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Keywords:

• matter element analysis;
• related function;
• weather forecast

Abstract

Matter element analysis is an emerging mathematical method that can resolve various actual problems in the objective world and has therefore been widely applied in many fields. However, no study to date has examined the use of matter element analysis to forecast change trends in weather. Prior predictors are used to conduct calculations, thus enabling the matter analysis to have a predictive function. By repeatedly adjusting the grade division value (classical domain and joint domain) of each factor, the fitting rate of the calculation grade of the meteorological element to the actual grade was maximized, resulting in more accurate results. This article discusses the method, steps and applications of matter element analysis and demonstrates matter element analysis as a new and efficient way to forecast future weather conditions. Copyright © 2012 Royal Meteorological Society

1. Introduction

Weather forecasting is a well-studied subject. However, because of the great volume and complexity of influential factors, the results of weather forecasting are far from accurate. One of the problems is how to determine the internal relationship (correlation) between changes in the weather and the factors that influence the changes (especially teleconnection influences) because the internal relationship has both certain and uncertain features (Awan and Awais, 2011).

Weather change is the result of the interaction of many physical factors, including movements of celestial bodies, atmospheric circulations and underlying surfaces (Bussel et al., 2011). These factors are related to weather change. To forecast trends in weather change using these factors, one needs a data-processing technology that can analyse relationships between physical factors, and matter element analysis is the right tool.

Matter element analysis, a newly-emerging method for studying contradictions, is an interdisciplinary subject, integrating cognitive science, systematic science and mathematics (Cai, 1994). That is to say, it is a theory of rules and methods for ‘thinking of ideas and solutions’ when one is solving contradictions. It is a new subject based on classical mathematics and fuzzy mathematics, but it is also different from them. The logical bases of classical and fuzzy mathematics are formal logic and fuzzy logic, respectively, but the basis of matter analysis is the integration of the two. Classical mathematics is a tool describing the way the human brain solves problems based on formal logic, and fuzzy mathematics is a tool describing the way the human brain solves problems based on fuzzy logic. Matter analysis, however, is a tool describing the way the human brain thinks of ideas and methods of solving contradictions, which is very intelligent (Cai, 1994).

The research object of matter element analysis is the contradictions in the real world, and its research direction is to discover the rules and means of dealing with contradictions. Matter element analysis is an application subject that cannot only be used in the ‘hard’ sciences but can also be used in the ‘soft’ sciences. It is a subject that spans the natural sciences and the social sciences. Matter element analysis is a model of human thinking that specializes in how to solve a problem. Therefore, it has been involved in artificial intelligence and artificial-intelligence-related subjects. It has also been applied in areas with many contradictions, such as economic planning, business management, process control, and military decision making (Cai, 1994; Feng, 1998). Matter element analysis is widely used in many fields at present. Gao (1995) first proposed using matter element analysis in the comprehensive assessment of eutrophication in water bodies and compared it to other methods based on actual examples and obtained satisfactory results. Yu and Yang (2005) put forward an assessment method based on matter element analysis. They used various indices of comprehensive assessments of groundwater quality based on the calculated synthetic relation degree to yield a groundwater quality grade. Xu and Zhu (2008) developed optimized alarm parameters based on measurements of synthetic relation degree and created an alarm optimization method suitable for the processing industry. Taking Wuhan City as an example, Ning et al. (2008) used matter element analysis to calculate and analyse water resource bearing capacity. Focusing on the problem of information shielding and subjectivity in most domestic comprehensive assessment research, Luo et al. (2009) tried to use matter element analysis to assess the ecological level of urban land. On the basis of an analysis of factors influencing the ecosystem of Minqin Oasis, Zhang et al. (2009) organically integrated fuzzy theory and an analytical hierarchy process with the matter element method to establish a fuzzy matter element model for a frangibility assessment of the ecosystem. Wang et al. (2011) presented a thermal image matter element used to design a circuit board signal fault diagnosis system. Extension theory is used to build several kinds of thermal image matter-element models with fault circuits. Feng and Gong (2004) used matter element analysis to forecast the runoff production for a drainage basin. However, no study has yet examined the use of matter element analysis to forecast change trends in weather. The present study examines the use of the methods and principles of matter element analysis to forecast the weather change (Liu and Liang, 2009; Yao et al., 2009).

It is feasible to forecast weather change by matter element analysis, as the prior predictors are used to conduct the calculation, thus enabling the matter analysis to have a predictive function. Here, advance period of forecasting for weather factors (predictand) lies on that of predictors, which can be from 3 to 6 months. This paper presents a novel approach to applying matter element analysis in weather forecasting. Because this approach has never been used in this area, the work is promising and provides valuable insights for researchers and practitioners.

