right]}\ &{\quad\quad=z^{T}(i)z(j)+\varphi^{T}(x(i))\varphi(x(j))}\ &{\quad\quad=z^{T}(i)z(j)+K(x(i),x(j)).}\end{array}```
FIGURE 8. Flowchart of the Grey Machine Learning.
图 8: 灰色机器学习流程图
Since it is identical to (65), the non parametric form (66) can be transformed into,
由于它与 (65) 相同，非参数形式 (66) 可以转换为，
\begin{array}{l}{\displaystyle\hat{y}(t)=\omega^{\prime T}\varphi^{\prime}(x(t))+b}\\ {\displaystyle\qquad=\sum_{k=1}^{m}\lambda_{k}\varphi^{\prime T}(x(k))\varphi^{\prime}(x(t))}\\ {\displaystyle\qquad=\sum_{k=1}^{m}\lambda_{k}z^{T}(k)z(t)+\sum_{k=1}^{m}\lambda_{k}K(x(k),x(t))+b,}\end{array}```
and \$\begin{array}{r}{\omega_{1}=\sum_{k=1}^{m}\lambda_{k}z(k)}\end{array}\$ . Therefore, (67) can also be written as,
且 \$\begin{array}{r}{\omega_{1}=\sum_{k=1}^{m}\lambda_{k}z(k)}\end{array}\$ 。因此，(67) 也可以写成，
\hat{y}(t)=\omega_{1}^{T}z(t)+\sum_{k=1}^{m}\lambda_{k}K(x(k),x(t))+b.```
From the description above, it can be shown that the nonlinear function $\phi(t)$ is mostly calculated as a kernel form,
从上述描述可以看出，非线性函数 $\phi(t)$ 大多以核形式计算，
\phi(t)=\sum_{k=1}^{m}\lambda_{k}K(x(k),x(t)).```
It guarantees the computational sustainability of grey models with an undefined nonlinear function and it is quite simple to calculate the estimation of the general forms (49) and (50).
它保证了具有未定义非线性函数的灰色模型的计算可持续性，并且计算一般形式 (49) 和 (50) 的估计值非常简单。
For instance, the following basic notations for computational formulations can easily be obtained by using (50),
例如，通过使用 (50) 可以轻松获得以下计算表达式的基本符号，
\left{\begin{array}{l}{y(k+1)=X_{1}^{(1)}(k+1)}\ {\quad\quad\omega_{1}=\alpha}\ {\quad z(k)=X_{1}^{(1)}(k)}\ {\quad\phi(k)=\omega_{2}^{T}\varphi(\chi(k))}\ {\quad\chi(k)=\left[X_{2}^{(1)}(k),X_{3}^{(1)}(k),\ldots,X_{n}^{(1)}(k)\right]^{T}.}\end{array}\right.```
Comprehensive derivations can be contained in the earlier works of Xin Ma et al. [154, 169, 181, 187].
详细的推导可以参考 Xin Ma 等人的早期工作 [154, 169, 181, 187]。
D. SOLUTIONS OF GENERAL FORMULATIONS Mathematically, the general formulation approaches use the same terminology as the linear formulations are as follows:
D. 通用公式的解决方案
数学上，通用公式方法使用与线性公式相同的术语，如下所示：
i. The continuous form of (49):
i. (49) 的连续形式：
\hat{X}_{1}^{(1)}(t)=X_{1}^{(0)}(1)\cdot e^{-a(t-1)}+\int_{1}^{t}e^{-a(t-\tau)}\phi(\tau)d\tau```
ii. Discrete form of (50):
ii. (50) 的离散形式：
\hat{X}{1}^{(1)}(k+1)=X{1}^{(0)}(1)\cdot\alpha^{k}+\sum_{\tau=2}^{k+1}\alpha^{(k+1-\tau)}\phi(\tau)```
Remember that, only modified the linear function $f(\cdot)$ in (3) and (4) into $\phi(\cdot)$ .
记住，仅将 (3) 和 (4) 中的线性函数 $f(\cdot)$ 修改为 $\phi(\cdot)$。
V. DISCUSSION
V. 讨论
Several promising Machine Learning models for long, medium, and short-term forecasting have been established using previous research findings data. It is noted that each of the mentioned strategies has a range of benefits and drawbacks. All have been carefully analysed and discussed in accordance to the study. Recently, the multivariate models gets particular consideration. The performance of the hybrid and ensemble models for time series forecasting was also proposed in recent review studies [20–24, 183, 188]. Such forecasting models are essential for time series forecasting as well as effective optimization. To explain GML framework, the non-parametric approximation is the traditional Least Square Support Vector Machines (SL-LSSVM) suggested by Johan et al. [189]. This is one of the most common architectures for Machine Learning techniques. A semiparametric approximation in the form of Slightly Linear Lowest Square Support Vector Machines (PL-LSSVM) proposed by Marcelo Espinoza et al. in 2004 [190], which has not been paid much more attention in recent years. In 2001, Bernhard et al., proved the Re presenter Theorem, which had very strong results and demonstrated the calculation still fits the form of (69) for any regularized formulation of traditional kernel approximation problems. These formulas are also valuable to design machines for the sake of applications [191].
