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Fourier transform pair correlation forex

fourier transform pair correlation forex

currency pair correlation structure. be used to invert the Fourier transform to obtain the probability density at T or the vanilla price. The pair with the same orientations has negative potential energy - ε while It may also help to understand the Fourier transform technique in terms of a. trading systems adapted to the Forex market can actually be e®ective.4 Taking The idea of using the Fourier transformation in the time-. INDIANA VS OHIO STATE FOOTBALL BETTING LINE

This behavior may indicate a power law decay. Figure 5A shows the behavior of correlation function on a double logarithmic plot. For small r the graph is approximately linear with slope —0. Analyzing such a data, one can easily dismiss the possibility of power law correlations on the basis that their range is so small. Indeed, in a finite system, the correlation length cannot be larger than the radius of the system.

This example demonstrates difficulties associated with correct identification of power law correlations in a finite system. It is illuminating to study also the anti-ferromagnetic Ising model, in which neighboring spins prefer to stay in the opposite direction, or be anti-correlated.

At low temperatures, an anti-ferromagnetic system looks like a checker board. Mathematically, ferromagnetic and anti-ferromagnetic Ising models are identical, so that any configuration of the anti-ferromagnetic model corresponds to exactly one configuration of the ferromagnetic model which can be obtained by flipping all the spins according to a simple deterministic rule. Thus in both models, correlation length has the same finite value at any temperature, except at the critical point at which the correlation length in both models diverges.

Nevertheless, the behaviors of correlation functions are totally different. In the anti-ferromagnetic case, correlation function is negative for all odd r and is positive for all even r Fig. Figure 6 A Correlation function for the two-dimensional anti-ferromagnetic Ising model. B Absolute values of the correlation function below Tc solid line , at Tc dotted line and above Tc, dashed line.

However, if one average odd and even values of the correlation function, this averaged correlation function decays exponentially to zero as expected. This shows that the correlation length is finite and that there is no true long range correlations.

For a totally uncorrelated sequence of codons in which codon frequencies are taken from real codon usage tables i. An alternative way to study the correlations is to compute a power spectrum S f which is the square of the absolute value of the Fourier transform of the function s k.

This technique goes back to X-ray crystallography, in which the intensity of scattered X-rays at certain angle, appears to be a Fourier transform of the density correlation function in the sample under study. Imagine that s k is a record of a melody. Now k is a continuum variable playing the role of time. Then S f tells how much energy is carried by frequency pitch f.

Unfortunately, applications of Fourier transform technique require substantial knowledge in mathematics involving complex numbers and trigonometry. In the following section, we give a brief review of the properties of Fourier transforms. Fourier transforms have many interesting functional properties which make them a useful tool in data analysis.

An important property of the Fourier transform is to turn a convolution of two functions into a product of their Fourier transforms: Due to this property, the power spectrum of a function with zero average is equal to the Fourier transform of its autocorrelation function. In case of long range anti-correlations as in the anti-ferromagnetic Ising model, Fig. Thus in case of anti-correlations, the graph of power spectrum does not look like a straight line on a simple log-log plot. Discrete Fourier Transform In reality, however, we never deal with infinitely long time series.

Usually we have a system of N equidistant measurements. Analogously, vector s can be restored by applying an inverse Fourier transform: If one assumes that the sequence s k is periodic, i. It is natural to define the discrete power spectrum S f to be exactly equal to the Fourier transform of the correlation function. The correlation function can be thus obtained as an inverse discrete Fourier transform of a power spectrum. This makes FFT a standard tool to analyze correlation properties of the time series.

Since the sequences we study are formed by random variables, the power spectra of such sequences are random variables themselves. Before proceeding further, it is important to calculate the power spectrum of a completely uncorrelated sequence of length N. Analogous conclusions can be made for S f. So the power spectrum of an uncorrected sequence has an extremely noisy graph. In the following, we will illustrate the usage of FFT computing power spectrum for a oneand two-dimensional Ising models near critical points.

The smooth bold lines represent exact discrete Fourier transform of the correlation function computed using Eqs. These analytical results give excellent agreement with the numerical data. Indeed, according to Eq. These two methods give consistent results for exponentially decaying correlations, but technically speaking they measure two different properties of the power spectrum.

