1 Introduction
Aggregation is a process of combining several numerical values into a single one which exists in many disciplines, such as image processing SoriaFrisch (2001); Liu et al. (2013)
Soda and Iannello (2009); Russo (1999), decision making Merigó and GilLafuente (2013); Merigó and Casanovas (2011a, b) and so forth Marek and Mayer (1996); Liu et al. (2010); Cheng et al. (2013); SmutnáHliněná and Vojtáš (2004); Wei et al. (2011); Wang et al. (2011). To obtain a consensus quantifiable judgements, some synthesizing functions have been proposed.For example, arithmetic mean, geometric mean, median can be regarded as a basic class, because they are often used and very classical. However, these operators is not able to model an interaction between criteria. For having a representation of interaction phenomena between criteria, Fuzzy measures have been proposed by Sugeno in 1974
Sugeno (1974). Two main classes of the fuzzy measures are Choquet and Sugeno integrals. Choquet and Sugeno integrals are idempotent, continuous and monotone operators. The ordered weighted average (OWA) operators are a particular case of discrete Choquet integrals. The OWA operators were introduced by Yager in Yager (1988) to provide an aggregation which lies in between the “and” and the “or ” operators. The “and” (tnorms) and the “or” (tconorms) operators are generalizations of the logical aggregation operators which are two specialized aggregation families. Above operators try to look for giving a “middle value”, but the tnorms and the tconorms can compute the intersection and union of fuzzy sets.However, to the best of our knowledge, these operators do not consider the influence of time specially and the time factor should not be ignored in some areas such as economics, space science, weather forecast and so forth. In this paper, a novel aggregation operator called visibility graph averaging (VGA) aggregation operator is proposed which can deal with the time series effectively.
This paper is inspired by the pioneering work the visibility graph Lacasa et al. (2008) which builts a natural bridge between complex network theory and time series. In the visibility graph, the values of a time series are plotted by using vertical bars. These vertical bars are regarded as landscapes, and every bar is linked with others that can be seen from the top of the considered one, then the associated graph is obtained. According to the study, it is found that the structure of the time series is conserved in the graph topology. For example, periodic series convert into regular graphs, random series convert into random graphs, and fractal series convert into scalefree graphs. Until now, the visibility graph has been applied in economic Wang et al. (2012), geology Telesca et al. (2013); Telesca and Lovallo (2012), praxiology Fan et al. (2012), biological system Ahmadlou et al. (2010, 2012) and so forth Yang et al. (2009); Gutin et al. (2011); Liu et al. (2010); Lacasa et al. (2009); Qian et al. (2010); Donner and Donges (2012). The proposed visibility graph averaging (VGA) operator is based on the visibility graph, hence, it conserves the time information likewise.
In some aggregation operator, how to decide the weight of each argument is a problem Xu (2005); Filev and Yager (1998); Beliakov (2005), but in this proposed VGA operator, while the time series is converted into graphs, the degree distribution is decided, and meanwhile the weights are decided. In general, if the degree of a node is bigger than others, this node will be more important, and in the visibility graph a node represents a data value of time series, so it offers a reasonable way to determine the weights of the corresponding data value.
The remainder of this paper is organized as follows. Section 2 briefly introduces some necessary preliminaries of the aggregation operators and the graph theory. Section 3 details the proposed visibility graph averaging (VGA) aggregation operator. The properties of the visibility graph averaging (VGA) aggregation operator will be discussed in the Section 4. In Section 5, VGA aggregation operator is applied in the economic and compared with OWA operators to show its advantage. Finally, some conclusions are given in Section 6.
2 Preliminaries
In this section, the aggregation operators problem and the graph theory are briefly introduced.
2.1 The aggregation operator
Aggregating values, a new value can be obtained, but this can be done in different ways. In other words, aggregation operators is various. In the following, the aggregation operator will be introduced in a formal way.
Let be the unit intervals. denotes the degree to which satisfies the criteria , denotes the set of the results of the aggregation.
Definition 1.
An aggregation operator is a function :
where represents the number of values to be aggregated.
Several fundamental conditions have been proposed to define the aggregation operators Mayor and Trillas (1986). The fundamental properties which generalize most of the precedent definitions are as follows Detyniecki (2001):
1)Identity when unary: If there is only one value needing to be aggregated, the result is itself.
