Grey data analysis methods, models and applications. Grey data analysis methods, models and applications. Grey data analysis methods, models and applications.
Appearance and Growth of Grey Systems Research
On the basis of dividing the spectrum of scientific and technological endeavors into
Across disciplines, the development of modern science shows a clear trend toward high-level synthesis This shift has given rise to systems science, bringing with it distinct methodological and epistemological significance Systems science reveals deep, intrinsic interconnections among objects and events, enriching the overall progress of science and technology and enabling many historically difficult problems in various scientific fields to be tackled through holistic, integrative approaches.
Fields have matured alongside the emergence of systems science and its specialized branches, deepening our understanding of nature and the laws governing objective evolution By the late 1940s, systems theory, information theory, and cybernetics had appeared, establishing foundational tools for holistic analysis In the late 1960s and early 1970s, the theories of dissipative structures, synergetics, catastrophe, and bifurcations broadened the reach of systems thinking By the mid to late 1970s, new transfield and interwoven approaches—such as ultracircular theory and dynamic systems theory—emerged, reflecting an increasingly integrated view of systems science.
Uncertainty and noise in information arise from internal and external disturbances and the limitations of our understanding As science, technology, and human knowledge advance, our grasp of system uncertainties deepens, ushering in a new era for studying complex systems In the second half of the twentieth century, a rapid proliferation of theories and methodologies for uncertain systems profoundly influenced systems science and systems engineering Prominent milestones include L A Zadeh’s development of fuzzy mathematics in the 1960s, J L Deng’s grey systems theory in 1982, and Z Pawlak’s rough set theory in the 1980s, collectively driving significant progress in this field.
S Liu et al., Grey Data Analysis , Computational Risk Management,
These works stand as among the most important contributions to uncertain systems research in this era, delivering foundational theories and practical methodologies for describing and managing uncertain information from multiple perspectives.
Grey systems theory, introduced by Julong Deng in 1982, is a modern approach for studying problems characterized by small samples and incomplete information It addresses uncertain systems that contain only partial knowledge by generating, excavating, and extracting useful information from the available data, enabling accurate descriptions of a system's operational behavior and its laws of evolution In the natural world, many systems are uncertain and data-poor, which makes grey systems theory highly applicable across diverse domains By focusing on how to infer dynamics from limited data, this theory provides practical tools for monitoring, predicting, and managing complex processes despite information gaps.
Development History and Current State
In 1982, Professor Julong Deng published "The Control Problems of Grey Systems," the first grey system paper to appear in Systems and Control Letters (Deng 1982a) In the same year, he released "Grey Control System" in Chinese, published by the Journal of Huazhong University of Science and Technology (Deng 1982b) The publication of these two seminal articles signaled the birth of grey system theory, a new cross-disciplinary field at the intersection of mathematics, control, and systems science.
Since its 1989 debut by Research Information Ltd in the UK, The Journal of Grey System has grown into a widely indexed publication, now listed by Mathematical Reviews, the Science Citation Index, and other major databases worldwide In 1997, Taiwan introduced a Chinese-language Journal of Grey System, which began publishing in English in 2004, expanding its international reach Emerald then launched Grey Systems: Theory and Application in 2011, edited by the Institute for Grey System Studies at Nanjing University of Aeronautics and Astronautics Today, more than a thousand journals globally have published papers on grey systems theory, with many leading publications and publishers—such as the Journal of the Association for Computing Machinery (USA), Communications in Fuzzy Mathematics (Taiwan), Kybernetes: The International Journal of Systems and Cybernetics, Transaction of Nanjing University of Aeronautics and Astronautics, China Ocean Press, Chinese Agricultural Science Press, Henan University Press, Huazhong University of Science and Technology Press Co Ltd, IEEE Press, and Springer-Verlag—featuring special issues or proceedings on grey system theory.
Across many universities worldwide, grey system theory has been integrated into formal curricula A prominent example is Nanjing University of Aeronautics and Astronautics (NUAA), where grey system theory is offered not only in PhD and Master’s programs but also as an elective within undergraduate studies across multiple disciplines In 2008, NUAA’s grey system theory course was designated a national-level model course in China, and in 2013 it was selected as a national excellent resource sharing course, evolving into a freely accessible open learning resource for grey system enthusiasts.