The structure of the paper includes presenting the method of matter element analysis, forecasting the precipitation from July to August at Beijing Station, China, and comparing this with the result calculated using neural networks.

2. Methodology and case analysis

During matter element analysis, the ordered combination comprises matter, N, its character, c, and quantity value, x, expressed as (Cai, 1994):

• (1)

This combination is known as the matter element. If matter N needs to be described by n characters c1, c2, ·, cn and corresponding quantity values x1, x2, ·, xn, then it is called an n-dimension matter element, which is expressed by the following matrix (Cai, 1994):

• (2)

In the present study, the precipitation forecast at Beijing Station, China, was taken as an example to illustrate the application of matter element analysis in weather forecasting. Table 1 lists the actual measured precipitation from July to August in 1951–1970 at Beijing Station and six prior forecast factors in accordance with their physical relation (Zhang, 1986). Here, the matter to be rated was precipitation y from July to August in 1951–1970 at Beijing Station (mm) (k = 1, 2, ·, 20). The characteristics are the six prior forecast factors of precipitation from July to August at Beijing Station: c1, precipitation in March at Beijing Station (mm); c2, day number of E-type circulation in January at 500 hPa over the Atlantic-Europe (d); c3, circulation exponential of average longitude direction in February in Asia (m s−1); c4, average intensity exponential of subtropical high in March at 500 hPa over the western Pacific (m2 s−2); c5, average intensity of East-Asia trough in March at 500 hPa (m2 s−2); and c6, intensity of polar high in January (hPa) (i = 1, 2, ·, 6). According to the long-term weather change and the forecast requirement, the precipitation from July to August at Beijing Station was separated into five grades (classes): 1 (dryness month, < 250 mm); 2 (relative dryness month, 251–370 mm); 3 (mid-month, 371–490 mm); 4 (relative wetness month, 491–610 mm); and 5 (wetness month, > 610 mm). The prior forecast factors were also divided into five grades (j = 1, 2, ·, 5) in accordance with their distributing intervals between maximum and minimum value. Based on the principle of the maximum fitting rate of calculated and actual grades of precipitation, the grade division values of factors in the calculation were repeatedly adjusted to obtain the grade division values with the maximum fitting rate (Table 2).

Table 1. Precipitation and prior forecast factors from July to August at Beijing Station
Yeary (mm)c1 (mm)c2 (d)c3 (m s−1)c4 (m2 s−2)c5 (m2 s−2)c6a
1. a

c6 = x − 1000, where the unit of x is hPa.

1951195.91.4150.54711617.2
1952382.516.190.90513619.0
1953399.74.0140.69014713.2
1954616.14.6150.55411714.4
1955567.89.440.63812814.6
1956607.111.1130.74514316.0
1957327.96.9100.6131096.6
1958415.36.530.78269815.4
19591086.119.6100.561616625.0
1960372.76.340.50312823.6
1961382.420.1130.571813319.0
1962235.02.300.6411919.4
1963653.712.400.62314921.4
1964482.510.660.72813510.2
1965218.90.170.62610816.6
1966494.617.850.69411722.2
1967459.48.710.62010313.4
1968276.05.200.51113717.6
1969630.216.5150.693310427.8
1970410.213.4120.58178723.4
Table 2. Grade division values of precipitation and forecast factors with maximum fitting rate
y< 250251–370371–490491–610> 610
c10.1–2.43.0–8.43.0–11.016.0–19.617.2–21.2
c22.0–3.03.2–3.46.0–11.010.0–13.013.8–16.2
c30.40–0.640.40–0.500.49–0.750.55–0.780.55–0.80
c41.1–3.08.5–14.012.2–23.821.0–28.028.0–34.0
c580–12490–155120–146130–160140–158
c66.0–10.010.0–15.016.0–21.820.0–22.018.0–24.8

The specific steps in matter element analysis include (Cai, 1994):

• Determine classical domain.

• (3)

In this matrix, N0j is the jth grade (j = 1, 2, ·, m); ci is the ith character of the jth grade; and x0ij is the quantity value of N0j with respect to ci, i.e., the classical domain describing the corresponding characteristics of each grade < a0ij, b0ij>.

Based on the volume of the precipitation from July to August at Beijing Station, the classical domains (grade division intervals) of five grades of prior forecast factors were determined. For example, the classical domain of the first grade (dryness month) is:

• (4)

For the classical domains of the other grades see Table 2.

• Determine joint domain.

• (5)

In this matrix, P is the collectivity (total historical data) of event grades; xpi is the quantity value of P with respect to ci—joint domain < api, bpi>. It requires x0ijxpi.