2)Boundary conditions: If all the values needing to be aggregated are completely bad, false or not satisfactory, the result has to be completely bad, false or not satisfactory. On the contrary, if all the values needing to be aggregated are completely good, true or satisfactory then the result has to be completely good, true or satisfactory.
3)Monotonicity: If the individual value increases the overall satisfaction should increase:
2.2 The graph theory
Graph theory is the study of graphs, which is made up of vertices and edges. Graphs can be used to deal with many types of relations and processes in computer science Narsingh (2004), biological Pavlopoulos et al. (2011); Haggarty et al. (2003), social Nastos and Gao (2013) and so forth Kim and Phillips (2013); Agosta et al. (2013); Liu and Zhan (2012). Recently, with the development of complex network research, the graph theory is widely used in the analysis of complex network Barabási and Albert (1999); Gao et al. (2013); Wei et al. (2013).
Definition 2.
A graph is formed by vertices and edges connecting the vertices.
Example 2.1.
A graph is a pair of sets (V,E), where V denotes the set of vertices and E denotes the set of edges. Figure 1 shows a graph with 5 vertices and 5 edges. The vertices are labeled as and the edges are labeled as .
Definition 3.
The degree of a vertex is the number of edges at .
Example 2.2.
In the Figure 1, , the sum of the degree is .
In general, if a vertex has a big degree, it illustrates that this vertex links with many other vertices and it will be important.
2.3 The visibility graph
The visibility graph method was first proposed by L. Lacasa et al. in 2008, which can convert a time series into a graph. The properties of the time series is conserved in the graph topology. In the visibility graph, the values of time series are plotted by using vertical bars. A vertical bar links with others which can be seen from the top of itself. The visibility criteria is established in the literature Lacasa et al. (2008) as follows:
Definition 4.
Two data value and have visibility, if any other value is placed between them fulfills:
Example 2.3.
In the Figure 2, the histogram shows a time series with 11 data values, and according to the visibility algorithm, the associated graph is obtained. In the histogram, if a bar can be seen from the top of considered one, they will be linked. If two bar are linked in the histogram, the vertices which represent them will be linked in the associated graph.
The associated graph derived from a time series has the following properties: 1)Connected: a node can see its nearest neighbors. 2)Undirected: the associated graph extracted from a time series is undirected. 3)Invariant under affine transformations of the series data: Although rescale both horizontal and vertical axes, the visibility criterion is invariant.
3 Visibility Graph Averaging Operator
As mentioned above, there is rarely aggregation operator considers the influence of time, but in many disciplines, time is a very important factor that can not be ignored. In this section, a new type of operator called visibility graph averaging (VGA) operator will be introduced which can be used to aggregate time series. This proposed VGA operator will be introduced in the following.
To a time series, if there are data values, the associated graph derived from the visibility algorithm will have vertices. VGA operator is defined as follows:
Definition 5.
VGA operator is a mapping F:
where
(1) 
and where is the value in the time series, and is the weight of the value satisfies:
Because the degree distribution can reflect the importance of a vertex. The weight can be obtained according to the degree distribution in the visibility graph which is defined as follows:
Definition 6.
(2) 
where is the value of the degree at the vertex .
It is necessary to notice that if there is only one value in the time series, in the associated visibility graph, there just exist one vertex and the degree of this vertex is equal to zero. Hence, the weight is revised as one, then we have:
Definition 7.
If there is only one value in the time series, the VGA operator is as:
The following simple example illustrates the use of this VGA operator.
Example 3.1.
According to the Eq.2, the weights of the values can be computed as follows:
So according to the Eq.1, we have
The weights of the VGA operator are obtained according to the distribution of the degree in the visibility graph, because in the network, if a node links with more nodes than others, its degree will bigger and it will be more important than others. In the histogram, Whether the considered one can see others is relevant to the order of the data. The order in our VGA operator is decided by time. The order of the data is not set artificially, and instead it is restricted by time. In other words, the time information is conserved in the VGA operator.
4 Properties of VGA Operator
In this section, we shall investigate some properties of this new operator. These properties will be divided into two parts: the mathematical properties and the time properties.
4.1 Mathematical property
Theorem 4.1.
Idempotency: if ,for all , then
Proof.
∎
Remark 1.
Particularly, there exist
The above equations illustrate that if every single value is small, the aggregated result will be small, and if every value is big, the aggregated will be big.
Theorem 4.2.
Stability for a linear function:
Where
Proof.