Across the globe, many universities are actively recruiting and funding doctoral and postdoctoral researchers in grey system theory and its applications Notable institutions include Huazhong University of Science and Technology, Nanjing University of Aeronautics and Astronautics, Southeast University, Wuhan University of Technology, Fuzhou University, Shantou University, the University of Central Florida, the University of Nebraska–Lincoln, the University of Waterloo, the University of Toronto, De Montfort University, Pablo de Olavide University, Boğaziçi University, the University of Cape Town, the Bucharest University of Economic Studies, Kanagawa University, and numerous universities across Taiwan Today, tens of thousands of graduate students and PhD researchers worldwide are conducting scientific research that applies grey system thinking and methods.
A wide range of publishing houses—Science Press, Defense Industries Press, Huazhong University of Science and Technology Press Co Ltd, Jiangsu Science and Technology Press, Shandong People’s Press, Science and Technology Literature Press of China, China Science and Technology Book Press of Taiwan, Gaoli Books Limited Company of Taiwan, ASE Press of Romania, Japan Polytechnic Press, IIGSS Academic Press, CRC of Taylor & Francis Group, Springer-Verlag, Springer-Verlag London Ltd, and John Wiley & Sons, Inc.—have released hundreds of scholarly works on grey systems in multiple languages, including Chinese, Traditional Chinese, English, Japanese, Korean, Romanian, and German.
Over the years, a cadre of cutting-edge disciplines—including grey hydrology, grey geology, grey breeding, and grey medical science—has emerged As this field gains prominence, national and local science funding agencies are actively supporting grey system research Today, hundreds of projects exploring grey systems and their applications are funded by major bodies such as the National Natural Science Foundation of China, the European Commission, the Royal Society, the Leverhulme Trust, and national funds from Canada, Spain, and Romania.
Since 2000, eighteen regional domestic conferences on grey system theory and its applications have been held, attracting increasing attention from researchers in the field These conferences have been supported by a consortium of prestigious institutions, including The Leverhulme Trust, the Institute for Grey System Studies, Nanjing University of Aeronautics and Astronautics, De Montfort University, Wuhan University of Technology, the Educational Society of Pudong in Shanghai, and the China Center of Advanced Science and Technology The China Center of Advanced Science and Technology is directed by Nobel Prize laureate Tsung-Dao Lee and by two former presidents of the Chinese Academy of Sciences, Zhou Guangzhao and Lu Yongxiang This strong backing has helped grey system theory attract a large number of young scholars to these events, fueling ongoing research and applications in this interdisciplinary field.
Numerous special sessions and dedicated tracks on grey system theory have been organized at major international conferences, reflecting the field's growing prominence Notable venues include the International Conference on Uncertain System Modeling and the International Conference on System Forecast and Control, where researchers share advances in grey system theory, uncertainty-aware modeling, and practical forecasting applications.
1.2 Development History and Current State 3
Grey systems theory has gained prominence through major international events, including the International Conference on General System Studies, the International Congress of the World Organization of Systems and Cybernetics, and the IEEE International Conference on Systems, Man and Cybernetics The topicality and rising popularity of grey systems theory at these high-profile conferences have actively advanced understanding and promoted the theory among peers in the systems science community.
From 2007 to 2015, the first five editions of the IEEE International Conference on Grey Systems and Intelligent Services were held in Nanjing, Macao, and Leicester, respectively, drawing a large international audience and a significant number of submissions from countries and regions including China, the United States, the United Kingdom, Germany, France, Spain, Switzerland, Hungary, Poland, Japan, South Africa, Russia, Turkey, Romania, the Netherlands, Malaysia, Iran, Ukraine, Kazakhstan, Pakistan, Taiwan, Macao, and Hong Kong In total, more than one thousand articles from these five conferences were indexed by the EI database, with more than three hundred outstanding papers published in Kybernetes, Grey Systems: Theory and Applications, The Journal of Grey System, Transactions of Nanjing University of Aeronautics and Astronautics (English version), and Springer-Verlag.
