Construction and empirical testing of comprehensive risk evaluation methods from a multi-dimensional risk matrix perspective: taking specific types of natural disasters risk in China as an example

Global climate change, continuous economic development and accelerated urbanisation have increased the frequency and complexity of various incidents. Risk assessment is an important means to deal with different crises and incidents. Based on the systematic analysis of the basic connotation, attribute relationship and main contents of risk assessment, this study extends the traditional two-dimensional risk matrix to multi-dimensional risk matrix. On this basis, this study further proposes four comprehensive risk assessment methods, including total ordinal, weighted average, Euclidean distance and two-norm methods. Empirical research was conducted by selecting three types of natural disasters, regional agro-meteorological disasters, geological disasters and forest fires, in 30 related regions in China between 2010 and 2019. The results indicated that compared to the two-dimensional risk matrix, the multi-dimensional risk matrix can intuitively characterise and compare the all-round risk situation of different regions, periods, types and dimensions, which is more helpful for a systematic analysis and risk comparison. All four evaluation methods proposed in this paper proved able to effectively conduct comprehensive regional risk evaluations with consistent results. However, when considering the fitting coefficient between the results obtained by various evaluation methods and the comprehensive evaluation results, it can be found that the accuracy (i.e. fitting coefficient) of Euclidean distance method is significantly higher than that of the other three methods. In the last 10 years, the temporal order of natural disaster risk in China has been more volatile, with Beijing, Shanxi, Tianjin, Hebei, Liaoning, Ningxia, Zhejiang, Shaanxi, Fujian and Gansu experiencing higher volatility in the ranking of natural disaster risk.


Introduction
With technological development and environmental changes, human society has entered a risk society where multiple risks coexist in almost every region. The evaluation of a particular disaster type alone can no longer meet the requirements of disaster risk management. The Sendai Framework (2015) specifically calls for "multi-hazard and solutions research in disaster risk management to address gaps, barriers, mutual constraints, and social, economic, educational, and environmental challenges and disaster risks".
Risk assessment is considered to be an easily understandable tool to compare the relative risk of economic and life loss in different regions and to compare different relative contributions of various factors (UNDRO 1991;Smith 1996;Cruz-Reyna et al. 1996;Ben and Wisner 2000;IPCC 2001;Tagliacozzo et al. 2022). Research on disaster assessment initially started with a single-hazard risk assessment and as the study progressed, the development of risk assessment methods has experienced a process from qualitative to quantitative, from certainty to uncertainty and from stochastic uncertainty to fuzzy uncertainty (Dempster 1967;Peng and Liu 2011;Khoshjavan et al. 2011;Rongtai et al. 2013). Researchers studied the risk assessment theory and gathered empirical evidence on single hazards, such as earthquake, flood, typhoon and geological hazards (Granger 2003;Manen et al. 2005;Romeoet al. 2006;Tian et al. 2015;Gao et al. 2020;Ratiranjan et al. 2021). Rikitake (1991) used the probability of earthquake occurrence and seismic intensity as the main indicators to assess the risk of seismic hazard in Tokyo over a 10-year period. Hall (2005) conducted a correlation analysis of rainfall, rainfall frequency and flood hazard losses in England and Wales. With the growing population and economy, the magnitude of disaster risk is increasingly related to the characteristics of a hazard-bearing body. Researchers began focusing on the impact of the vulnerability level of risk carriers on disaster risk, and five evaluation models were proposed, including the pressure and release model (Blaikie et al. 1994), vulnerability place model (Cutter et al. 2015), human dimensions of vulnerability model (Tobin and Montz 1998), human ecology of endangerment model (Hewitt 1997) and the circle of vulnerability model (Alexander 2000). Vulnerability assessments based on specific hazard types and consequences are best represented by the disaster risk index program and the Hotspots Fire Project (Pelling 2003;Dilley et al. 2005), both of which are conducted globally. While risk is usually defined as the possibility of suffering harm or loss in the future, the data used for the assessment must be accessible. Currently, due to current data acquisition techniques and methods, assessment based on historical data is the main and common approach in the field of risk assessment (Paul et al. 1998;Li et al. 2012;Jin et al. 2012).
Due to the frequent occurrence of multi-hazard disasters worldwide in recent years, effective multi-hazard risk analysis is imperative for all the phases of disaster risk manage. Comprehensive multi-hazard risk assessment has become a key aspect of disaster management and a fundamental necessity for disaster prevention and mitigation planning. Comprehensive risk assessment is generally realised through indicator preference, weight assignment, evaluation method selection and calculation. For example, the pervasive vulnerability index and risk management index are constructed in the Americas Programme (Cardona 2005) to express the vulnerability characteristics of the hazard-bearing body. Aksha (2020) conducted an integrated hazard assessment of three hazards-landslides, floods, and earthquakes-using statistical methods and hierarchical analysis. The first ROADMAP thematic paper emphasised the importance of good practices in multihazard risk scenarios for disaster risk management (Capone et al. 2022). Multi-hazard risk scenario analysis is not only a frontier challenge for scientific research, but also a phenomenon that has a huge impact on national economy, politics and people's lives (Ba et al. 2021). However, due to the different principles of various evaluation methods and different analysis perspectives, the results of different comprehensive evaluation methods may vary greatly when evaluating the same research object. Therefore, multi-objective integration, methodological robustness and consistency of results are also important issues to be addressed when researching comprehensive evaluation methods.
In view of this, this study reanalyses the risk system with the goal of multi-hazard risk analysis, constructs unified feature indicators and multi-dimensional matrix expressions applicable to multi-hazard, multi-region and multi-time points. The main contributions of this paper are summarised as follows: First, this is the first study to construct a multidimensional risk matrix containing regions, periods, types and indicators; Second, we propose four evaluation methods, including total ordinal, weighted average, Euclidean distance and two-norm methods, while their consistency and accuracy are tested; Finally, this is the first comprehensive regional assessment study of natural disaster risk in 30 regions of China over a 10-year period, and the relevant findings and methodology can serve as a direction for practice and research.

