Exploring the Principle of Multi-Dimensional Risk Analysis and a Case Study in Two-Dimensional Risk

Abstract

By examining the significant flaws in multivariate risk analysis and integrated risk analysis, this article introduces a new approach to evaluating the total risk within complex risk systems: the principle of multi-dimensional risk (MDR) analysis. Under this framework, the scope of each individual risk is first defined, and the risk-bearing entity is identified. Each risk is then analyzed independently, and the results are subsequently integrated to provide a comprehensive view of MDR. Multivariate risk analysis becomes increasingly impractical as the number of factors grows, due to the correspondingly large sample size required—often unattainable in real-world conditions. Integrated risk analysis methods, such as weighted combinations and Copula techniques, are heavily influenced by subjective factors, which compromise the reliability of their results. In contrast, MDR analysis involves fewer variables per individual risk, reducing the sample size requirement and making data collection more feasible. Individual risks can be quantified using objective physical indicators such as economic loss or physical injury, enabling more accurate calculations of the total risk across the system. A case study involving two-dimensional risks—flood and earthquake—demonstrated that these events often have vastly different occurrence cycles. When these risks are entangled in conventional analysis, the resulting annual total risk value can be severely distorted. By analyzing individual risks separately, maintaining the focus on overall system risk, and treating the total risk as an MDR problem, a more reliable foundation for policy-making and risk management can be established. There are at least three types of MDR relationships: independent, compounding, and negatively correlated. As a result, no universal MDR analysis model exists.

1. Introduction

Risk is a simple yet complex concept, defined differently across various fields, with no universally accepted definition. The World Health Organization (WHO) defines public health risk as “the likelihood of an event that may adversely affect the health of human populations” (WHO 2005). In contrast, the International Organization for Standardization (ISO) defines risk as “the effect of uncertainty on objectives” (ISO 2018). The WHO’s definition is limited in scope, as risks extend beyond health-related concerns. Meanwhile, the ISO’s broader and more neutral definition—stating that risk “can be positive, negative, or both, and can address, create, or result in opportunities and threats”—conflicts with the public’s predominantly negative perception of risk.

After filtering out duplicate and overlapping definitions, at least 40 distinct definitions of risk can be identified. Most of these emphasize future uncertainties and adverse events. For for-profit organizations, whose primary aim is to maximize gains or minimize losses, the risks encountered in management strategies typically refer to potential future losses. These include negative impacts, business setbacks, and even bankruptcy. In this context, “potential” denotes uncertainty. Probability is generally used to describe random uncertainty, while fuzzy sets are often employed to represent uncertainty arising from incomplete information.

In this paper, we adopt the following definition provided by Huang and Ruan (2008) to define the risks under discussion.

Definition 1.

Risk is a scene in the future associated with some adverse incident.

Risk analysis is a distinct and multidisciplinary field that encompasses risk assessment, perception, communication, management, governance, and policy. It addresses risks relevant to individuals, organizations (both public and private), and society as a whole—on local, regional, national, and global scales (SRA 2025). At its core, risk analysis aims to build a comprehensive understanding of risk. This understanding serves as an input for risk assessments and supports informed decision-making about whether and how risks should be managed, as well as the selection of appropriate treatment strategies.

The methods used in risk analysis can vary widely—ranging from qualitative to semi-quantitative and fully quantitative approaches. The level of detail required depends on several factors, such as the specific application, the quality and availability of data, and the decision-making needs of the organization involved. In some contexts, the chosen methodology and depth of analysis may also be shaped by regulatory or legislative requirements (IEC/ISO 2019). For example, in the energy sector, quantitative risk assessments might be mandated to ensure public safety, while a tech startup may opt for qualitative methods to assess product development risks more flexibly.

The purpose of this study is to explore a simpler yet more robust approach to analyzing the total risk of complex systems. This is motivated by a critical examination of the serious limitations found in current multivariate and integrated risk analysis methods. The structure of this article is as follows: Section 2 and Section 3 address the major flaws of multivariate risk analysis and integrated risk analysis, respectively. In Section 4, we delve into the challenges posed by multi-dimensional risk scenarios. Section 5 introduces the principle of multi-dimensional risk analysis, offering a conceptual alternative. A practical demonstration of this principle, through a two-dimensional risk case study, is provided in Section 6 to show that the proposed method is both simpler and more reliable. Finally, Section 7 concludes this article.

2. Multivariate Risk Analysis

Multivariate risk refers to risk influenced by multiple interrelated factors that do not operate in isolation but interact in complex and dynamic ways. For example, stock market risk is a type of multivariate risk shaped by variables such as interest rates, economic growth, and competitor behavior. Likewise, the risk of developing a disease is influenced by a combination of genetics, lifestyle choices, environmental conditions, and social determinants. The concept of multivariate risk underscores the need for a holistic approach to understanding and managing risks involving multiple contributing factors.

Multivariate risk analysis is a statistical method used to assess and understand potential risks by examining the interactions among multiple variables. Rather than evaluating each risk factor independently, this approach considers how various factors influence one another and contribute collectively to overall risk. It seeks to identify correlations and dependencies between variables and to quantify the cumulative impact of these relationships. By analyzing historical data and identifying patterns, multivariate risk analysis supports more accurate predictions of future outcomes and their associated risks.

The most common multivariate risk analysis methods include:

• Regression analysis, which examines the relationship between a dependent variable and one or more independent variables.
• Principal component analysis, which reduces the dimensionality of data by identifying the most important underlying factors.
• Monte Carlo simulation, which uses random sampling to simulate potential outcomes and assess the probabilities of different scenarios.

By accounting for the interplay among multiple factors, multivariate risk analysis offers a more comprehensive and accurate understanding of risk than approaches that evaluate individual variables in isolation.

One of the most well-known applications of multivariate risk analysis is in assessing the risk of coronary heart disease (CHD). CHD is a multifactorial condition, and its risk must be estimated by evaluating all relevant cardiovascular risk factors simultaneously. Simply counting the number of “at-risk” values for individual factors often fails to identify individuals at high risk—particularly those with several moderately elevated risk factors. A more effective approach is to apply a quantitative multivariate risk score.

Using longitudinal data on CHD risk factors and outcomes—including morbidity and mortality—from participants in the town of Busselton, Western Australia, Knuiman et al. (1998) developed a multivariate CHD risk-scoring system. This system incorporates variables such as age, blood pressure, use of antihypertensive medication, total and HDL cholesterol levels, smoking status, diabetes, left ventricular hypertrophy, and previous CHD history. The authors employed the proportional hazard regression model (Cox 1972) to generate a multivariate CHD risk score table specifically tailored for Busselton men and women (see

Table 1. Coronary heart disease (CHD) risks for men and women in Busselton, Australia (Knuiman et al. 1998).