Grey system theory has earned the praise of numerous prominent scholars, underscoring its growing recognition in the scientific community Notable supporters include Lotfi A Zadeh (US), founder of fuzzy mathematics; Herman Haken (Germany), founder of synergetics; James M Tien (US), former vice-president of IEEE and member of the National Academy of Engineering; Robert Valee (France), president of the World Organization of Systems and Cybernetics; and Alex Andrew (UK), Secretary General of the World Organization of Systems and Cybernetics and President of the Canadian Royal Academy of Sciences In addition, many academicians from the Chinese Academy of Sciences and the Chinese Academy of Engineering have endorsed grey system theory, including Qian Xuesen, Yang Shuzi, Xiong Youlun, Lin Qun, Chen Da, Zhao Chunsheng, Hu Haiyan, Xu Guozhi, Wang Zhongtuo, and Yang Shanlin.
In 2005, the Grey System Society of China (CSOOPEM) was approved by the China Association for Science and Technology and the Ministry of Civil Affairs, China At the beginning of 2008, the Technical Committee of IEEE SMC on Grey Systems was established In 2012, De Montfort University hosted the first Workshop of the European Grey System Research Collaboration Network, with delegates from twelve European Union member states attending.
In 2013, Professor Sifeng Liu was selected for the Marie Curie International Incoming Fellowship (FP7-PEOPLE-IIF-GA-2013-629051) under the European Commission’s 7th Framework Programme, marking notable international recognition of his work in grey system theory In 2014, the Leverhulme Trust funded an international network project titled “Grey Systems and Its Applications” (IN-2014-020), which supported cooperative research and academic exchanges on grey system theory across Europe, North America, and China This collaborative effort paved the way for the establishment of the International Association of Grey Systems and Uncertainty Analysis (GSUA) in 2015 Today, grey system theory is emerging as a recognized discipline with growing influence in the scientific community.
Characteristics of Uncertain System
Incomplete Information
Incompleteness in information is one of the fundamental characteristics of uncertain systems The most common situations involving incomplete system information include cases where:
(1) Information about system elements (parameters) is incomplete;
(2) Information on the structure of the system is incomplete;
(3) Information about the boundaries of the system is incomplete; and
(4) Information on the system’s behaviors is incomplete.
Incomplete information is a common phenomenon in our social, economic, and scientific research activities For instance, in agricultural production, even if we have exact information regarding plantation, seeds, fertilizers, and irrigation, uncertainties in areas such as labor quality, natural environment characteristics, weather conditions, and the commodity markets make it extremely difficult to precisely predict the production output and consequent economic value of agri- culturalfields For biological prevention systems, even if we know the relationship between insects and their natural enemies, it is still really difficult to achieve the expected prevention effects due to uncertainty regarding the relationships between insects and their baits, insects’natural enemies and their baits, and a specific kind of natural enemy with another kind of natural enemy As for the adjustment and reform of pricing systems, it is often difficult for policy makers to take actions because of the lack of information regarding price elasticity of demand and how price changes on a certain commodity would affect the prices of other commodities.
In security markets, even the brightest market analysts cannot be assured of winning constantly due to their inability to correctly predict economic policy and interest rate changes, management changes at various companies, the direction of political changes, investors’behavioral changes in international markets, and the effects of price changes in one block of commodities on another As for the general economic system, because there are no clear relationships between the “inside” and the
Because the boundary between a system and its environment is not clearly defined, the “outside” can extend beyond the system itself into its surroundings This blur between internal and external borders makes it difficult to analyze how economic inputs translate into economic outputs, since effects propagate across ill‑defined interfaces and are shaped by external factors To meaningfully assess input–output relationships, analysts must acknowledge these blurred boundaries and incorporate environmental interactions into their models, recognizing that external conditions and feedback processes influence observed economic performance.
Information is inherently incomplete, while completeness is always relative Humans use limited cognitive ability to observe the infinite universe in an attempt to obtain complete information, yet such completeness remains unattainable In statistics, the concept of a large sample reflects how tolerant we are of incompleteness; a common rule of thumb says a sample with at least 30 objects is large But even when a sample contains thousands or tens of thousands of observations, the true statistical laws of a system can still remain unrevealed This tension highlights the gap between theoretical models and empirical data, and the ongoing challenge of drawing definitive conclusions from partial information.