Introduction of multi-dimensional risk matrix
A risk matrix is a structural method used to identify and evaluate the level of risk (risk set). Based on the array structure of data storage in computer science, abstract theory of spatial geometry, matrix application of image recognition and processing and characteristics of disaster risk, this paper proposes the concept of a multi-dimensional risk matrix to include all factors, such as time, space, risk types and risk indicators.
Based on the need for a comprehensive risk evaluation, a multi-dimensional risk matrix refers to the continuous enrichment and enhancement of an existing two-dimensional risk matrix. The existing two-dimensional risk matrix can be summarised into two categories: (1) a result-based risk matrix, in which all obtained evaluation results are placed in four quadrants, which is only a 'form' matrix but not a real matrix, and (2) an indicator-based risk matrix, in which the individual indicators used to obtain the evaluation results are expressed in the form of a matrix; this type of risk matrix can be considered as a real application of matrices in the field of risk research. The current widely used risk matrix is a two-dimensional risk matrix comprising two indicator dimensions, likelihood and severity, which was proposed by the Electronic Systems Center in 1995 (Johnson 2016), with the core objective of combining the probability of occurrence and the resulting severity of a risk and then, deriving a rating for that risk.
However, given the complexity of risk characteristics and the diversification of risk types, the traditional risk matrix, which evaluates only the two dimensions of risk likelihood and severity, can no longer comprehensively characterise the components of the risk system. In addition, the traditional risk matrix can no longer present this information simultaneously when evaluating risks in a comprehensive manner for multiple objects and at multiple time points. Therefore, the traditional risk matrix needs to be extended to reflect and express multiple indicators, objects and time points in an integrated manner.

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A multi-dimensional risk matrix can simultaneously incorporate the four dimensions of information-region, time, risk type and risk indicator-into one evaluation system. Constructing each multi-dimensional risk matrix is similar to generating a unique 'cube ID' ('magic cube ID'). Each evaluation object is assigned a unique 'cube ID', which contains the risk profile of the region at any time, for any category and for any indicator. In this evaluation system, multi-dimensional risk information can be extracted at will, and the comprehensive risk can be assessed from different dimensions.

Functional attributes of the multi-dimensional risk matrix
According to systems theory (UN/ISDR 2004;Marzocchi et al.2012;Chen et al.2017), the risk system can be divided into three types of elements: risk sources, receptors and managers (Fig. 1). Risk sources include hazardous substances, energy and their carriers, which objectively lead to inevitable emergencies. Risk receptors refer to objects that generate value losses, including those related to people, property, objects and environment. Risk manager is the manager of the risk receptor, the actor who identifies, analyses, evaluates and controls the risk implementation. Risk evaluation aims to manage and prevent risk, so the evaluation results should provide a basis for determining the necessity and feasibility of work implementation. Therefore, risk assessment indicators can be conducted according to three types of elements in terms of both feasibility and necessity.
Necessity analysis of risk prevention and control is an analysis of the risk managers 'should or should not' take measures. The more serious the loss caused by the risk and the lower the tolerance of risk controllers, the more it should be prevented and controlled. Therefore, the necessity analysis should be measured primarily in terms of the destructive capacity of the risk sources, the fluctuation of the risk sources and the exposure of the risk receptors (Adger 2006). Feasibility analysis of risk prevention and control is conducted to analyse whether the risk managers 'can or cannot' prevent and control the risk. The higher the concentration of risk receptors and the higher the mitigation capacity of risk managers, the higher the feasibility of risk prevention and control (Adger and Vincent 2005;Comfort et al.1999). Therefore, the feasibility analysis should be performed in terms of the concentration of risk receptors and the mitigation capacity of risk managers. Based on the evaluation purpose of feasibility and necessity, this study selects five evaluation indicators-saturation of risk source, fluctuation of risk source, exposure of risk receptor, concentration of risk receptor and mitigation of risk manager.

Risk Receptor
Risk Manager Saturation refers to the extreme value of the loss caused by the risk source in a certain cycle. The greater the damage capacity of the risk source and the higher the saturation, the stronger the necessity of risk prevention and control.
Fluctuation measures the degree to which risk is uneven over time. The greater the fluctuation of risk, the higher the risk.
Exposure is measured as the ratio of the value of the receptors exposed to the risk to the value of all receptors of that type in the area, with higher exposure implying a higher need for risk prevention and control.
Concentration is an indicator of the unevenness of risk in space. A low concentration indicates that risk receptors are scattered and that the prevention and control of risks are more difficult and less feasible.
Mitigation refers to the degree to which risks can be reduced or mitigated, mainly measured in terms of emergency resources and preparedness, with stronger mitigation capacity representing higher feasibility of risk prevention and control.