利用先前的研究数据，已经建立了几种有前景的机器学习模型，用于长期、中期和短期预测。值得注意的是，每种提到的策略都有一系列优点和缺点。根据研究，所有这些策略都经过了仔细的分析和讨论。最近，多元模型得到了特别的关注。最近的综述研究也提出了混合模型和集成模型在时间序列预测中的性能 [20–24, 183, 188]。这些预测模型对于时间序列预测以及有效优化至关重要。为了解释 GML 框架，非参数近似是 Johan 等人提出的传统最小二乘支持向量机 (SL-LSSVM) [189]。这是机器学习技术中最常见的架构之一。Marcelo Espinoza 等人在 2004 年提出的半参数近似形式的略微线性最小二乘支持向量机 (PL-LSSVM) [190]，近年来并未得到太多关注。2001 年，Bernhard 等人证明了表示定理，该定理具有非常强的结果，并证明了对于任何正则化的传统核近似问题，计算仍然符合 (69) 的形式。这些公式对于设计应用机器也很有价值 [191]。
Furthermore, the linear differential in (49) and (50) is the partly understood dimension, similar to the general formulation of GML models poposed by [182] as described above. These differentials usually imply the state or performance sequence of models which are decreasing or increasing over time. Such mechanisms are also called dynamic systems which are widely used in real-world applications-including oil and gas prediction, weather forecasting, biomedical engineering, and so on. Whereas, the nonlinear function determined by the kernels represents a nonlinear aspect of the input sequence or the machine dependency series. These functions are clear and simply illustrate the nonlinear relationship between dynamic systems. As a result, GML models are nonlinear dynamic systems. Based on these attributes, the GML models were found to be much more efficient than the traditional linear grey models.
此外，(49) 和 (50) 中的线性微分是部分理解的维度，类似于 [182] 提出的 GML 模型的一般公式。这些微分通常暗示模型的状态或性能序列随时间递减或递增。这种机制也被称为动态系统，广泛应用于现实世界的应用中，包括石油和天然气预测、天气预报、生物医学工程等。而由核函数确定的非线性函数则代表了输入序列或机器依赖序列的非线性方面。这些函数清晰且简单地说明了动态系统之间的非线性关系。因此，GML 模型是非线性动态系统。基于这些属性，GML 模型被发现比传统的线性灰色模型更为高效。
A. FUTURE PERSPECTIVES
A. 未来展望
From the point of view of GML methodology, it can be demonstrated that the approaches of grey models and kernelbased methods can be successfully merged. The families of grey models and kernel-based models are very broad. However, we highlight some limitations of the GML framework addressed by the authors in their articles [154, 182]. It may lead the researchers to enhance the GML concept efficiently, which are as under:
从GML方法论的角度来看，可以证明灰色模型和基于核的方法可以成功融合。灰色模型和基于核的模型家族非常广泛。然而，我们强调了作者在其文章[154, 182]中提到的GML框架的一些局限性。这些局限性可能会引导研究人员有效地增强GML概念，具体如下：
1) Based on Dataset
1) 基于数据集
To discuss the modality of the dataset, GML framework specifically for the time series datasets. Therefore, it is not suitable for multimedia datasets or time-invariant situations. However, it is more efficient compared with other grey models for the big datasets, but still has some limitations of dataset size. Moreover, the GML framework is suitable for the single-time-line situation. Therefore, still there exist a need for improvement for multi-time-line datasets.
为了讨论数据集的模态，GML框架专门针对时间序列数据集。因此，它不适用于多媒体数据集或时间不变的情况。然而，与其他灰色模型相比，它在处理大数据集时更为高效，但在数据集大小方面仍存在一些限制。此外，GML框架适用于单时间线的情况。因此，对于多时间线数据集，仍存在改进的需求。
2) Based on Over-fitting
2) 基于过拟合
GML concept is a mixture of grey models and kernel-based models, ensuring that the shortcomings of such models can often be expressed in the applications. For example, LSSVM over-fitting happens in certain situations where the LSSVM has extremely high measuring performance but very low predicting accuracy. However, it does not happen with the GML, but the risk also exists.