Smooth lines show analytical result Eq. B Inverse power spectrum of the same data more The figure shows a remarkable straight line indicating long range power law correlations. The discrepancy shows that the power spectrum analysis of a finite system may often give inaccurate values of the correlation exponents.

Detrended Fluctuation Analysis DFA A somewhat more intuitive way to study correlations was proposed in the studies of the fluctuations of environmental records by Hurst in The idea is based on comparison of the behavior of the standard deviation of the record averaged over increasing periods of time with the analogous behavior for an uncorrelated record. According to the law of large numbers, the standard deviation of the averaged uncorrelated time series must decrease as the square root of the number of measurements in the averaging interval.

This method naturally emerges when the goal is to determine an average value of a quantity e. Since the average is equal to the sum divided by the number of measurements, the same analysis can be performed in terms of the sum. In addition to its analytical merits, this method provides a useful graphical description of a time series which otherwise is difficult to see due to high frequency fluctuations.

A variant of Hurst analysis was developed in reference 64 under the name of detrended fluctuation analysis DFA. Since these trends are subtracted in each observation box, this analysis is called detrended. Any deviation from the straight line behavior indicates the presence of correlations or anti-correlations.

A Low frequency Fourier approximation of the detrended landscape yD n for the one-dimensional Ising model. Fourier DFA computes the average square deviation of this approximation from the landscape. B more All these different types of DFA have certain advantages and disadvantages. A visual comparison of Figures 9A and 9B , suggests that these two procedures of subtracting local trends are equivalent.

Thus we can define a Fourier detrended fluctuation as According to Eq. Thus, according to the L-dimensional analogy of the Pythagorean theorem, the square of the vector yD k — yr k is equal to the sum of the squares of its orthogonal components and therefore, The latter sum is nothing but the sum of all the high frequency components of the power spectrum Sy f of the integrated signal.

These two methods are graphically introduced in Figure 9. Li proposed a duplication-mutation model of DNA evolution which predicted long-range power law correlations among nucleotides. So in order to produce a power law decay of correlations, one must assume long-range interactions among nucleotides. In the model of W. Li, such interactions are provided by the fact that the time axes serves as an additional spatial dimension which connects distant segments of DNA developed from a single ancestor.

The model is based on two assumptions both of which are well biologically motivated: Every nucleotide can mutate with certain probability. Every nucleotide can be duplicated or deleted with certain probability. First phenomenon is known as point mutation which can be caused by random chemical reactions such as methylation. If the exchanging segments are of identical length the duplication does not happen. However, if two segments differ in length by n nucleotides, the chromosome that acquires larger segment obtains an extra sequence of length n which is identical to its neighbor, while another chromosome loses this sequence.

In many cases, duplications can be more evolutionary advantageous than deletions. This process leads to creation of large families of genes developed from the same ancestor. Next, we will discuss the implications of deletions. Schematically, this model can be illustrated by Figure For simplicity, we assume only two types of nucleotides X and Y say purine vs. Each level of the tree-like structure represents one step of the evolutionary process during which every nucleotide duplicates, a nucleotide X can mutate with probability pY into Y, and a nucleotide Y can mutate with probability pX into X.

Figure 12 Mutation duplication model of W. Mutations are indicated by dashed lines. The correlations can spread along solid lines. Thus nucleotides more After k duplication steps, this process will lead to a sequence of total 2k nucleotides. The frequencies of nucleotides X and Y in this sequence can be computed using the theory of Markovian processes.

Let us compute the dependence coefficients between two nucleotides which are at distance r from each other in the resulting sequence. The reason of why the correlations are now long-range is obvious. This simple example shows that the exponent of the power law crucially depends on the parameters of the model. In real DNA sequences, the duplication unit is rather a gene or a part of a gene coding for a protein domain.

One can generalize this model assuming that coding sequences X and Y can duplicate, and with some probability jump from place to place effectively mimicking mutations X to Y and Y to X in the above scheme. One can also introduce various point mutation rates for nucleotides in the sequences X and Y.

Alternation of Nucleotide Frequencies Let us assume that a nucleotide sequence consists of two types of patches, 57 in one of which the frequency of nucleotide X is fX1 while in the other it is fX2. Let us assume that the lengths of these patches l are distributed according to the same probability distribution P l. The motivation for this model could be the insertion of transposable elements, 53 , 66 e. The overall frequency of purines Of course, much more complex models with many parameters can be introduced.