Multivariate CHD Risk Score for Busselton Men:
Risk score	= + + − + + + + +	0.530 × Age (years) 0.055 × Systolic BP (mm Hg) 0.43 if on drug treatment for hypertension 56.26 × HDL/total cholesterol ratio 3.01 if smoke 1–14 cigs/day or smoke pipe/cigar 3.20 if smoke at least 15 cigs/day 3.83 if have left ventricular hypertrophy on ECG 11.48 if have previous history of CHD 10
Multivariate CHD Risk Score for Busselton Women:
Risk score	= + + − + + + + + +	0.846 × Age (years) 0.122 × DiastolicBP (mm Hg) 5.86 if on drug treatment for hypertension 33.25 × HDL/total cholesterol ratio 2.20 if smoke 1–14 cigs/day or smoke pipe/cigar 6.16 if smoke at least 15 cigs/day 2.97 if have left ventricular hypertrophy on ECG 7.85 if have diabetes. 7.99 if have previous history of CHD 25

).

$$h\left(t\right)=h_0\left(t\right)\times \exp ({\beta }_1{X}_1+{\beta }_2{X}_2+\cdots )$$

(1)

where h(t) is hazard rate for an individual at time t; h₀(t) is the baseline hazard rate; β₁, β₂, … are regression coefficients for each predictor variable; and X₁, X₂, … are values of the predictor variables for the individual. The model estimates the coefficients (β) that best fit the observed data. The coefficients are then used to calculate hazard ratios.

Moreover, it is worth noting that, in many studies, risks influenced by multiple factors are often referred to as multi-dimensional risk (MDR) or multivariate risk. For instance, multivariate probability distributions in ruin theory are treated as MDR models (Burnecki et al. 2022). Similarly, Viana et al. (2021) assessed the MDR of a 26,800 m natural gas pipeline by examining operational, technical, consequence-related, and preferential properties of pipeline sections, using tools such as Monte Carlo simulation. Gong et al. (2012) described insurance claim risk modeled through a multivariate process as MDR. Additionally, Escobar-Anel (2022) applied the concept of multivariate risk to associate different levels of risk aversion with diverse sources of wealth—such as sectors, stocks, and asset classes.

In the existing literature, methods for analyzing the complexity of a single risk are well established. In this context, complexity refers to the challenge of identifying and quantifying causal links between a wide array of potential causes and a specific observed effect (Renn 2005). Some analysts consider each influencing factor as a separate “dimension”, thereby classifying any risk influenced by multiple factors as MDR. However, this classification is conceptually flawed.

For example, if disaster risk influenced by multiple factors was considered MDR, then earthquake risk—affected by variables such as seismic intensity and building integrity—would be classified as MDR. Yet, this is misleading. A more accurate example of MDR would involve multiple distinct disaster types, such as earthquakes, floods, and typhoons, contributing simultaneously to the overall risk.

This distinction—between defining MDR based on the number of contributing factors versus the number of distinct risk types—can lead to confusion in risk classification. To improve clarity in the context of natural disaster risk analysis, we propose that the number of dimensions in MDR should correspond to the number of distinct disasters, not merely the number of influencing factors.

3. Integrated Risk Analysis

Integrated risk refers to risk caused by multiple hazards—also known as multi-hazard risk (Huang and Huang 2018). Integrated risk analysis primarily focuses on the compounding effects of these hazards, which may interact in ways that amplify overall risk. Researchers also study disaster chains, where one hazard triggers another, and explore how interdependencies between systems increase overall vulnerability. This type of analysis can span across sectors, identifying risks not only within individual departments or business units but also in the weaknesses of systems, processes, or human factors that can be exploited by cascading threats.

In theory, integrated risk assessment should be the quantitative component of integrated risk analysis. However, in practice, it is often treated more holistically, emphasizing the integration of multiple disciplines (Munns et al. 2003; Sekizawa and Tanabe 2005) and the use of integrated databases (Fedra 1998). Meanwhile, integrated risk management tends to focus less on analytical methods and more on strengthening disaster risk governance through better coordination and resource mobilization (UNDRR 2015).

One typical application of integrated risk analysis is the multi-hazard risk assessment of water supply systems. These systems are often exposed to a variety of climate-related hazards such as droughts, floods, and cyclones. For example, Becher et al. (2023) developed a broadly applicable framework for assessing multi-hazard risk and applied it to evaluate the effects of both present and future climate extremes on Jamaica’s national water supply network.

The framework involves stress-testing a model of the water supply system using a large set of spatially coherent hazard events—including drought, cyclone, pluvial flooding, and fluvial flooding. The analysis quantifies the number of customers whose water supply would be disrupted during an event, measured in Customer Disruption Days (CDD). Under the current conditions, this study estimated the average annual multi-hazard disruption to be approximately five days per customer. The expected annual loss—representing integrated multi-hazard risk—was calculated by integrating the inverse cumulative probability function of disruption across annual exceedance probabilities using numerical quadrature methods (see Equation (2)).

$$CD{D}_{av,j}={\displaystyle {\int }_0^{1}QCD{D}_j}(p)dp.$$

(2)

i.e., the expected annual with respect to the jth supply source CDD_av,j is obtained by Quadrature and the function QCDD(p) is constructed by using the simulation-based method described in their article with a large sample of disruption events.

It is evident that most risk events involve multiple hazards, and scholars have long recognized the presence of multi-hazard or complex risks. In response, integrated risk assessment approaches have been developed, often leveraging interdisciplinary databases to capture the diverse nature of these risks.

One typical example involves the accumulation of acute risks from the accidental release of hazardous chemicals alongside chronic risks from long-term exposure to toxic substances in the surrounding environment—a case of multi-hazard risk documented by Gurjar and Mohan (2003). Another approach integrates data from hazardous devices, hazardous materials, and accident databases, combining them with simulation models to assess a wide range of potential accident scenarios (Fedra 1998). Additionally, the widely used “probability–loss” model serves as an integrated risk assessment model with an integration degree of 2, indicating the inclusion of two types of variables or perspectives (Huang 2009).

To improve the monitoring and early identification of integrated risk events, the European Emerging Risk Radar was developed by Jovanovic et al. (2012). This system aims to identify, locate, and assess risks across five key dimensions: environmental, social, economic, regulatory, and technological. Similarly, Huang et al. (2016) proposed an integrated risk radar driven by the Internet of Intelligences, designed to support emergency management within communities. This system monitors risks across six aspects:

• Natural hazards;
• Fire, explosion, and public health;
• Security and public order;
• Public opinion and sentiment analysis;
• Construction and equipment accidents;
• Resident activity.

Although integrated risk analysis offers a systematic overview of risks and hazard sources, it remains uncertain whether the risks identified through such integration accurately reflect real-world risk conditions. Furthermore, there is no clear consensus on how many degrees of integration are appropriate or effective. Neither a simple patchwork-style approach—such as the weighted integration of risks from earthquakes, windstorms, floods, volcanic eruptions, bushfires, and frost (Munich Re 2003)—nor a complex fuzzy-style approach—such as an integrated assessment model based on evidential reasoning algorithms and fuzzy set theory (Chen et al. 2014)—can fully address these challenges.

One of the most significant limitations of integrated risk analysis is its susceptibility to subjective influence. While decision support methods such as PRISM (Partial Risk Map) and TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) have been proposed to improve the Analytic Hierarchy Process (AHP) in integrated risk analysis, their effectiveness remains constrained by human judgment. For instance, Bognár et al. (2022) applied these methods to assess strategic incident groups in the incoming logistics business processes of a nuclear power plant, but the results were heavily influenced by the subjective evaluations of a 10-member expert risk assessment committee.