Inaccuracies in Data
A fundamental characteristic of uncertain systems is the inherent inaccuracy in the data available In grey systems theory, the terms uncertain and inaccurate are used interchangeably to describe errors or deviations from actual data values These inaccuracies arise from how uncertainties are generated and can be categorized into three types: conceptual inaccuracies, level inaccuracies, and prediction-type inaccuracies.
Conceptual inaccuracies arise when language expresses events, objects, concepts, or wishes with insufficient precision Everyday descriptors such as large, small, many, few, high, low, fat, thin, and good can introduce ambiguity unless their meaning is clearly defined By specifying criteria, providing thresholds, and using measurable attributes alongside qualitative terms, writers can reduce misinterpretation and deliver clearer, more SEO-friendly content Recognizing these semantic pitfalls is essential for accurate communication across disciplines and audiences.
“bad,” “young,”and“beautiful”are inaccurate due to lack of clear definition.
Expressing these concepts with exact quantities is challenging For instance, a job seeker with an MBA who demands a minimum annual salary of $150,000 illustrates a conceptual precision issue Likewise, a manufacturing firm aiming to keep its defective-product rate below 0.1% highlights another form of conceptual inaccuracy These examples show how precise numeric targets can misrepresent the underlying ideas in business and quality control.
Data accuracy depends on the observational scale: measurements can be reliable at the macroscopic or cognitive conceptual level, but when data are examined at a microscopic or localized subcomponent, they often become inaccurate For example, a person’s height can be measured accurately to the centimeter or millimeter, yet if precision must reach one ten-thousandth of a micrometer, the previously reliable reading becomes extremely inaccurate This illustrates scale-dependent accuracy: reliability can hold at higher levels but degrade when data are broken down into finer, more detailed levels.
(3) The Prediction (or Estimation) Type
Because understanding the laws of evolution is inherently incomplete, any prediction of the future tends to be inaccurate For example, forecasts claim that in two years a country’s GDP will surpass $10 billion; a bank may be expected to attract residents’ savings between $70,000 and $90,000 for 2017; and Leicester’s June temperatures are predicted not to exceed 30°C These examples show the uncertainty inherent in prediction numbers In statistics, samples are often used to estimate population quantities, so much data are imprecise Indeed, regardless of the method, it is very difficult to obtain any absolutely accurate value When planning for the future and deciding what actions to take, we generally have to rely on imperfect predictions and estimates.
The Scienti fi c Principle of Simplicity
Throughout the history of science, the pursuit of simplicity has guided thinkers who hoped to explain the material world with a few universal elements By the sixth century BCE, natural philosophers sought to uncover the laws of nature using a small set of basic substances Around 500 BCE, Pythagoras and the Greeks proposed the theory of four elements—earth, water, fire, and air—as the fundamental building blocks of all matter At roughly the same time, ancient Chinese thinkers developed a five-element framework that included water, fire, wood, gold, and earth These early ideas stand as the most primitive and elementary attempts to understand nature through simple, foundational components.
The scientific principle of simplicity grows from the habit of clear, minimal thinking used to understand nature, and as the natural sciences mature it becomes the core guiding principle of research Newton’s laws show how broad macroscopic phenomena can be captured by remarkably simple relations, and Newton himself suggested that nature does not waste effort or rely on unnecessary explanations to boast itself In the era of relativity, Einstein proposed external confirmation and internal coherence—rooted in logical simplicity—as essential criteria for a true theory, arguing that harmony and order in nature demand simple explanations Earlier debates in electromagnetism among Ampère, Weber, and others culminated in Maxwell’s theory, which best conforms to simplicity and thus gained wide acceptance Kepler’s third law, T^2 = D^3, stands as another example where a simple mathematical form captures planetary motion Together these cases illustrate how the principle of simplicity underpins robust scientific theories by reflecting the orderly structure of the natural world.
According to the slaving principle of synergetics proposed by H Haken (1978), a high-dimensional nonlinear system can be reduced to a low-dimensional evolution equation for its order parameters by eliminating fast-relaxing variables Since order parameters dominate the system’s dynamics near boundary points, solving their evolution equations reveals the system’s time, space, or time-space structure, enabling efficient control over its behavior.
