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Distance correlation detecting Lyapunov instabilities, noise-induced escape times and mixing
Accepted Manuscript Distance correlation detecting Lyapunov instabilities, noise-induced escape times and mixing Carlos F.O. Mendes, Marcus W. Beims
PII: DOI: Reference:
S0378-4371(18)30977-4 https://doi.org/10.1016/j.physa.2018.08.028 PHYSA 19914
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Physica A
Received date : 17 April 2018 Please cite this article as:, Distance correlation detecting Lyapunov instabilities, noise-induced escape times and mixing, Physica A (2018), https://doi.org/10.1016/j.physa.2018.08.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights • Distance correlation measuring nonlinearities in dynamical systems. • Describing noise-induced escape times using the distance correlation. • Lyapunov exponents and correlation decay.
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Distance correlation detecting Lyapunov instabilities, noise-induced escape times and mixing Carlos F. O. Mendes, Marcus W. Beims Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, PR, Brazil
Abstract The properties of the statistical method of distance correlation between multivariate data are analyzed in the context of nonlinear dynamical systems. The distance correlation between the noisy and the noiseless quadratic maps are studied in periodic and chaotic regimes. Results are compared to the classical method of Pearson’s correlation. While distance and Pearson’s correlations are affected by the Lyapunov exponent from the noiseless orbits, only the distance correlation is able to recognize the correct qualitative behaviour of noise-induced escape times decays and the mixing of chaotic trajectories. In addition, the distance correlation is capable of detecting distinct size of attractors. The main goal of this work is to establish the validity of the distance correlation as an method of correlation between multivariate data in dynamical systems. Keywords: Distance correlation, chaos, noise, escape times, mixing.
1. Introduction Distance covariance and distance correlation were proposed around a decade ago [1] for testing joint independence of random vectors in arbitrary dimensions. Of most interest for applied statisticians, independence emerge in several applications and it is essential to measure intricate dependence structures in bivariate and multivariate data. The classical Pearson correlation [2] measures the linear dependence between two random variables and it is zero for bivariate normal data, charaterizing independence. It has been widely used in modelling financial data, synchronization in neurons, atmospheric research, time series analysis, pattern recognition, among many others. However, for more general multivariate data, nonlinear dependence between variables may exist and ρ is not be able to characterize independence anymore. In such cases the distance correlation becomes essential since it is zero if and only if the random vectors are independent [1, 3]. After the distance correlation was proposed, additional recent works have shown its relevance in applied statistics [3, 4, 5, 6, 7]. Distance correlation has already become significant in other areas. We mention some examples. It is used to analyse the quantum-classical transition regarding the ratchet current [8], applyed in astrophysical data analysis, where the distance correlation is applied in the analysis of COMBO-17 database variables [9], used for characteristic screening procedures in genetic risk problems: in blue cell tumor data and in ovarian cancer data [10], and also in the formulation of the correlation between serum cholesterol and blood pressure [11], in the study of functional brain connectivity and structural covariance regions of interest. In this work the authors found that the similarity between functional connectivity and structural covariance estimates was greater for distance correlation compared to Pearson’s correlation [12]. The method is also used in the parameter identification in nonlinear systems [13] and to analyse correlations between the rivers Solimões and Negro in the Amazonas forest [14]. Other measures have been developed, as the Rényi’s Email address: [email protected] (Marcus W. Beims) Preprint submitted to Elsevier
April 17, 2018
maximal correlation [15], rank correlation [16, 17], maximal linear correlation [18], but they are beyond the scope of the present work and, contrary to the distance correlation, more complicated to be implemented in numerical simulations. The present work analyses the ability of the distance correlation to describe relevant properties in dynamical systems. For this we use the paradigmatic quadratic map, which has a simple form but exhibits high complexity of phenomena in the chaotic regime. In turns out that the distance correlation is a relevant statistical method capable in describing the main properties regarding nonlinear dynamics. Surprisingly the distance correlations correctly describes the qualitative decay of noise-induced escape times from attractors in one-dimensional maps [19]. In addition, results show that the distance correlation is able to describe mixing [20] of chaotic trajectories. The result for mixing is compared to the intensity of segregation [21, 22], an established method to measure mixing. The paper is organized as follows. In Sec. 2 we reproduce the formal definition of the Pearson and Distance correlations. While in Sec. 3 both correlations are compared in simple cases under the effect of noise, in Sec. 4 they are compared for the case of the quadratic map under noise. Also in Sec. 4 the quantification of mixing in the chaotic regime of the quadratic map is studied. Finally in Sec. 6 we summarize our conclusions. 2. Correlation methods 2.1. Pearson correlation method The Pearson correlation coefficient (ρ), also known as the product-moment correlation coefficient, is a linear measure of dependence between variables [2]. The coefficient is the covariance of the two variables divided by the product of their standard deviations. Let us start defining some important statistical quantities. Consider two variables X and Y , we denote the mean of X and Y by E[X] and E[Y ], respectively. The variance of X is given by σ 2 (X) = E[X 2 ] − [E[X]]2 and analogous definition for the variance of Y . The covariance between the two variables that is given by σ(X, Y ) = E[XY ] − E[X]E[Y ]. The Pearson correlation is then defined by the expression σ(X, Y ) , ρ(X, Y ) = p σ(X)σ(Y )
(1)
and its values is inside the interval [−1, +1]. The value ρ(X, Y ) = 1 is the positive linear correlation, ρ(X, Y ) = −1 the negative linear correlation and ρ(X, Y ) = 0 means that there is no linear correlation between the variables. In such cases nonlinear correlations may be present which are not detected by the Pearson coefficient and other measures are needed. Assuming to have a sample (X, Y) = {(Xk , Yk ) : k = 1, ..., N } with N ≥ 2, the statistical measure can be computed from the set of data using SXX ≡
SXY ≡
N X
k=1
k=1
(Xk − X)2 ,
N X
(2)
(Yk − Y )2 ,
(3)
(Xk − X)(Yk − Y ).
