Method for generating two coupled Gaussian stochastic processes

Method for generating two coupled Gaussian stochastic processes Tayeb Jamali∗ and G. R. Jafari Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran (Dated: February 16, 2016)

arXiv:1602.04697v1 [math.PR] 25 Jan 2016

Most processes in nature are coupled; however, extensive null models for generating such processes still lacks. We present a new method to generate two coupled Gaussian stochastic processes with arbitrary correlation functions. This method is developed by modifying the Fourier filtering method. The robustness of this method is proved by generating two coupled fractional Brownian motions and extending its range of application to Gaussian random fields. PACS numbers: 05.40.-a, 02.50.Ey, 05.45.Tp

I.

INTRODUCTION

Evolution of real systems mainly contains correlated noises in which some valuable information may be hidden. To study such noises numerically, a number of methods have been developed such as the matrix decomposition methods [1, 2], the Fourier filtering method (FFM) [3–6], and the circulant embedding method [7–9]. All these methods are capable of generating realizations of only one stochastic process. Therefore, when facing multiple processes simultaneously, these methods can only be used to the individual study of each process. But, since the evolution of a process in reality depends more or less on the evolution of other processes, an individual study of a process would lead to vague results. In order to measure the coupling between two or more processes, some various techniques have been introduced, e.g. the detrended cross-correlation analysis (DXA) [10–12], partial-DXA [13], coupled-DXA [14], the cross-wavelet analysis [15, 16], the random matrix theory [17–19], the coupled level-crossing method [20], the cross-visibility algorithm [21] and etc. There also exist special methods for generating two coupled processes but only for power-law correlations [22, 23]. However, there is still lack of a general method. In this paper, we aim to present a method capable of generating two Gaussian coupled stochastic processes with any arbitrary correlations. To do this, we modify the Fourier filtering method. The established FFM can be thought of as a machine which takes an uncorrelated sequence of random numbers and returns a correlated sequence of random numbers. We modify this machinery so that it takes two uncorrelated sequences of random numbers and returns two correlated ones. To be more precise, linear combinations of two Gaussian white noises in the Fourier space is written, such that when returning to the real space, our desired correlations would be injected in the processes. Then, the method is implemented for modeling two coupled Brownian

∗

Email: [email protected]

motions, and two coupled fractional Brownian motions (FBM). We also show how to extend this method to generate two coupled Gaussian random fields.

II.

METHOD

Consider two independent sequences of L uncorrelated L random numbers {ui }L i=1 and {vi }i=1 , with Gaussian distributions. Their correlation functions are hui ui+n i ∼ δn,0 and hvi vi+n i ∼ δn,0 , where δn,0 is the Kronecker delta, and h. . . i denotes an ensemble average. The aim of this study is to provide an algorithm by which one could construct two stationary sequences {xi } and {yi } with desired autocorrelations Cxx (n) = hxi xi+n i Cyy (n) = hyi yi+n i,

(1a) (1b)

and the cross-correlation Cxy (n) = hxi yi+n i,

(1c)

starting from the two sequences {ui } and {vi }. In this line, we take the advantage of Fourier filtering method for generating two coupled processes, {xi } and {yi }. Working in the Fourier space brings the need to deal with spectral densities instead of correlation functions. Here, the corresponding autospectral densities are Sxx (q) = hxq x−q i Syy (q) = hyq y−q i,

(2a) (2b)

and a cross-spectral density is Sxy (q) = hxq y−q i,

(2c)

where {xq } and {yq } are Fourier transforms of {xi } and {yi } respectively. The autospectral densities Sxx (q) and Syy (q) are real-valued even functions while the cross-spectral density Sxy (q) is a complex-valued function [24]. It is noteworthy to recall that for a stationary random process, due to the Wiener-Khintchine theorem, the spectral density is the Fourier transform of the correlation function [25].