One-dimensional risk matrix
A one-dimensional risk matrix indicates that this risk matrix contains information in only one dimension. For example, the one-dimensional risk evaluation result vector contains the ranking results of risk evaluation in each region, which intuitively shows the risk degree of each region. As described in the following paragraph, Figure 2 shows a spatial graph corresponding to the one-dimensional risk matrix.
Here, i represents the final result of the risk evaluation. A one-dimensional risk matrix is suitable for visualising data arrangement. The onedimensional risk matrix defined in this paper is mainly used to store and express the results of risk evaluation. Considering the regional one-dimensional risk matrix as an example, the corresponding formula is shown below.
where R R×1 refers to the ranking of risk assessment results of region R,R r refers to the ranking of risk assessment results of the rth province or city. (1)

Two-dimensional risk matrix
A two-dimensional risk matrix contains information regarding two dimensions. The spatial graph corresponding to the two-dimensional risk matrix can be presented as follows (Fig. 3): In this figure, i indicates the types of regional risks and j indicators to an indicator of regional risk.
Considering the two-dimensional risk matrix of a region at a certain point in time as an example, the two-dimensional risk matrix in formula (2) contains N indicators of risk category C, which can represent the situation of all risks and indicators in a certain region. The corresponding formula is expressed as follows: where R C×N refers to N indicators of C risk in a region at a certain point in time, r cn is the risk information of the Nth indicator in class C risk.
Take, for example, the two-dimensional risk matrix of a certain type of risk in a certain region that contains N indicators of a certain type of risk for T years, where the change in indicators of this type of risk in this region over time can be observed. The corresponding formula is expressed as follows: where R T×N refers to N indicators of a certain type of risk in a certain region in T years, r tn represents the risk information of the tth indicator in the type of risk in the tth year.
Considering the two-dimensional risk matrix of a certain type of risk in a certain year as an example, the two-dimensional risk matrix contains N indicators in R regions under a certain risk type in a certain year. Accordingly, the movement of indicators of this risk type can be observed in different regions in that year. The corresponding formula is expressed as follows: where R R×N refers to N risk indicators of a certain type of risk in R regions in a certain year, r rn is the risk information of the nth indicator of that type of risk in the rth region. (2)

Three-dimensional risk matrix
A three-dimensional risk matrix is a risk matrix that contains information about the three dimensions simultaneously. The corresponding spatial graph of a three-dimensional risk matrix can be represented as follows (Fig. 4): In this graph, i refers to the type of risk, j refers to the indicators of risk and k refers to the object or time point of the risk assessment.
Considering the three-dimensional risk matrix of a certain region as an example, the three-dimensional risk matrix contains N indicators of C types of risks in the region in T years, which can express the indicators of different risks in a certain region at different times. The corresponding formula is expressed as follows: where R C×N×T refers to N indicators of C types of risks in the region in T years, R c represents the information of cth type of risks, r cn is the risk information of the nth indicator in cth type of risk, r cnt represents the risk information of the nth indicator of type c risks in the tth year.

Four-dimensional risk matrix
A four-dimensional risk matrix is a risk matrix containing information in four dimensions: region, time, type and indicator. The spatial graph corresponding to a four-dimensional risk matrix can be presented as follows (Fig. 5): Here, i indicates the type of risk, j indicates the indicators of risk, k indicates the object of risk assessment and l indicates the time point of risk assessment.
The four-dimensional risk matrix contains all information regarding risk, including the C class risk indicators N of R regions in T years, which can express all different risks and indicator information of all regions in all periods. The corresponding formula is expressed as follows: where R C×N×R×T refers to N indicators of C types of risks in all R regions in T years, R c represents the information of cth type of risks, r cn is the risk information of the nth indicator in cth type of risks, r cnr represents the risk information of the nth indicator of type c risks in the rth region, r cnrt is the risk information of the nth indicator of the cth type in the rth region in the tth year.

Construction of comprehensive risk evaluation methods based on a multi-dimensional risk matrix
On May 31, 2020, the central Committee of the Communist Party of China (CPC) and The State Council decided to deploy the national natural disaster risk survey, which included earthquake disasters, geological disasters, meteorological disasters, floods and droughts, Marine disasters, and forest and grassland fires. And most of the existing research on risk is going in various directions (Silva et al. 2020). As a result, the construction of a multidimensional risk matrix can not only represent the comprehensive risk status of the region but also specifically describe the risk magnitude of different times, regions, types and indicators. Hence, by analysing existing studies, searching for relevant mathematical information and accounting for factors such as evaluation quality, calculation complexity and innovation significance, four types of comprehensive risk evaluation methods-total ordinal, weighted average, Euclidean distance and two-norm methods-are proposed in this paper. The weighted average method, which calculates the final weighted value by assigning weights to all elements, is the most commonly used evaluation method in existing research. The total ordinal method is a comprehensive evaluation method originally proposed in this paper, and the Euclidean distance and two-norm methods are evaluation methods proposed with reference to methods for calculating spatial distances in mathematics.

Total ordinal method
The total ordinal method uses the simplest principle and an ambiguous evaluation. Let us consider the risk matrix of R regions in a fixed time dimension as an example. First, the C × N two-dimensional risk matrix of a region for T years is calculated by applying the principle of matrix addition. Then, the added risk matrix is then averaged by applying the principle of matrix multiplication (multiplied by the reciprocal of the number of years, 1/T) to obtain the average risk matrix of a region for T years. Finally, the above matrix addition and averaging steps are repeated R times to obtain the average risk matrix of R regions for T years (i.e. the R × C × N three-dimensional risk matrix that represents the average risk level of each region for T years).
To better understand the calculation principle of the total ordinal method, the abovementioned R × C × N-order three-dimensional risk matrix is imaged as a cube and placed in the three-dimensional coordinate system, where the z-axis represents R regions, x-axis represents C-type risks and y-axis represents N indicators. The specific calculation steps are as follows: In the first step, the elements of the R × C × N-order three-dimensional risk matrix are normalised. Then, R elements of an indicator of a certain type of risk ( R regions) are standardised in the same column (i.e. the element of each region minus the minimum value in that column divided by the difference between the maximum and minimum values of the elements in that column, which is denoted as 1 time standardisation). C × N times standardisation is performed to obtain the standardised three-dimensional risk matrix. Furthermore, based on the standardisation of the risk matrix, the factors at each position are assigned the same indicator and the same risk category.
In the second step, the elements of each column (i.e. elements at the same position on each C × N-order two-dimensional matrix) parallel to the z-axis (i.e. R regions) are sorted for size, which can be interpreted as puncturing the cube perpendicular to the xy-plane. This sorting does not alter the positions of the elements in the matrix, but numbers the elements at each position in the range [1,R].
In the third step, the size sorting in the previous step is defined as a 1-group operation, and as each region has C classes of risks and N indicators in the average risk matrix, a total of C × N groups are required for this sorting. When the numbering of the elements in all positions is completed, the elements in all positions of the cube are assigned a number corresponding to the range [1,R ]. By replacing the elements in the original positions of the cube with the sorted numbers, a sorting matrix of order R × C × N is obtained.
In the fourth step, the R × C × N-order ranking matrix is formed from the average risk level ranking results of the R regions. Then, each number on each layer of the two-dimensional sorting matrix of order C × N is added to obtain the final value, and the three-dimensional sorting matrix of order R × C × N is converted into a two-dimensional vector of order R × 1. Comparing the size of the ordinal numbers at each row position in the vector provides the final ranking of the average risk level of each region of the country in T years.