4. Multi-Dimensional Risk Issues

Clearly, whether approached from the perspective of multiple contributing factors or multiple hazards, it remains impossible to fully capture the composite mechanisms underlying systemic risk—defined as adverse effects that impact an entire system. As a result, risk assessment in such contexts is typically conducted using statistical models or experience-based methods. However, the reliability of these approaches is often questionable: statistical models rely heavily on large, high-quality datasets, while experience-based methods are inherently subjective, leading to potential inconsistencies and reduced credibility in assessment outcomes.

The risk–risk tradeoff framework offers a structured method for confronting, comparing, and weighing multiple impacts, with the goal of identifying risk-superior options—i.e., strategies that mitigate multiple risks simultaneously (Felgenhauer et al. 2022). Despite its promise, this framework struggles to evolve into a robust tool for systemic risk analysis, largely due to the lack of understanding surrounding risk dimensions and the composite mechanisms that drive such complex interactions. As a result, it also fails to substantially improve the reliability of risk assessments in these contexts.

To address this gap, we introduce a refined concept based on the four foundational elements outlined in

Table 2. Definitions of risk-bearing body, risk system, systemic risk, and single risk.

Concept	Definition	Example
Risk-bearing body	An object that would take some adverse incident in the future is called a risk-bearing body.	Buildings in a seismic area. Patients with coronary heart disease People who may be infected with COVID-19. Banks with loans.
Risk system	A system consisting of risk sources, risk-bearing bodies, time and geographical elements is called a risk system.	{Next five years, San Francisco, San Andreas fault} {Next 5 days, John Smith, Influenzas, Travelling in California}
Systemic risk	Systemic risk refers to the total risk as the adverse effects for the entire system.	Total possible loss caused by floods and strong winds in village A next year. Risk in European sovereign debt markets.
Single risk	The risk related to a specific hazard is called a single risk.	Earthquake risk. Risk of cardiovascular disease related to the use of a specific estrogen. Credit risk related to unstable employment.

. Using these concepts, we define multi-dimensional risk (MDR) as presented in Definition 2. The key distinctions between MDR, multivariate risk, and integrated risk are illustrated in Figure 1.

Definition 2.

Multi-dimensional risk refers to the total risk of a risk system including more than one single risk.

When analyzing the risk of multiple natural disasters in Türkiye, earthquake risk is typically regarded as a one-dimensional risk. However, if we distinguish between earthquakes originating from the North Anatolian Fault and the East Anatolian Fault, the analysis evolves into a two-dimensional risk. Similarly, at a global level, the COVID-19 pandemic and regional conflict together represent a two-dimensional risk. For a city surrounded by n high-infection zones, the COVID-19 risk alone can be considered n-dimensional due to its multiple sources of exposure.

Furthermore, COVID-19 risk interacts with other stressors, such as air pollution and pre-existing comorbidities. The impacts of the pandemic are also compounded by indirect effects: measures taken to mitigate COVID-19, such as lockdowns or medical triage protocols, can inadvertently amplify other risks, including delays in treatment for conditions like heart disease (Wiener 2020).

This highlights an important conceptual requirement: all individual risks that constitute a multi-dimensional risk (MDR) must conform to a consistent definition of risk, as outlined in Definition 1. Just as each axis in a Cartesian coordinate system must represent a real number for the system to remain valid, so too must each component of MDR share a common conceptual foundation. If, for instance, one risk is defined as “the effect of uncertainty on objectives” (ISO 2018) and another as “the probability of loss” (Crichton 1999), it becomes difficult—if not impossible—to coherently determine their combined total risk.

There are at least three types of MDR issues:

Type I: Independent Risk Issues

In this category, different risks arise and develop independently, with no causal relationship between them. However, they affect the same risk-bearing entity, leading to multi-dimensional risk (MDR) issues. For example, in a given region, flood risk and earthquake risk are independent in their causes and development, yet both impact the region, forming a two-dimensional risk and resulting in independent risk issues.

Type II: Compounding (Worsening) Risk Issues

Here, the origins of different risks may be independent, but their mutual impact on the risk-bearing entity leads to worsened consequences. For instance, heart disease and gastric ulcers have no direct causal link, but their interaction in a single patient can lead to more severe health complications. Thus, heart disease and gastric ulcers form compounding (worsening) risk issues.

Type III: Negatively Correlated Risk Issues

In this type, risks are negatively correlated—meaning that as one risk increases, another (or more) risks decrease, and vice versa. For example, in a region, flood risk and drought risk exhibit a negative correlation: an increase in flood risk results in a decrease in drought risk, and vice versa.

Another example is citywide lockdowns during the COVID-19 pandemic. Lockdowns reduce the risk of infection, but they simultaneously increase the risks of mental health issues (e.g., depression) and food shortages. In this case, public health safety, depression, and hunger form a three-dimensional negatively correlated risk issue.

In the real world, an issue involving multiple risks is often a hybrid of Type I, Type II, and Type III risks. For example, if economic recession and energy crisis are multi-dimensional risks (MDRs) resulting from COVID-19, a regional war, population aging, and natural disasters, they can be expressed as follows:

• Economic recession = COVID-19 + regional war + population aging + natural disasters
• Energy crisis = regional war − (COVID-19 + population aging) ⊖ natural disasters

In a “multi-risk world” (Wiener 2002), we must not only address the complexities of individual risks but also the challenges posed by multiple interrelated risks. A narrow focus on a single risk at a time can lead to unintended consequences in interconnected systems (Felgenhauer et al. 2022).

Analyzing multi-dimensional risk (MDR) is a highly challenging task. Based on the intention of MDR as defined in Definition 2, we will explore the principles of MDR analysis in Section 5 and apply them to examine a two-dimensional risk case in Section 6.

5. Principle of Multi-Dimensional Risk Analysis

All contributing factors within each single risk in a multi-dimensional risk (MDR) framework collectively determine the total risk of the MDR system. At first glance, multivariate risk analysis may appear suitable for MDR—provided that sufficient historical data on systemic risk events exist to satisfy the law of large numbers in probability theory. However, as noted by Huang (2009), a sample size smaller than 30 is generally considered insufficient to reliably estimate even a univariate probability distribution. Consequently, for MDR involving m variables, a multivariate dataset would need to exceed 30× m observations to yield statistically reliable results.

In statistical theory, to evaluate analysis reliability under small-sample conditions, tools such as Student’s t-test or non-parametric methods are often employed. These techniques, however, do not enhance the reliability of the estimators themselves—they merely provide a confidence interval around potentially unstable estimates. In contrast, empirical Bayesian methods and related approaches leverage prior distributions to improve estimator reliability. Yet, in many practical scenarios, historical data may be insufficient or unreliable, making it difficult to construct meaningful priors.

The Monte Carlo method has been proposed as a way to address data scarcity by generating large numbers of virtual samples. In some contexts—such as credit risk assessment in financial institutions—this approach has shown promise in improving predictive performance and reducing potential economic losses (Zhang et al. 2022). However, because these virtual samples are generated from probability distributions derived from the same limited datasets, the Monte Carlo method does not fundamentally resolve the small sample problem.

In the case study presented in this article, we address this limitation by employing the information diffusion technique (Huang 1997), which enhances analytical reliability under conditions of data scarcity.

