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Meteorological service is the place where meteorological undertaking starts and ends (Han and Pu [1]). Meteorological services can be classified into two categories: public meteorological services and private meteorological services. A public meteorological service consists of meteorological forecasts, information, technology and engineering services made freely and widely available to the public by meteorological service organizations affiliated to meteorological authorities at all levels through radio, television, newspapers, the Internet, telephone, and other media (Chapman [2]; Nicholls[3]). In recent years, the evaluation of the level of satisfaction regarding the benefit provided by both public and private meteorological services has been assessed on a regular basis. In an evaluation index system for a public meteorological service, it is first necessary to establish evaluation indicators, which is the primary objective of this work, and the basis for the scientific evaluation of meteorological service satisfaction. However, existing evaluation index systems are based mainly on subjective analysis and judgment, lacking rigorous testing and identification of theoretical methods. The effectiveness of evaluation indicators cannot be judged, affecting evaluation results. Therefore, it is fundamental that an effective scientific technical method be determined for the establishment of meteorological service evaluation indicators.
Fuzzy clustering is one of the major fields of fuzzy mathematics. It can determine quantitatively the degree of similarity between samples according to their attributes or characteristics and thus it can be used to objectively categorize a classification (Wang et al. [4]; Li et al. [5]; Xu and Wunsch[6]; Wu and Zhou[7]). Clustering analysis has been applied increasingly not only in the fields of weather forecasting, earthquake prediction, geological exploration, environmental protection, language recognition, fault diagnosis, and data evaluation(Yan et al. [8]), but also in computer networks, comprehensive risk evaluation(Li et al. [9]; Celikel et al.[10, 11]; Molly et al[12]; Zhao et al. [13]; Wang et al. [14]), score management, and even user sentiment analysis (Wang et al. [15]; Yang et al. [16]; Zhong et al. [17]). In clustering analysis, similarity scales are used to determine the degree of similarity between elements to achieve clustering. The role of clustering analysis is to measure a set of samples in the feature space according to the similarity measure (distance or similarity) between all sample points (also called model), between a sample point and a sample point subset, and between all sample point subsets to obtain the relationship system between the sample points and the subsets. This method can be used to determine both qualitatively and quantitatively the closeness of the relationship between research subjects to achieve correct and reasonable classification (Wang [18]). The clustering analysis method is a quantitative method that can scientifically determine categories. The fuzzy clustering analysis method is a multivariate statistical quantification method based on the clustering analysis method in combination with fuzzy mathematics. It not only improves the level of classification considerably, but also enhances the accuracy of the classification by matrix calculation and analysis of data.
Fuzzy clustering analysis has been used widely in meteorological researches. Yan et al. used fuzzy clustering analysis in their research into classification of numerical rainstorm forecasting products [19]. It has also been applied to the analysis of the differences of the latitudinal, meridional, and water vapor transport in the vertical layer during rich and rare Meiyu years in the Jianghuai region (China), and to analyze the regional characteristics of rainstorms during Meiyu years in the Jianghuai region over the past 50 years (Simmonds et al.[20]; Ninomiya[21]; Ninomiya and Kobayashi[22]; Li et al. [23]). In relation to the analysis of the classification of climatic regions, the climate change characteristics of Xinjiang (China) and relevant climatic zoning methods were studied with the use of fuzzy clustering analysis by Mao et al. [24]. He et al. [25] used fuzzy clustering in zoning precipitation in China. Zou et al. [26] used mixed clustering analysis to analyze various precipitation areas, and Kong et al. [27] used hybrid clustering analysis to analyze precipitation classification. Fuzzy clustering has also been widely used in the analysis of various meteorological data. For example, based on a classification of gridded 500-hPa geopotential height data, Li et al. applied the fuzzy clustering method to the prediction of tropical cyclone paths in summer in the South China Sea, and introduced the fuzzy dynamic clustering analysis method from fuzzy mathematics theory [28]. Hong et al. used fuzzy C-means clustering, a genetic algorithm, fuzzy subtractive clustering cross-fusion, and complementary thinking to perform comprehensive clustering analysis of subtropical high impact factors and diagnostic prediction of a subtropical high index [29]. Then, they applied the fuzzy clustering technique to diagnostic prediction of changes of intensity of the subtropical high. In addition, Koffler et al. and Seze et al. combined fuzzy clustering and a genetic algorithm to study the cloud classification using satellite imagery [30, 31]. Shenk et al. [32], Rossow et al. [33] and Desbois et al. [34] applied the multispectral cloud classification techniques of stepwise clustering and fuzzy clustering to the identification of frontal cyclone cloud systems. Furthermore, Geleyn et al. also conducted a study on the comparative performance of stepwise clustering and fuzzy clustering in cloud classification applications [35]. Foreign scholars have also done research on the classification of clouds by cluster analysis.
