Discovery of time-inconsecutive co-movement patterns of foreign currencies using an evolutionary biclustering method

Applied Mathematics and Computation 218 (2011) 4353–4364

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Discovery of time-inconsecutive co-movement patterns of foreign currencies using an evolutionary biclustering method Qing-Hua Huang ⇑ School of Electronic and Information Engineering, South China University of Technology, Wushan, Guangzhou, PR China Guangdong Provincial Key Laboratory of Short-Range Detection and Communication, Guangzhou, PR China

a r t i c l e

i n f o

Keywords: Foreign exchange rate Inconsecutive co-movement pattern Biclustering Genetic algorithm Geometric interpretation

a b s t r a c t This paper proposes an evolutionary biclustering algorithm to discover inconsecutive comovement patterns of different foreign exchange rates. The rows/columns of a bicluster (i.e. a submatrix with a subset of rows and a subset of columns) are not necessarily consecutive. A typical bicluster with constant values on rows and/or columns is represented as a hyperplane in a high-dimensional space and the coefficients of the hyperplane are determined using a genetic algorithm. A detected bicluster demonstrates the co-moving behaviors of a subset of currencies in inconsecutive time periods, indicating that the currencies moved in different manners in some specific time periods. In our experiments, we relate these patterns to the geographically close economic connections and find out the correspondence between the nominal exchange rates and the economic conditions. The findings are useful as a guide for investing foreign currencies. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Over the past decades, foreign currency exchange rate determination based on international economics and finance has become increasing important since the collapse of the Bretton Woods regime in 1971 [1–4]. Many researchers have worked on modeling and predicting the movement and future trend of floating exchange rates. It is known conventionally that macroeconomic fundamentals significantly influence the exchange rate behavior over the long-run [5–7]. Therefore, earlier models for analyzing the relationship between exchange rate and international macroeconomics consider the power purchasing parity (PPP) and some fundamental variables such as the interest rate and inflation as the core factors which force the exchange rates to reach a final equilibrium [3,7]. However, those models are difficult to validate strictly because of the large variability of economic variables and financial conditions over a long period. Furthermore, an exchange rate often appears to be disconnected from its fundamentals [5,6] due to deviations from PPP and the non-linear relationship between them [8]. The increased volatility in the short-run has thus made the modeling of floating exchange rates a difficult problem [6]. On the other hand, the nominal exchange rate regimes, as well as economic policies, can also affect the variability of macroeconomic quantities [9], resulting in huge macroeconomic changes. Though it is not easy to correctly model exchange rate economics, much attention has been paid to characterization of the natural relationship between exchange rate changes and domestic as well as international fundamentals in recent years [4,5,10]. Nowadays, more and more regional and international economic cooperation organizations are established to strengthen multi-literal trade and coordinate economic policies. A typical and successful example is the European Union (EU) in which 25 countries share a common market and 13 of the 25 countries share the common currency, the EURO. Improved financial ⇑ Address: School of Electronic and Information Engineering, South China University of Technology, Wushan, Guangzhou, PR China. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.011