For example, in a two-dimensional risk issue, if one risk has two factors and another has three factors, then multivariate risk analysis would require a sample size of more than 30⁵ = 24,300,000—a condition that is nearly impossible to meet. Thus, multivariate risk analysis remains mostly theoretical, with limited practical value.

Integrated risk analysis flowcharts can help clarify the internal structure and interdependencies within a multi-dimensional risk (MDR) framework, but they do not provide a means of calculating actual risk values. Moreover, the weighted combination methods frequently used in integrated risk analysis are highly subjective and lack physical meaning, making them unsuitable for MDR assessment.

The Copula approach (Su and Furman 2017) offers an alternative by linking multiple univariate marginal distributions to construct a joint probability distribution, potentially circumventing the small-sample problem. However, identifying a suitable Copula function for a given integrated risk scenario remains a significant challenge. For example, Lei and Shemyakin (2023) applied the Copula method to assess COVID-19-related mortality in Minnesota and Wisconsin, exploring several classes of copulas—both elliptical and Archimedean families. Their study highlights the complexity and ambiguity in selecting an appropriate Copula, especially when data are limited or the dependencies between variables are not well understood.

In summary, the principle of MDR analysis is fundamentally different from both multivariate risk analysis and integrated risk analysis. Due to the unique structure and characteristics of MDR, it is essential to develop a dedicated analytical framework. As a foundation for this framework, it is necessary to clarify the core tasks of MDR analysis.

To illustrate one such task, consider the combined risks of COVID-19 and a major earthquake.

In April 2020, COVID-19 was rapidly spreading across the globe, affecting nearly every city along seismic belts. If a strong earthquake were to occur during such a pandemic, the number of earthquake-related fatalities could significantly increase due to heightened vulnerability and disrupted emergency response systems. In this context, death risk refers to a future scenario involving fatalities caused by either disease or disaster. For instance, the death risk from COVID-19 can be estimated using the infection rate and the mortality rate among infected individuals (Huang 2020).

Similarly, earthquake death risk can be assessed using various estimation models, with the projected number of fatalities as a key output metric.

At first glance, it may appear that simply summing the number of expected deaths from COVID-19 and those from a hypothetical earthquake would yield the total MDR. However, this approach is flawed. Risk assessment is inherently forward-looking, and thus, the actual number of deaths is not yet known at the time of assessment. Moreover, in scenarios where disasters coincide—such as an earthquake striking during a COVID-19 outbreak—the risks are not independent. For instance, a person cannot die twice, so the combined death toll is not simply additive. Instead, MDR must account for overlapping vulnerabilities, interdependent impacts, and constrained system responses.

Given that the duration of an earthquake is negligible compared to that of COVID-19, the MDR analysis follows these steps:

• Estimate the number of potential deaths due to earthquake risk.
• Subtract the estimated earthquake-related deaths from the city’s total population, as this determines the remaining risk-bearing body.
• Use the adjusted population size to estimate potential COVID-19 deaths.

This method accounts for overlap, ensuring that the sum of the two estimated death tolls accurately reflects the total number of deaths caused by the overall risk.

Overlapping fatalities represent just one of the simpler challenges in multi-dimensional risk (MDR) analysis—a challenge that can be addressed by carefully defining the risk scenario and analyzing the characteristics and interdependencies of each individual risk. However, it is neither necessary nor feasible to precisely quantify overall risk values in most MDR contexts.

In general, estimating the total risk within an MDR framework is difficult due to the complex and often nonlinear mechanisms that underlie the formation of composite risk. For instance:

Over meteorological timescales (hours to weeks), hazards such as extreme winds and heavy precipitation can interact, compounding the overall risk through physical or operational feedbacks.

Over climatological timescales (seasonal to annual), entities such as infrastructure operators, government agencies, reinsurers, and public health systems are more concerned with aggregated risks, where interactions and long-term uncertainties become even more complex.

As such, effective MDR analysis requires careful attention to temporal scales, risk interactions, and system-level impacts in order to yield meaningful insights for multi-hazard and multi-sectoral risk management.

Moreover, it is not always appropriate—or even meaningful—to attempt to unify risk indicators across dimensions to produce a singular overall risk value. For example, attempting to standardize five types of risk—market, technical, financial, environmental, and management—onto a normalized scale (e.g., [0, 1]) and then computing a weighted sum is inherently problematic. Not only is the weighting process highly subjective, but the resulting aggregate score lacks robust interpretability or decision-making value.

These challenges are magnified when MDR involves conflicting risk-bearing entities. For example, reducing industrial emissions may improve public health by lowering respiratory risks, but it could simultaneously increase the concentration of harmful gases within enclosed workspaces, thereby raising risks for factory workers. In such situations, how should one define overall risk—public health vs. occupational health? There is no universally meaningful way to resolve this trade-off through simple aggregation.

One pragmatic approach to MDR analysis is dimensionality reduction—transforming a complex multi-dimensional risk scenario into a simplified single-risk framework, which can then be analyzed using conventional risk assessment methods. For instance, in the Jamaican water supply case study (Becher et al. 2023), the risks posed by drought, cyclones, rainfall, and flooding were effectively condensed into a single disruption metric—Customer Disruption Days (CDDs)—facilitating tractable risk analysis.

The process involves:

• Using historical data and the Climate Forecast System Reanalysis (Saha et al. 2010) to build spatial statistical models of disaster events.
• Running random simulation tests on these models to generate 10,000 years of synthetic events.
• Achieving dimensionality reduction by transforming the four-dimensional risk issue into a single-risk issue, simplifying the analysis.

However, in most cases, dimensionality reduction is not a viable approach, as it results in significant information loss, leading to severe distortions in risk assessment.

Thus, the task of MDR analysis can be summarized as follows:

• Identify the risk-bearing bodies involved.
• Determine the sources of multiple risks affecting the system.
• Define the MDR issue within a specific future time frame.
• Analyze the characteristics of each individual risk and their potential linkages.
• Combine the risks into an overall assessment that has clear physical significance and minimal subjectivity.

By following these principles, MDR analysis can provide a more structured and objective approach to evaluating complex, multi-risk environments.

There are various types of multi-dimensional risks (MDRs), but all of them are composed of multiple single risks. The principles of analyzing a single risk are relatively straightforward (Huang 2011):

• Define the intension of the risk and the system involved.
• Quantitatively or qualitatively analyze the risk sources, exposure fields, and risk-bearing bodies in the context of uncertainty.
• Combine the analysis results to describe the adverse incident scenario associated with the risk’s intension.

The term “intension” originates from logical terminology, where it is used in natural language processing and pre-formal reasoning (Peregrin 2007), specifically referring to the meaning of propositions. In modern usage, “intension” is a broader concept, referring to the unique attributes of an object or entity. The set of all objects or entities that share these unique attributes is called “extension”.

When analyzing risk, the intension of a studied risk refers to its unique attributes, rather than the general attributes of risk. This distinction is important because the definition of risk itself is not universally agreed upon.

Some researchers define risk purely in terms of uncertainty. However, all possible meanings of risk contribute to its intension. For example:

• Random uncertainty of a risk source is an intension of risk.
• The financial losses from bank failures are an intension of risk.
• Depression as an emotional consequence of risk is also an intension of risk.