2 In order to construct two stationary sequences {xi } and {yi } with the desired correlations we suggest combining two independent and uncorrelated sequences {ui } and {vi } in the Fourier space as xq = Aq uq + Bq vq yq = Cq uq + Dq vq ,

(3)

where {uq } and {vq } are the Fourier transforms of {ui } and {vi } respectively. The coefficients Aq , Bq , Cq , and Dq are functions in the Fourier space which are of essential importance for specifying xq and yq . The coefficients Aq , Bq , Cq , and Dq are well named as being of essential importance because having these coefficients would lead us to obtain two sequences {xi } and {yi } possessing the desired correlations. In this regard, after substituting xq and yq from Eq. (3) into the expressions for spectral densities (Eqs. (2a)-(2c)), the relation of these coefficients with the desired spectral densities is obtained Sxx (q) = Aq A−q + Bq B−q Syy (q) = Cq C−q + Dq D−q Sxy (q) = Aq C−q + Bq D−q .

(4a) (4b) (4c)

where αq and βq should satisfy αq −

B−q = Bq∗

C−q = Cq∗ ,

D−q = Dq∗ .

2

2

Syy (q) = |Cq | + |Dq | Sxy (q) = Aq Cq∗ + Bq Dq∗ .

III.

p Sxx (q) cos αq , q Cq = Syy (q) cos βq ,

p Sxx (q) sin αq q Dq = Syy (q) sin βq , Bq =

,

(8)

ALGORITHM

We propose a numerical algorithm using the Eqs. (7) and (8) in order to generate two stationary sequences with the desired correlations. The algorithm is as follows: a) Generate two independent sequences of uncorrelated random numbers {ui } and {vi } with a Gaussian distribution, then calculate their Fourier transform coefficients {uq } and {vq }. b) Calculate the fourier transforms Sxx , Syy and Sxy of the desired correlation functions Cxx , Cyy and Cxy . c) Obtain the coefficients Aq , Bq , Cq and Dq using Eq. (7) & Eq. (8) and substitute in Eq. (3) to get {xq } and {yq }. d) Calculate the inverse Fourier transform of {xq } and {yq } in order to obtain two sequences {xi } and {yi } with the desired correlations.

IV. A.

(6a) (6b) (6c)

Before solving this system of equations it is instructive to interpret it algebraically. Equations. (6a) and (6b) are nothing but the square of Euclidean length of two complex vectors Φ ≡ (Aq , Bq ) and Ψ ≡ (Cq , Dq ). Equation. (6c) also represents the scalar product hΨ, Φi. Hence, Eqs. (6a)-(6c) talk about the length of two complex vectors and the angle between them [26, 27]. This algebraic interpretation leads us to the following answer to the system of equations (6a)-(6c) which also satisfies the condition of Eq. (5) Aq =

!

APPLICATIONS

Two coupled Brownian motions

(5)

This exactly resembles the characteristics of the spectral densities [24]. This assumption does not result in any loss of information, because in the end, we find a class of coefficients by which two stationary sequences with desired correlations can be generated. The Eq. (5) leads to Sxx (q) = |Aq |2 + |Bq |2

= arccos

Sxy (q) p Sxx (q)Syy (q)

and βq∗ is the complex conjugate of βq .

To derive these relations we use the fact that the spectral density of a white noise is constant [24]. From all possible answers for the coefficients we only take into account the case A−q = A∗q ,

βq∗

(7)

The method proposed in the present study provides two Gaussian time series with any sort of autocorrelation and cross-correlation. For instance, this method enables the construction of two coupled time series without any autocorrelations. In other words, it allows us to produce two Gaussian white noises {xi } and {yi } which are coupled. If we consider {xi } and {yi } as the steps of two Pt Pt random walks, then, X(t) = i=1 xi and Y (t) = i=1 yi would represent the positions of two coupled Brownian motions at time t. In Fig. 1, two coupled Brownian motions are shown for three different kind of couplings. The top panel of Fig. 1 refers to a Gaussian coupling, the middle panel corresponds to an exponential coupling, and the bottom panel shows a damped harmonic coupling. Note that in generating the two series, {xi } and {yi }, the sequences {ui } and {vi } have been considered initially equal for all three panels because the coupling shows its effects in the pattern of two series X(t) and Y (t) in the right panels.