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and economic connections among the countries within an economic cooperation organization lead to more and more trades and capital flows, hence the closer links of nominal currency exchange rates among the member countries. Discovering the correlations of different currencies would be beneficial to studying the economic and financial connections between different member countries and provide economists and governments with verification evidences that are important for making financial policies. In addition, the correlations are potentially important for guiding investment in the market of foreign exchange rates. Sopraseuth [9] demonstrated that the floating exchange rates of industrialized countries in a mutual business cycle were associated with high interdependence using a bootstrap test. The similar conclusions can also be drawn by using a conventional clustering technique. However, the economic or political events may often influence the nominal exchange rates in a specific short time period instead of a long time period. Because the specific period associated with a specific event is unknown, statistics based clustering analysis can only be performed with a long time period, hence ignoring important clues for relating short-term fluctuations to the specific economical or political events. Some attentions have been paid to making use of biclustering methods to analyze the co-movement patterns of exchange rates [11,12]. However, there are drawbacks in those previous studies. In [11], the biclustering method called plaid models detected a predefined small number of biclusters, hence often failing in detection of enough significant patterns. In [12], the authors used Hough transform (HT) to perform the search of biclusters. With a high dimensional data, HT is extremely computational expensive, making this technique impractical for real applications. Furthermore, in those analyses, the biclustering methods were actually regarded as alternatives of conventional clustering methods and took no advantage of their capability of clustering the instances and features simultaneously. As a result, the analysis of co-movement patterns that were not time-consecutive was ignored. Accordingly, we propose a new biclustering algorithm using a genetic algorithm [13] to discover time-inconsecutive fluctuation patterns among different foreign exchange rates. The algorithm is developed with respect to a geometrical perspective and can find a typical bicluster by detecting a single hyperplane. Because enumerating the currencies is very timeconsuming, we use a genetic algorithm to search for a subset of currencies for defining the dimensions of the hyperplane. In the preliminary experiments, the results demonstrate good performance of our algorithm in discovering biclusters from both of synthetic and real data sets with an acceptable computation time. This paper is organized as follows. The next section provides a brief state-of-the-art discussion of biclustering algorithms. Our biclustering algorithm is detailed in Section 3. We illustrate the preliminary results and corresponding analysis in Section 4, and finally present conclusions in Section 5.

2. A brief state-of-the-art discussion on biclustering algorithms In conventional clustering methods, a data matrix is partitioned in either the row or the column direction to classify the data into different groups. In biclustering, however, the data are portioned in both row and column directions to discover local subpatterns in the data matrix. The concept of biclustering goes back to the work of Hartigan [14], who called it direct clustering. The term biclustering was first proposed by Cheng and Church [15] in gene microarray data analysis. Because it can overcome the limitation of traditional clustering algorithms in which the rows or columns are clustered globally, biclustering has attracted a lot of attention from researchers in various fields. In the following context, we present the definitions of several types of biclusters. In a data matrix X with Nr rows and Nc columns, a bicluster is defined as a coherent pattern consisting of a subset of rows and a subset of columns. It can be expressed as a pair (R, C) where R # {1 . . . Nr} is a subset of rows and C # {1 . . . Nc} is a subset of columns. The goal of a biclustering algorithm is to extract all biclusters meeting some evaluation standards. In order to evaluate the quality of a bicluster, Cheng and Church [15] proposed a homogeneity constraint, i.e. the mean squared residue score (MSRS), formulated as:

hðR; CÞ ¼ eiC ¼

X 1 ðeij  eiC  eRj þ eRC Þ2 ; jRj  jCj i2R;j2C

1 X eij ; jCj j2C

eGj ¼

1 X eij ; jRj i2R

eRC ¼

X 1 eij ; jRj  jCj i2R;j2C

ð1Þ

where eij denotes the element value at the ith row and jth column in the bicluster, and h(R, C) the value of MSRS for the bicluster. With a homogeneity threshold d defining the maximum allowable dissimilarity within the elements of the bicluster, a valid bicluster can be determined if h(R, C) 6 d. The homogeneity threshold is set by users according to their respective applications. According to Madeira and Oliveira [16], biclusters can be grouped into four major types: 1. 2. 3. 4.

Biclusters Biclusters Biclusters Biclusters

with with with with

constant values, constant values in rows or columns, coherent values, coherent evolutions.