Scientific models achieve simplicity by using straightforward expressions and by ignoring nonessential factors in the system under study In economics, tools such as the Gini coefficient for describing income differences (Gini, 1921) and the Cobb-Douglas production function for measuring how technological progress contributes to economic growth (Cobb and Douglas, 1928) illustrate this simplification approach Following this tradition, Modigliani and Brumberg (1954) proposed a model to describe the average propensity to consume, showing how a compact, interpretable framework can yield insights into consumer behavior and macroeconomic dynamics.
The curve Phillips (1958) employs to describe the relationship between the rate of inflation Dp p and the unemployment ratexis:
Additionally, the well-known capital asset pricing model (CAPM, Sharpe 1964) can be seen below:
Essentially, all of these equations can be reduced to the simplest linear regres- sion model with a few straightforward transformations.
Precise Models Suffer from Inaccuracies
Whenever information is incomplete and data are inaccurate, pursuing precise models becomes meaningless, a truth Lao Tzu articulated more than two thousand years ago L A Zadeh’s principle of incompatibility, introduced by the founder of fuzzy mathematics, states that as a system’s complexity increases, our ability to describe its characteristics precisely diminishes until precision and meaningfulness become mutually exclusive (Zadeh 1994) This inherent antagonism shows that chasing precision can undermine the practicality and usefulness of cognitive outcomes, so exact models are not always the best tool for addressing complex matters.
In 1994, Jiangping Qiu and Xisheng Hua developed two modeling approaches based on deformation and leakage data from a large-scale hydraulic dam: a theoretically delicate statistical regression model and a relatively coarse grey model Hua’s grey model provided a better fit than Qiu’s regression model Comparison of the prediction errors with actual observations showed that the grey model generally achieved higher accuracy than the regression model, with details in Table 1.1.
In 2001, Dr Haiqing Guo, together with Zhongru Wu and colleagues, developed two models for the vertical displacement data of a large clay-rock-filled dam with inclined walls: a statistical regression model and a grey time-series combined model They evaluated each model’s data fit and predictive performance against actual observations and found that the grey time-series combined model provided a significantly better data fit and more accurate predictions than the statistical regression model.
Xiaobing Li, Haiyan Sun, and their colleagues employed fuzzy prediction functions—an uncertainty-prediction technique—to dynamically track and precisely regulate the fuel oil feeding temperature for anode baking This fuzzy-prediction–based control significantly outperformed conventional PID control, as demonstrated in Li and Sun (2009).
Caixing Sun and his research team applied grey incidence analysis, grey clustering, and a range of grey prediction models to diagnose and predict insulation-related incidents in electric transformers Their findings indicate that these relatively coarse, data-driven methods are more practical and yield more efficient results than traditional models, as demonstrated by Sun et al (2002, 2003) and Li et al (2002).
Table 1.1 Comparison between the prediction errors of a statistical model and a grey model
Order No Type Average error
5 Water level of pressure measurement hole 6.297 3.842
6 Water level of pressure measurement hole 0.204 0.023
Comparison of Several Studies of Uncertain Systems
Probability and statistics, fuzzy mathematics, grey system theory, and rough set theory are four of the most widely used approaches for investigating uncertain systems They share a focus on modeling different kinds of uncertainty, but the exact nature of the uncertainty in each research object is what distinguishes them In particular, probability and statistics address stochastic uncertainty and aim to reveal historical statistical laws by evaluating the likelihood of each possible outcome, with their analyses grounded in large samples that conform to a typical distribution.
Fuzzy mathematics deals with problems of cognitive uncertainty where the intention (inner meaning) of a concept is clear, but its extension (the set of all objects it applies to) is not For example, the notion of a “young man” has a precise intension, yet there is no universally agreed age boundary that cleanly separates who is young from who is not, making the extension fuzzy In fuzzy mathematics, this uncertainty is addressed by leveraging experience and by using the membership function to assign degrees of belonging to a concept.