Weighted average method
Previous studies have commonly used the weighted average method for risk evaluation. This method aims to assign certain weights to the influencing factors and then, multiply the influencing factors and weights to obtain the final value for comparison. In this paper, the regional risk matrix calculated using the weighted average method is divided into three parts: determining the set of factors, determining the set of comments and building a comprehensive evaluation model. Similar to the total ordinal method, the average risk matrix of R regions in a fixed time dimension is used as an example. The specific calculation steps are as follows: In the first step, the factor set is determined, which is a common set composed of various factors as elements affecting the evaluation object, usually denoted by U . The factor set U is divided into C subsets according to the type of attributes, which is the C class risk in the multi-dimensional risk matrix (i.e. the first-level indicators affecting the regional risk, denoted as U = (u 1 , u 2 , … , u c ) ). There are several secondlevel indicators under the first-level of indicators affecting the regional risk, which in this paper refers to the N indicators in the multi-dimensional risk matrix, denoted as U = (u 11 , u 12 , … , u 1n , u 21 , u 22 , … , u 2n , … , u cn ).
In the second step, the set of ratings is determined. A rubric set is a collection of the various results that an evaluator may obtain for the evaluation object, which is the weight in the weighted average method, usually denoted by V . Corresponding to the factor set, the rubric set also has two layers. The first layer is the factor set U divided into C subsets according to the type of attributes. That is, C categories of risks in the multidimensional risk matrix, with each category corresponding to the first level of indicator weights, denoted as V = (v 1 , v 2 , … , v c ) . The second layer is each category of risks in the multi-dimensional risk matrix that also contains N indicators under each category, and each indicator also corresponds to the second level of indicator weights, denoted as To determine the weights of the influencing factors, this research integrated the various considerations and adopted the entropy weight method. This method aims to determine the objective weight according to the size of the variability of the indicators. To evaluate regional risk, if the information entropy E j of a certain type of risk or an indicator is smaller, the factor value's degree of variability is greater, and the more information it can provide, the greater is the role it can play in comprehensively evaluating regional risk and the greater is its weight. Following are the specific steps of the entropy weighting method.
(1) Standardise the factors. The factors in the multi-dimensional risk matrix of this paper contain C categories of risk, each of which contains N indicators. Let us assume that k factors X 1 , X 2 , … , X k are given, where X i = x 1 , x 2 , … , x n , and assuming that the normalised values of each factor are Y 1 , Y 2 , … , Y k , where, in the risk category Ith, Y ij refers to the standardised value of the Jth indicator, X ij refers to the Jth indicator, max X i refers to the maximum risk indicator value and min(X i ) refers to the minimum risk indicator value.
(2) Calculate the information entropy of each factor.
According to the definition of information entropy in information theory, the information entropy of a dataset can be expressed as p ij lnp ij = 0 . E j refers to the information entropy value of class I risk, p ij represents the proportion of the Jth indicator in the Ith region in all regional risk indicators, and n is the total number of regions.
(3) Determine the weight of each factor.
According to the formula of information entropy, the information entropy of each factor is calculated as E 1 , E 2 , … , E k , and the weight of each factor is calculated by information entropy as where W i refers to the weight of risk type I ; E i is the information entropy value of risk type I and k is the number of risk types.
In the third step, a comprehensive evaluation model is established. After determining the factor set and comment set affecting regional risk, this research establishes a comprehensive evaluation model to weigh the average of each indicator affecting regional risk and obtain the weighted average of risk in different regions. The comprehensive evaluation model involves the following two steps.
(1) The second-level index values are multiplied by the second-level weight values and added to obtain the first-level index values, which can be calculated as where U i refers to the first-level indicator value risk type, u ij refers to the second-level indicator value risk indicator in risk type, v ij refers to the risk weights corresponding to different risk indicator and c is the number of risk types.
(2) The first-level indicators are multiplied by the first-level weight values and added to obtain the integrated risk-weighted value for each region as follows: where R i refers to the comprehensive weighted value of regional risk for the i th region, u j refers to the first-level indicator risk type, v j refers to the risk weight corresponding to different risk type and r is the number of regions.