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Fig. 1 illustrates several typical examples for these biclusters. Types 1, 2 and 3 can be identified by direct observation and easily interpreted with the numerical values, as illustrated in Fig. 1a–e. Most attention has been paid to discovering the three types of biclusters [15–19]. The final type is also analyzed to reveal coevolution properties [20,21]. Fig. 1f shows that the columns correspond to the same order, which is one of coevolution modes. For different types of bicluster, the evaluation function should be different. The introduced MSRS has been widely applied for evaluating the first three types of biclusters. In this paper, we focus on the first two types. 3. Methods 3.1. Geometric interpretation of biclusters In this study, we make use of a novel perspective for interpreting biclusters in a data matrix [22]. The biclusters with constant values in rows/columns, additive model, multiplicative model and linear model were addressed in their paper. As illustrated in Fig. 1, each column can be regarded as a variable denoting an axis in the 4-dimensional (4D) space [x, y, z, w] and each row is regarded as a point in the space. Without considering the coevolution biclusters, this geometric perspective can discover biclusters in the first five patterns in Fig. 1 in the following way: (a) the constant bicluster (Fig. 1a) can be expressed as four points at the same location (1.5, 1.5, 1.5, 1.5), (b) the bicluster with constant values in rows (Fig. 1b) can be expressed as four points on the same line x = y = z = w, (c) the bicluster with constant columns (Fig. 1c) is expressed as four points at (1.5, 3.5, 1.0, 4.5), (d) the bicluster with an additive model (Fig. 1d) is defined by the line x = y  1.5 = z  0.5 = w + 0.5, and (e) the bicluster with a multiplicative model (Fig. 1e) is defined by the line x = y/1.5 = z/2 = w  2. For biclusters with more complicated linear models, the points (rows) can be located on the same hyperplane in the high-dimensional space. The five typical bicluster patterns in Fig. 1 can be regarded as the special cases of the linear model and hence we treat each of the biclusters as a number of points lying on a single hyperplane in the space. If we select three columns, i.e. x, y, and z from the examples in Fig. 1, three-dimensional (3D) views for the corresponding hyperplanes of biclusters covering x and z can be seen in Fig. 2. When the column number selected is more than 3, the visualization of the hyperplanes is not feasible but the geometric structures are similar. With high-dimensional space and the points from the data matrix, the task of finding biclusters can be realized by finding hyperplanes on each of which a minimum number of points are lying. In a N-dimensional space, a hyperplane can be defined by a constant c0 and a normal vector [c1, c2, . . . , cN], as follows, N X i¼1

ci  xi þ c0 ¼ 0;

N X

c2i ¼ 1;

ð2Þ

i¼1

where ci(i = 1 . . . N) denotes the weight for the ith selected column. These hyperplanes can be detected using Hough transform (HT) [22]. However, the computation of HT is extremely expensive with a column number of more than 4. Although a fast HT method was employed in [12], the computational speed cannot be acceptable yet. Accordingly, we put forward an alternative method to rapidly determine a hyperplane for biclusters with constant rows/columns or additive coherent values based on a genetic algorithm. 3.2. Identification of biclusters In this study, we address only the biclusters with a constant model, constant rows or columns and an additive model. As these bicluster models may provide enough evidence to discover the connections between exchange rates, the multiplicative and coevolution models are not involved. From the example patterns shown in Fig. 1, it can be concluded that the constant

Fig. 1. Examples of different biclusters, (a) Constant bicluster, (b) Constant rows, (c) Constant columns, (d) Coherent values with an additive model, (e) Coherent values with a multiplicative model, and (f) Coherent evolution values in columns.

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Fig. 2. Different geometries (lines or planes) in a 3D space for general additive bicluster patterns, (a) A bicluster with constant values (x = z) is represented as a line on the grey plane that can represent a bicluster with constant rows, and (b) a bicluster with coherent values with an additive model that is represented by one of the planes parallel to the T-plane.

model and constant rows/columns are actually special cases of the additive model. Therefore, we consider two general additive models in this study. Given two additive biclusters Y1 (m  n) and Y2 (m  n) with added constants on rows and columns, respectively, they can be expressed as follows:

2

y1 þ a1

6y þa 2 6 1 Y1 ¼ 6 4  y1 þ am

y2 þ a1



y2 þ a2







y2 þ am

yn þ a1

3

2

y1 þ a1

6y þa yn þ a2 7 1 7 6 2 7 and Y 2 ¼ 6 4   5 ym þ a1

   yn þ am

y1 þ a2



y2 þ a2







ym þ a2

y1 þ an

3

y2 þ an 7 7 7;  5

ð3Þ

   ym þ an

where a1, a2, . . . , am/n are constant values. Further, an arbitrary yj (j = 2 . . . m/n) can be rewritten as yj = y1 + bj (j = 2 . . . m/n). Therefore, we obtain two different expressions for the two additive models,

2

y1 þ a1

6y þa 2 6 Y1 ¼ 6 1 4  y1 þ am

y1 þ a1 þ b2



y1 þ a2 þ b2







y1 þ am þ b2

y1 þ a1 þ bn

3

2

y1 þ a1

6y þa þb y1 þ a2 þ bn 7 1 2 6 1 7 7 and Y 2 ¼ 6 5 4  

   y1 þ am þ bn

y1 þ a1 þ bm

y1 þ a2



y1 þ an

y1 þ a2 þ b2



 y1 þ a2 þ bm

    y1 þ an þ bm

3

y1 þ an þ b2 7 7 7: 5  ð4Þ

From the above expressions, we note that a bicluster with constant values on columns can be obtained if all the columns subtract from the first column, i.e. Y1,2(i, j) = Y1,2 (i, j)  Y1,2 (i, 1), i = 1 . . . m, j = 1 . . . n. The new expressions of the two additive models become

2

0

b2



bn

3

2

0

a2  a1

6 0 b 6 0 a a    bn 7 2 1 2 7 6 6 Y1 ¼ 6 7; and Y 2 ¼ 6 4   5 4  0 a2  a1 0 b2    bn

   an  a1

3

   an  a1 7 7 7:   5

ð5Þ

   an  a1

Obviously, the biclusters with the general additive models become those with constant values on columns. With respect to the geometric perspective introduced above, the new patterns can be regarded as a number of points at the same location in the space. Therefore, the task of finding hyperplanes can be replaced with that of finding a number of overlapping points in the space, which significantly simplifies the searching procedure. For a target hyperplane that represents a bicluster, the points included in the bicluster should be lying on the hyperplane. Theoretically, there may be infinite hyperplanes corresponding to a specific bicluster with the general additive model as illustrated in Fig. 2. In this study, given n selected columns that denote n dimensions and m rows of a data matrix, according P to Eq. (2), we then let ni¼1 ci ¼ 0 and ci – 0 for i = 1, . . . , n to remove the component of y1 in Eq. (4) and let c0 = 0 to uniquely determine the mathematical form of the hyperplanes for the biclusters with the general additive model. Given each point (row) pi = [p1, p2, . . . , pn], i = 1 . . . m, we calculate the distances of all the points to the defined hyperplane using the following method,

dk ¼

n X

ci  pi þ c0 ;

k ¼ 1 . . . m;

c0 ¼ 0:

ð6Þ

i¼1

For all distances, we use a traditional agglomerative hierarchical clustering method based on the average linkage to find the points that have similar distances, which indicate that these points in a cluster may lie on a hyperplane parallel to the hyperplane defined by ci, i = 0, . . . , n, as illustrated in Fig. 3. With the predefined n columns, the rows (ms) are grouped into one cluster and treated as a preliminary bicluster.

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Fig. 3. Detection of a set of points that lie on a single hyperplane. The points that have similar distances to the defined hyperplane (A) may lie on another hyperplane (B) which is parallel to A. A hierarchical clustering method can find the set of points.