According to the requirement of x0ijxpi, the joint domain (division intervals including maximum and minimum value) of prior forecast factors was determined (Table 2):

• (6)
• List matter elements to be rated. The matter elements to be rated can be expressed as:

• (7)

In this matrix, Pk is the matter to be rated (k = 1, 2, ·, l) and xi is the quantity value of Pk with respect to ci, i.e., the actual data of each character.

Based on the data in Table 1, the six factors of the precipitation from July to August in 1951 were expressed with matter elements as (others are similar):

• (8)
• Calculate weight co-efficient. Here, pairwise comparisons are used to determine the set of weights. In other words, based on expert consultation, pairwise comparison of the importance of any two of the n indices is conducted, which gives the ratio dij (this can be the average of the values provided by experts) (i, j = 1, 2, ·, n) and the judgment matrix D:

• (9)

Every row of elements of matrix D is multiplied, after which the nth roots are defined, resulting in β = (β1, β2, ·, βn)T, and ). After normalization treatment, , which gives the weight set A = (a1, a2, ·, an)T and satisfies .

Using Equation ((9)) and based on suggestions of experts, the weight co-efficients of six factors were obtained: A = [0.22(c1), 0.21(c2), 0.19(c3), 0.17(c4), 0.11(c5), 0.10(c6)].

• Calculate correlation function value. Let:

• (10)

In this equation:

• (11)
• (12)

The correlation function of matter to be rated Pk with regard to the jth grade is then given by:

• (13)

Using Equations ((10))–(12), Yj(xi) of each grade of the precipitation from July to August in 1951 was calculated: Y1(x1) = 2.5000, Y2(x1) = − 0.5333, Y3(x1) = − 0.5333, Y4(x1) = − 0.9125, Y5(x1) = − 0.9186; for other Yj(xi) see Table 3.

Table 3. Yj(xi) of each grade of precipitation from July to August in 1951
Factor12345
c12.5000− 0.5333− 0.5333− 0.9125− 0.9186
c2− 0.8571− 0.8529− 0.6667− 0.50001.5000
c30.2778− 0.08000.1220− 0.0213− 0.0213
c4− 0.3636− 0.1765− 0.4262− 0.6667− 0.7500
c50.14290.6842− 0.0588− 0.1795− 0.2727
c6− 0.3789− 0.15710.1132− 0.1918− 0.0635

Next, Equation ((13)) was used to calculate the correlated function value Yj(P1) of each grade of the precipitation from July to August in 1951:

• (14)

For the correlated function values of each grade in other years see Table 4.

Table 4. Correlated function value, calculated and actual grades of precipitation from July to August at Beijing Station
19510.34− 0.28− 0.30− 0.46− 0.0511
1952− 0.48− 0.36− 0.15− 0.25− 0.3433
1953− 0.49− 0.28− 0.18− 0.27− 0.2233
1954− 0.25− 0.08− 0.12− 0.45− 0.0455
1955− 0.230.010.02− 0.30− 0.3634×
1956− 0.42− 0.31− 0.15− 0.13− 0.2044
1957− 0.14− 0.190.20− 0.33− 0.4232×
1958− 0.28− 0.120.07− 0.29− 0.3733
1959− 0.47− 0.41− 0.12− 0.11− 0.0955
1960− 0.160.030.04− 0.45− 0.4533
1961− 0.43− 0.290.00− 0.04− 0.0333
1962− 0.25− 0.27− 0.26− 0.50− 0.5311
1963− 0.41− 0.44− 0.27− 0.29− 0.2555
1964− 0.30− 0.10− 0.03− 0.25− 0.3033
1965− 0.13− 0.35− 0.18− 0.41− 0.4611
1966− 0.32− 0.31− 0.22− 0.11− 0.1244
1967− 0.34− 0.33− 0.20− 0.48− 0.4933
1968− 0.31− 0.14− 0.15− 0.53− 0.5522
1969− 0.56− 0.58− 0.42− 0.250.4755
1970− 0.35− 0.32− 0.01− 0.06− 0.1733
• Rate matter grade. Based on the maximum subordination principle in fuzzy mathematics (Wang et al., 2005), the maximum correlated function is determined by Yj(Pk):

• (15)

Pk then belongs to the jth grade (Cai, 1994). The correlated function value of each grade of the precipitation from July to August in 1951 was as follows:

• (16)

For j′ = 1, the precipitation from July to August in 1951 belonged to the first grade (dryness month, < 250 mm). The actual precipitation from July to August in 1951 was 195.9 mm and belonged to the first grade, which matched the actual conditions. As shown in Table 4, there were 18 years with calculated and actual grades matching the precipitation from July to August at Beijing Station for the 20 years from 1951 to 1970. Calculated and actual grades in 1955 and 1957 were discrepant with 1 grade, so the results were sound.