(4)
SY Y ≡ and
N X
k=1
Then the Pearson correlation coefficient can be obtained by replacing the relations (2), (3) and (4) in the expression below SXY ρN (X, Y) = √ , (5) SXX SY Y 2
and results in
Here X =
1 N
PN (Xk − X)(Yk − Y ) ρN (X, Y) = qP k=1 . PN N 2. 2 (X − X) (Y − Y ) k k=1 k=1 k PN P N 1 k=1 Xk and Y = N k=1 Yk are the arithmetic means of the samples [9, 23].
(6)
2.2. Distance correlation method Distance correlation (DC) is a statistical measure of the dependence between variables and is based on Euclidean distances between these elements. It is derived from the other quantities known, such as the distance variance and distance covariance. While the Pearson correlation is sensitive to a linear relation between variables and can be easily zero if the dependencies are nonlinear, the DC is equal zero if and only if the variables are independent. This is one fundamental characteristic of the distance correlation and one advantage when compared to the Pearson correlation. The distance correlation can be defined by using samples X and Y in arbitrary dimensions and is equal zero when X and Y are independent [1, 3, 4, 5, 6]. Assuming p and q positive integers we can define vectors as X ∈ Rp and Y ∈ Rq . Considering the fact that the vectors X and Y can be defined in arbitrary dimensions, it gives another advantage of the distance correlation compared to the Pearson correlation. The distance correlation is defined inside the interval [0, 1]. The distance covariance between two vectors X and Y is defined by σ 2 (X, Y ) =k ϕX,Y (t, s) − ϕX (t)ϕY (s) k2
(7)
with ϕX (t) and ϕY (s) are characteristics functions of X and Y , respectively, and ϕX,Y (t, s) the joint characteristic function of X and Y . The joint characteristic function under the independence of two random vectors leads to the property σ 2 (X, Y ) = 0 if and only if X and Y are independent, that is, ϕXY (t, s) = ϕX (t)ϕY (s). We can obtain similarly the distance variance σ 2 (X) and σ 2 (Y ). Therefore, the distance correlation coefficient between two vectors is given by σ(X, Y ) , DC(X, Y ) = p σ(X)σ(Y )
(8)
where we can see the similarity with Eq.(6). In the sequence we will see the empirical form of the distance correlation that involves distances between pairs of a data set. The computational method defined below will be used to obtain our results discussed later. The corresponding statistics of distance covariance and DC(X, Y ) are defined by the substitution of the empirical characteristics functions in Eq.(7). The statistics distance dependency are defined as follows: For an observed sample (X, Y) = {(Xk , Yk ) : k = 1, ..., N } with X ∈ Rp and N ≥ 2, is defined by i = 1, ..., N and j = 1, ..., N , the matrix is Aij = aij − a ¯i. − a ¯.j + a ¯.. , (9)
where aij = |Xi − Xj |p is the Euclidean norm of the distance between the elements of the sample, a ¯i. = PN PN 1 1 a and a ¯ = a are the arithmetic mean of the rows and columns, respectively, and general .j j=1 ij i=1 ij N N PN mean is defined by a ¯.. = N12 i,j=1 aij . Similarly to Y ∈ Rq defined by i = 1, ..., N and j = 1, ..., N , we also define the matrix Bij = bij − ¯bi. − ¯b.j + ¯b.. ,
(10)
where the terms bij , ¯bi. , ¯b.j and ¯b.. are similar to shown for the matrix Aij . From these matrices we can compute the empirical distance covariance for a sample, which is defined by the expression 1/2 N 1 X σN (X, Y) = Aij Bij , N i,j=1 3
(11)
which is similar to Eq.(7), as shown in [1]. We can also calculate the empirical distance variance given by 1/2 N 1 X 2 σN (X) = A , N i,j=1 ij
and
σN (Y) =
1/2
N X
1 B2 N i,j=1 ij
.
(12)
(13)
Therefore, the empirical distance correlation for a random sample is given by the relation σN (X, Y) DCN (X, Y) = p . σN (X)σN (Y)
(14)
If all elements of the observable samples X or Y are identical, then we have σN (X)σN (Y) = 0 and consequently DCN (X, Y) = 0. In addition, it is easy to check that DCN is scale independent. In other words, the samples X and Y, can be multiplied by a and b (real numbers) and DC remains unaltered. 3. Comparing the correlations in simple cases under noise As already mentioned, ρ characterizes the linear dependence between variables while DC is a more general measure of correlation between variables. Here we compare both correlations in very simple examples, already studied before using the Pearson correlation [13]. We start with the simplest relations y+ = x + δξ,
(15)
plotted in Fig.1(a) and for (x ≤ 4)
y+
= x + δξ,
y−
= (8 − x) + δξ,
(16)
for (x > 4)
displayed in Fig.1(b), where ξ is a Gaussian noise with zero mean and variance 1, and δ provides the intensity of the noise. For reference, the black line in Fig.1(a) is the noiseless case (δ = 0) for which ρ = DC = 1.0. When noise is added (δ = 0.1), see red circles in Fig.1(a), both correlations are equal ρ = DC = 0.99. To obtain these curves, we divide the x-axis into 100 parts equally spaced. Then we obtain the data sequence (X, Y) = {(x1 , y1 ), (x2 , y2 ), ..., (x100 , y100 )}, from which we can apply the methods given in the previous Section. This procedure is used for all plots presented in this Section. Figure 1(b) nicely shows one distinction between both correlations. The function (16) for δ = 0, black line in Fig.1(b), is linear by parts with equal positive and negative inclinations. Thus, Pearson’s correlation is close to one in the positive part and close to −1 in the second part, leading to ρ = 0.02 in the noiseless case. Adding a noise with intensity δ = 0.1 such correlation value remains roughly unaltered. The distance correlation in this case is DC ∼ 0.50 for the noiseless case and DC ∼ 0.49 for δ = 0.1. The next cases are given by the equations p (17) y± = ± 1 − x2 + δξ, which is a circular curve with unitary radius when δ = 0, shown in Fig.1(c), and y = sin(x) + δξ
(18)
which is a sinusoidal curve for δ = 0 in Fig.1(d). First observation is that in both cases ρ ∼ 0 since positive and negative inclination of the curves are symmetric. Small deviation from zero is observed when noise is 4
10
6 DC=0.50, ρ=0.02, δ=0.0 DC=0.49, ρ=0.02, δ∼0.1
DC=1.00, ρ=1.00, δ=0.0 DC=0.99, ρ=0.99, δ∼0.1
8
4
y
6 4
2
2
(a)
0 0
2
4
6
(b)
0 0
8
2
4
2
2
DC=0.15, ρ= 0.00, δ=0.0 DC=0.14, ρ=-0.02, δ∼0.1
8
DC=0.42, ρ=0.00, δ=0.0 DC=0.41, ρ=0.00, δ∼0.1
1
1
y
6
x
x
0
0
-1 -1
(d)
(c) -1.0
-0.5
0.0
0.5
-2 0 0
1.0
0
180
360
0
0
540
x
x
Figure 1: Plotted are the curves obtained from (a) Eq.(15), (b) Eq.(16), (c) Eq.(17) and (d) Eq.(18) in the plane xy. DC and ρ are given.