However, some points grouped into the cluster may not truly satisfy the patterns shown in Fig. 1 and Eq. (3). According to Eqs. (4) and (5), we apply Y⁄(i, j) = Y(i, j)  Y(i, 1), i = 1 . . . m, j = 1 . . . n, in the submatrix (ms  n) of the preliminary bicluster. The same hierarchical clustering method is performed again on the rows of the updated submatrix Y⁄. Eventually, the rows that can be grouped into one cluster are constructed as a true bicluster with the selected n columns. Note that two distance threshold parameters (T1 and T2) for the two hierarchical clustering procedures need to be carefully set. They can significantly affect the quality of the detected biclusters, and can also be used for noise tolerance. The algorithm for the identification of biclusters with the general additive model is summarized into Algorithm 1, as follows. Algorithm 1 Step 1. Given a data matrix (m  n) and a number of selected N columns (X), find N coefficients ci that satisfy and ci – 0 for i = 1, . . . , N. Step 2. For j = 1 to m do

d½j ¼

N X

PN

i¼1 ci

¼0

ci  pi ;

i¼1

end For Step 3. Perform the hierarchical clustering algorithm on d[j] for j = 1, . . . , m. Step 4. If no cluster is obtained, stop. Otherwise, each cluster Y is regarded as a preliminary bicluster. Do Y⁄(i, j) = Y(i, j)  Y(i, 1), i = 1 . . . ms, j = 1 . . . ns, in the submatrix (ms  ns) of each preliminary bicluster Y. Then, use the hierarchical clustering algorithm to cluster the rows of Y⁄ to get the final bicluster.

Fig. 4. The procedure to embed a bicluster into a randomly generated data matrix, (a) A generated bicluster (25  8) with constant values on the rows, (b) the bicluster embedded into a data matrix (100  30), and (c) the background of the data matrix randomly generated.

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3.3. Genetic algorithm based framework for finding biclusters Using the geometric perspective, the rows of a bicluster can be detected using the above method. However, the dimensions (columns) are difficult to extract from a large number of columns because the columns associated with a specific bicluster are unknown a priori. Tewfik et al. [19] proposed a biclustering algorithm by enumerating subsets of conditions (columns) of a microarray data matrix. However, their method is of high complexity and not feasible for real applications. Generally, most previous biclustering algorithms make use of greedy heuristics that iteratively refine a number of biclusters. These techniques also exhaustively search local optimal solutions, but the user cannot decide how much computation time to accept for improving the biclustering outcome. In comparison, evolutionary computation techniques are able to find the global or approximate global optimum for problems which usually have NP-hard complexity and need to search an exponentially large solution space. These algorithms maintain and manipulate a population of solutions and perform a ‘‘survival of the fittest’’ strategy for searching better solutions [13]. The computation of simulated evolution can stop when an optimal solution is found or the number of generations has reached a maximum limit. That means that the user can fully control the computation time as well as the quality of biclusters. They have been successfully used for optimizing clustering and classification algorithms [23,24]. Thus, we use a genetic algorithm to search for related columns of biclusters in this study. In our genetic algorithm, the binary encoding is chosen and each individual represents one binary chromosome with the bit length the same as the columns. For each individual in the population, the bits that equal 1 indicate that the corresponding columns (X) are chosen to build up a high-dimensional space. Each column in X corresponds to one axis in the high-dimensional space. Using the method introduced in Section 3.2, a hyperplane passing through the origin can be defined for each individual. To evaluate the performance of discovering biclusters for each individual, the following fitness function is used.