Combination of similar prior forecast factors often results in similar precipitation (Zhou, 2011). Therefore, to forecast the precipitation from July to August in 1971 at Beijing Station, the following six prior forecast factors were added: c1 = 23.5 mm, c2 = 5 day, c3 = 0.52, c4 = 0, c5 = 112, c6 = 16.6. Matter element analysis was then conducted, and the following was finally obtained:

• (17)

For j′ = 1, the precipitation from July to August in 1971 at Beijing Station is predicted to be a dryness month (precipitation y < 250 mm). The actual precipitation in 1971 was 247.0 mm, indicating a dryness month: therefore the forecast is correct.

3. Discussion

In the above, a novel approach to applying matter element analysis in weather forecasting is presented. The work is promising and provides valuable ideas for researchers and practitioners. Figure 1 is a framework of applying matter element analysis in climate prediction, which helps to make this process more understandable for those who are new to this approach.

In fact, it is feasible to forecast the change trend of weather factors (including precipitation, temperature, humidity, air pressure, wind and similar factors) by matter element analysis. Matter element analysis in general application is used mostly to classify and evaluate and does not have a forecasting function. In the present case, however, prior predictors are used to conduct the calculation, thus enabling the matter analysis to have a predictive function. Certainly the physical basis of this prediction is that combination of similar prior forecast factors often results in a similar weather trend (Zhou, 2011). To conduct accurate weather forecasts, the physical relationship between the prior predictors and predictions must be analysed. Here, in addition to judgment based on experience from physical causes, the prior predictors can also be chosen using the relationship analysis.

Neural networks have the advantages of self-learning, self-organizing and self-adapting (Feng and Hong, 2008). This precipitation forecast at Beijing Station was taken as an example to illustrate the application of neural networks in weather forecasting. Because the prior forecast factors (c1, c2, ·, c6) are the inputs while the precipitation y is the output, there are six nodes in the input layer and one node in the output layer. It follows from Kolmogorov's law that there are 15 nodes in the hidden layer. Hence, neural networks have the topological structure (6, 15, 1).

The six prior forecast factors are input into the input layer of the BP algorithm and select training data to start the training and learning process. The learning rate η = 0.85 and the momentum value α = 0.60 are chosen. To test the BP algorithm after each training and learning step, take the precipitation values for 1951–1970 as the training samples and the precipitation values for 1971 as the testing samples. After 100 000 cycles of training and learning from the training samples, the network error E = 0.0004, which is less than the expected error; thus, the BP algorithm is convergent. Because the trained network has imitated and memorized the functional relationship between input and output, it can be used to forecast the precipitation. The testing result shows that the precipitation from July to August in 1971 at Beijing Station was 223.9 mm, and the actual precipitation was 247.0 mm. The result of neural networks, therefore, is consistent with that of matter element analysis (y < 250 mm). Compared with neural networks, the calculating method of matter element analysis is briefer, and its calculating precision can satisfy the general forecast requirement.

4. Conclusion

Matter element analysis is an emerging mathematical method that can resolve various actual problems in the objective world and has therefore been widely applied in many fields.

In fact, it is feasible to forecast the change trend of weather factors by matter element analysis. Matter element analysis in general application is used mostly to classify and evaluate and does not have a forecasting function. However, in the present case, prior predictors are used to conduct the calculation, thus enabling the matter analysis to have a predictive function. Certainly the physical basis of this prediction is that a combination of similar prior forecast factors often results in a similar weather trend. To conduct accurate weather forecasts, the physical relationship between the prior predictors and the predictions must be analysed. The testing result of neural networks shows that the precipitation from July to August in 1971 at Beijing Station was 223.9 mm, and the actual precipitation was 247.0 mm. The result of neural networks, therefore, is consistent with that of matter element analysis (y < 250 mm). Compared with neural networks, the calculating method of matter element analysis is briefer, and its calculating precision can satisfy the general forecast requirement.

The prior predictors are used to conduct the calculation, thus enabling the matter analysis to have a predictive function. By repeatedly adjusting the grade division value (classical domain and joint domain) of each factor, the fitting rate of the calculation grade of the precipitation to the actual grade was maximized, resulting in more accurate results. The results presented here indicate that this technique is a new method of forecasting weather change. This paper presents a novel approach to applying matter element analysis in weather forecasting. Because this approach has never been used in this area, the work promises to provide valuable ideas for researchers and practitioners.

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 41171430, 40771044).

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