These types of models are similar to hidden Markov processes. Let us compute the correlation function for this model. One can see good agreement with Eq. This means that the sequence must be very long so that the long range correlations can be seen on top of random noise. The lower and upper horizontal lines show random noise levels for the long and short sequences, respectively.

B The power spectrum for the long more If the white noise level is subtracted, the long-range correlations become apparent Fig. Indeed the graph of S f — C 0 on a log-log scale is a perfect straight line with slope —0. Similar situation is observed in coding DNA, in which the long range correlations may exist but are weak comparatively to the white noise level. These correlations are limited to the third nucleotide in each codon 72 and can be detected if the white noise level is subtracted.

The average frequency we obtain will be always the frequency of the largest patch. This behavior known as non-stationarity is observed in many natural systems in which different parts are formed under different conditions. Earlier 73 he applied the same type of analysis to the music of different composers from J. No matter how intriguing this observation might seem, the explanation is somewhat trivial. In DNA, these patches may represent different structural elements of 3D chromosome organization, e.

Such hierarchical structure of several length scales may produce effective long-range power law correlations. An interesting model can reproduce some feature of the human genome, namely the abundance of interspersed repeats or retroposons, 68 virus-like sequences that can insert themselves into different places of the chromosomes by reverse transcriptase. An example of such a sequence is LINE-1, which we discussed earlier in this section. In order to keep the length of the chromosome constant, let us delete exactly l nucleotides selected at random after each insertion.

One can easily see Fig. This example shows that the presence of many copies of interspersed repeats some of which have partially degraded can lead to the characteristic peaks at high frequencies larger than the inverse length of the retroposons and strong power-law like correlations at low frequencies comparable with the inverse length of the retroposons. Dotted lines indicate peaks more The reading frame is a non-interrupted sequence of codons each consisting of three nucleotides.

One of the most fundamental discoveries of all time, is the discovery of the universal genetic code, i. In the different codons used for coding the same amino acid, the first letter is usually preserved. Since the amino acid usage is non-uniform, the same is true for the codon usage, particularly for the frequency of the first letter in the codon.

This preference exists for any organism in the entire phylogenetic spectrum and is the basis for the species independence of mutual information. Here c is a random offset which is constant within each patch and can take values 0,1,2 with equal probabilities. Following Herzel and Grosse, 43 we will call this construction a random exon model. All the correlation properties of the random exon model can be computed analytically. One can see approximate straight line behavior with the slope 0.

B The log-log plot of the absolute value of the correlation function for the same sequence. The examples of the previous sections show us that among different methods of analysis, the power spectrum usually gives the best results. Thus, the power spectrum restores useful information which cannot be seen from C r quickly sinking below the white noise level for large r. On the other hand, the power spectrum does not smooth out the details on the short length scales corresponding to high frequencies as DFA does.

Also it is much less computationally intensive than the two other methods. Once the intuition on how to use the power spectrum analysis is developed, it can be applied to DNA sequences with the same success as in X-ray crystallography, especially, today when the length of the available DNA sequences becomes comparable with the number of atoms in the nano-scale experimental systems.

Not surprisingly, power spectra of the DNA from different organisms have distinct characteristic peaks, 81 similarly to the X-ray diffraction patterns of different substances. Accordingly, in this section, we will use only the power spectrum analysis. The problem was not only to determine genes, i. Only the information from exons is translated into proteins by the ribosomes. Later these results were confirmed by the DFA method, the wavelet, 55 , 72 , 82 the power spectrum 80 and modified standard deviation analyses.

In Figure 17 we present the results 80 of the analysis of coding and noncoding sequences of the eukaryotic organisms. These were all the genomic DNA sequences published in the GenBank release of August 15th, whose length was at least nucleotides. The conclusions hold not only for the average power spectrum of all eukaryotes but also for the average power spectra of each organism analyzed separately. Figure 17 The RY power spectrum obtained by averaging power spectra of all eukaryotic sequences longer than bp, obtained by FFT with window size Upper curve is average over 29, coding sequences; lower curve is average over 33, noncoding sequences.

The presence of the weak peak in the noncoding regions can be attributed to the nonidentified genes or to pseudo-genes which have recently on the evolutionary time scale become inactive like olfactory genes for humans. Presently, when several complete or almost complete genomes are just a mouse-click away, it is easy to test if the true power-law long-range correlations do exist in the chromosomes of different species. A very interesting feature of the human genome is the presence of the strong peaks at high frequencies.