( FðXÞ ¼

Pnb nb i¼1

0

MSRSi =si

if nb > 0

ð7Þ

if nb ¼ 0

where, nb denotes the number of found biclusters for each individual, si is the size of the ith bicluster and MSRSi is the mean square residue score for the ith bicluster in the individual. It is demonstrated that smaller MSRS and larger bicluster numbers and their sizes result in a more fitted individual that has a larger probability to reproduce and survive in the next generation. Under the evolutionary computation strategy, the surviving individuals can produce more fitted offspring that are able to expand in columns or merge several small biclusters discovered by their ancestors into a larger bicluster after enough generations. In this study, the normalized geometric selection method with a selection rate of 0.1 is used as the selection function. The simple crossover method with a crossover rate of 0.6 and binary mutation method with a mutation rate of 0.05 are employed for reproducing the offspring. This genetic algorithm is summarized into several major steps as shown below. Algorithm 2 Step 1. Give an initial pollution P0 with Ng individuals and set the parameters for crossover and mutation functions. Let i = 1. Step 2. Use selection, crossover and mutation functions to improve successive generations as follows,

Pi ¼ selection functionðPi  1Þ; Pi ¼ crossov er functionðPi Þ; Pi ¼ mutation functionðPi Þ: Step 3. Perform the proposed biclustering algorithm (Algorithm 1) to discover biclusters for each individual; record the discovered biclusters; and evaluate individuals in Pi using the fitness function. Step 4. Let i = i + 1. Step 5. Repeat step 2 until the termination conditions are satisfied.

3.4. Experiments with synthetic and real data sets The performance of our algorithm is evaluated using synthetic and real exchange rate data sets. The algorithm is programmed using Matlab and runs on a PC with a microprocessor of 3.0 GHz and a RAM of 1 GB.

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3.4.1. Synthetic data We embed four biclusters with the same patterns as shown in Fig. 1a–d into a data matrix of size 100  30. The four typical biclusters have the same size of 25  8. The values of the biclusters are generated from the uniform distribution U (5, 5). The background of the matrix is also randomly generated from U. Fig. 4 illustrates an example of a bicluster with constant values on rows embedded into a 100  30 data matrix. The two threshold parameters (T1 and T2) are empirically set at 1.0 and 0.1. The population size is set at 40. The simple crossover method with a probability of 50% and a binary mutation method with a probability of 5% are employed in the genetic algorithm. Once the embedded bicluster is detected, the simulated evolution procedure is terminated. We evaluate our algorithm by conducting it for 10 runs and averaging the corresponding generations in which the biclusters are detected. The maximum of generations for each run is set at 10000. 3.4.2. An application to real exchange rate data sets In this study, we choose 15 floating foreign exchange rates from Europe, Asia and Pacific regions, Africa and America to test the proposed biclustering algorithm. They were AUD (Australian Dollar), CAD (Canadian Dollar), CHF (Swiss Franc), DKK (Danish Kroner), EUR (European Euros), GBP (British Pounds), JPY (Japanese Yen), KRW (South Korean Won), NOK (Norwegian Kroner), NZD (New Zealand Dollar), SEK (Swedish Krona), SGD (Singapore Dollar), THB (Thai Baht), TWD (Taiwan Dollar), and ZAR (South African Rand). The historical data series is freely available from http://fx.sauder.ubc.ca/data.html. Because the USD (United States Dollar) is the most important currency in the international financial and economic system, we regard it as the reference currency. The data sets for the 15 currencies are expressed as the nominal exchange rates vis-àvis one USD. Because of the great disconnections from the fundamentals to the nominal exchange rates in the short-run, we use averaged monthly data instead of high frequency data to reveal the connections among different currencies from January 1996 to June 2007. Fig. 5 demonstrates the monthly returns with respect to the nominal exchange rates in January 1996 for the 15 currencies. Thus, we have a data matrix with 138 rows and 15 columns. With reference to [11], the following monthly logarithmic returns are used for each currency in this paper.