As mentioned in the preliminaries, the visibility criterion is invariant under affine transformations, hence, the weight distribution of time series is same as the weight distribution of time series . We have:
∎
4.2 Time property
Because the visibility graph averaging (VGA) operator decides the weights according to the degree distribution of the associated graph derived from the visibility algorithm, and the visibility graph conserves the structure of the time series, certainly the VGA will conserve the structure of the time series likewise.
It is obvious that if a series is periodic, the weights of data in this series will be periodic except for weights of the first period and the last period. The Figure 4 shows the weights of a periodic series . If a series is random, the weights will be random. If the series is fractal, the weights will be scalefree. The Figure 5 shows the weights distribution of the Conway series.
Here we consider some special cases to show its special time properties. Suppose that there is a periodic series with one special big value which is like the Figure 6 showing. In this figure, if a data value is bigger than others, it will be more important (In the VGA operator, its weight will be big), because it can see more other bars in the histogram, but with time passing, the number of bars that it can see will get smaller (In the VGA operator, its weight will get smaller). In the figure, the periodic series on the left and right sides of the special big value are divided into 4 groups. In the group 1, in total values can be seen from the top of the special one. In the group 2, there are 6 values being seen. In the group 3, there are 4 values, but in the group, there are just 2 values. In other words, time will decrease the influence of a bigger value. This property reflects the reality. In the real world, some important events will influence other events, but with time passing, this influence will be decreased (In the VGA operator, the weights will be decreased), and the number of the events which are influenced by the important event will be decreased too.
Moreover, if one argument is far away from the considered one argument
, the probability that
can be seen from the top of will decrease, because with other data appearing between them, these data may block the sight. With the time passing, the distance between and will get bigger and bigger, then more and more other data will block between them, so the probability that they can see each other will get smaller. It means that time will decrease the probability of the connection between data. This situation is shown in the figure 8.In addition, if an argument with big value lies in between two arguments which are just a little smaller than it, it can not connect many other arguments and its weight will not be great. It illustrates that at what time the argument with big value appears in the histogram is important. All of the time properties will influence the weights distribution in the VGA operator.
5 Case Study
In this section, a case is used to illustrate the feasibility and the efficiency of the VGA operator. The VGA operator will be used in the analysis of the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) database. In economic, forecasting can provide a superior investment strategy for investors. The problem is that it is difficult to handle highorder stock data, because it is hard to determine an appropriate weight to each period of past stock price. VGA operator will solve this problem conveniently.
The Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) of the year 20002012 is chosen as the forecasting basic data. The data of December from 2000 to 2012 which are listed in the Table 2 will be aggregated by VGA operator and OWA operators, respectively.
5,342.06  5,870.17  5,798.62  6,179.82  7,613.57  4,518.43  7,649.23  8,520.11  7,178.69  
5,277.35  4,683.18  5,911.45  5,867.95  6,228.95  4,356.98  7,677.62  8,585.77  7,140.68  
4,646.61  4,793.93  5,884.97  5,893.27  8,583.84  4,307.26  7,684.67  8,624.01  7,599.91  
5,174.02  4,766.43  4,727.49  5,920.46  7,647.01  8,651.28  4,254.96  7,650.91  7,600.98  
5,199.20  4,924.56  4,755.40  5,900.05  6,348.31  7,609.90  8,676.95  4,225.07  7,098.08  7,649.05  
5,170.62  5,208.86  4,738.98  5,919.17  6,350.52  7,693.33  8,694.41  8,702.23  6,956.28  7,623.26  