It is plausible that these peaks are due to the hundreds of thousands almost identical copies of the SINE and LINE repeats, 87 which constitute a major portion of human genome. The absence of these peaks in the genomes of primitive organisms see Fig. It is clear that the long-range correlations lack universality, i.

The slopes of the power spectra change with frequency and undergo sharp crossovers which do not coincide for different organisms. The middle frequency regime which can be particularly well approximated by power law correlations in C. Li in which duplications and mutations occur on the level genes, consisting of several hundred base pairs.

The high frequency correlations, sometimes characterized by small positive slopes of the power spectra can be attributed to the presence of simple sequence repeats see next section. In contrast, the high frequency spectrum of the bacterium E. Bacterial DNA practically does not have noncoding regions, thus in agreement with refs. The spectrum of E. Distribution of Simple Repeats The origin, evolution, and biological role of tandem repeats in DNA, also known as microsatellites or simple sequence repeats SSR , are presently one of the most intriguing puzzles of molecular biology.

The expansion of such SSR has recently become of great interest due to their role in genome organization and evolutionary processes. SSR are of considerable practical and theoretical interest due to their high polymorphism. Among such diseases are fragile X syndrome, myotonic dystrophy, and Huntington's disease 94 , SSR of the type CA l are also known to expand due to slippage in the replication process. These errors are usually eliminated by the mismatch-repair enzyme MSH2.

However, a mutation in the MSH2 gene leads to an uncontrolled expansion of repeats—a common cause of ovarian cancers. Dimeric tandem repeats are so abundant in noncoding DNA that their presence can even be observed by global statistical methods such as the power spectrum. The abundance of dimeric tandem repeats in noncoding DNA suggests that these repeats may play a special role in the organization and evolution of noncoding DNA.

First, let us compute the number of repeats in an uncorrelated sequence. Suppose that we have a random uncorrelated sequence of length 2L which is a mixture of all 16 possible types of dimers XY, each with a given frequency fXY, The probability that a randomly selected dimer belongs to a dimeric tandem repeat XY l of length l can be written as where 1 — fXY is the terminating factor responsible for not producing an additional unit XY at the beginning or end of the repeating sequences and the factor l takes into account l possible positions of a dimer XY in a repeat XY l.

Finally, which decreases exponentially with the length of the tandem repeat. Thus, a semi-logarithmic plot of NXY l versus l must be a straight line with the slope In order to compare the prediction of this simple model with real DNA data, we estimate fXY for the real DNA as follows: i divide the DNA sequence into L non-overlapping dimers, ii count nXY, the total number of occurrences of a dimer XY in this sequence, and calculate Indeed, most dimeric tandem repeats in coding DNA produce linear semilogarithmic plots, Fig.

Let us assume that a trader holds initially euros EUR. That is we could end up with more currency EUR than we had initially. In practice, however, this is difficult on real markets and in fact after the first leg of such multiple transactions, remaining trades would not be possible to complete or the price will be changed by the time they will be completed. Again its value compared to one would indicate a theoretical possibility of executing arbitrage opportunity. Multifractal statistical methodology Let us consider multiple time series of exchange rates recorded simultaneously.

Both time series are synchronized in time and have the same number N of data points. Following the idea of a new cross-correlation coefficient defined in terms of detrended fluctuation analysis DFA and detrended cross-correlation analysis DCCA —[ 32 ], which has been put forward in [ 33 ], we use in the present study a multifractal detrended cross-correlation analysis MFCCA [ 34 ] with q-dependent cross-correlation coefficient.

The cross-correlations of stock markets have been also investigated with a time-delay variant of DCCA method [ 35 ]. Multiscale multifractal detrended cross-correlation analysis MSMF—DXA has been proposed and subsequently employed to study dynamics of interactions in the stock market [ 36 ].

Other methods, including weighted multifractal analysis of financial time series [ 37 ] and multiscale properties of time series based on the segmentation [ 38 ], allow for multifractal and multiscale nonlinear effects investigations. The approach adopted in the present study has been introduced by [ 41 ].

In what follows, we briefly state main points of this approach. Detrended cross-correlation methodology We define a new time series X k of partial sums of the original time series elements x i. In an analogous way, the second time series of interest, Y k , is obtained out of the original time series y i. Depending on the nature of the signal, we may expect in time series trends and seasonal periodicities.