present in case from Fig.1(c). On the other hand, DC changes in both cases. In the circular case from Fig.1(c) it is astonishingly small ∼ 0.15, while in Fig.1(d) it is ∼ 0.42. These values are almost independent of the presence of noise. Even though in all geometric shapes there is obviously a dependence between horizontal and vertical variables, the Pearson correlation is essentially zero (exception is the straight line in Fig. 1(a)). The distance correlation, on the other hand, was able to detect some dependence between the variables, even in the presence of noise. For larger values of the noise intensity δ, ρ and DC decay. 4. Correlation in the quadratic map To understand the role of the distance correlation in the context of nonlinear systems, in this Section we compare the correlation between one quadratic map and a Gaussian noise and between two quadratic maps, one is the usual map and for the other one we add a Gaussian noise. We start describing the basic properties of both maps which are relevant for later discussions. 4.1. The usual map The quadratic map is a one-dimensional dynamical system given by the relation [24, 25] xn+1 = r − x2n ,
(19)
where r is a parameter, n = 0, 1, 2, 3, ..., N an integer which represents the number of iterates and xn ∈ [−2, 2] is the state of the system at time n. For reference we plot the well known bifurcation diagram in Fig.2(a) (see black line). Periodic orbits with period-1, 2, 4 and 8 are clearly visible. For r smaller than −0.25, all initial conditions diverge to −∞. A tangent bifurcation occurs at r = −0.25, where a periodic orbit-1 is created. It persists from r = −0.25 to r = 0.75. At r = 0.75, the period-1 orbit loses stability and is replaced by periodic-2 orbits. Increasing the values of r, a sequence of period doubling bifurcations occurs until the chaotic motion is reached with some periodic windows. For r > 2.0 all initial conditions diverge. Figure 2(b) the Lyapunov exponent (LE) λδ is presented. For positive Lyapunov exponents the dynamics 5
Figure 2: (a) Bifurcation diagram of the quadratic map for the noiseless case in black. Red dots is the bifurcation diagram including noise (see text). For 2250 equally spaced r values we have computed 1050 iterates with initial condition x0 = 0.1. For each r only the values of the iterates from x1001 to x1050 were plotted. (b) The Lyapunov exponent λδ in black for the map (19) and in red for the noise map (20).
is chaotic. It is easy to observe that all periodic orbits are regular (negative LE) and that LE is zero at the bifurcation points. 4.2. The map with noise The quadratic map with noise is given by yn+1 = r − yn2 + δξn ,
(20)
where ξn is the Gaussian noise with zero mean and variance 1, and δ gives the intensity of the noise. This map was studied many years ago [19] in the context of noise-induced escape from attractors. The author used Monte-Carlos simulations and analytical results to show that the escape rate follows R ≈ τ10 exp(−E0 /δ), PN −1 where E0 , proportional to n=1 ξn2 , is the minimum escape energy and τ0 , the time between attempts is proportional to the number of paths which are close to the path with minimum escape energy. The paths begin on the attractor and end on the boundary of the basin of attraction. Red points in Fig. 2(a) exemplify the bifurcation diagram obtained using the map (20) with δ = 0.05. There are no periodic orbits anymore but some relation (correlation) to the usual map is expected. In the simulations we discard 103 iterations (transient) before the noise is added. For increasing values of δ many ICs may diverge. Here we consider that an IC diverges when |yn | ≥ 103 . It is also possible to consider divergency when orbits are outside the corresponding basin of attraction from the noiseless case, as done in [19]. We observed that both methods provide similar results and in our case we do not need to determine the boundaries of the basins of attraction. In Fig. 2(b) the Lyapunov exponent λδ is plotted in red for the noisy map. Figure 3 displays the number of ICs which diverge after N = 10, 20, 30 and 40 iterations as a function of δ. Panel (a) shows r = 1, which is the regular period-2 dynamics from the noiseless case and (b) the chaotic case with r = 2. Clearly there is a critical value δcrit , above which many ICs diverge after N iterations and a noise-induced escape from the attractors occurs. Analysing the curves from Fig. 3 for the other values of r (not shown), we estimate that δcrit ≈ 0.2 represents well all periodic cases. This critical value decreases as r increases and for the chaotic case it approaches δcrit → 0. Similar behaviour was 6
10
10
10
10
(b)
(a) 8
10
6
10
Nº ICs - Diverge
10
10
10
10
4
10
N = 10 20 30 40
2
10
8
6
4
2
0
0
100.0
0.2
0.4
0.6
0.8
1.0
10 0.0
0.2
0.4
0.8
0.6
1.0
δ
δ
Figure 3: Number of ICs which diverge after N iterations as a function of δ for (a) r = 1 and (b) r = 2.