Rough set theory provides a mathematical framework for studying uncertain systems by focusing on accuracy and the limits of knowledge Its central idea is to describe and manage imprecision using a known knowledge base Zdzisław Pawlak introduced the concept of boundary areas, defining a boundary as the difference between the upper and lower approximations of a set The boundary reflects the elements that cannot be decisively classified, and it is described by the degree to which the upper and lower approximations converge toward each other.
Grey system theory addresses uncertainty arising from small data sets and limited information, setting it apart from probability, fuzzy mathematics, and rough set theory It seeks to uncover the realistic laws governing evolution and events through information coverage by a possibility function and the use of sequence operators, enabling model building with minimal data A key distinction from fuzzy mathematics is that grey system theory emphasizes objects with clear extension but unclear intension For example, by 2050 China’s population is expected to lie within the 1.5–1.6 billion range—a grey concept where the extension is definite, yet the exact value within that range is not determinable The possibility function describes the likelihood that a given value is a grey number The differences among these four main uncertainty research methods—probability, fuzzy mathematics, rough set theory, and grey system theory—are often summarized in comparative tables.
Most Actively Studied Uncertain Systems Theories
Fuzzy mathematics, grey system theory, and rough set theory are currently the most actively studied uncertainty theories Analysis of the ISI and EI Compendex databases reveals a large number of research papers that include 'fuzzy set' as a keyword, highlighting the prominence and rapid growth of fuzzy set theory within this field.
“grey system,”and “rough set”has increased rapidly (see Table1.3).
A CNKI search indicates that, from 1990 to 2015, the number of scholarly publications containing at least one of the keywords fuzzy mathematics, grey system, or rough set exhibits an upward trend, as shown in Tables 1.4, 1.5, and 1.6; this pattern suggests increasing research interest in these areas over the period.
Research on uncertain (fuzzy, grey, and rough) systems can be categorized into the following three aspects:
(1) The mathematical foundation of uncertain systems theories;
(2) The modeling of uncertain systems and computational schemes, including various uncertain systems modeling, modeling combined with other relevant methods, as well as related computational methods; and
(3) The wide-range of applications of uncertain systems theories in natural and social sciences.
Uncertain (fuzzy, grey, rough) systems theories have been widely applied across natural sciences, social sciences, and engineering Their applications span aviation, spaceflight, civil aviation, information, metallurgy, machinery, petroleum, chemical industry, electrical power, electronics, light industries, energy resources, transportation, medicine and health, agriculture, forestry, geography, hydrology, seismology, meteorology, environmental protection, architecture, and behavioral science.
Table 1.2 Comparison among the four methods of uncertainty research
Grey system Prob statistics Fuzzy math Rough set
Basic set Grey number set
Cantor set Fuzzy set Approximate set
Density func Membership func Upper, lower appr.
Emphasis Intension Intension Extension Intension
Objective Law of reality Historical law Cognitive expression
Approx. approaching Characteristics Small data Large sample Depend on experience
1.5 Most Actively Studied Uncertain Systems Theories 11 management science, law, education and military science These practical appli- cations of uncertain systems theories have brought significant social and economic benefits.
Uncertain systems research remains highly active in both theory and application, but the field shows a bias toward practical use at the expense of theoretical development and methodological innovation There is limited exploration of the differences and commonalities among the diverse uncertain systems theories, and few efforts to integrate traditional approaches with emerging methods in fuzzy, grey, and rough systems Bridging this gap is essential to advance coherent uncertain systems theory and to develop robust, integrative tools that connect rigorous theory with real-world applications.
Table 1.3 Search outcomes of ISI and EI Compendex databases(2010 – 2015)
Keyword Fuzzy set Grey system Rough set
# in the EI Compendex database 9670 5087 6588
Table 1.4 Search outcomes of “ fuzzy mathematics ”
Table 1.5 Search outcomes of “ grey system ”
Table 1.6 Search outcomes of “ rough set ”
Traditional and emerging uncertain systems theories are not separate rivals but complementary frameworks, each with unique strengths suited to different kinds of uncertainty They supplement one another and are not theoretically incommensurable Many complex, dynamic uncertainty problems exceed the capacity of any single theory, making it essential to combine various traditional theories with new uncertain systems approaches Therefore, advancing science requires ongoing research that encourages interaction, exchange, and integration of these theories and methods.