5,212.73  5,333.93  5,925.28  6,329.52  7,686.52  8,722.38  7,775.64  8,704.39  7,033.00  7,642.26  
5,252.83  5,847.15  5,892.51  6,249.19  7,636.30  4,418.33  7,768.71  8,703.79  6,982.90  
4,823.67  5,859.56  5,913.97  6,264.36  4,472.66  7,797.42  8,753.84  6,893.30  
5,321.28  4,755.01  5,803.42  5,911.63  8,598.03  4,658.87  7,677.91  8,718.83  7,609.50  
5,284.41  5,273.97  4,699.41  7,612.12  8,638.33  4,655.57  7,795.07  7,613.69  
5,380.09  5,539.31  4,669.70  5,858.32  6,266.29  7,458.56  8,490.84  4,481.27  6,949.04  7,690.19  
5,384.36  5,407.54  4,588.14  5,878.89  6,261.18  7,450.30  8,187.95  8,736.59  6,896.31  7,757.09  
5,320.16  5,486.73  5,909.65  6,235.35  7,480.41  8,118.08  7,819.13  8,740.43  6,922.57  7,698.77  
5,224.74  5,924.24  6,002.58  6,258.47  7,538.82  4,613.72  8,756.71  6,764.59  
5,134.10  4,582.05  5,887.23  6,019.23  6,350.69  4,616.89  7,751.60  8,782.20  6,785.09  
5,456.15  4,545.62  5,752.01  6,009.32  7,830.85  4,648.02  7,742.17  8,817.90  7,631.28  
5,055.20  5,329.19  4,535.93  5,768.76  7,624.62  7,807.39  4,694.81  7,753.63  7,643.74  
5,040.25  5,221.96  4,549.23  5,759.23  6,431.42  7,598.88  8,014.31  4,694.52  6,633.33  7,677.47  
4,947.89  5,309.10  4,595.67  5,985.94  6,427.84  7,648.35  7,857.08  8,768.72  6,662.64  7,595.46  
4,817.22  5,109.24  5,987.85  6,471.89  7,620.94  7,941.44  7,787.27  8,827.79  6,966.48  7,519.93  
4,811.22  5,835.11  6,001.52  6,417.20  7,652.47  4,535.54  7,856.00  8,860.49  6,966.35  7,540.14  
4,572.77  5,845.51  5,997.67  6,512.63  4,405.86  7,901.50  8,898.87  7,110.73  
5,164.73  4,544.50  5,857.87  6,019.42  8,135.48  4,423.09  7,963.54  8,861.10  7,535.52  
5,372.81  4,484.43  5,853.70  7,646.81  8,167.07  4,413.45  7,972.59  7,636.57  
4,721.36  5,392.43  4,567.37  5,857.21  6,534.77  7,727.59  8,156.39  4,425.08  7,092.58  7,634.19  
4,614.63  5,332.98  4,547.32  5,985.94  6,531.59  7,733.18  8,313.72  8,892.31  7,085.03  7,648.41  
4,797.14  5,398.28  6,000.57  6,524.40  7,732.93  8,396.95  8,057.49  8,870.76  7,056.67  7,699.50  
4,743.94  5,804.89  6,088.49  6,575.53  7,823.72  4,416.16  8,053.83  8,866.35  
4,739.09  4,457.75  5,866.75  6,100.86  6,548.34  4,589.04  8,112.28  8,907.91  7,072.08  
5,551.24  4,452.45  5,890.69  6,139.69  8,506.28  4,591.22  8,188.11  8,972.50 
5.1 Aggregating with OWA operators
In this paper, the weights of the OWA operators can be identified by using Fuller and Majlender’s approach. Fuller and Majlender transform Yager’s OWA equation by using Lagrange multipliers. The main calculation process is as follows:
(1)If , then .
(2)If or , then or , respectively.
(3)If and then
(3) 
(4) 
(5) 
Where characterizes the degree to which the aggregation is like an or operation.
In the Table 2, some data do not exist. For example, to December 1, the data of the year 2001, 2002, 2007 and 2012 do not exist, so the number of data which need to be aggregated is 9. However, to December 5, the number of data which need to be aggregated is 10. Hence, in the OWA operator, the parameter is different. In this case, when , and , the corresponding weights need to be computed which are shown in the Table 3, Table 4 and Table 5.
0.0012  0.0030  0.0071  0.0173  0.0417  0.1006  0.2421  0.5864  
0.1250  0.1250  0.1250  0.1250  0.1250  0.1250  0.1250  0.1250  
0.1917  0.1674  0.1461  0.1275  0.1113  0.0972  0.0848  0.0740  
0.5864  0.2421  0.1006  0.0417  0.0173  0.0071  0.0030  0.0012 
0.0009  0.0020  0.0044  0.0098  0.0220  0.0493  0.1104  0.2473  0.5540  
0.1111  0.1111  0.1111  0.1111  0.1111  0.1111  0.1111  0.1111  0.1111  
0.1726  0.1527  0.1350  0.1195  0.1057  0.0936  0.0828  0.0732  0.0648  
0.5540  0.2473  0.1104  0.0493  0.0220  0.0098  0.0044  0.0020  0.0009 
0.0007  0.0014  0.0029  0.0061  0.0127  0.0268  0.0564  0.1186  0.2495  0.5250  
0.1000  0.1000  0.1000  0.1000  0.1000  0.1000  0.1000  0.1000  0.1000  0.1000  
0.1569  0.1404  0.1256  0.1123  0.1005  0.0899  0.0804  0.0720  0.0644  0.0576  
0.5250  0.2495  0.1186  0.0564  0.0268  0.0127  0.0061  0.0029  0.0014  0.0007 
The thirteen periods of stock prices, will be aggregated. According to the OWA operators, the aggregated value is computed as follows:
(6) 
Where is the largest element of the price. The aggregated results are listed in the Table 6.