Z ij ¼ log

Ei;j ; Ei1;j

i ¼ 1; . . . ; nt ;

j ¼ 1; . . . ; nc ;

ð8Þ

where, Ei,j is the nominal exchange rate of the jth currency at the ith month, nt denotes the number of months, nc the number of currencies and Zij the logarithm of monthly return. Therefore, there are a total of 137 monthly returns and a new matrix is obtained with 137 rows and 15 columns. The two threshold parameters (T1 and T2) are empirically set at 0.05 and 0.015. The population size is set at 30. The simple crossover method with a probability of 60% and binary mutation method with a probability of 5% are employed in the genetic algorithm. The maximum generations are set at 10,000. The algorithm is conducted for five runs and the total number of found biclusters for each run is averaged. 4. Results and discussions Table 1 demonstrates the averaged generations at which the embedded biclusters are found. It can be seen that the evolutionary strategy used in this study successfully detect the target bicluster. The average computation time is also acceptable in comparison with HT based method [22] by which the average computation time is more than 20 h. Therefore, the usefulness of the proposed algorithm can be proved for improving the computation time.

Fig. 5. The monthly return for the 15 foreign exchange rates with respect to January 1996.

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For the real exchange rate data series, an average of 289 biclusters with a standard deviation (SD) of 34 is obtained in this experiment, indicating that the proposed algorithm is capable of discovering more significant connections among different exchange rates during specific periods compared with plaid models [11]. In this section, we present two typical biclusters that reveal important connections between the nominal exchange rates and the fundamentals, and emphasize the comparisons between the previous studies [11,12] and ours. Fig. 6 provides the first bicluster that demonstrates the connections among five European currencies. They are the CHF, EUR, GBP, SEK and DKK. The sequence of the rows included in the bicluster are 1, 22, 24–26, 31, 35, 40, 41, 44, 46, 62, 64, 65, 67, 72, 74, 76, 80, 82, 84, 91, 92, 94, 95, 99, 103, 108–113, 117, 122, 127, 129, 130, 132, and 137. Three continuous periods are

Fig. 6. The first bicluster including five European currencies and 40 inconsecutive months. (a) The logarithm of monthly returns for the five currencies over the 40 month, (b) the monthly return with respect to January 1996 for the five currencies over the total 137 month, (c) the first selected period from the 22nd to the 46th month, (d) the second selected period from the 62nd to the 103rd month, and (e) the third selected period from the 108th to the 137th month.

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Fig. 6 (continued)

Table 1 Parameters set for the biclustering algorithms.

a

Mean (Gen. ) SD (Gen. a) Average time (min) a

Constant model

Constant rows

Constant columns

Additive model

1557 773 53.7

2824 721 91.3

1251 806 41.1

5375 1786 179.5

Gen. denotes the Generation number when the embedded biclusters are found .

selected, i.e. 22–46, 62–103 and 108–137 months to further demonstrate the co-movement of the five currencies, as shown in Fig. 6c–e. This result reveals a truth that the nominal exchange rates for these industrialized European countries often fluctuate in a similar way against the USD. This conclusion is identical to that of [9]. It can be explained by the close relationships among these countries in macroeconomics and international finance. The five countries are geographically close to each other and share similar economic structures and the development modes after World War II. In particular, recent official policies for promoting stronger economic integration of European countries have led to much closer economic and financial relations. Accordingly, these relations result in a very close linkage for the real exchange rates and hence the significant co-movements of nominal exchange rates for the five European countries. Therefore, as illustrated in Fig. 6b, when the EUR depreciates or appreciates against the USD, the CHF, DKK and SEK follow the EUR closely. In the three selected periods (Fig. 6c–e), the GBP also follows a similar trend to the EUR. However, in the 1–21st weeks, it is worth noting that the GBP presents a significantly different movement patterns from its neighbors and its nominal exchange rate against USD looks much weaker. It can be interpreted by a fact that United Kingdom was suffering a long duration of economic recession beginning at early of 1990s. In comparison, previous authors [11,12] neither reported such an important bicluster that illustrates the close connections of European currencies, nor noticed the connections between fundamental factors and various movement patterns.