Elementary Concepts of Grey System
Across social, economic, agricultural, industrial, ecological, and biological systems, naming conventions typically reflect the distinctive features of the research object classes, while grey systems are identified through color coding that signals the system of concern.
Control theory uses color codes to describe the clearness of information: black boxes denote unknown internal information, white information denotes complete visibility, and grey information denotes partial knowledge This terminology, widely accepted in science, mirrors democratic societies where citizens demand greater policy transparency and more white information Consequently, a system with fully known information is white, one with fully unknown information is black, and one with partially known and partially unknown information is grey.
In this context, incompleteness in information is the fundamental meaning of
“grey.” However, the meaning of“grey”can be expanded or stretched from dif- ferent angles and in varied situations (see Table1.7).
Table 1.7 Extensions of the concept of “ grey ”
Situation/concept Black Grey White
Information Unknown Incomplete Completely known
Methods Negation Change for the better Con fi rmation
Attitude Letting go Tolerant Rigorous
Outcomes No solution Multi-solutions Unique solution
1.5 Most Actively Studied Uncertain Systems Theories 13
At this point, the difference between“system”and “box”Must be highlighted.
On the one hand, the term 'box' is used when researchers overlook interior details and focus on the object's external characteristics, assessing its properties through the input-output relationship This black-box approach analyzes what goes in and out without disclosing the internal structure On the other hand, the term also refers to white-box or glass-box analyses that examine internal features and mechanisms to explain how internal processes produce observed outputs.
System analysis refers to studying an object's structure and functions by examining the intrinsic connections among the object, its relevant factors, the surrounding environment, and the governing laws of change This systems thinking approach highlights how components interact within a larger whole, how context and external forces influence behavior, and how dynamic patterns emerge from the network of relationships By focusing on these interconnected relationships, researchers can gain a holistic understanding of complex phenomena, explain performance, and predict outcomes across disciplines.
Grey systems theory studies uncertain systems that are known only partially, often with small sample sizes and limited information The framework focuses on generating and extracting partially known information using grey sequence operators and possibility functions to achieve accurate descriptions and a deeper understanding of real-world phenomena.
Fundamental Principles of Grey Systems
In the process of developing grey systems theory, Julong Deng established six fundamental principles containing intrinsic philosophical intensions, as discussed below (Deng 1985).
Axiom 1.7.1 The Principle of Informational Differences “Difference” implies the existence of information Each piece of information must carry some kind of
When we say that object A is different from object B, we mean that object A contains some distinct information that does not hold for object B All differences between natural objects and events provide us with fundamental information that helps us understand their nature.
Whenever new information reshapes our understanding of a complex issue, it departs from what we initially believed Great breakthroughs in science and technology provide essential knowledge and tools that enable us to understand and change the world around us, and such advanced information is clearly different from pre-scientific ideas The more substantive content a piece of information contains, the more its differences from earlier versions of that information become evident.
Axiom 1.7.2 The Principle of Non-Uniqueness The solution to any problem with incomplete and indeterminate information is not unique.
By embracing the principle of non-uniqueness—the fundamental rule of grey systems theory—you can view problems with greater flexibility This adaptive mindset increases your effectiveness in reaching your goals.
Strategically, the non-uniqueness principle is realized through the grey target concept, a unification of non-unique targets and non-restrainable targets For instance, if a high school graduate intends to enroll in only one institution, his university admission chances are greatly constrained; conversely, a similarly qualified graduate who applies to multiple universities beyond his preferred one increases the probability of success, because having multiple targets raises the odds of hitting at least one This multi-target approach within the grey target framework thus improves overall admission prospects and demonstrates how flexible targeting can outperform single-target strategies in higher education admissions.
The principle of non-uniqueness encompasses the idea that every target can be approached from multiple angles, that any available information can be supplemented, that earlier plans can be revised and improved, that relationships can be harmonized, that thinking can proceed in multiple directions, that understanding can be deepened, and that each path can be optimized When confronted with the possibility of multiple solutions, one can identify one or several satisfactory options through deterministic analysis and information supplementation Therefore, the method of finding solutions relies on integrating structured analysis with ongoing information enhancement to select optimal outcomes.