2 observed in [19], i. e. an almost zero minimum escape energy E0 (proportional to δcrit ) is needed to obtain the noise-induced escape from the chaotic attractor.
4.3. Correlations decay between the map (19) and noise For reference we start analysing correlation coefficients ρ and DC between the states xn generated by the quadratic map (19) with the Gaussian noise. For this, the sample {X = (xn+1 ) : n = 0, 1, 2, ..., N } is obtained from (19) and the sample {Y = (δξn+1 ) : n = 0, 1, 2, ..., N } from the Gaussian generator. We use 2 × 104 ICs randomly selected and equally spaced in the interval [−2, 2]. In this way we obtain the average 0.35
1.00
10
-3
0
-10
2
0.30
0.28
-3
0.01
0.33
〈DC20 〉
0.10
〈ρn 〉
〈DCn 〉
r = 1.0 1.3 1.38 2.0
2
1000
500
10
100
(a)
(b) 1000
n
0.25 0.0
0.2
0.4
0.6
0.8
1.0
δ
Figure 4: Plot showing the (a) time dependence of the average distance correlation and the average Pearson’s (inset) for δ = 1 and (b) noise dependence of the average distance correlation for n = 20.
Pearson’s correlation hρn (X, Y)i and the average distance correlation hDCn (X, Y)i. For simplicity we write hρn i ≡ hρn (X, Y)i and hDCn i ≡ hDCn (X, Y)i. The mean quantities hρn i and hDCn i are plotted in Fig.4(a) as a function of the time for four distinct dynamical regimes from the map (19), namely r = 1.0 for which the state xn has period-2, r = 1.3 with period-4, r = 1.38 with period-8 and r = 2.0 with a chaotic attractor. We use red colours for period-2, green for period-4, blue for period-8 and magenta for the chaotic motion. Results show in Fig.4(a) that while hρn i oscillates around zero, hDCn i decays with ∝ n−0.5 . Curiously, for very small times (n . 5), hDCn i decays faster for smaller values of r, so that for n = 20, for example, the value of hDC20 i is different for each r and remains independent of δ, as shown in Fig.4(b). The fact that the distance correlation is constant is a consequence of the scale invariance in {Y} mentioned in Sec. 2.2. The interesting point is that the distance correlation between the chaotic motion and the noise is larger than between the regular motion and the noise. In addition, for smaller periods hDC20 i decreases. This shows that hDC20 i 7
between the noise and two regular points (period-2), is smaller than hDC20 i between the noise and four regular points (period-4), and so on. In other words, hDC20 (2) i < hDC20 (4) i < hDC20 (8) i < . . . < hDC20 (20) i, where the superscript refers to the period-p. In the chaotic case the period is infinity, but since we are just iterating until N = 20, we say the period is 20. Therefore, for a given n & 5, hDCn i assumes values depending on the number of points from the attractors, and could be used to compare and quantify the size of distinct attractors. These results show the first distinction between both correlations and the relevance of the distance correlation in the context of dynamical systems. It is worth to mention that DC decays to zero asymptotically, as expected. However, from the physical and dynamical point of view, it is very interesting to know the qualitative behaviour of such decay. It can elucidate the underline dynamics, as demonstrated next. 4.4. Correlations decay between maps (19) and (20) Now we wish to analyse the correlation coefficients ρ and DC between the states xn generated by the quadratic map (19) with states yn generated by the map (20) with noise. For this, the sample {X = (xn+1 ) : n = 0, 1, 2, ..., N } is obtained from (19) and the sample {Y = (yn+1 ) : n = 0, 1, 2, ..., N } is generated from (20). In other words, we determine the correlations between the noiseless case (black lines in Fig. 2(a)) and the noisy case (red points in Fig. 2(a) for δ = 0.05) for specific values of r and increasing values of δ. After eliminating a transient of 103 iterations, for each IC we determine the coefficients ρN (X, Y) and DCN (X, Y) between both time series until N = 20. Due to noise, the states yn may change attractors and diverge. In such cases we choose another IC until we have a total of 2 × 104 ICs which did not diverge for N = 20. Also here the ICs are randomly selected and equally spaced in the interval [−2, 2] and we look at the mean quantities hρ20 i and hDC20 i. 0.50
1.00
〈ρ20 〉
(a)
(b) 0.10
0.10
0.01
r = 1.00 1.30 1.38 2.00
0.01
0.01 0.10
1.00
0.20
δ 0.65
1.00
(c) 〈DC20 〉
1.00
0.50
δ
(d)
0.50
0.50
0.25
0.20 0.01
0.10 δ
1.00
0.20
0.50
1.00
δ
Figure 5: Log-log plots showing the noise dependence of the (a) average Pearson’s correlation, (b) adjusted fitted curves from (a) inside the δ interval [0.2, 1], (c) average distance correlation and (d) adjusted fitted curves from (c) inside the same δ interval from (b). Functions for the fits are presented in Table 1.