B.

Two coupled fractional Brownian motions

Although there are many techniques developed to generate a time series with a power-law autocorrelation [6, 28–30], but in reality we sometimes

3

0.1 0 0

5

10

-50 -100

15

0

2000

n

t

Cyy(n)

102

10-2 10-4 100

104

102

104

n

0

5

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-50 -100

15

0

2000

n

4000

t 50

0.3

X(t) , Y(t)

0.2 0.1 0 0

5

10

15

10-2 Cxx(n)

10-4 100

Cyy(n)

102

B

100 10-2 10-4 100

104

n

102

104

n

0 -50 -100

0

2000

n

4000

t

Figure 1. Left panels show the coupling between two white noises {xi } and {yi } of the length L = 210 and the right panels show coupled Brownian motions Ptthe realizations of twoP t X(t) = Top panels are i=1 xi and Y (t) = i=1 yi . related to Gaussian coupling, middle panels is for exponential coupling, and bottom panels represent damped harmonic coupling. Note that the circles show couplings obtained numerically from our algorithm which very well coincided with the expected couplings shown by the solid lines.

face a couple of power-law autocorrelated time series which posses a power-law cross-correlation e.g. fluctuations of stock prices [31]. Podobnik et. al. modeled two power-law autocorrelated time series with long-range cross-correlations [22, 23]. In this section we aim to generate such power-law correlated time series by our algorithm where we use the special form of correlation function proposed by Makse et. al. [6] C(l) = (1 + l2 )−γ/2 ,

(9)

where 0 < γ < 1 is the correlation exponent. The function C(l) which is well defined at l = 0 shows the desired power-law behavior for large l. This special form for power-law correlation function has two advantages. First, it enables generating a power-law correlation throughout the system. Second, its Fourier transform, spectral density function, has an analytic form stated by S(q) =

B

100

Cxy(n)

0

2π 1/2  q β Kβ (q), Γ(β + 1) 2

(10)

where Kβ (q) is the modified Bessel function of order β = (γ − 1)/2, and Γ is the gamma function, for more details see ref [6]. Here, we use Eq. (10) for spectral densities Sxx , Syy , and Sxy with exponents γxx , γyy , and γxy

C

100

10

-2 Cxx(n)

10

10

10

-2

10

-4

Cyy(n)

-4 0

10

n

2

C

100

Cxy(n)

0.1 0

Cxy(n)

Cxx(n)

n

Autocorrelations

X(t) , Y(t)

Cxy(n)

0.2

-0.1

10-2 10-4 100

A

100

50

0.3

-0.1

4000

Autocorrelations

-0.1

0

A

100

Cxy(n)

X(t) , Y(t)

Cxy(n)

0.2

Autocorrelations

50

0.3

10

4

10

0

10

2

10

4

n

Figure 2. The log-log plots of the average correlations Cxx (n), Cyy (n), and Cxy (n) for three different cases labeled by ’A’, ’B’, and ’C’. Each case is the average of 100 power-law correlated samples for L = 221 . For cases ’A’, ’B’, and ’C’ the desired exponents (γxx , γyy , γxy ) are respectively equal to (0.7, 0.8, 0.6), (0.6, 0.8, 0.7), and (0.6, 0.7, 0.8). The dashed lines represent the best fits for the exponents (γxx , γyy , γxy ) which yeald values of (0.70 ± 0.01, 0.80 ± 0.01, 0.60 ± 0.03), (0.61 ± 0.01, 0.80 ± 0.02, 0.71 ± 0.02) and (0.60 ± 0.01, 0.70 ± 0.01, 0.79 ± 0.05) for cases ’A’, ’B’, and ’C’ respectively.