For a given timescale s, we may repeat the partitioning procedure from the other end of the time series, thus obtaining in total 2M s time intervals, each of which will contain s data points. In general, it could be a polynomial of any finite degree, depending on the nature of the signal. The definition of the family of fluctuation functions given by Eq.

In such a case, we obtain, as it should be, the q-th order fluctuation function, which follows from the multifractal detrended fluctuation analysis MFDFA for a single time series [ 42 ]. Detrended cross-correlation q-coefficient The family of fluctuation functions of order q, defined by Eq. The definition of the cross-correlation given by Eq. The parameter q helps to identify the range of detrended fluctuation amplitudes corresponding to the most significant correlations for these two time series [ 41 ].

Hence, by choosing a range of values in q we may filter out correlation coefficient for either small or large fluctuations. Analysis and results Based on the methodology described, we will investigate multiscale properties for cross-correlations among time series corresponding to currency exchange rates for the whole period — as well as for some sub-periods.

We also apply standard methods of MATLAB source codes validation and surrogate data checking against artefacts or robustness of nonlinear correlations within our data sets [ 41 ]. Although in our study we focus on different signatures and statistical properties of multivariate time series with respect to triangular arbitrage, one may envisage a broader picture of such analysis, whereby one would like to uncover a specific kind of cross-correlations in these time series which would help us to detect underlying interconnections useful for the system behavior prediction in future.

Logarithmic returns and the inverse cubic law In this subsection, we will present a general picture for the financial time series dynamics with an emphasis on events which have an impact on the Forex market. For a reference, some important global political and economic events have been indicated over the timescale Full size image The currency index given by Eq. With such a single averaged characteristics, one may have a general overview of a global temporal behavior and performance of any currency in the Forex market.

In Fig. For this plot, we take logarithmic returns arising from average bid and ask exchange rates. In the figure, we have indicated some political or economic events on the timescale with dotted vertical line which in principle could have impact on the Forex market performance during that period of time. These labels may serve as an intuitive explanation of features observed on the curves related to the currency indexes. In general, one observes significant variations of considered currency indexes over the period of 8 years.

In the following, we will explore in some more details statistical properties of the Forex market data in the vicinity of these events. We will discuss to what extent our proposed statistical analysis corroborates these features, when looking from the hindsight with the help of historical data from the Forex market.

Each solid line of different color demonstrates the tail behavior for the corresponding currency. The inset shows these distributions when a short period of an extreme volatility in currency exchange rates has been removed. This removed period a half an hour corresponds to the wake of the SNB intervention on January 15, Full size image We note that all currency exchange rates with the Swiss franc CHF as base currency yield higher probability of larger absolute logarithmic returns than other exchange rates.

These outliers could be attributed to two instances of the SNB interventions in and In order to demonstrate the origin of these deviations, we remove from our data sets a period of a half an hour in the morning on January 15, , when a significant volatility of currencies exchange rates has been observed in the wake of the SNB intervention [ 9 ]. The resultant tails of the probability distributions are shown in the inset of Fig.

Hence without this single, short-termed event on the market, the tails of the distributions approximately follow the inverse-cubic behavior. Multifractality and scaling behavior of cross-correlation fluctuation functions We already know that the outliers of the cumulative distributions document an increased level of larger fluctuations in absolute log-returns. This means also a higher chance to encounter fluctuations yielding larger returns. The question arises to what extent these fluctuations are cross-correlated among exchange rates.

Such cross-correlations at least between two exchange rate time series would offer a potential opportunity of triangular arbitrage. The graph illustrates the cross-correlation fluctuation functions scaling over the range of timescales s from 5 min up to 2 weeks for different q-coefficients. We verify indeed that the scaling according to Eq. Additionally, for the reference both insets in Fig.

Two examples of the cross-correlation between two series of returns for exchange rates are shown in Fig. This seems to be expected as in the former case there is a common base currency JPY. In such a way, a pair of returns is intrinsically correlated by JPY currency performance due to the triangular constraint in the exchange rates.

This is an example of cross-correlations among 3 currencies. In this case, the cross-correlations are in the triangular relation the top panel of Fig. In the case shown in the bottom panel of Fig. The magnitude of this cross-correlation measure is weakly dependent on the timescale and only slightly grows with time. Its growth is more pronounced for larger fluctuations cf.