4.4.1. Concordances of hρ20 i and hDC20 i In Fig.5(a) the log-log plot of hρ20 i is displayed and is to be compared with Fig.5(c), where hDC20 i is plotted, also in a log-log scale. For δ = 0 both correlations are maximal (not shown) and they are 8
independent of the dynamics, even in the chaotic regime (for identical ICs in both maps). For the periodic motion we observe that both correlations are equal to 1 for values of δ smaller than ∼ 0.03. In the chaotic regime the mean Pearson’s correlation is ∼ 0.33 and the mean distance correlation ∼ 0.46. This already shows that the distance correlation is more sensitive to existing nonlinear correlations between the chaotic data. For a noise intensity δ = 0.1 we have for period-2: hDC20 (2) i = 0.88 and hρ20 (2) i = 0.82, for period-4: hDC20 (4) i = 0.76 and hρ20 (4) i = 0.63, period-8: hDC20 (8) i = 0.67. and hρ20 (8) i = 0.50 and for the chaotic case: hDC20 (20) i = 0.35 hρ20 (20) i = 0.17. Thus, in all cases hDC20 i > hρ20 i showing that the distance correlations is more resistant to noise effects, independent of the dynamical regime. In addition, both correlation are Table 1: For distinct values of r, the Table summarizes the associated period, Lyapunov exponent λδ and the functions used to adjust the curves of the mean quantities hDC20 i and hρ20 i. In the chaotic case the period is infinity, but since we are just iterating until N = 20, we say the period is 20.
r 1 1.3 1.38 2.0
Period-p 2 4 8 20
λδ=0 -∞ -0.42 -0.33 0.69
hDC20 (p) i ∼ α eβ/δ ∼ 0.2178 e0.169/δ ∼ 0.2551 e0.133/δ ∼ 0.2757 e0.110/δ ∼ 0.3239 e0.006/δ
hρ20 (p) i ∼ 0.0158 δ −1.78 ∼ 0.0263 δ −1.46 ∼ 0.0285 δ −1.36 ∼ 0.1680 e−1.73 δ
more resistant to noise for lower periods, since they start to decrease from 1 at distinct values of δ. In other words, hDC20 (2) i > hDC20 (4) i > hDC20 (8) i > . . . > hDC20 (20) i. This is directly related to the Lyapunov stability exponent λδ=0 (see Table 1), lower periods have smaller negative Lyapunov exponent are also more stable under noise. Chaotic attractors with λδ=0 > 0 are less stable under noise. For values δ > δcrit ≈ 0.2, both correlations cross around ∼ 0.24 and for δ > 0.24 their values interchange for regular and chaotic regimes when compared to the δ = 0.1 case. In other words, for δ > 0.24 we have hDC20 (2) i < hDC20 (4) i < hDC20 (8) i < . . . < hDC20 (20) i which is the inequality observed in Sec. 4.3, and the numerical values of hDC20 (p) i at δ = 1 are very similar to those from Fig. 4(b). This means that for large values of δ the distinct dynamical regimes of the noisy map (20) are not detected anymore in the distance correlation. Just the dynamical regimes from map (19) are still relevant. From the adjusted curves in Table 1 we conclude that for δ → ∞, while Pearson’s correlation approaches zero, the distance correlation approaches α, which clearly agrees with the values of hDC20 (p) i from Fig. 4(b). 4.4.2. Discordances of hρ20 i and hDC20 i After discussing some similarities of both correlations, we analyse now the differences, which are mainly related to their qualitative behaviour as a function of δ. Figures 5(b) and (d) display the best adjusted curves related to the data. Table 1 presents a summary. The curves for the mean Pearson correlation can be well adjusted by an inverse power-law in the periodic regime and an exponential decay in the chaotic regime. On the other hand, the mean distance correlation follows α exp (β/δ) for regular and chaotic regimes, where α and β are adjusting parameters. Astonishingly, this is exactly the qualitative behaviour of the escape time decay 1/R observed in [19] (see also above). By comparing the values of β from Table 1 with the values of E0 from Fig. 3 in [19], we can affirm that β ≈ E0 . In addition, by comparing α with Fig. 5 from [19], we observe roughly that log10 (p/α) ≈ log10 (τ0 ), with p being the period of the orbit. From this the dimensionless adjusting parameter α ≈ p/τ0 is the period of the orbit divided by the escape time. Thus, we conclude that the mean distance correlation properly describes the noise-induced escape times through τ02 hDCN i ≈ 1/R. p
(21)
In other words, the mean distance correlation is able to reproduce correctly the qualitative behaviour of noise-induced escape times, while the Pearson correlation is not. 9
5. Measuring mixing in chaotic regimes There are several models to evaluate mixing process in chaotic dynamics, such as, recurrence, transitivity, ergodicity and entropy. The purpose of the present Section is to check the ability of the distance correlation to describe mixing in chaotic regimes and compare it to other mixing measure, namely the intensity of segregation [22, 21]. We call to attention that while the distance correlation is determined comparing pairs of data X and Y, the intensity of segregation is obtained from just one set of data. 5.1. Distance correlation As described above, to reckon DC we need a pair of samples X and Y. They can be obtained by starting two distinct ICs, x01 = 0.1 and x02 = 0.2 in the chaotic regime and iterate them for N = 2 × 103 . Thus, we obtain the data sequence from (X, Y) = {(x01 , y01 = x02 ), (x2 , y2 ), . . . , (xn , yn ), . . . , (xN , yN )}, and can determine DC. Figure 6(a) displays the the log-log plot of DC as a function of time for distinct values of r = 1.6, 1.7, 1.8, 1.9 and 2.0, all in the chaotic regime. If DC tends to zero in time, it can be considered as a valid statistical measure of mixed states. The strong oscillations observed in the behaviour of DC disappear
DCn (x01,x02)
1.00
0.10
0.01 2
(b)
1
(a)
0.1
r = 1.6 1.7 1.8 1.9 2.0
10
0.01 1000 2
100
10
n
100
1000
n
Figure 6: (a) Log-log plot of the time evolution of DCn (x01 , x02 ) between two trajectories with initial conditions x01 = 0.1 and x02 = 0.2 in chaotic regimes and (b) hDCn (x01 , x02 )i for many ICs starting distinct values of x02 .
when more ICs are used. This is shown in the log-log plot in Fig.6(b) by starting ICs with x01 = 0.1 and x02 chosen randomly and equally spaced in the interval [−2, 2]. It clearly shows that hDCn (x01 , x02 )i decays as a power-law with ∝ n−0.5 . 5.2. Intensity of Segregation The first step to obtain the intensity of segregation [26] is to determine Dj =
measured quantity , q
(22)
where q is the total number of observations. Dividing the interval [−2, 2] of states from the one-dimensional quadratic map in S subintervals of length ∆xj (j = 1, ..., S), we calculate the probability that each subinterval is visited using Eq.(22). The mean of Dj by the sum over subintervals is hDi =
S 1X Dj , S j=1
(23)
and the mean of the square density Dj2 is given by hD2 i =
S 1X 2 D . S j=1 j
10
(24)
1.0000
S=103 S=104 S=105 n-1
I
0.1000
0.0100
0.0010
0.0001 0 10
101
102
n
103
104
Figure 7: Log-log plot of the intensity segregation I as a function of the iteration.