Euclidean distance method
Euclidean distance refers to the distance between two or more points. This research compares the risk profiles of different regions by calculating the Euclidean distance between the risk matrices of different regions and taking the average risk matrix of R regions in a fixed time dimension as an example. Following are the calculation steps involved.
In the first step, the elements of the R × C × N-order three-dimensional risk matrix are normalised. The normalisation process is consistent with other evaluation methods mentioned above.
In the second step, the maximum value TOP is selected for each column element in the same position in each region, which is the maximum value of a certain indicator for a certain type of nationwide risk. As the three-dimensional risk matrix entails C types of risks and N indicators, screening C × N times the maximum value is necessary to obtain a two-dimensional TOP risk matrix of order C × N , which represents the worst situation of different risk types and indicators across the country in T years.
In the third step, the R × C × N-order three-dimensional risk matrix is considered to be composed of the C × N-order two-dimensional risk matrix of R layers. The Euclidean distance between the two-dimensional risk matrix and the TOP risk matrix for each layer (i.e. each region) is calculated separately as follows.
where d (A, B) indicates the Euclidean distance between the two-dimensional risk matrix of each region and the TOP risk matrix, a ij indicates the jth indicator of the ith risk category in the evaluation region, b ij indicates the jth indicator of the ith risk category in the TOP risk matrix, m indicates the cth risk category and n refers to the Nth risk indicator.
In the fourth step, the elements in the two-dimensional Euclidean distance vector that was calculated in the previous step, which formed a two-dimensional Euclidean distance vector of order R × 1 , are sorted to obtain the ranking of the average risk level of each region in T years.

Two-norm method
Norm is the face of a complex space and a multi-dimensional array through the selection of a unified quantitative standard to facilitate measurement and comparison. There are several norms, and this research selects two norms. The average risk matrix of R regions in the fixed time dimension is also taken as an example, which is calculated as follows.
In the first step, the elements of the R × C × N-order three-dimensional risk matrix are normalised. The normalisation process is consistent with other evaluation methods mentioned above.
In the second step, the R × C × N-order three-dimensional risk matrix is regarded as the C × N-order two-dimensional risk matrix of R layers. The two-norm values are calculated separately for each layer (i.e. each region) as follows.
where ||X|| 2 refers to the regional risk matrix in different areas of the two-norm numerical, xi indicates the information on each risk indicator of each type of risk and M is the product of class C risk and N risk indicators.
In the third step, the two-norm numerical vectors of order R × 1 for all regions are obtained through the calculation in the previous step. By sorting the elements in the calculated two-norm numerical vector, the ranking of the average risk level of each region in T years can be obtained.

Data and research area
China is one of the countries with disasters that are the most serious and varied, have high occurrence frequency, exhibit wide geographical distribution and incur heavy losses. In 2020, the Ministry of Emergency Management found that natural disasters affected approximately 1.38 billion people in China, affecting 19.96 million hectares of crops and yielding direct economic losses of 370.15 billion yuan. This research selected three types of natural disasters, namely agro-meteorological disasters, geological disasters and forest fires, which are typical natural disasters in China and cause heavy losses every year. The sample included 30 provinces and cities, of which 22 were provinces, 5 were autonomous regions and 3 were municipalities directly under the control of the central government in China. Data on Hong Kong, Macao and Taiwan were removed from the sample because they were difficult to obtain. Shanghai was also excluded because it has a relatively low share of agricultural sown area, mountainous area and forest area; moreover, there were inadequate relevant data, so comprehensive risk evaluation could not be conducted properly. All data were gathered from the China Statistical Yearbook, and the missing data on some provinces and cities were obtained through the government information disclosure application process or 'sliding average algorithm'.
To select the indicators, this paper uses the results obtained by related researchers (Murthy 2015; Kayet 2020; Huangfu 2021) combined with expert experience and comprehensive consideration of data acquisition feasibility, based on the following five risk evaluation indexes: saturation, fluctuation, exposure, concentration and mitigation, were used to develop the risk evaluation index system of agro-meteorological disasters, geological disasters and forest fires ( Table 1).
As for the data sources, the China Statistical Yearbook was used to obtain the data for agricultural meteorological disasters including affected area, agricultural sown area, total reservoir volume and total area, the data on 'geological disasters, economic losses and forest area' during 2010-2019 and 'investment in geological disaster prevention and control' during 2010-2017. The data on 'investment in geological disaster prevention and control' for Hubei, Henan, Yunnan, Guangdong, Beijing, Hebei, Zhejiang, Anhui, Jiangxi, Shaanxi, Gansu and Jilin for 2018 and 2019 were gathered from the responses to government information disclosure requests, and the data for other provinces were calculated by the sliding average algorithm. The 'mountain area' data during 2010-2019 were collected from the official website of each region. The data on 'forest fires and forest cover' during 2010-2019 and those on 'forestry investment completion' for 2010 and 2012-2019 were collected from the China Statistical Yearbook, whereas the data on 'forestry investment completion' for 2011 were obtained from the sliding average algorithm.

Results of risk assessment
On the basis of the multi-dimensional risk matrix, this research proposed four types of comprehensive risk evaluation methods: total ordinal, weighted average, Euclidean distance and two-norm models. Furthermore, the available quantitative data were used to identify the risks of different regions, to emphasise the uncertainty of the occurrence and consequences of specific types of natural hazards and to fuzzy rank the risks of different regions. Thus, the priority and non-priority regions for specific types of natural hazard prevention are identified. This research systematically analysed the comprehensive risk status of natural disasters in 30 provinces and cities in China based on three types of risks-agro-meteorological disasters, geological disasters and forest fires-and considering five major indicators: saturation, exposure, concentration, mitigation and fluctuation. The following results of the comprehensive risk ranking of these regions were obtained under the four evaluation methods (the top ranking of provincial and municipal risks indicates a larger comprehensive risk of the region and vice versa).
As shown in Table 2, the results of the comprehensive risk ranking of different regions obtained by the four types of evaluation methods have certain similarities as well as differences. The ranking principle of the risk assessment results is as follows: if the ranking level of a region is smaller, it means that the risk of the specific natural disaster in that region is higher; otherwise, it has a lower risk. For example, the risk ranking results of Chongqing, Zhejiang, Sichuan, Gansu, Guizhou, Shandong and Guangdong provinces under the four types of evaluation methods are very stable, and the ranking difference does not exceed ± 2; Beijing ranks 28th under the total ordinal method, which indicates a small comprehensive risk, whereas the ranking under the other three evaluation methods is within 10 and indicates a larger comprehensive risk, and the ranking difference between different evaluation methods The ranking difference between different evaluation methods reaches ± 26; the comprehensive risk rankings of Gansu Province, Heilongjiang, Inner Mongolia, Sichuan Province, Tibet, Xinjiang Province and Yunnan Province under the four methods are in the top 10, indicating larger comprehensive risk of the region, and must be focused on for prevention; the comprehensive risk rankings of Guangdong Province, Jiangsu Province, Zhejiang Province and Chongqing City under the four methods are all in the bottom 10, indicating smaller comprehensive risk of the region. In summary, the four comprehensive types of risk evaluation results show that it is necessary to focus on risky regions with two types of characteristics: (1) a region that ranks in the top 10 under all four types of evaluation methods, such as Xinjiang, Inner Mongolia, Heilongjiang, Gansu, Sichuan and Yunnan, indicating that the comprehensive risk of this region is greater than that of other regions and, thus, needs greater focus for taking precautions, and (2) a region with large fluctuations in ranking under the four evaluation methods, such as Beijing (± 26), Guangxi Province (± 16), Hainan Province (± 10) and Hebei Province (± 12). The large difference in the risk ranking of these regions indicates that the region has a greater potential risk than other regions and should also be focused on.