Time  

5016.64  6518.31  6890.91  8039.99  
4689.42  6191.60  6598.81  7974.68  
4617.58  6445.85  6934.81  8345.14  
4569.12  6265.32  6728.68  8119.05  
4602.82  6238.66  6678.87  8067.38  
5075.26  6705.77  7135.36  8409.16  
5500.49  7036.57  7410.04  8467.75  
4945.69  6527.31  6920.95  8165.84  
4817.88  6347.35  6757.01  8139.45  
4893.96  6560.95  7022.27  8403.36  
4838.17  6446.57  6887.57  8184.62  
4785.57  6278.36  6677.09  7954.31  
5065.98  6654.83  7060.31  8296.42  
5582.23  6973.13  7318.53  8330.72  
5006.11  6385.48  6755.32  8068.70  
4785.18  6211.50  6621.63  8070.54  
4818.32  6491.94  6935.00  8278.61  
4752.56  6245.29  6639.31  7677.38  
4766.81  6162.06  6545.92  7700.12  
4957.02  6579.87  6986.89  8221.64  
5247.54  6905.01  7273.98  8326.75  
4957.93  6647.60  7051.87  8276.31  
4726.36  6405.69  6847.39  8282.43  
4698.79  6499.93  6978.75  8397.26  
4678.45  6443.43  6871.73  7965.21  
4648.71  6210.90  6625.86  7821.17  
4860.76  6668.51  7116.24  8425.71  
5320.42  7053.47  7439.94  8508.03  
4780.11  6546.61  7000.28  8376.40  
4670.64  6265.38  6724.67  8270.49  
4727.19  6536.52  7037.10  8561.03 
5.2 Aggregating with VGA operator
According to the visibility graph and Eq.2, the weights of the VGA operator can be obtained which are shown in the Table 7. The aggregated results are shown in the Table 8.
0.0714  0.1429  0.1071  0.1071  0.2500  0.0714  0.1071  0.1071  0.0357  
0.1111  0.0556  0.1667  0.1111  0.1389  0.0556  0.1667  0.1667  0.0278  
0.1154  0.0769  0.1538  0.0769  0.2308  0.0769  0.1154  0.1154  0.0385  
0.1471  0.1176  0.0882  0.1471  0.0882  0.2059  0.0588  0.0882  0.0588  
0.1250  0.1250  0.0750  0.1500  0.0500  0.1250  0.1750  0.0500  0.0750  0.0500  
0.1136  0.1364  0.0909  0.1364  0.0909  0.1364  0.1364  0.0682  0.0455  0.0455  
0.1250  0.1250  0.1250  0.1000  0.1250  0.1500  0.0500  0.1000  0.0500  0.0500  
0.1000  0.1333  0.1000  0.1333  0.2333  0.0667  0.1000  0.1000  0.0333  
0.0357  0.1786  0.1429  0.1786  0.0714  0.1786  0.1786  0.0357  
0.1154  0.0769  0.1538  0.0769  0.2308  0.0769  0.1154  0.1154  0.0385  
0.1071  0.1429  0.1071  0.1429  0.2500  0.0714  0.1071  0.0714  
0.1190  0.1429  0.0476  0.1429  0.0952  0.1190  0.1667  0.0476  0.0714  0.0476  
0.1190  0.1429  0.0952  0.1190  0.0952  0.1429  0.1190  0.0714  0.0476  0.0476  
0.1111  0.1389  0.0833  0.1111  0.1389  0.1389  0.0556  0.1111  0.0556  0.0556  
0.0833  0.1667  0.1250  0.1250  0.2500  0.0833  0.1250  0.0417  
0.1111  0.0556  0.1667  0.1111  0.1389  0.0556  0.1667  0.1667  0.0278  
0.1429  0.0714  0.1429  0.1071  0.2143  0.0714  0.1071  0.1071  0.0357  
0.0833  0.1667  0.0833  0.1250  0.1667  0.1250  0.0833  0.1250  0.0417  
0.1053  0.1316  0.0526  0.1316  0.1053  0.1316  0.1053  0.0789  0.0789  0.0789  
0.0588  0.1471  0.0882  0.1176  0.0882  0.2059  0.0588  0.1176  0.0588  0.0588  
0.1176  0.1176  0.1176  0.1176  0.1471  0.0882  0.0588  0.1176  0.0588  0.0588  
0.0625  0.1250  0.0938  0.0938  0.2188  0.0625  0.0938  0.1250  0.0625  0.0625  
0.0417  0.2083  0.0833  0.2083  0.0833  0.1667  0.1667  0.0417  
0.1154  0.0769  0.1538  0.0769  0.2308  0.0769  0.1154  0.1154  0.0385  
0.1818  0.0909  0.1364  0.1364  0.1818  0.0909  0.1364  0.0455  
0.0333  0.1667  0.0667  0.1333  0.1000  0.1333  0.1333  0.0667  0.1000  0.0667  
0.0313  0.1875  0.0938  0.1250  0.0938  0.1563  0.0938  0.0938  0.0625  0.0625  