For the case of mixing in a trajectory, the final state of mixing is characterized by a minimum statistical square density h(Dj − hDi)2 i ≡ hD2 i − hDi2 → 0. (25)
The rate of decrease of hD2 i/hDi2 characterizes the mixing process and depends on time and also on the number of subinterval. From the above quantities it is possible to determine the the intensity of segregation (I), which is a statistical measure [26]. For comparison with DC, we use the modified intensity of segregation (normalized between 0 and 1) defined as [22, 21] h(Dj − hDi)2 i I= . (26) hDi(1 − hDi)
When I tends to zero in time, the result is a good mixing state. In Fig.7 we display the decrease of the intensity of segregation for one trajectory in the chaotic regime from the quadratic map (19) with r = 2.0 and S = 103 , 104 and 105 . For this we consider N = 104 iterations of the map and discard a transient of 4 × 103 . Until iterations around n = 400 we observe a linear decay in the log-log scale. For the case of S = 105 the linear decay exist for √ all interval of iterations. The regime with a power-law decay of I satisfies ∝ n−1.0 . Thus, the square-root I decays in time as the mean distance correlation shown in Fig. 6(b), showing that the later is capable of correctly describing the qualitative of chaotic mixing. 6. Conclusions
The distance correlation [1] is a well defined statistical quantity able to measure dependence between random vectors of multivariate data. The relevant property is that it is zero if and only if the random vectors are independent (see [1, 3]). In addition, the random vectors can have any dimension. As detailed in the Introduction, the distance correlation has recently being applied in distinct areas. This work analyses properties of the distance correlation to describe dynamical systems. More specifically, it is studied for the quadratic map in the periodic and chaotic regimes subjected to a Gaussian noise. In other words, we determine the distance correlation (i) between the usual quadratic map (19) and a Gaussian noisy signal, and (ii) between the map (19) and the quadratic map (20) with the Gaussian noise. Results are compared to the classical Pearson’s correlation which only detects linear dependences between sample data. In general, we found that the distance correlation value is always larger than the Pearson’s correlation. In case (i), while Pearson’s correlation is almost zero for all times, the distance correlation decays in time roughly with n−0.5 and it is able to detect distinct size of the attractors from the map (19). In case (ii) both correlations are more resistant to noise for trajectories with smaller Lyapunov exponents from the noiseless case. Thus, the local linear stability of trajectories affects both correlations for small noise values. For larger noise values, for which almost all ICs from the map (20) diverge very fast, both correlations are sensitive to the number 11
of points which contain the attractors, i. e. the size of attractors. Also in this case the distance correlation is better than Pearson’s correlation to detect nonlinearities. In distinction to Pearson’s correlation, we found that the qualitative decay of the distance correlation as a function of the noise intensity is remarkably equivalent to the noise-induced escape times decay from attractors [19]. This means that only the distance correlation correctly detects the complicated nonlinear dynamics involved in escape processes. In addition, for the noiseless case of the quadratic map, the distance correlation is able to quantify the degree of mixing between chaotic trajectories. This is confirmed by comparing the decay of the distance correlation with the intensity of segregation. Future investigations may focus on the relevance of the distance correlation in describing other properties in dynamical systems, like decays in leaking chaotic systems [27], in open billiards [28], in stickiness [29], in high-dimensional non-integrable systems [30] in coupled oscillators [31], in chimera states [32], in the evolution of scientific disciplines [33], to mention a few. acknowledgments CFOM thanks FAPEAM and MWB thanks CNPq for financial support. The authors also acknowledge computational support from Professor Carlos M. de Carvalho at LFTC-DFis-UFPR. 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] [25] [26] [27] [28] [29] [30] [31] [32] [33]
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PII: DOI: Reference:
S0378-4371(18)30977-4 https://doi.org/10.1016/j.physa.2018.08.028 PHYSA 19914
To appear in:
Physica A
Received date : 17 April 2018 Please cite this article as:, Distance correlation detecting Lyapunov instabilities, noise-induced escape times and mixing, Physica A (2018), https://doi.org/10.1016/j.physa.2018.08.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights • Distance correlation measuring nonlinearities in dynamical systems. • Describing noise-induced escape times using the distance correlation. • Lyapunov exponents and correlation decay.
*Manuscript Click here to view linked References
Distance correlation detecting Lyapunov instabilities, noise-induced escape times and mixing Carlos F. O. Mendes, Marcus W. Beims Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, PR, Brazil
Abstract The properties of the statistical method of distance correlation between multivariate data are analyzed in the context of nonlinear dynamical systems. The distance correlation between the noisy and the noiseless quadratic maps are studied in periodic and chaotic regimes. Results are compared to the classical method of Pearson’s correlation. While distance and Pearson’s correlations are affected by the Lyapunov exponent from the noiseless orbits, only the distance correlation is able to recognize the correct qualitative behaviour of noise-induced escape times decays and the mixing of chaotic trajectories. In addition, the distance correlation is capable of detecting distinct size of attractors. The main goal of this work is to establish the validity of the distance correlation as an method of correlation between multivariate data in dynamical systems. Keywords: Distance correlation, chaos, noise, escape times, mixing.