respectively. As an example, we applied our algorithm to generate two power-law autocorrelated series with a power-law cross-correlation for three different classes of exponents γxx , γyy , and γxy . In each row of Fig. 2 the autocorrelations Cxx & Cyy and a cross-correlation Cxy is illustrated for 100 samples of the length 221 . Note that the values for γxx , γyy , and γxy are obtained according to the best fit for every curve in the panels of Fig. 2. Similarly to section IV A, we can consider the series {xi } and {yi } which are power law correlated as the Pt steps of two random walks. X(t) = i=1 xi and Pt Y (t) = i=1 yi represent the positions of two fractional Brownian motions at time t with a power-law coupling. V.

TWO COUPLED RANDOM FIELDS

Since, sometimes, we may need to generate two coupled random fields [32], generalizing this method to random fields would be very useful. Consider two discrete d-dimensional Gaussian random fields namely, {xi } and

4 hx

hy

200

400

100

200

0 0

-100

-200

-200

140

240

Cyy(n)

γxx=1.32±0.04 γyy=1.53±0.04

-6

102

103

Cxx(n) Cyy(n)

10-2

10-4

γxx=1.13±0.04 γyy=1.55±0.06

10

-6

100

101

n

102

103

10-4

10

-6

100

101

2

10

3

103

Cxy(n)

10-2

10

-4

γxy=1.35±0.07 10

102

C

100

-4

γxy=1.09±0.05 1

γxx=1.14±0.04 γyy=1.32±0.03

n

Cxy(n)

Cxy(n)

10

10

Cyy(n)

10-2

10-2

-4

10-6 0 10

Cxx(n)

B

100

10-2

10

C

n

A

100

240

100

Autocorrelation functions

Cxx(n)

101

B

100

Autocorrelation functions

Autocorrelation functions

140

A

10-2

220

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10-6 0 10

10

γxy=1.51±0.1

1

n

10

n

2

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10-6 0 10

10

1

10

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10

3

n

Figure 3. The top row represents our numerical results obtained for two coupled self-affine surfaces hx and hy of size 256 × 256. The power-law exponents of the underlying fields {xi } & {yi } are γxx = 0.7, γyy = 1.5, and γxy = 1. The middle and the bottom panels are log-log plots of average correlations Cxx , Cyy , and Cxy for three different cases which is labeled by ’A’, ’B’, and ’C’. Each case is the average of 100 samples in a square lattice of 212 × 212 . The middle panels show the log-log autocorrelation functions for different desired values of (γxx , γyy ) respectively from left to right equal to (1.3, 1.5), (1.1, 1.5), and (1.1, 1.3). The bottom panels show the cross-correlation function for different desired values of γxy respectively from left to right equal to 1.1, 1.3, and 1.5. The best fit for each exponent is labeled in each panel.

{yi } which are defined over a d-dimensional cube of volume Ld with desired correlations Cxx (n), Cyy (n), and Cxy (n) where i ≡ (i1 , i2 , . . . , id ) and n ≡ (n1 , n2 , . . . , nd ). By the definition, the autospectral densities are Sxx (q) = hxq x−q i Syy (q) = hyq y−q i,

(11a) (11b)

and cross-spectral density is Sxy (q) = hxq y−q i,

(11c)

where {xq } and {yq } are the Fourier transforms of {xi } and {yi } respectively [33]. Assuming that the fields are homogeneous and isotropic, the correlations Cxx (n), Cyy (n) & Cxy (n) only depend on n = |n| and the spectral densities Sxx (q), Syy (q) & Sxy (q) which are the Fourier transforms of corresponding correlation functions only depend on q = |q|. This property, the dependency of spectral densities to only one parameter (q), would allow us to apply the procedure of stochastic processes to random fields. In other words, for generating two coupled