The principle outlined above underscores the importance of analyzing risks in specific, well-defined terms, rather than relying on generalized or abstract concepts. When the intension of a risk—its clearly defined attributes and conditions—is unclear, targeted risk analysis becomes infeasible. Unlike connotation, which is inherently subjective and shaped by cultural or individual interpretation, intension is intended to be objective and precisely defined, enabling rigorous assessment.

Despite the diverse range of adverse events encompassed within MDR, we can establish a foundational principle for MDR analysis by recognizing that concepts such as “overall risk”, “association”, and “trade-off” are simply different lenses for evaluating total system risk.

Unless explicitly stated otherwise, we assume that each single risk within an MDR framework can be analyzed independently. That is, ambiguous risks—those influenced by interpretative, cultural, or normative differences (Renn 2005)—are excluded from MDR analysis due to their inherent subjectivity and inconsistency.

Based on a review of MDR issue types, the relevant theoretical foundations, and established methods of single-risk analysis, we now tentatively propose the following core principles for the analysis of multi-dimensional risks:

• Specify the intension of each single risk and identify the risk-bearing bodies involved in the MDR.
• Select appropriate theories and methods to analyze each single risk individually.
• Combine the analysis results to describe the adverse incident scenarios affecting the risk-bearing bodies within a specific future time period.

While the principles of MDR analysis and single-risk analysis appear similar, their essences are fundamentally different. Both frameworks follow a three-step process: First, “Specify the intension of the risk”. Second, “Analyze the risk based on its fundamental elements”. Finally, “Combine the analysis results to describe the risk”.

However, MDR analysis differs from single-risk analysis in the following key ways:

• The risk-bearing bodies in MDRs are often more complex.
• The fundamental elements of MDRs are no longer just risk sources, exposure fields, and risk-bearing bodies but rather multiple interacting single risks.
• There is no universal combinatorial algorithm for integrating single risks into an overall MDR assessment.

In theory, a functional relationship exists between a multi-dimensional risk (MDR) and the various factors influencing its individual risks. However, the existence of such a function does not guarantee that it can be precisely determined or meaningfully approximated. Functions derived from theoretical assumptions often fail to capture the full complexity of real-world systems. As such, MDR analysis is not simply an extension of multivariate risk analysis with additional variables.

To illustrate this, consider a 4 × 400 m relay race. While a team’s final result theoretically depends on a variety of physiological and psychological attributes of each runner, the actual function connecting these factors to the final performance is practically impossible to specify. Instead, analysts rely on past team performances, cohesion, and contextual factors to estimate winning probabilities. In the same way, a single-risk factor in MDR is comparable to an individual athlete’s trait, while MDR as a whole represents the team’s aggregate performance—a complex outcome that cannot be reliably derived from a straightforward combination of inputs.

Moreover, the timing and interaction of risk events critically affect MDR outcomes. Whether multiple risk sources—such as epidemics, armed conflict, and natural disasters—erupt or evolve simultaneously has a major influence on the associated MDR. These interactions are inherently dynamic, meaning that MDR levels can vary significantly across time periods, defying fixed or stable characterization.

Most existing risk theories and models, especially probabilistic approaches, are designed for static risk analysis, which assumes a relatively stable set of conditions. However, the challenge of analyzing multi-dimensional dynamic risk—where conditions shift and interact over time—remains largely unaddressed in current frameworks. Developing robust methodologies to assess and manage dynamic MDR represents a critical frontier for future research.

6. Analyzing System Risk of Santai County with Respect to Natural Disasters

The purpose of this case study is to demonstrate that multi-dimensional risk (MDR) analysis is not only simpler than both multivariate risk analysis and integrated risk analysis, but also more reliable. MDR avoids the significant limitations inherent in these traditional approaches—most notably, issues related to subjectivity, data dependency, and flawed aggregation methods. Moreover, MDR addresses a critical conceptual gap, often overlooked in conventional models. This case study focuses on a two-dimensional risk scenario involving flood and earthquake hazards, utilizing historical disaster data to estimate the annual total risk.

6.1. Disaster Data Collection

The data used in this case study (see

Table 3. Flood and earthquake disasters in Santai County from 2008 to 2016.

Year	x	y	m	l	z
2008	910.1	120	8	162,200	162,320
2009	959.4	1235	5.6	400	1635
2010	755.7	65	0	0	65
2011	1152.9	1465	0	0	1465
2012	929.1	735	0	0	735
2013	988.4	1089	4.7	300	1389
2014	630.7	90	4.6	800	890
2015	1048.6	1214	0	0	1214
2016	720.9	129.3	0	0	129.3

Note: x—Annual Rainfall (mm); y—Property loss caused by flood. m—Earthquake magnitude (M_L); l—Property loss caused by earthquake. z—Total loss (y + l). Property loss is in ten thousand RMB. Data Sources: x, y, l—Civil Affairs Bureau of Santai County, Sichuan, China, m—China Earthquake Information Network.

) were collected from the Civil Affairs Bureau of Santai County and the China Earthquake Information Network, covering records of floods and earthquakes that occurred in Santai County, Sichuan Province, China, between 2008 and 2016 (Wang and Huang 2018).

6.2. Methods Employed to Analyze Risks

The risk analysis based on Table 3 is constrained to using the expected value of property losses to describe future flood and earthquake scenarios. In this context, the intension of flood risk and earthquake risk is defined as the expected property loss caused by floods and earthquakes, respectively. Clearly, in this case, the risk-bearing body for the two-dimensional risk (comprising flood and earthquake risks) is Santai County, Sichuan Province.

Traditionally, there are at least two common approaches for processing such data and assessing annual total risk:

• The average loss method;
• The expected value calculated from an assumed probability distribution.

However, Table 3 contains data spanning only nine years, yielding nine sample points. Statistically, this sample size is too small to yield reliable estimates using either traditional method. When using averages or probability distributions based on assumed types (e.g., normal or exponential), the resulting estimates are highly sensitive to outliers and distributional assumptions, leading to low reliability.

To address this, Huang and Shi (2002) demonstrated that the information diffusion technique provides more reliable estimations under small-sample conditions. In this study, we apply the normal diffusion model to assess both flood risk and earthquake risk in Santai County. The annual total risk is then computed as the sum of the annual flood risk and the annual earthquake risk, as estimated through diffusion.

6.2.1. Using Average Losses to Estimate Risks

Calculating the averages of loss data in columns y and l in Table 3, respectively, we estimate the two risks:

$${\overline{\mathrm{Risk}}}_{\hspace{0.17em}\mathrm{F}}=\overline{y}={\displaystyle \frac{120+1235+65+1465+735+1089+90+1214+129.3}{9}}=682.5.$$

(3)

$${\overline{\mathrm{Risk}\hspace{0.17em}}}_{\mathrm{E}}=\overline{l}={\displaystyle \frac{\mathrm{162,200}+400+300+800}{9}}=\mathrm{18,189}.$$

(4)

In these two formulas, subscripts F and E (not in italics, but in regular font) indicate that the risk values are related to flood and earthquakes, respectively. In other words, they are identifiers rather than mathematical variables. The following content also applies.