1. Introduction Distance covariance and distance correlation were proposed around a decade ago [1] for testing joint independence of random vectors in arbitrary dimensions. Of most interest for applied statisticians, independence emerge in several applications and it is essential to measure intricate dependence structures in bivariate and multivariate data. The classical Pearson correlation [2] measures the linear dependence between two random variables and it is zero for bivariate normal data, charaterizing independence. It has been widely used in modelling financial data, synchronization in neurons, atmospheric research, time series analysis, pattern recognition, among many others. However, for more general multivariate data, nonlinear dependence between variables may exist and ρ is not be able to characterize independence anymore. In such cases the distance correlation becomes essential since it is zero if and only if the random vectors are independent [1, 3]. After the distance correlation was proposed, additional recent works have shown its relevance in applied statistics [3, 4, 5, 6, 7]. Distance correlation has already become significant in other areas. We mention some examples. It is used to analyse the quantum-classical transition regarding the ratchet current [8], applyed in astrophysical data analysis, where the distance correlation is applied in the analysis of COMBO-17 database variables [9], used for characteristic screening procedures in genetic risk problems: in blue cell tumor data and in ovarian cancer data [10], and also in the formulation of the correlation between serum cholesterol and blood pressure [11], in the study of functional brain connectivity and structural covariance regions of interest. In this work the authors found that the similarity between functional connectivity and structural covariance estimates was greater for distance correlation compared to Pearson’s correlation [12]. The method is also used in the parameter identification in nonlinear systems [13] and to analyse correlations between the rivers Solimões and Negro in the Amazonas forest [14]. Other measures have been developed, as the Rényi’s Email address: [email protected] (Marcus W. Beims) Preprint submitted to Elsevier
April 17, 2018
maximal correlation [15], rank correlation [16, 17], maximal linear correlation [18], but they are beyond the scope of the present work and, contrary to the distance correlation, more complicated to be implemented in numerical simulations. The present work analyses the ability of the distance correlation to describe relevant properties in dynamical systems. For this we use the paradigmatic quadratic map, which has a simple form but exhibits high complexity of phenomena in the chaotic regime. In turns out that the distance correlation is a relevant statistical method capable in describing the main properties regarding nonlinear dynamics. Surprisingly the distance correlations correctly describes the qualitative decay of noise-induced escape times from attractors in one-dimensional maps [19]. In addition, results show that the distance correlation is able to describe mixing [20] of chaotic trajectories. The result for mixing is compared to the intensity of segregation [21, 22], an established method to measure mixing. The paper is organized as follows. In Sec. 2 we reproduce the formal definition of the Pearson and Distance correlations. While in Sec. 3 both correlations are compared in simple cases under the effect of noise, in Sec. 4 they are compared for the case of the quadratic map under noise. Also in Sec. 4 the quantification of mixing in the chaotic regime of the quadratic map is studied. Finally in Sec. 6 we summarize our conclusions. 2. Correlation methods 2.1. Pearson correlation method The Pearson correlation coefficient (ρ), also known as the product-moment correlation coefficient, is a linear measure of dependence between variables [2]. The coefficient is the covariance of the two variables divided by the product of their standard deviations. Let us start defining some important statistical quantities. Consider two variables X and Y , we denote the mean of X and Y by E[X] and E[Y ], respectively. The variance of X is given by σ 2 (X) = E[X 2 ] − [E[X]]2 and analogous definition for the variance of Y . The covariance between the two variables that is given by σ(X, Y ) = E[XY ] − E[X]E[Y ]. The Pearson correlation is then defined by the expression σ(X, Y ) , ρ(X, Y ) = p σ(X)σ(Y )
(1)
and its values is inside the interval [−1, +1]. The value ρ(X, Y ) = 1 is the positive linear correlation, ρ(X, Y ) = −1 the negative linear correlation and ρ(X, Y ) = 0 means that there is no linear correlation between the variables. In such cases nonlinear correlations may be present which are not detected by the Pearson coefficient and other measures are needed. Assuming to have a sample (X, Y) = {(Xk , Yk ) : k = 1, ..., N } with N ≥ 2, the statistical measure can be computed from the set of data using SXX ≡
SXY ≡
N X
k=1
k=1
(Xk − X)2 ,
N X
(2)
(Yk − Y )2 ,
(3)
(Xk − X)(Yk − Y ).
(4)
SY Y ≡ and
N X
k=1
Then the Pearson correlation coefficient can be obtained by replacing the relations (2), (3) and (4) in the expression below SXY ρN (X, Y) = √ , (5) SXX SY Y 2
and results in
Here X =
1 N
PN (Xk − X)(Yk − Y ) ρN (X, Y) = qP k=1 . PN N 2. 2 (X − X) (Y − Y ) k k=1 k=1 k PN P N 1 k=1 Xk and Y = N k=1 Yk are the arithmetic means of the samples [9, 23].