In summary, fuzzy clustering analysis is a scientific quantitative multivariate statistical analysis method that has been used widely in many fields. However, in the field of meteorology, most previous applications of the method focus on analyses of climatic zoning, regional rainstorm characteristics, classification of various weather conditions, and classification of cloud maps. The clustering methods described above belong to Q-type clustering; however, R-type clustering is used in this study for the determination and correction of meteorological service evaluation indicators, which is the first use in this field and an innovaive attempt. The objectives of this study are to analyze the applicability of the R-type fuzzy clustering technique, and to provide the basis for a technical method for the establishment of a robust scientific evaluation system in the future. This method can also be used for analysis and improvement of similar evaluation indicators.
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The objective of fuzzy analysis is to categorize samples or variables with multiple attribute indicators in a certain way. There are two types of classification method: Q-type clustering that categorizes different sample objects with multiple attribute indicators and R-type clustering that classifies different attribute indicators of multiple sample objects. The clustering analysis adopted for the evaluation index of public meteorological services belongs to R-type clustering.
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Multilayer clustering is required for samples containing different levels of evaluation indicators. In the proposed public meteorological service evaluation system, there are two levels of evaluation indicators suitable for clustering analysis. Multilayer fuzzy clustering analysis first classifies the features of the samples based on the degree of influence on the samples at different levels. Then, the weights of each feature are determined at each level and a fuzzy equivalence matrix is established for clustering analysis (Wang and Shu [47]). The expert evaluation method can be used to determine the weight of an evaluation indicator. Suppose there are M characteristics on level L. Then, we compare the importance of these M features, and either the expert evaluation method can be used to determine their weight vectors (ω11, ω12, ω13, …, ω1m) or another technical method can be implemented to determine their weights. Fuzzy equivalent matrices on the same level can be determined using corresponding fuzzy equivalent matrices and their respective weight coefficients:
$$ R_{i}=\sum\limits_{j=1}^{M} \omega_{i j} R_{j} $$ (1) where Rj is the fuzzy equivalent matrix corresponding to the j characteristics of the sample on level L. If the same, the final fuzzy equivalent matrix can be determined:
$$ R=\sum\limits_{i=1}^{n} R_{i} \omega_{i} $$ (2) where Ri is the fuzzy equivalent matrix corresponding to the i characteristic on the first level, and ωi is the product of the weights of each layer on the first level of i from the initial layer to the final one.
It can be seen that the principal idea of multilayer fuzzy clustering is to take into account the factors of the weight coefficients of each layer, and the total weight coefficients are the product of the weight coefficients of each layer. In this paper, the expert evaluation method is not used to determine the weight coefficient; instead, it is based directly on the importance of the data in the satisfaction survey. The weight coefficient is determined using the analytic hierarchy process.
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The data used in this paper is from a public satisfaction survey of Wafangdian City in 2010. A total of 100 questionnaires were spread to collect data from employees of enterprises and institutions, civil servants and ordinary citizens. A total of 75 valid questionnaires were returned. Satisfaction survey involves three aspects: meteorological service information content, meteorological service information release and meteorological knowledge dissemination, each of which includes three or five questions. Wafangdian Satisfaction Survey results in 2010 are on the average level of the overall satisfaction of the whole country and thus are representative. Therefore, based on 75 groups of scoring results, multi-level fuzzy clustering analysis was carried out.
To facilitate rapid evaluation of the survey data, for each item in the questionnaire, the respondents were requested to select one of five possible responses: "Satisfactory", "Reasonably satisfactory", "Adequate", "Unsatisfactory"and"Not sure". The numeric scores corresponding to these five options were 100, 80, 60, 0 and average scores. Therefore, the first step in the treatment of the original data was to transfer all 75 sets of data into specific scores, which are the basis for a data matrix.
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R-type clustering differs from Q-type clustering because the classification is performed according to different attribute indicators of the sample objects rather than different samples with multiple attributes. Thus, the row-column construction of the data matrix in R-type clustering is opposite to that of Q-type clustering. To perform R-type clustering for the evaluation index of meteorological service satisfaction, the 13 elements of the second-level evaluation indicators shown in Fig. 1 were used as matrix rows and the 75 sets of evaluation data were used as matrix columns.
Figure 1. Hierarchical model of meteorological service evaluation system.
To undertake multilayer fuzzy clustering, the weight coefficients of each layer should be determined first. In this paper, analytic hierarchy process (AHP) is used to determine the weight of meteorological service evaluation index adopted by Yan et al. [48]. Through the steps of constructing judgment matrix, matrix consistency test, and determining the maximum eigenvalue, i.e. weight vector, the weight coefficients of the first and second indexes are finally obtained as shown in Tables 1 and 2, respectively. Indicators 1-13 in the first row of Table 2 represent the 13 secondary evaluation indicators illustrated from left to right in Fig. 1.