6.2.2. Estimating Risks by Using Expected Values of Hypothetical Probability Distributions

Assuming that the annual rainfall in Santai County follows a normal distribution, using the sample mean $\overline{x}$ in Equation (5) and the sample variance s² in Equation (6), we can estimate the population mean µ and population variance σ², and obtain the probability distribution of the annual rainfall x in Equation (7).

$$\overline{x} = (x_1 + x_2 + \dots + x_9)/9 = (910.1 + 959.4 + \dots +720.9)/9 = 899.53,$$

(5)

$$s^2 = [{(x_1 - \overline{x})}^2 + {(x_2 - \overline{x})}^2 + \dots + {(x_9 - \overline{x})}^2]/9 = \mathrm{24,861.32},$$

(6)

$$p_{\mathrm{F}}(x)={\displaystyle \frac{1}{157.67\sqrt{2\pi }}}\exp [-{\displaystyle \frac{{(x-899.53)}^2}{2\times \mathrm{24,861.32}}}]$$

(7)

Assuming that the earthquake magnitude m in Santai County follows an exponential distribution, using the parameters calculated by Equation (8), we obtain Equation (9) as an estimate of the probability distribution of magnitude m.

$$\lambda ={\displaystyle \frac{1}{\overline{m}}}={\displaystyle \frac{9}{m_1+m_2+\cdots +m_9}}={\displaystyle \frac{9}{8+5.6+\cdots +0}}=0.394,$$

(8)

$$p_{\mathrm{E}}(m)=0.394\hspace{0.17em}{e}^{-0.394\hspace{0.17em}m}.$$

(9)

By the linear regression method to process the data in Table 3, we obtain the relationship f (x) between flood loss y and rainfall x, and the relationship g(m) between earthquake loss l and magnitude m:

f(x) = −2088.34 + 3.08028x.

(10)

g(m) = −9946.95 + 11,057.8m.

(11)

So, the flood risk and earthquake risk estimated by using expected values are as follows:

$${\mu }_{\mathrm{F}}={\displaystyle {\int }_a^b{p}_{\mathrm{F}}(x)f(x)}dx={\displaystyle {\int }_{630.7}^{1152.9}\frac{1}{157.67\hspace{0.17em}\sqrt{2\pi }}\exp \hspace{0.17em}[-\frac{{(x-899.53)}^2}{2\times \mathrm{24,861.32}}]}(-2088.34+3.08028\hspace{0.17em}\mathrm{x})\mathrm{d}x=607.5$$

(12)

$${\mu }_{\mathrm{E}}={\displaystyle {\int }_a^b{p}_{\mathrm{E}}\left(m\right)g\left(\mathrm{m}\right)}dm={\displaystyle {\int }_0^{8}0.394\hspace{0.17em}{e}^{-0.394\hspace{0.17em}m}}(-9946.95+\mathrm{11,057.8}\hspace{0.17em}\mathrm{m})\mathrm{dm}=\mathrm{13,570}$$

(13)

6.2.3. Information Diffusion Technique

The concept of information diffusion was proposed when studying learn functions (including probability distribution functions) from small samples (Huang 1997). Let X be a sample in size n drawn from a population with density p(x), written as follows:

X = {x₁, x₂,…, x_n}.

(14)

Let U be the universe of discourse of sample X, which is the range of all possible values of random variable x, written as follows:

U = {u}.

(15)

A set consisting of some discrete points of the universe of discourse is called a discrete universe of X. When the points are u₁, u₂,⋯⋯, u_m, the discrete universe is written as follows:

U = {u₁, u₂,⋯⋯, u_m}.

(16)

The most commonly used information diffusion technique is the normal diffusion model in Equation (17), through which the information carried by the sample point x_i is diffused to all points in U. It is also called the information gain obtained by the discrete point u_j from an observation value x_i.

$$\mu (x_i,u_j)=\exp [-{\displaystyle \frac{{(x_i-u_j)}^2}{2h^2}}],\hspace{0.17em}\hspace{0.17em}{x}_i\in X,u_j\in U.$$

(17)

The asymptotically optimized diffusion coefficient h can be calculated using Equation (18).

$$h=\left\{\begin{array}{l}0.8146 (b-a),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=5;\\ 0.5690 (b-a),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=6;\\ 0.4560 (b-a),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=7;\\ 0.3860 (b-a),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=8;\\ 0.3362 (b-a),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=9;\\ 0.2986 (b-a),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=10;\\ 2.6851(b-a)/(n-1),n\ge 11.\end{array}\right.$$

(18)

where $b=\underset{1\le i\le n}{\max }\{x_i\},\hspace{1em}a=\underset{1\le i\le n}{\min }\{x_i\}.$

6.3. Estimating Flood Risk by Information Diffusion Technique

The annual rainfall sample can be obtained from the rainfall data x in Table 3:

X = {x₁, x₂,…, x₉} = {910.1, 959.4, 755.7, 1152.9, 929.1, 988.4, 630.7, 1048.6, 720.9}.

(19)

According to the size of X and its maximum and minimum, we select the discrete universe U in Equation (20) to estimate the discrete probability distribution $p_1\left(x\right)$ of the annual rainfall in Santai County.

U = {u₁, u₂, ⋯⋯, u₇} = {500, 630, 760,890, 1020, 1150, 1280}.

(20)

With the diffusion coefficient h_x = 175.56 calculated from Equation (18), we obtain the information gain distribution on U, shown in Equation (21).

Q₁ = {q₁₁, q₁₂, ⋯⋯, q₁₇} = {0.556, 1.110, 1.654, 1.974, 1.844, 1.261, 0.602}.

(21)

where

$$q_{1j}={\displaystyle \sum _{i=1}^9\mu (x_i,u_j), }\hspace{0.33em}{u}_j\in U.$$

(22)

The information gain q_1j means that if the points in Equation (20) are used to monitor the sample X in Equation (19), the “number” of sample points monitored by u_j will be q_1j. In general, q_1j, j = 1, 2, … , m, are not integer, but their sum is equal to the size of the given sample.

Let

$$p_1(u_j)={\displaystyle \frac{q_{1j}}{q_{11}+q_{12}+\cdots +q_{17},}},\hspace{1em}\hspace{0.33em}{u}_j\in U.$$

(23)

Thus, we can obtain an estimate of the probability distribution of annual rainfall x:

P_F = (p₁(u₁), p₁(u₂), …, p₁(u₇)) = (0.062, 0.123, 0.184, 0.219, 0.205, 0.140, 0.067)

(24)

There are obvious flaws in f(x) of Equation (10) when we use it to calculate flood loss. For example, substituting the rainfall u₁ = 500 in Equation (20) into this linear regression model, we obtain:

f(u₁) = −2088.34 + 3.08028 u₁ = −548.2.

That is, a flood with a rainfall of 500 mm, registers no loss but a benefit of CNY 5.482 million. This is clearly inaccurate, and the calculation result is completely distorted. This example highlights that a linear regression model based on a small sample size is not suitable for risk assessment.

To address this issue, the information diffusion technique, which optimizes the processing of small sample data, provides a more reliable alternative. Using a two-dimensional information diffusion technique combined with fuzzy approximate reasoning (Huang and Shi 2002), we process the rainfall (x) and loss (y) data from Table 3. As a result, we derive the relationship between y and x within the discrete domain U in Equation (20), leading to the formulation of Equation (25).