Evaluation method testing
Regional aggregate risk indicates uncertainty. Hence, even if Inner Mongolia is found to have a higher risk ranking and a higher risk of specific types of natural disasters, it cannot be said that natural disasters will definitely occur in Inner Mongolia or that the losses caused by natural disasters in Inner Mongolia will be greater than those in other regions. The accuracy of the risk evaluation method cannot be verified through real disaster data. Comprehensive evaluation aims to predict present and future risk situations through existing data, and as several factors affect the occurrence of disasters, all of them cannot be quantified and considered. Therefore, we can only indirectly verify the accuracy of the different evaluation methods through a comparison among them (Fig. 6).
First of all, by utilising the multi-dimensional risk matrix, this research conducted fuzzy ranking of the average risk of specific natural disasters in 30 provinces and cities in China (excluding Hong Kong, Macao, Taiwan and Shanghai) from 2010 to 2019 through the four evaluation methods proposed above (Table 2)-total ordinal, weighted average, Euclidean distance and two-norm models, and retained the ranking results under the four evaluation methods.
Secondly, the regional risk ranking data calculated under four different evaluation methods were divided into six levels according to the same criteria, namely ultra-high risk level, high risk level, medium-high risk level, medium risk level, medium-low risk level, and low risk level.
Then, on this basis, this research further uses ArcGIS visualisation and spatial analysis tools to draw regional risk maps of China based on four evaluation methods. Among them, the red area indicates that the area belongs to ultra-high risk level area, the light yellow area indicates that the area belongs to low risk level area, and other colour areas are between ultra-high risk level and low risk level. In addition, Hong Kong, Macao, Taiwan, and Shanghai without data are represented by white in this study.
Lastly, from a visualisation intuitive viewpoint, the risk conclusions under the four evaluation methods are roughly in the same direction according to the risk maps drawn based on four ranking results-total ordinal (Fig. 7), weighted average (Fig. 7), Euclidean distance (Fig. 7) and two-norm models (Fig. 7).
Furthermore, in terms of the individual regions, the accuracy of different evaluation methods was validated by calculating the average deviation coefficients of the rankings of different regions under these methods. The formula for calculating the deviation coefficients is Step 1: Construct multi-dimensional risk matrix and calculate the ranking results under four evaluation methods.
Step 4: Analyze the consistency of the results of four evaluation methods according to the risk maps.
Step 2: Divide the regional risk ranking results of four evaluation methods into six levels.
Step 3: Use ArcGIS visualization and spatial analysis tools to draw regional risk maps of China based on four evaluation results. where D i represents the deviation coefficients of the four evaluation methods, r is the number of regions, R ij represents the ranking of region j under the i th evaluation method, Ave R i represents the average value of ranking obtained for each region under the four types of evaluation methods and i is the number of evaluation methods. The following figure depicts the deviation coefficients of each region. As shown in Fig. 8, among the four types of risk evaluation methods, the deviation coefficients of the two-norm method and Euclidean distance method are significantly smaller than those of the weighted average method and total ordinal method in terms of data magnitude and variation. It can thus be tentatively concluded that the evaluation accuracy of the two-norm method and Euclidean distance method is better than that of the weighted average method and total ordinal method.
Average deviation coefficients results of the four evaluation methods show that the average deviation coefficient of the Euclidean distance method is 14.44%, which is the lowest among the four methods, and that of the total ordinal method is 31.72%, which is the highest among the four methods. Hence, the evaluation accuracy of the two-norm method and Euclidean distance method can be considered to be better than that of the weighted average method and total ordinal method. The accuracy magnitude of the four types of evaluation methods can be roughly summarised as Euclidean distance method > two-norm method > weighted average method > total ordinal method.

Comparison of risk evaluation based on a multi-dimensional risk matrix and comparison of four evaluation methods
Examining the regional risk evaluation method and empirical test based on the multidimensional risk matrix is a typical cross-scientific study that combines qualitative and quantitative aspects. Such a study can not only focus on the multi-dimensional comprehensive representation form of regional risks from different dimensions of natural disaster risk and analyse the risk status of natural disasters in different regions, types and stages in China but also test the accuracy of different methods for natural disaster risk evaluation by comparing the comprehensive regional risk evaluation results of the four types of risk evaluation methods. Based on the above findings, this study further analyses the evaluation validity, computational complexity and research innovation of the above four types of methods in regional risk management.