“non-uniqueness”is one that combines both quantitative and qualitative analysis.
Axiom 1.7.3 Principle of Minimal Information One characteristic of grey system theory is that it makes the most and best use of the“minimal amount of available information.”
The“principle of minimal information”can be seen as a dialectic unification of
Grey system theory excels at handling uncertain problems with small data or poor information Its foundation lies in the concept of spaces of limited information, where the minimal amount of information defines the operating territory and powers the theory The line between grey and not-grey is determined by how much information can be acquired Consequently, the essential problem‑solving logic of grey system theory is to make sufficient discoveries and apply any available minimal information to solve problems.
Axiom 1.7.4 Principle of Recognition Base Information is the foundation on which people recognize and understand (nature).
This principle argues that all recognition must be based on information; without information, there is no way for people to know anything When information is complete and deterministic, we can achieve a clear and firm understanding of nature; with incomplete and non-deterministic information, we can only obtain a grey, partial, and non-deterministic understanding of specific phenomena.
Axiom 1.7.5 Principle of New Information Priority The function of new pieces of information is greater than that of old pieces of information.
The“principle of new information priority”is the key idea behind information application in grey system theory That is, by applying additional weights to new
Fundamental Principles of Grey Systems Theory show that new information improves outcomes across grey modeling, grey prediction, grey analysis, grey evaluation, and grey decision making The saying "the new replaces the old" captures the principle of new information priority, which assigns greater weight to recent data to boost accuracy and relevance With each new data point, the motivation to whiten or update grey elements strengthens, emphasizing that information is time sensitive and must drive modeling and decision processes.
Under the Principle of Absolute Greyness, information incompleteness is absolute, and incompleteness and non-determinism are general features of information Completeness is relative and temporary, representing the moment when the original non-determinism has just vanished and a new non-determinism is about to emerge Human understanding of the objective world improves through continual supplementation of information, and with an endless stream of information, our knowledge and comprehension become limitless Consequently, the greyness of information is absolute and will never disappear.
The Grey Systems Theory Framework
Over the past three decades, grey systems theory has emerged as a distinct scientific discipline, with its own robust theoretical framework that encompasses systems analysis, evaluation, modeling, prediction, decision-making, control, and optimization techniques.
Grey Models and Framework
A grey number denotes a value that spans a range rather than a single precise value when the exact value is unknown; the range can be an interval or a general number set It is typically represented by the symbol ⊗, referred to as grey, and it conveys the degree of information uncertainty in a given system As the foundation of grey systems theory, research on grey numbers and grey measures has attracted increasing attention in recent years.
Julong Deng introduced the core ideas and models of Grey System Theory (GST) in his 1982 Systems and Control Letters paper, then formalized the GST framework in 1985 across two books published by National Defense Industry Press and Huazhong University of Science and Technology Press These works cover grey numbers and the operations on interval grey numbers, grey incidence analysis, grey generation, grey clustering, grey forecasting models, grey decision-making, and grey control He further solidified the GST framework in later publications, including A Course on Grey System Theory (1990) and The Basis of Grey System Theory (2002).
S Liu et al., Grey Data Analysis , Computational Risk Management,
The Thinking, Models and Framework of Grey Systems
Grey Numbers and Its Operations
In 2004, Sifeng Liu and Yi Lin proposed an axiomatic definition of the grey degree of a grey number, grounded in a measure of the grey number and its background or domain This definition satisfies the standardability requirement, providing a solid bedrock for cognizing the uncertainty of grey information.
In 2010, the unreduction axiom and a new definition of the degree of greyness for grey numbers were introduced, enabling the construction of operations for grey numbers and the grey algebraic system based on the grey kernel and greyness degree (Liu et al., 2010) Consequently, grey-number operations were transformed into real-number operations, and to a significant extent the problem of establishing a functional framework for grey numbers and the grey algebraic system has been resolved.
In 2012, Sifeng Liu, Zhigeng Fang, Yingjie Yang and others developed the concept of general grey number as follows: g 2[ n iẳ 1 a i ;a i
Among them, any interval grey numbers i2 ẵa i ;ai S n iẳ 1a i ;a i
, which satisfy a i ;a i 2