Y = (y(u₁), y(u₂), …, y(u₇)) = (99, 212, 385, 642, 768, 684, 372)

(25)

The flood risk value can be calculated by P_F in Equation (24) and Y in Equation (25):

$$R_{\mathrm{F}}=P_{\mathrm{F}}\times Y={\displaystyle \sum _{i=1}^7{p}_1\left(u_i\right)}\hspace{0.17em}y(u_i)=522$$

(26)

6.4. Estimating Earthquake Risk by Information Diffusion Technique

From the magnitude data m in Table 3, we can obtain the earthquake magnitude sample:

M = {m₁, m₂, … , m₉} = {8, 5.6, 0, 0, 0, 4.7, 4.6, 0, 0}.

(27)

According to the size of M and its maximum and minimum, we select the discrete universe V in Equation (28) to estimate the discrete probability distribution $p_2\left(x\right)$ of earthquake magnitude in Santai County.

V = {v₁, v₂, ⋯⋯, v₇} = {0, 5, 5.75, 6.5, 7.25, 8, 8.75}.

(28)

With the diffusion coefficient h_m = 1.14 calculated from Equation (18), we obtain the information gain distribution on V, shown in Equation (29).

$$Q_2=\{q_{21},q_{22},\dots ,q_{27}\}=\{5, 1.277, 1.014, 0.646, 0.442, 0.361, 0.260\}.$$

(29)

where

$$q_{2j}={\displaystyle \sum _{i=1}^9\mu (m_i,v_j), }\hspace{0.33em}{v}_j\in V.$$

(30)

Let

$$p_2(v_j)={\displaystyle \frac{q_{2j}}{q_{21}+q_{22}+\cdots +q_{27},}},\hspace{1em}\hspace{0.33em}{v}_j\in V.$$

(31)

Thus, we can obtain an estimate of the probability distribution of magnitude m:

$$P_{\mathrm{E}}=\left\{p_2\right(v_1),p_2(v_2),\dots ,p_2(v_7)\}=\{0.556, 0.142, 0.113, 0.072, 0.049, 0.040, 0.029\}.$$

(32)

Just as we used a two-dimensional information diffusion technique and fuzzy approximate reasoning to construct the relationship between y and x, we use the same method to process the loss l and magnitude m data in Table 3 to obtain the relationship between l and m on the discrete domain V in Equation (28), shown in Equation (33).

$$L=\left(l\right(v_1),l(v_2),\dots ,l(v_7))=(0, \mathrm{30,442}, \mathrm{35,083}, \mathrm{43,885}, \mathrm{59,685}, \mathrm{82,453}, \mathrm{107,606}).$$

(33)

The earthquake risk value can be calculated by P_E and L:

$$R_{\mathrm{E}}=P_{\mathrm{E}}\times L={\displaystyle \sum _{i=1}^7{p}_2\left(m_i\right)}l(m_i)=\mathrm{20,790}$$

(34)

6.5. Assessing Annual Total Risk of Natural Disasters

Table 3 shows that earthquakes did not exacerbate flood disasters, nor did floods exacerbate earthquake disasters in Santai County. Therefore, the correlation between flood and earthquake events in this region can be reasonably disregarded. This implies that there is no overlap between the expected value of flood loss (R_F) and the expected value of earthquake loss (R_E). Under traditional risk analysis frameworks, the sum R_F + R_E is typically considered the total risk from flood and earthquake events.

However, due to the fundamentally different statistical characteristics of R_F and R_E, simply summing them does not accurately represent the true total risk—especially not the annual total risk. Neither multivariate risk analysis nor integrated risk analysis adequately addressed this critical flaw.

Flood disasters occur almost annually in Santai County, which means that R_F, and its associated probability distribution, closely reflects the annual distribution of rainfall-related damage. In contrast, the exceedance probability of a magnitude 8.0 (M_L = 8.0) earthquake in Sichuan Province—including Santai County—is approximately 0.03, indicating such an event is expected to occur only once every 33 to 34 years (Zhang and Huang 2009). Therefore, the probability distribution P_E, used to evaluate R_E, does not represent an annual distribution of earthquake risk.

This stark difference in event frequency and distribution structure invalidates the assumption that R_F and R_E can be aggregated linearly without further adjustment. It underscores the need for MDR-specific methods that can account for asymmetrical frequency, intensity, and systemic relevance of each risk component.

Since the probability distributions of flood and earthquake events do not share the same temporal framework, directly summing R_E (from Equation (34)) and R_F (from Equation (26)) lacks physical significance and may mislead risk management decisions. This discrepancy highlights a core tenet of multi-dimensional risk (MDR) analysis: the importance of defining a specific future time period for risk evaluation, rather than relying on simplistic additive methods.

To address this issue, we recalibrate the earthquake probability distribution P_E in Equation (32) using seismic activity data from the broader Sichuan Province, thereby correcting for estimation errors in total risk assessment. The earthquake data provided in Table 3, though useful, span only a limited nine-year period, which is insufficient for predicting long-term seismic trends or accurately estimating annual earthquake losses.

In other words, the annual earthquake risk cannot be directly inferred from this small dataset. Originally, P_E in Equation (32) was calculated by normalizing the frequency distribution Q₂ (from Equation (29)) over nine years (2008–2016). However, based on the estimated recurrence interval of 33.5 years for M_L = 8.0 earthquakes in Sichuan Province, we recalibrate P_E to reflect the annualized probability distribution of earthquake magnitudes. This adjustment enables a more accurate and time-consistent assessment of annual earthquake risk, aligning it more appropriately with the flood risk framework.

Let

c = 33.5/9 = 3.722.

(35)

Then, the annual probability of an earthquake with maximum magnitude v_j is:

$$p_2^{\prime }(v_j)={\displaystyle \frac{p_2\left(v_j\right)}{c,}},\hspace{1em}\hspace{0.33em}{v}_j\in V.$$

(36)

Because

p_v>0 = p′₂(v₂) + p′₂(v₃) + … + p′₂(v₇) = 0.1194,

(37)

Let v′₁ = 0 represent all non-destructive earthquakes; then, the annual probability of occurrence is:

p′₂ (v′₁) = 1 − p_v>0 = 0.8806.

(38)

Therefore, the calibrated annual magnitude probability distribution is P′_E in Equation (39), which is very different from P_E in Equation (32). P′_E shows the fact that destructive earthquakes rarely occur in Santai County.

P′_E = (p′₂(v′₁), p′₂(v₂), …, p′₂(v₇)) = (0.8806, 0.0381, 0.0303, 0.0193, 0.0132, 0.0108, 0.0078)

(39)

So, the annual earthquake risk value can be calculated by the P’_E and L in Equation (33).

$$R_{\mathrm{E}}^{\prime }=P_{\mathrm{E}}^{\prime }\times L={\displaystyle \sum _{i=1}^7{p}_2^{\prime }\left(m_i\right)}{l}_2(m_i)=5587.$$

(40)

Finally, the annual total risk of flood and earthquake is:

R_Total = R_F + R′_E = 522 + 5587 = 6109.

(41)

The estimated value of annual earthquake risk R′_E = 5587 in Equation (40), in accordance with the local seismic activity, is much smaller than the estimates of ${\overline{\mathrm{Risk}\hspace{0.17em}}}_{\mathrm{E}}=\mathrm{18,189}$ in Equation (4) and ${\mu }_{\mathrm{E}}=\mathrm{13,570}$ in Equation (13), which are simply estimated from the scarce data in Table 3. This fact shows that MDR analysis must pay more attention to the differences in risk characteristics in order to reasonably estimate the overall risk of MDR.