First-grade evaluation indicators Meteorological service information content Meteorological services information release Meteorological knowledge dissemination Weight coefficients 0.1634 0.2970 0.5376 Table 1. Weight coefficients for first-level evaluation indicators.
Indicators 1 2 3 4 5 6 7 8 9 10 11 12 13 Weight 0.1267 0.2972 0.4380 0.0798 0.0583 0.7703 0.0679 0.1618 0.2213 0.0628 0.0698 0.2645 0.3817 Table 2. Weight coefficients for the second-level evaluation indicators.
The total weight coefficient of the target layer was obtained by multiplying the weight coefficients of the corresponding first-level and second-level evaluation indicators, from which the final score of the 75 groups was obtained. Thus, the construction of the raw data matrix was completed; however, because of the sheer volume of data, there is no specific data matrix.
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Various dimensions can be used to describe the characteristics of things. Therefore, to eliminate interference in the calculation process, the matrix must be standardized. There are two principal methods for standardization.
The first method is the translation of the standard deviation, which can be implemented by using the following formula:
$$ {{\rm{x}}_{ij}}^\prime = \frac{{{{\rm{x}}_{ij}} - {{\rm{x}}_i}}}{{{{\rm{s}}_j}}}(i = 1, \ldots , n, j = 1, \ldots , {\rm{m}}) $$ (3) where
$$ \bar{x}_{j}=\frac{1}{n} \sum\limits_{i=1}^{n} x_{i j} $$ (4) and
$$ s_{j}=\sqrt{\frac{1}{n} \sum\limits_{i=1}^{n}\left(x_{i j}-\bar{x}_{j}\right)^{2}} $$ (5) The second method is the translation of displacement-extreme difference, for which the formula of transformation is as follows:
$$ {{\rm{x}}_{ij}}^\prime = \frac{{{{\rm{x}}_{ij}} - \min \left\{ {{{\rm{x}}_{ij}}\mid 1 \le i \le n} \right\}}}{{\max \left\{ {{{\rm{x}}_{ij}}\mid 1 \le i \le n} \right\} - \min \left\{ {{{\rm{x}}_{ij}}\mid 1 \le i \le n} \right\}}}(j = 1, \cdots , m) $$ (6) The method of translation of displacement-extreme difference was used in this study for standardization during data processing.
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For the standardized processing matrix above, the degree of similarity among the classification objects was calculated to establish a model similarity matrix (Matrix R= (rij)n × n), which is a process also known as calibration. After data standardization, the matrix was transformed into an n-order array after calibration, where the elements in the matrix fall within the specified range. Various methods can be used to calculate and demarcate the model and the similarity matrix, and many methods are available for the construction of the model and the similarity matrix. In this study, we used the Euclidean distance method:
$$ r_{i j}=1-c \sqrt{\sum _{k=1}^{m}\left(x_{i k}-x_{j k}\right)^{2}}, i, j=1 \ldots \mathrm{n} $$ (7) where c is a constant to ensure 0≤ rij ≤1.
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The final model and aggregation analysis can be produced using various methods, i. e., (1) clustering methods based on the modular equivalent matrix, including the transfer closure method and the Boole matrix method; (2) direct clustering methods based on model similarity, which include the maximum tree method and the network method; (3) the modular clustering method based on the model c-mean value. In this study, the method used most widely for solving the model and the equivalent matrix, i.e., Matrix tr(R), was used. Given R2=R·R, R4= R2·R2, and R8= R4·R4, i.e., R→ R2→R4→R8→…R2k, then after such a limited operation, when R2k= Rk is available, Matrixtr(R)= Rk, which is the equivalent modulus of Matrix R, can be considered transitive. Thus, different truncated fuzzy matrices can be obtained by choosing the appropriate confidence level [0, 1], i.e., the results of fuzzy clustering analysis at different levels of λ.
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All the above steps involved large amounts of matrix data calculation, which was performed conveniently using the Matlab program. The main ideas and methods were as follows. First, the main program file jl.m was established. This not only specified CS = 2, i. e., the use of translation of displacement-extreme difference for data standardization processing, and used the original data A established in"2.4.1, "as input, but also cnamed three. m files: F_JISjBzh. m, coesimeu. m, and F_JlDtjl. m. Second, the. m file (F_JISjBzh. m) of standardized data was established. Third, the.m file for the constructed fuzzy similarity matrix (coesimeu. m) was established. Fourth, the. m file for the fuzzy clustering graph (F_JlDtjl. m) was established. The specific codes in each.m file are omitted here.