Fluctuations of different magnitude and cross-correlations Let us investigate in some more detail the cross-correlations between relatively small and large fluctuations of two exchange rate return series. Note that the cross-correlation pairs are grouped into two classes. One class of exchange rate pairs in black, left top and bottom panels which pertain to the triangular relation and the second class, where cross-correlated pairs are outside the triangular relation in red, right top and bottom panels.

This gives an idea about the range of obtained values of cross-correlation coefficient distributions for the currency pairs which are in or out the triangular relation. The black dotted horizontal line on the top-left panel shows the average cross-correlation of different pairs pertaining to the triangular relation.

The value of that overall average is about 0. It indicates a possibility of observing stronger correlations in exchange rates among four currencies in comparison with what we would expect on average in the case of exchange rates linked with the triangular relations. This somewhat unexpected result could be ascribed to mechanisms coupling economies of these two countries. Hence, from our study it follows that indeed some cross-correlations of the pairs, which are not linked by a common currency and are traded on the Forex market, may reach that overall average cross-correlation of exchange rates with a common base.

This seems to be a surprising conclusion, since typically we would expect stronger correlations between explicitly correlated two series by means of a common, base currency rather than in a case where there is no such common base. However, we have to appreciate the fact that cross-correlations between any pair of exchange rates will have some impact on the cross-correlations of other pairs through mutual connections arising from different combinations of currencies being exchanged.

Nonetheless, the small fluctuations in logarithmic returns would be difficult to use in viable trading strategies, mainly due to finite spreads in bid and ask rates. From the practical point of view, correlations of large fluctuations seem to be more promising in finding and exploiting arbitrage opportunities. The order of currency pairs in the bottom panels is kept the same as in the top panels. For the case of the larger fluctuations, the level of overall average of cross-correlations is marked again with horizontal black dotted line at a value of 0.

The cross-correlations of the large fluctuations are therefore approximately two times smaller than in the case of small correlations. In the case of cross-correlations outside the triangular relations, the strong cross-correlations arise when we take AUD as base currency on the one side and NZD on the other. This is in agreement with previous findings [ 25 , 41 , 54 ] for other financial instruments like stocks and commodities.

The time evolution of averaged cross-correlations over currency pairs with the common base for large fluctuations will be discussed later cf. Dendrograms—agglomerative hierarchical trees from cross-correlations of exchange rates As we have already seen, studying quantitative levels of cross-correlations may uncover some less obvious connections among currencies than just the explicit link through a common base currency. In order to uncover a hierarchy of currencies in terms of logarithmic returns from exchange rates, one may consider cross-correlation coefficients as a measure of the distance d i, j between different exchange rate pairs.

We define the distance d i, j similarly to the definition introduced by [ 55 ], but instead of correlation coefficient, we use q-dependent cross-correlation coefficient [ 41 , 56 ]. As a result of adopting the distance given by Eq. An interesting observation follows that Australlian AUD and New Zealand NZD dollars are strongly correlated—they appear together in the same clusters of exchange rates for both the small and large fluctuations.

This indicates a possibility of building strong cross-correlations between exchange rate pairs which do not have the same common base. Such findings are important when designing the trading strategies, both for optimizing portfolio and for its hedging. We would like to stress the fact that our method is not limited only to time series from the Forex and it may well be applied to the signals in a form of time series arising in other fields of research and applications.

Let us now investigate the cross-correlations in the time domain. The results are shown in Fig. For each q value, we consider 4 values of s corresponding to 5 min, 1 h, 24 h and 1 week Full size image We also show the overall average for the currency exchange rate pairs complying to the triangular relation the black dotted line and for currency exchange rate pairs which are not bounded by the triangular relation red dotted line.

The most striking feature when comparing the small and large fluctuation cross-correlations over different timescales is that in the former case little is happening over different timescales considered. The plots indicate nearly static cross-correlations, almost independent on the timescale for the small fluctuations. Specifically, the overall average denoted by the black dotted horizontal line grows from a value which is less than 0. The growth of the overall average of cross-correlation is even more convincing for the class of pairs of currency exchange rates which are not in a strict triangular relation.

What is more, for the shortest timescale shown here, the difference between cross-correlations for pairs that are in the triangular relations and those that are not, is the biggest.

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Fourier Analysis Technique

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