This study comprehensively characterises the regional disaster risk situation in terms of risk saturation, exposure, concentration, mitigation and fluctuation
As per the literature review, traditional disaster risk is mainly characterised in terms of the likelihood of occurrence and hazard. However, this research refines the levels of different types of disaster risk and comprehensively characterises its main elements and basic logic in terms of saturation, exposure, concentration, mitigation and volatility. It also extends the above two-dimensional risk matrix to further characterise and analyse the full range of risk conditions in different regions and stages and extends the two-dimensional risk matrix into an abstract four-dimensional risk matrix. This fourdimensional risk matrix includes all risk information, with each element having its corresponding meaning that helps compare risks intuitively and concretely from different   Fig. 8 Changes in deviation coefficients of the four types of regional risk assessment methods dimensions and conducts empirical testing and comparative analysis of various risk evaluation methods.

Assessment results of the four proposed regional risk evaluation methods are consistent
Based on the multi-dimensional risk matrix developed in this study, the risk ranking trends of natural hazards in 30 provinces, cities and autonomous regions of China (excluding Hong Kong, Macao, Taiwan and Shanghai) derived from the above four types of risk evaluation methods were found to be consistent, which indicates that all four methods can effectively help conduct comprehensive regional risk evaluation (Fig. 9). A combination of the above four methods showed that the provinces with higher risk rankings for natural hazards, such as agricultural meteorological hazards, geological hazards and forest fires, in China are Gansu, Heilongjiang, Inner Mongolia, Sichuan, Tibet, Xinjiang and Yunnan (in the four methods, the provinces are ranked in the top 10 in terms of risk), whereas the provinces with lower risk rankings are Guangdong, Jiangsu, Zhejiang and Chongqing (in the four methods, the provinces are ranked in the bottom 10 in terms of risk).

The four types of risk evaluation methods have their own advantages and disadvantages, among which the accuracy of the risk assessment results of the Euclidean distance method is significantly higher than that of the other methods
To further test the consistency of the assessment results of different risk evaluation methods, a comparative study was conducted using the average evaluation results of the four types of evaluation methods as a benchmark (fitted curve). Among them, the fitting  Fig. 9 Natural hazard assessment ranking of Chinese provinces and municipalities based on four types of risk assessment methods coefficient R 2 of the Euclidean distance method was found to be the highest (0.930935), followed by those of the two-norm method (~0.885148), traditional weighted average method (~0.875130), and total ordinal method (0.626898). Furthermore, the computational complexity of these methods was found to be low, with the total ordinal method having the highest computational complexity. Thus, these methods have their respective advantages and disadvantages: the total ordinal method showed a lower risk evaluation accuracy and relatively higher evaluation computational complexity, the Euclidean distance method showed a higher risk evaluation accuracy and relatively lower computational complexity and the weighted average method and two-norm method showed comparable evaluation accuracy and computational complexity. Therefore, considering the different risk evaluation requirements, the two-norm method, weighted average method and Euclidean distance method can all be used as ideal methods for risk evaluation (Fig. 10).

This study found that natural disaster risk is highly volatile in time series, with higher risk volatility in Beijing, Shanxi, Tianjin, Hebei and Liaoning
In view of the high efficiency of natural disaster risk assessment by the Euclidean distance method, this research further carried out natural disaster risk assessment in 30 provinces, cities and autonomous regions in China (excluding Hong Kong, Macao, Taiwan and Shanghai) from 2010 to 2019 from the perspective of the Euclidean distance method. The assessment results showed that the natural disaster risk ranking of Beijing, Shanxi, Tianjin, Hebei, Liaoning, Ningxia, Zhejiang, Shaanxi, Fujian and Gansu in the past 10 years was more volatile. These regions had both high and low risk ranking years, and the standard deviation of risk ranking exceeded 6.37 in the 10 years, indicating uncertainty in regional risk management. Moreover, in the past 10 years, Sichuan, Hunan, Yunnan, Inner Mongolia and other provinces and cities had higher overall risk ranking despite the lower volatility of natural disaster risk and, thus, they were typical  Fig. 10 Schematic of the accuracy comparison of the four types of risk evaluation methods areas with high incidence of natural disaster risk. Furthermore, in the last 10 years, Henan, Hubei, Anhui, Chongqing and other provinces and cities had lower natural disaster risk volatility and overall risk ranking, indicating that they were typical areas with low incidence of natural disaster risk (Fig. 11).

Research implications of regional risk evaluation comparison and method comparison based on multi-dimensional risk matrix
The comprehensive risk evaluation of emergencies has been extensively researched. The present research systematically elaborates the connotations of saturation, exposure, concentration, mitigation and fluctuation of risk, establishes a multi-dimensional risk matrix of emergencies and proposes four types of comprehensive regional risk evaluation methods: total ordinal, weighted average, Euclidean distance and two-norm methods. The study thus conducted a comprehensive risk assessment of specific types of natural hazards and compared the advantages and disadvantages of the four risk evaluation methods. Given the conclusions of regional risk evaluation based on the multi-dimensional risk matrix and comparative analysis of methods, the following are the insights and recommendations of this study.

The multi-dimensional risk matrix can comprehensively characterise the comprehensive risk situation and provide a decision basis for specific types of natural disaster risk prevention
The influencing factors of comprehensive risk evaluation of emergencies involve uncertainties, such as multiplicity, randomness and fuzziness. The proposed multi-dimensional risk matrix completely reflects the objective reality of comprehensive risk evaluation of sudden-onset disasters affected by multiple uncertainties. The data array of each module, data matrix of each interface, data vector of each column, data vector of each row and data value of each point of the four-dimensional risk matrix indicated in this research can reflect Risk Ranking

Fig. 11
Time-series volatility of natural disaster risk in some Chinese provinces from the perspective of the Euclidean distance method the status or value of the risk of a specific region, for a specific time and for a specific type of emergency. Furthermore, comparing the relevant risk values can provide a decision basis for regional defence against relevant types of emergencies and help improve comprehensive disaster resilience.