0.0333  0.1667  0.1000  0.1000  0.1333  0.1333  0.0667  0.1333  0.0667  0.0667  
0.1071  0.1429  0.1071  0.1429  0.2500  0.0714  0.1071  0.0714  
0.0938  0.0625  0.1875  0.0938  0.1563  0.0625  0.1250  0.1875  0.0313  
0.1538  0.0769  0.1538  0.1154  0.2308  0.0769  0.1154  0.0769 
Time  Aggregated Results  Time  Aggregated Results 

6718.52  12.17  6626.59  
6499.89  12.18  6334.17  
6749.91  12.19  6235.76  
6437.80  12.20  6655.86  
6420.31  12.21  6803.89  
6606.90  12.22  6902.51  
6954.10  12.23  6728.45  
6683.75  12.24  6758.54  
6703.36  12.25  6545.96  
6854.48  12.26  6426.12  
6781.71  12.27  6673.25  
6466.89  12.28  7166.44  
6507.55  12.29  6696.84  
6922.77  12.30  6610.13  
6601.03  12.31  6762.35  
6557.66 
5.3 Discussion
The aggregated results derived from the OWA operators and the VGA operator are described in the Figure 9. When set different values of , the different aggregated results will be obtained by OWA operators which means all of these aggregated results are possible. The aggregated results obtained by the VGA operator are neutral that lie in between the aggregated results obtained by the OWA operators when and when , and graphically, the change tendency of the aggregated results obtained by VGA operator is mainly same as OWA operators’ which illustrates VGA operator aggregating data is correct and feasible.
Different from the OWA operators, the VGA operator gets weights without other complicated calculation and the weights obtained by VGA operator are varied. Suppose that there are several time series, if the order of arguments in each time series is different, then the weights will be different. Because the weights are determined according to the degree distribution in the associated graph that means the determination of the weights are relevant with when the argument appears in the histogram. In other words, the VGA operator considers time factor when determining the weights. The VGA operator offers a convenient method to effectively aggregate time series and conserve time information.
6 Conclusion
In this paper, a visibility graph averaging (VGA) aggregation operator is proposed. This aggregation operator converts data into the visibility graph, then decides the weights according to the degree distribution. The experimental result illustrates that the VGA operator is practical and compared with OWA operators, it shows its advantage that it does not need to calculate weights by other complicated methods and it can aggregate time series effectively, hence, we believe that it can be used in some areas such as economics, space science, weather forecast and so forth where it needs to analyze abundant data of time series.
7 Acknowledgments
The work is partially supported by National Natural Science Foundation of China, Grant No. 61174022, Chongqing Natural Science Foundation (for Distinguished Young Scholars), Grant No. CSCT, 2010BA2003, National High Technology Research and Development Program of China (863 Program), Grant No. 2013AA013801.
References
 SoriaFrisch (2001) A. SoriaFrisch, A new paradigm for fuzzy aggregation in multisensorial image processing, in: Computational Intelligence. Theory and Applications, Springer, 2001, pp. 59–67.
 Liu et al. (2013) H. Liu, X. Wang, A. Kadir, Color image encryption using choquet fuzzy integral and hyper chaotic system, OptikInternational Journal for Light and Electron Optics (2013).