Clearly, if the intension of flood risk or earthquake risk is unclear, discussing their two-dimensional risk in general terms becomes meaningless. This example also highlights an important fact: there is no universal combination algorithm for calculating total risks in multi-dimensional risks (MDRs). Different types of MDRs require different combination methods, depending on the characteristics of all single risks and interactions among the single risks.

6.6. Results

The natural disaster risks in Santai County estimated using different methods are shown in

Table 4. Estimation of natural disaster risk in Santai County using different methods.

Method	Flood Risk	Earthquake Risk	Annual Total Risk	Remark
AL	682.5	18,189	18,871.5	The average value is highly sensitive to outliers.
HPD	607.5	13,570	14,177.5	Difficult to verify hypothesis.
IDT	522	20,790	21,312	Optimize the processing of small samples.
CSA	522	5587	6109	Solved the problem of different the flood cycle and earthquake activity cycle in a region.

Note: AL—Average Loss; HPD—Hypothetical Probability Distribution; IDT—Information Diffusion Technique; CSA—Calibrating earthquake risk using Seismic Activity. The risks are measured by property losses in CNY ten thousand.

. Obviously, if the intension of flood risk or earthquake risk is unclear, it is meaningless to talk about their two-dimensional risk in general.

By applying Equations (36)–(38), we recalibrated the seismic activity data from a very limited local timeframe in Santai County to more accurately reflect the true regional seismic behavior. This recalibration enabled a more realistic estimation of the annual earthquake risk, thereby reducing the risk of numerical distortion in the overall assessment of annual total risk. Notably, this issue—arising from inconsistencies in temporal scales and data insufficiency—cannot be addressed using traditional multivariate risk analysis or integrated risk analysis. It is a challenge that requires the distinct methodological perspective of MDR analysis.

Although it may appear that small-sample-based risk assessments could be improved through the application of the empirical Bayes formula, this is seldom feasible in practice. In most real-world situations, the required empirical distribution is either unavailable or unreliable, making such adjustments impractical and reinforcing the need for alternative techniques like the one presented here.

7. Conclusions and Discussions

The serious limitations of both multivariate risk analysis and integrated risk analysis highlight the need for a new approach to evaluating the total risk of complex systems. Because each individual risk is typically influenced by a relatively small number of factors, and the required sample size is correspondingly low, multi-dimensional risk (MDR) analysis offers a simple yet reliable alternative.

By examining the fundamental properties of MDR and drawing from the principles of single-risk analysis, we propose the following guiding principle for MDR analysis:

After clearly defining the connotation and risk tolerance of each single risk, each risk should be analyzed independently. The final step is to synthesize the individual results to characterize the overall MDR.

Applying this principle to a practical case, we used nine years of flood and earthquake observations, supplemented with regional seismic activity data, and applied information diffusion techniques to conduct an MDR analysis for Santai County. Using expected property loss as the risk metric, the estimated annual total risk from floods and earthquakes is approximately CNY 61 million, which accounts for 0.27% of the county’s 2016 GDP (CNY 584.5 million).

The three major advantages of multi-dimensional risk (MDR) analysis are as follows:

• Resolution of multivariate factor entanglement:

MDR analysis addresses the challenges posed by the entanglement of numerous factors with differing properties in multivariate risk analysis. In MDR, the factors influencing each individual risk are used solely to analyze that specific risk. This ensures clear physical meaning and high reliability of the results. Moreover, it avoids the issue of geometric growth in required sample size that occurs as more variables are introduced in multivariate models.

2.

Elimination of subjectivity in integration methods:

MDR avoids the subjective weighting and lack of physical significance often found in integrated risk analysis. It also circumvents the need to subjectively choose correlation functions, such as in Copula-based approaches, thereby enhancing the objectivity and reliability of the results.

3.

Enhanced transparency and bug detection in combined risk systems:

MDR makes it easier to identify inconsistencies in combined risk scenarios and to take targeted actions to eliminate them, thus reducing the risk of severe distortions in the final analysis. Because single-risk analysis is generally simpler and more transparent, MDR allows analysts to derive risk values using physically meaningful indicators (e.g., property loss, casualties), which facilitates accurate total risk estimation and greatly simplifies the analysis of complex risk systems.

While existing studies in this field often focus on specific MDR-related issues—such as risk–risk tradeoffs—this work is, to our knowledge, the first to propose a formal principle of MDR analysis. The application of this principle opens promising future research directions, including the analysis of compounding (worsening) MDR and negatively correlated MDR. The case study presented here, based on independent MDR, represents a relatively simple scenario compared to these more complex dynamics.

From the COVID-19 pandemic to the ongoing Russia–Ukraine war, and even to global tariff conflicts or investment project success/failure, these all represent complex MDR scenarios that go well beyond single-domain risks. A key limitation of this study, however, is that it assumes all single risks within the MDR framework are quantifiable. In reality, many MDR systems involve qualitative or semi-quantitative risks, which present a significant analytical challenge. Addressing such systems clearly extends beyond a simple weighted sum approach and demands more nuanced methodologies.

In a world of increasing uncertainty and rapidly evolving complex risks, decision-making frameworks must adapt accordingly. By tracking individual risks, maintaining a focus on overall systemic risk, and treating the total risk of complex systems as an instance of MDR, we provide a more reliable foundation for policy-making and risk management strategies. This represents a departure from traditional multivariate or integrated risk perspectives, offering a simplified yet more meaningful approach.

Ultimately, the original intention behind MDR analysis is to simplify complex problems without oversimplifying them—to clarify structure, reduce distortion, and grasp the essential dynamics of systemic risk.

Risk analysis is increasingly recognized as an information technology (Goble and Bier 2013). With the exponential growth of the Internet, both the volume and accessibility of risk information have expanded dramatically. This digital transformation has created qualitatively new pathways for risk communication, including through platforms such as Google, which has become a primary channel for public access to risk-related information (Krimsky 2007).

The emergence of artificial intelligence (AI) is expected to accelerate the development of risk analysis theories and methodologies (Guikema 2020; Stødle et al. 2024; Kong and Yuen 2024). Technologies such as unsupervised machine learning and natural language processing (NLP) offer unique advantages in supporting safety inspections and in reorganizing accident databases at the state or regional level (Chokor et al. 2016). NLP, in particular, is widely regarded as a positive force in enhancing both the efficiency and accuracy of modern risk assessment processes (Kalogiannidis et al. 2024). Meanwhile, granule-state intelligent mathematics offers AI the capacity to draw inferences from individual instances to general cases, paving the way for conscious AI systems capable of analyzing previously unseen risks (Huang and Huang 2024).

With the continued advancement of AI, probabilistic risk analysis supported by big data is expected to be increasingly automated. However, due to the inherent complexity of multi-dimensional risk (MDR)—particularly the absence of a universal algorithm for aggregating diverse risk types—AI systems relying solely on big data are currently incapable of accurately analyzing MDR. This highlights the unique value of the MDR framework proposed in this work and reinforces the need for human-guided, principle-based approaches in complex risk environments.