Specific types of natural disaster risk assessment and its grade evolution analysis are an important basis for developing regional risk classification and management measures
This research conducted a natural disaster risk assessment and determined the risk-level classification method and the risk-level evolution trend. The comprehensive risk assessment showed that the regions with higher disaster risk levels of specific types of natural in China are mainly concentrated in the western and northeastern regions, whereas those with lower risk levels are mainly concentrated in the eastern region. Based on the variability of comprehensive natural disaster risks in different regions of China, a follow-up study for analysing the dynamic risk grade evolution trends in the eastern, central, western and northeastern regions should be urgently conducted so that appropriate regional risk classification and grading management measures can be proposed to improve the accuracy of natural disaster risk management in different regions.

Euclidean distance method has obvious advantages in carrying out risk evaluation of emergencies and is an important direction and basis for the improvement of regional risk evaluation methods
The comparative study of regional risk evaluation methods found that the traditional weighted average method to carry out comprehensive risk evaluation is not the most effective method and that comprehensive risk evaluation is significantly less accurate than the two-norm method and Euclidean distance method. For natural hazard risk evaluation scenarios, such as agro-meteorological hazards, geological hazards and forest fires, the Euclidean distance method not only has relatively lower computational complexity of evaluation but also has the highest accuracy rate of risk evaluation. Therefore, the subsequent research can use the Euclidean distance method to improve the comprehensive risk evaluation method of natural disasters, specifically to standardise the risk values of different dimensions, optimise how different elements of the risk matrix are compared and improve how the Euclidean distance between different elements is calculated.

Specific types of natural disaster risk are highly random and volatile, and there is an urgent need to strengthen the construction of disaster prevention and mitigation infrastructure in key regions to ensure equalisation
The results of natural disaster risk assessment in different regions conducted in the present research using the Euclidean distance method were found to be highly volatile in time series, thus resulting in several uncertainties concerning regional risk management. For Beijing, Shanxi, Tianjin, Hebei, Liaoning, Ningxia, Zhejiang, Shaanxi, Fujian, Gansu and other areas with higher volatility in the national natural disaster risk ranking in the last 10 years, regional risk prevention should aim to strengthen the disaster risk classification and grading management. Hence, it is also necessary to account for natural disaster prevention and mitigation infrastructure construction, focusing on key areas such as power supply, water supply and drainage, gas supply, heat supply and communications to build a longterm mechanism for infrastructure construction, improve infrastructure carrying capacity, improve infrastructure operation and maintenance and enhance the level of infrastructure emergency protection. In addition, the 'short board' of urban and rural infrastructure in relevant areas should be increased to enhance the ability of key areas to prevent and respond to natural disasters.

Research conclusions and future prospects of comprehensive regional risk evaluation research
Improving the form or method of risk evaluation has important theoretical and practical significance for strengthening the modernisation of the risk governance system and governance capacity. Based on the systematic analysis of the connotation, main contents and specific characteristics of risk, this paper summarises the basic connotation of risk as risk saturation, exposure, concentration, mitigation and fluctuation and further expands the traditional two-dimensional risk matrix into a multi-dimensional risk matrix. Subsequently, this paper proposes four types of comprehensive risk evaluation methods-total ordinate, weighted average, Euclidean distance and two-norm-and considers regional agricultural meteorological disasters, geological disasters, forest fires and other natural disasters as the entry point to comprehensively characterise the disaster risk situation in 30 provinces and autonomous regions of China (excluding Hong Kong, Macao, Taiwan and Shanghai). The research also analyses the strengths and weaknesses of different risk evaluation methods and proposes strategies for specific types of natural disaster risk management in China.
The main conclusions of this study are as follows: (1) Compared with the traditional twodimensional risk matrix, the multi-dimensional risk matrix can intuitively, comprehensively and comprehensively characterise and compare the risk status of natural disasters in different regions, different periods, different types and different dimensions.
(2) The four evaluation methods proposed in this paper can effectively evaluate the risk status of regional specific natural disasters, and the evaluation results are highly consistent with the risk status of natural disasters in China.
(3) When considering the fitting coefficient between the results obtained by various evaluation methods and the comprehensive evaluation results, the fitting coefficient of the evaluation results of Euclidean distance method is significantly higher than that of the other three methods, indicating that the accuracy of the evaluation results is relatively high. (4) Over the past 10 years, the natural disaster risk ranking of provinces and cities in China has fluctuated greatly, including Beijing, Shanxi, Tianjin, Hebei, Liaoning, Ningxia, Zhejiang, Shaanxi, Fujian and Gansu. This study also has some limitations. First, China has a vast territory and a wide variety of natural disasters, involving agricultural disasters, meteorological disasters, geological disasters, forest fires, etc. It is difficult to simply obtain the evaluation conclusion of natural disaster risk level of each province by using an evaluation method. This study has tried to use a variety of methods to comprehensively evaluate the risk level of natural disasters in various provinces and cities, but some evaluation methods are still subjective. Second, in view of the availability of data, this study mainly selects regional agro-meteorological disasters, geological disasters and forest fires as the evaluation object to evaluate the risk level of specific natural disasters in 30 regions in China. The relevant conclusions cannot fully reflect the risk status of natural disasters in China. Based on these limitations, in future work, we will consider the following: (1) In the theoretical research link, we will further refine the representation of multi-dimensional risk matrix, develop and improve other comprehensive risk assessment methods, and compare and analyse them with the four assessment methods proposed in this paper. (2) In the empirical research link, we will try to use the way of network big data mining to obtain other natural disaster risk data and incorporate it into the existing research framework to enrich the natural disaster assessment conclusions of provinces and cities.