Soda and Iannello (2009)
P. Soda, G. Iannello,
Aggregation of classifiers for staining pattern recognition in antinuclear autoantibodies analysis,
Information Technology in Biomedicine, IEEE Transactions on 13 (2009) 322–329.  Russo (1999) F. Russo, Fire operators for image processing, Fuzzy Sets and Systems 103 (1999) 265–275.
 Merigó and GilLafuente (2013) J. M. Merigó, A. M. GilLafuente, Induced 2tuple linguistic generalized aggregation operators and their application in decisionmaking, Information Sciences (2013).
 Merigó and Casanovas (2011a) J. M. Merigó, M. Casanovas, Decisionmaking with distance measures and induced aggregation operators, Computers & Industrial Engineering 60 (2011a) 66–76.
 Merigó and Casanovas (2011b) J. M. Merigó, M. Casanovas, Induced aggregation operators in the euclidean distance and its application in financial decision making, Expert Systems with Applications 38 (2011b) 7603–7608.

Marek and Mayer (1996)
I. Marek, P. Mayer,
Iterative aggregation/disaggregation method for computing stationary probability vectors of markov type operators,
Computers & Mathematics with Applications 31 (1996) 27–40.  Liu et al. (2010) J.W. Liu, C.H. Cheng, Y.H. Chen, T.L. Chen, Owa rough set model for forecasting the revenues growth rate of the electronic industry, Expert Systems with Applications 37 (2010) 610–617.
 Cheng et al. (2013) C.H. Cheng, L.Y. Wei, J.W. Liu, T.L. Chen, Owabased anfis model for taiex forecasting, Economic Modelling 30 (2013) 442–448.
 SmutnáHliněná and Vojtáš (2004) D. SmutnáHliněná, P. Vojtáš, Graded manyvalued resolution with aggregation, Fuzzy sets and systems 143 (2004) 157–168.

Wei et al. (2011)
G. Wei, Y. Ling, B. Guo,
B. Xiao, A. V. Vasilakos,
Predictionbased data aggregation in wireless sensor networks: combining grey model and kalman filter,
Computer Communications 34 (2011) 793–802.  Wang et al. (2011) L. Wang, L. Wang, Y. Pan, Z. Zhang, Y. Yang, Discrete logarithm based additively homomorphic encryption and secure data aggregation, Information Sciences 181 (2011) 3308–3322.
 Sugeno (1974) M. Sugeno, Theory of fuzzy integrals and its applications, Tokyo Institute of Technology, 1974.
 Yager (1988) R. R. Yager, On ordered weighted averaging aggregation operators in multicriteria decisionmaking, Systems, Man and Cybernetics, IEEE Transactions on 18 (1988) 183–190.
 Lacasa et al. (2008) L. Lacasa, B. Luque, F. Ballesteros, J. Luque, J. C. Nuño, From time series to complex networks: The visibility graph, Proceedings of the National Academy of Sciences 105 (2008) 4972–4975.
 Wang et al. (2012) N. Wang, D. Li, Q. Wang, Visibility graph analysis on quarterly macroeconomic series of china based on complex network theory, Physica A: Statistical Mechanics and its Applications (2012).
 Telesca et al. (2013) L. Telesca, M. Lovallo, A. RamirezRojas, L. FloresMarquez, Investigating the time dynamics of seismicity by using the visibility graph approach: Application to seismicity of mexican subduction zone, Physica A: Statistical Mechanics and its Applications (2013).
 Telesca and Lovallo (2012) L. Telesca, M. Lovallo, Analysis of seismic sequences by using the method of visibility graph, EPL (Europhysics Letters) 97 (2012) 50002.
 Fan et al. (2012) C. Fan, J.L. Guo, Y.L. Zha, Fractal analysis on human dynamics of library loans, Physica A: Statistical Mechanics and its Applications (2012).
 Ahmadlou et al. (2010) M. Ahmadlou, H. Adeli, A. Adeli, New diagnostic eeg markers of the alzheimer s disease using visibility graph, Journal of neural transmission 117 (2010) 1099–1109.
 Ahmadlou et al. (2012) M. Ahmadlou, H. Adeli, A. Adeli, Improved visibility graph fractality with application for the diagnosis of autism spectrum disorder, Physica A: Statistical Mechanics and its Applications 391 (2012) 4720–4726.