2.1. Q-type clustering and R-type clustering
2.2. Multilayer fuzzy clustering analysis
2.3. Data description
2.4. Steps and realization of clustering analysis
2.4.1. ESTABLISHMENT OF THE RAW DATA MATRIX
2.4.2. DATA STANDARDIZATION
2.4.3. CALIBRATION TO ESTABLISH A MODEL SIMILARITY MATRIX
2.4.4. FUZZY CLUSTERING ANALYSIS
2.4.5. IMPLEMENTATION OF MATLAB PROGRAM
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Following the series of Matlab program operations, a graph of the fuzzy clustering analysis of the evaluation indictors of meteorological service satisfaction was obtained, as shown in Fig. 2.
Figure 2. Fuzzy clustering result of evaluation indicators of meteorological service satisfaction.
The numbers at the top of Fig. 2 refer to the 13 evaluation indicators of the secondary evaluation index (from left to right) in Fig. 1, and λ is the reliability value. It can be seen from Fig. 2 that when λ =1, each indicator is classified into a separate class. When λ = 0.971, the evaluation indicators are divided into 12 categories, where the second-level indicators"species richness"for the"meteorological service information content" and "channel diversity" for the "meteorological service information release" are clustered into a single category. When λ = 0.967, two further types of secondary evaluation indicator "readability" and "interesting" of "promotion of meteorological knowledge popularization"are grouped into a single category. With further decline of the λ value, additional evaluation factors are clustered together until they are all grouped together when λ = 0.4773.
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The two indicators clustered together first are "species richness"and"channel diversity, "which are secondary indicators for "meteorological service information content" and "meteorological service information release", respectively. These two secondary indicators have similarity because the "diversity"of "meteorological service information content"includes different types of forecast product with meteorological information content, and the "channel diversity" of "meteorological service information release"includes different methods and channels for the release of meteorological information. Therefore, the clustering results are consistent with the facts. However, because these two secondary indicators belong to different primary evaluation indicators, it is inappropriate to replace or remove either one.
The second two indicators clustered together are "readability" and "attractivenes", which are both secondary indicators of "meteorological knowledge dissemination". The similarity relationship between these two evaluation indicators is reasonably close. For popularization of meteorological knowledge in scientific literature, if it is readable, it will be interesting in a certain sense, and the interesting science will improve its readability. Therefore, these two indicators can be grouped together, and it is possible that they could be combined or one or the other removed.
The later the evaluation indicators are clustered together, the better the independence relationship. The final indicators classified are the "timeliness" of "meteorological service information release", "timely release of meteorological disaster prevention knowledge""meteorological knowledge dissemination", and "usefulness"and "accuracy"of "meteorological service information content".These secondary indicators are divided among the three different first-level indicators, and their degree of independence is reasonably good. Although"usefulness"and"accuracy" belong to the same first-level category, they have good independence and one cannot be replaced by each other. Thus, it can be seen that the results of the fuzzy clustering analysis are consistent with the real situation.
Finally, the results of the clustering analysis are compared with the weight coefficients of each indicator to evaluate the overall rationality of the indicator system. It can be seen from Table 2 that the weight coefficients of the four evaluation indicators with the best degree of independence, i. e., those with numerical codes 6, 13, 3, and 2 (Fig. 2) are 0.7703, 0.3817, 0.4380, and 0.2972, respectively. The weight coefficients of these evaluation indicators are the highest, indicating that they are the most important evaluation indicators and that their degree of independence is good. This proves that the evaluation system is reasonable and that only individual evaluation indicators need be adjusted.
3.1. Multilayer clustering result graph
3.2. Analysis of applicability of multilayer clustering in establishing a meteorological service satisfaction index
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Through the application of fuzzy clustering analysis to the evaluation indicators of meteorological service satisfaction, our study has reached the following conclusions.
(1) When the multi-layer fuzzy clustering method is used to analyze the evaluation index of meteorological service satisfaction, the similarity relation of the evaluation index is positively correlated with the clustering order. For the evaluation indicators clustered together, if they are in the same sub-evaluation layer, one of them can be removed or the two indicators can be merged; if they are not in the same sub-evaluation layer, they cannot be replaced each other.
(2) Cluster analysis can not only revise the evaluation index, but also evaluate the performance of the index. The more independent the evaluation index is, the better its evaluation performance is. In the analysis of clustering method, the performance of evaluation index is better when the order of clustering is lower.
(3) The analysis of the applicability of the fuzzy clustering method shows that this technical method could play a certain role in the establishment and revision of an evaluation index.
(4) Comparison of the results of the clustering analysis with the weight factors of meteorological service satisfaction evaluation reveals that the evaluation system has overall rationality and that only individual evaluation indicators need be adjusted.
Acknowledgments: We thank James Buxton MSc fromLiwenBianji, Edanz Group China, for reviewing the English language quality of this paper.
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