5 random fields which are homogeneous and isotropic, the path taken is similar to that of two coupled sequences. Hence, from this point on, we focus on homogeneous and isotropic Gaussian random fields. The goal here is to construct two coupled random fields {xi } and {yi } with the desired correlations starting from two independent and uncorrelated Gaussian random fields {ui } and {vi }. To do so, we combine {ui } and {vi } in the Fourier space as xq = Aq uq + Bq vq yq = Cq uq + Dq vq ,

(12)

where {uq } and {vq } are the Fourier transforms of {ui } and {vi } respectively. Since xq and yq should satisfy Eqs. (11a)-(11c), finally, the functional forms of the above coefficients in terms of spectral densities would be obtained. Because of dealing with homogeneous and isotropic random fields, we assume that the coefficients Aq , Bq , Cq , and Dq are only functions of q = |q|. Moreover, we assume that each coefficients has the property that its value at (−q) is equal to its conjugate at (q), similar to Eq. (5). These two assumptions ultimately leads to equations Sxx (q) = |Aq |2 + |Bq |2 2

2

Syy (q) = |Cq | + |Dq | Sxy (q) = Aq Cq∗ + Bq Dq∗ .

(13a) (13b) (13c)

The above equations are exactly similar to those in Eqs. (6a-6c) and so the coefficients Aq , Bq , Cq , and Dq are given by the Eqs. (7) and (8). As a result, one can generate two Gaussian random fields which are homogeneous and isotropic with the algorithm proposed for two Gaussian random sequences in section III. The question that may arise here is; why are the coefficients and the algorithm similar for random sequences and random fields? The answer to this question lies in the assumption of homogeneity and isotropicity for random fields. To be more precise, when the random fields are assumed to be homogeneous and isotropic, the correlation functions and spectral densities become functions of only one variable, C(n) = C(n) and S(q) = S(q). Our algorithm for generating two coupled random fields is similar to when generating two coupled random sequences. However, the only difference is related to the form of spectral densities. For instance, consider a special case of d-dimensional random fields with power-law correlations. Makse et.al. showed that the spectral density of a d-dimensional random field with

the power-law exponent γ has the following form [6] S(q) =

2π d/2  q βd Kβd (q), Γ(βd + 1) 2

(14)

where Kβd (q) is the modified Bessel function of order βd = (γ − d)/2, Γ is the gamma function. Note that, qi = 2πmi /L, −L/2 ≤ mi ≤ L/2, i = 1, . . . , d. Equation. (14) shows explicitly the dependency on the dimension of space, d. In order to test our algorithm, two-dimensional random fields with power-law correlations is considered. Before proceeding to test the algorithm it is worthy to say that, given a power-law correlated random field {xi,j } with 0 < γ < 2, one can construct a two-dimensional FBM by defining [6] hx (s, t) =

s X i=1

xi,t +

t X

xs,j ,

(15)

j=1

where in the physics literature it is also called a self-affine surface. The top row of Fig. 3 is our numerical representation of two self-affine surfaces hx and hy of size 256 × 256 with power-law coupling. The power-law exponents of the underlying fields are γxx = 0.7, γyy = 1.5, and γxy = 1. The middle and the bottom panels of Fig. 3 show the results for autocorrelation functions together with the cross-correlation function for the square lattice of size 212 × 212 . Note that these plots are done by considering three various cases ’A’, ’B’, and ’C’.

VI.

CONCLUSION

We developed an algorithm based on the modified version of the Fourier filtering method to generate two coupled stochastic processes. This algorithm is general because of leaving us free in choosing any desired correlations either autocorrelations or cross-correlation. Moreover, the method proposed in this work can also be used to generate two coupled random fields of any dimensionality with any sort of correlations. The method has been applied to one and two-dimensional models namely times series and rough surfaces. A random media is a three dimensional application of the present study.

VII.

ACKNOWLEDGEMENT

The authors would like to thank Dr. Soheil Vasheghani Farahani, Dr. Mohammad Reza Dehghani, and Dr. Hossein Bayani who assisted in editing the manuscript.

6

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