 Yang et al. (2009) Y. Yang, J. Wang, H. Yang, J. Mang, Visibility graph approach to exchange rate series, Physica A: Statistical Mechanics and its Applications 388 (2009) 4431–4437.
 Gutin et al. (2011) G. Gutin, T. Mansour, S. Severini, A characterization of horizontal visibility graphs and combinatorics on words, Physica A: Statistical Mechanics and its Applications 390 (2011) 2421–2428.
 Liu et al. (2010) C. Liu, W.X. Zhou, W.K. Yuan, Statistical properties of visibility graph of energy dissipation rates in threedimensional fully developed turbulence, Physica A: Statistical Mechanics and its Applications 389 (2010) 2675–2681.

Lacasa et al. (2009)
L. Lacasa, B. Luque,
J. Luque, J. C. Nuno,
The visibility graph: A new method for estimating the hurst exponent of fractional brownian motion,
EPL (Europhysics Letters) 86 (2009) 30001.  Qian et al. (2010) M.C. Qian, Z.Q. Jiang, W.X. Zhou, Universal and nonuniversal allometric scaling behaviors in the visibility graphs of world stock market indices, Journal of Physics A: Mathematical and Theoretical 43 (2010) 335002.
 Donner and Donges (2012) R. V. Donner, J. F. Donges, Visibility graph analysis of geophysical time series: Potentials and possible pitfalls, Acta Geophysica 60 (2012) 589–623.
 Xu (2005) Z. Xu, An overview of methods for determining owa weights, International Journal of Intelligent Systems 20 (2005) 843–865.
 Filev and Yager (1998) D. Filev, R. R. Yager, On the issue of obtaining owa operator weights, Fuzzy Sets and Systems 94 (1998) 157–169.
 Beliakov (2005) G. Beliakov, Learning weights in the generalized owa operators, Fuzzy Optimization and Decision Making 4 (2005) 119–130.
 Mayor and Trillas (1986) G. Mayor, E. Trillas, On the representation of some aggregation functions, in: Proceeding of ISMVL, pp. 111–114.
 Detyniecki (2001) M. Detyniecki, Fundamentals on aggregation operators, This manuscript is based on Detyniecki s doctoral thesis and can be downloaded from (2001).
 Narsingh (2004) D. Narsingh, Graph Theory with Applications to Enginnering with Computer Science, PHI Learning Pvt. Ltd., 2004.
 Pavlopoulos et al. (2011) G. A. Pavlopoulos, M. Secrier, C. N. Moschopoulos, T. G. Soldatos, S. Kossida, J. Aerts, R. Schneider, P. G. Bagos, et al., Using graph theory to analyze biological networks., BioData mining 4 (2011).
 Haggarty et al. (2003) S. J. Haggarty, P. A. Clemons, S. L. Schreiber, Chemical genomic profiling of biological networks using graph theory and combinations of small molecule perturbations, Journal of the American Chemical Society 125 (2003) 10543–10545.
 Nastos and Gao (2013) J. Nastos, Y. Gao, Familial groups in social networks, Social Networks (2013).
 Kim and Phillips (2013) D. Kim, J. D. Phillips, Predicting the structure and mode of vegetation dynamics: An application of graph theory to stateandtransition models, Ecological Modelling 265 (2013) 64–73.
 Agosta et al. (2013) F. Agosta, S. Sala, P. Valsasina, A. Meani, E. Canu, G. Magnani, S. F. Cappa, E. Scola, P. Quatto, M. A. Horsfield, et al., Brain network connectivity assessed using graph theory in frontotemporal dementia, Neurology (2013).
 Liu and Zhan (2012) X. Liu, Q. Zhan, Description of the human hand grasp using graph theory, Medical engineering & physics (2012).
 Barabási and Albert (1999) A.L. Barabási, R. Albert, Emergence of scaling in random networks, science 286 (1999) 509–512.
 Gao et al. (2013) C. Gao, D. Wei, Y. Hu, S. Mahadevan, Y. Deng, A modified evidential methodology of identifying influential nodes in weighted networks, Physica A: Statistical Mechanics and its Applications (2013).
 Wei et al. (2013) D. Wei, X. Deng, X. Zhang, Y. Deng, S. Mahadevan, Identifying influential nodes in weighted networks based on evidence theory, Physica A: Statistical Mechanics and its Applications (2013).
Comments
There are no comments yet.