GML 概念是灰色模型和基于核的模型的混合体，确保这些模型的缺点通常可以在应用中表现出来。例如，LSSVM 在某些情况下会出现过拟合，即 LSSVM 的测量性能极高，但预测精度却很低。然而，GML 不会出现这种情况，但风险仍然存在。
3) Based on Prediction
3) 基于预测
GML requires more data to yield more reliable results and ideal for short-term predictions, whereas LSSVM and linear grey models that in certain cases be useful for medium or long-term predictions. Research to solve these shortcomings should often be taken into consideration in prospective studies [182].
GML 需要更多数据以产生更可靠的结果，并且适合短期预测，而 LSSVM 和线性灰色模型在某些情况下可能对中长期预测有用。在未来的研究中，应经常考虑解决这些缺点的研究 [182]。
Finally, it can be helpful to use these approaches to develop grey models or kernel-based models to enhance the GML framework in the future. Certain Machine Learning techniques can also be known to be useful for building GML models, including Multilayer Perce ptr on s (MP), Deep Learning Neural Networks, Gaussian Process Regression (GPR), etc. In addition to the range of other impressive Machine Learning, state-of-the-art approaches and the integration of such methods with grey models can also be useful in future research.
最后，利用这些方法开发灰色模型或基于核的模型，以增强未来的GML框架可能会有所帮助。某些机器学习技术也被认为对构建GML模型有用，包括多层感知器 (MP)、深度学习神经网络、高斯过程回归 (GPR) 等。除了其他令人印象深刻的机器学习技术外，最先进的方法以及这些方法与灰色模型的结合也可能在未来的研究中发挥作用。
VI. CONCLUSION
VI. 结论
A comprehensive analysis of Machine Learning, Grey Forecasting Models, and GML was presented in this paper. Moreover, the increasing performance of the new combination models can be anticipated to be produced by the methods outlined in this article. Throughout our survey, a primer overview of Machine Learning was discussed based on different algorithms and applications by using big datasets. On the other hand, conventional grey models are specially developed for forecasting by using small time series datasets, which are divided into two types: Univariate and Multivariate models. From the traditional GM(1,1) model to the GML framework, each model is working for corresponding applications. Whereas, GML framework is derived from multivariate grey models. However, when there is some limited dataset of time series, GML has been demonstrated to outperform traditional LSSVM and grey models, which is the basic static framework, and showing the dynamic properties of GML models. The LSSVM analysis also indicated that the usage of established knowledge can significantly increase the efficiency of conventional Machine Learning models.
本文对机器学习 (Machine Learning)、灰色预测模型 (Grey Forecasting Models) 和 GML 进行了全面分析。此外，本文所概述的方法有望产生性能不断提升的新组合模型。在我们的调查中，基于不同算法和应用，使用大数据集对机器学习进行了初步概述。另一方面，传统的灰色模型专门用于使用小时间序列数据集进行预测，分为单变量和多变量模型两种类型。从传统的 GM(1,1) 模型到 GML 框架，每个模型都适用于相应的应用。而 GML 框架则源自多变量灰色模型。然而，当时间序列数据集有限时，GML 已被证明优于传统的 LSSVM 和灰色模型，这是基本的静态框架，并展示了 GML 模型的动态特性。LSSVM 分析还表明，使用已有知识可以显著提高传统机器学习模型的效率。
Developing new GML frameworks will increase more attention from different fields of scientists. More efficient GML models can be established in the future with the general formulation of GML. It is clear from this survey that GML not only develops many modern grey models but rather provides the possibilities of integrating the combination of traditional grey models as well as the dynamic nature of nonlinear Machine Learning models. Eventually, with the growing existence of a nonlinear dynamic system, more studies should also be carried out across a broader variety of real-world applications. Nonetheless, this survey illustrates the GML framework as well as the general perspective of Machine Learning and grey models, making it very useful for other applications as well as developing new GML models. This indicates that this framework allows researchers to use it for both theoretical and practical research.
开发新的GML框架将吸引来自不同领域科学家的更多关注。随着GML的一般化表述，未来可以建立更高效的GML模型。从本次调查中可以清楚地看出，GML不仅发展了许多现代灰色模型，还提供了将传统灰色模型与非线性机器学习模型的动态特性相结合的可能性。最终，随着非线性动态系统的日益增多，应在更广泛的现实世界应用中进行更多研究。尽管如此，本次调查展示了GML框架以及机器学习和灰色模型的一般视角，使其对其他应用以及开发新的GML模型非常有用。这表明该框架允许研究人员将其用于理论和实践研究。
ACKNOWLEDGMENT
致谢
The authors would like to thank Dr. Xin Ma and Dr. Wenqing Wu at Southwest University of Science and Technology, China for their insightful suggestions and guidance of the theoretical and numerical analysis.
作者感谢西南科技大学的马鑫博士和吴文清博士在理论和数值分析方面提供的深刻建议和指导。