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This bicluster discovers these connected currencies and their relationships during specific periods, which would provide strong evidence for further analysis in international finance and economics. In addition, the discovered connected currencies

Fig. 7. The second bicluster including three East-Asian currencies and 36 inconsecutive months. (a) The logarithm of monthly returns for the three currencies over the 36 month, (b) the monthly return with respect to January 1996 for the three currencies over the total 137 month, (c) the first selected period from the 1st to the 15th month, (d) the second selected period from the 37th to the 53rd month, and (e) the third selected period from the 75th to the 101st month.

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Fig. 7 (continued)

provide useful information for investors who plan to invest their funds in the foreign exchange rate market. As these European currencies often move towards the same target in a similar behavior, the investors can only focus on one of the currencies, for example the EUR, and distribute their capital into other disconnected currencies or different markets. Furthermore, the circles marked in Fig. 6 show that the SEK has the largest fluctuation among all the five currencies. This indicates that investing in the SEK may carry the largest risk owing to the great volatility of exchange rates, which further shows the usefulness of the proposed biclustering analysis for investment. Fig. 7 presents the second typical bicluster. In this bicluster, three currencies (i.e. the SGD, THB and TWD) are shown to behave in a similar way. The sequence of the rows are 1–3, 6–9, 11–13, 15, 37, 39, 41, 42, 50–53, 75–78, 81, 82, 85, 91, 96, 98–101, 111, 118, 124, and 136. Three continuous periods are selected, i.e. 1–15, 37–53, and 75–101 months, as illustrated in Fig. 7c–e. During the three periods, it can be seen that the three currencies present similar fluctuation behaviors. Similarly, the connections for the three currencies can be interpreted by the close economic relations among the East-Asian countries and districts. Furthermore, as marked by the circles in Fig. 7a, the THB presented much larger volatility indicating the larger instability of the domestic fundamentals of Thailand. It is well known that Thailand was the most affected country by the 1997 East Asian financial crisis, suffering financial collapse and difficult economic problems. The bicluster discovers not only the co-movement of connected currencies in East Asia but also the volatilities of exchange rates, which provide important clues for studying international and domestic economics and their influence on the nominal exchange rate. In Li and Yan’s work [12], the correlations of the SGD, THB and TWD are also found in a typical bicluster. However, they detected the correlation using a simplified data set in which the downward change is denoted as 1, the upward change 1, and no change 0. Therefore, the detected bicluster was a constant row model. Although the similarity of the movement directions could be discovered, one might want to know more about the fluctuation amplitudes and patterns for these currencies. In this study, we use the data set with real differences and present more detailed co-movement patterns for the three currencies, which further validate the usefulness of the proposed biclustering algorithm. 5. Conclusions In this study, we propose a new biclustering algorithm based on an evolutionary computation strategy. This algorithm can detect biclusters with general additive models. In comparison with previous work [11,12], our algorithm is relatively more

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tolerant of noises and can discovery more useful patterns. The preliminary results demonstrate that the biclustering algorithm is able to extract biclusters from the synthetic and real foreign exchange rate data series. For the exchange rate data, each bicluster presents the correlations among several currencies during specific short-term periods. Two typical biclusters are analyzed, illustrating that there are significant connections among several European currencies and East-Asian currencies. The connections can be interpreted well using regional economic relations. In addition, our method can reveal the significance of the periods that are not included in the detected co-movement patterns by relating the fundamental events to the nominal exchange rates. Therefore, our method can detect time-inconsecutive co-movement patterns. Although we do not provide a full picture for deeply relating the fundamentals to the fluctuating patterns of exchange rates, this study offers a useful tool for researchers to further identify more clues and principles in the fields of finance and economics. Finally, the discovered co-movement of exchange rates should be useful for the guidance of investment. Acknowledgments This work is partially supported by the National Natural Science Foundation of China (Nos. 61001181 and 60902035), the Program for New Century Excellent Talents in University, Ministry of Education, China (No. NCET-10-0363), and the Specialized Research Funds for the Doctoral Program of Higher Education of China (No. 20100172120010). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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