(6)
2.2. Distance correlation method Distance correlation (DC) is a statistical measure of the dependence between variables and is based on Euclidean distances between these elements. It is derived from the other quantities known, such as the distance variance and distance covariance. While the Pearson correlation is sensitive to a linear relation between variables and can be easily zero if the dependencies are nonlinear, the DC is equal zero if and only if the variables are independent. This is one fundamental characteristic of the distance correlation and one advantage when compared to the Pearson correlation. The distance correlation can be defined by using samples X and Y in arbitrary dimensions and is equal zero when X and Y are independent [1, 3, 4, 5, 6]. Assuming p and q positive integers we can define vectors as X ∈ Rp and Y ∈ Rq . Considering the fact that the vectors X and Y can be defined in arbitrary dimensions, it gives another advantage of the distance correlation compared to the Pearson correlation. The distance correlation is defined inside the interval [0, 1]. The distance covariance between two vectors X and Y is defined by σ 2 (X, Y ) =k ϕX,Y (t, s) − ϕX (t)ϕY (s) k2
(7)
with ϕX (t) and ϕY (s) are characteristics functions of X and Y , respectively, and ϕX,Y (t, s) the joint characteristic function of X and Y . The joint characteristic function under the independence of two random vectors leads to the property σ 2 (X, Y ) = 0 if and only if X and Y are independent, that is, ϕXY (t, s) = ϕX (t)ϕY (s). We can obtain similarly the distance variance σ 2 (X) and σ 2 (Y ). Therefore, the distance correlation coefficient between two vectors is given by σ(X, Y ) , DC(X, Y ) = p σ(X)σ(Y )
(8)
where we can see the similarity with Eq.(6). In the sequence we will see the empirical form of the distance correlation that involves distances between pairs of a data set. The computational method defined below will be used to obtain our results discussed later. The corresponding statistics of distance covariance and DC(X, Y ) are defined by the substitution of the empirical characteristics functions in Eq.(7). The statistics distance dependency are defined as follows: For an observed sample (X, Y) = {(Xk , Yk ) : k = 1, ..., N } with X ∈ Rp and N ≥ 2, is defined by i = 1, ..., N and j = 1, ..., N , the matrix is Aij = aij − a ¯i. − a ¯.j + a ¯.. , (9)
where aij = |Xi − Xj |p is the Euclidean norm of the distance between the elements of the sample, a ¯i. = PN PN 1 1 a and a ¯ = a are the arithmetic mean of the rows and columns, respectively, and general .j j=1 ij i=1 ij N N PN mean is defined by a ¯.. = N12 i,j=1 aij . Similarly to Y ∈ Rq defined by i = 1, ..., N and j = 1, ..., N , we also define the matrix Bij = bij − ¯bi. − ¯b.j + ¯b.. ,
(10)
where the terms bij , ¯bi. , ¯b.j and ¯b.. are similar to shown for the matrix Aij . From these matrices we can compute the empirical distance covariance for a sample, which is defined by the expression 1/2 N 1 X σN (X, Y) = Aij Bij , N i,j=1 3
(11)
which is similar to Eq.(7), as shown in [1]. We can also calculate the empirical distance variance given by 1/2 N 1 X 2 σN (X) = A , N i,j=1 ij
and
σN (Y) =
1/2
N X
1 B2 N i,j=1 ij
.
(12)
(13)
Therefore, the empirical distance correlation for a random sample is given by the relation σN (X, Y) DCN (X, Y) = p . σN (X)σN (Y)
(14)
If all elements of the observable samples X or Y are identical, then we have σN (X)σN (Y) = 0 and consequently DCN (X, Y) = 0. In addition, it is easy to check that DCN is scale independent. In other words, the samples X and Y, can be multiplied by a and b (real numbers) and DC remains unaltered. 3. Comparing the correlations in simple cases under noise As already mentioned, ρ characterizes the linear dependence between variables while DC is a more general measure of correlation between variables. Here we compare both correlations in very simple examples, already studied before using the Pearson correlation [13]. We start with the simplest relations y+ = x + δξ,
(15)
plotted in Fig.1(a) and for (x ≤ 4)
y+
= x + δξ,
y−
= (8 − x) + δξ,
(16)
for (x > 4)
displayed in Fig.1(b), where ξ is a Gaussian noise with zero mean and variance 1, and δ provides the intensity of the noise. For reference, the black line in Fig.1(a) is the noiseless case (δ = 0) for which ρ = DC = 1.0. When noise is added (δ = 0.1), see red circles in Fig.1(a), both correlations are equal ρ = DC = 0.99. To obtain these curves, we divide the x-axis into 100 parts equally spaced. Then we obtain the data sequence (X, Y) = {(x1 , y1 ), (x2 , y2 ), ..., (x100 , y100 )}, from which we can apply the methods given in the previous Section. This procedure is used for all plots presented in this Section. Figure 1(b) nicely shows one distinction between both correlations. The function (16) for δ = 0, black line in Fig.1(b), is linear by parts with equal positive and negative inclinations. Thus, Pearson’s correlation is close to one in the positive part and close to −1 in the second part, leading to ρ = 0.02 in the noiseless case. Adding a noise with intensity δ = 0.1 such correlation value remains roughly unaltered. The distance correlation in this case is DC ∼ 0.50 for the noiseless case and DC ∼ 0.49 for δ = 0.1. The next cases are given by the equations p (17) y± = ± 1 − x2 + δξ, which is a circular curve with unitary radius when δ = 0, shown in Fig.1(c), and y = sin(x) + δξ
(18)
which is a sinusoidal curve for δ = 0 in Fig.1(d). First observation is that in both cases ρ ∼ 0 since positive and negative inclination of the curves are symmetric. Small deviation from zero is observed when noise is 4
10
6 DC=0.50, ρ=0.02, δ=0.0 DC=0.49, ρ=0.02, δ∼0.1
DC=1.00, ρ=1.00, δ=0.0 DC=0.99, ρ=0.99, δ∼0.1
8
4
y
6 4
2
2
(a)
0 0
2
4
6
(b)
0 0
8
2
4
2
2
DC=0.15, ρ= 0.00, δ=0.0 DC=0.14, ρ=-0.02, δ∼0.1
8
DC=0.42, ρ=0.00, δ=0.0 DC=0.41, ρ=0.00, δ∼0.1
1
1
y
6
x
x
0
0
-1 -1
(d)
(c) -1.0
-0.5
0.0
0.5
-2 0 0
1.0
0
180
360
0
0
540
x
x
Figure 1: Plotted are the curves obtained from (a) Eq.(15), (b) Eq.(16), (c) Eq.(17) and (d) Eq.(18) in the plane xy. DC and ρ are given.
present in case from Fig.1(c). On the other hand, DC changes in both cases. In the circular case from Fig.1(c) it is astonishingly small ∼ 0.15, while in Fig.1(d) it is ∼ 0.42. These values are almost independent of the presence of noise. Even though in all geometric shapes there is obviously a dependence between horizontal and vertical variables, the Pearson correlation is essentially zero (exception is the straight line in Fig. 1(a)). The distance correlation, on the other hand, was able to detect some dependence between the variables, even in the presence of noise. For larger values of the noise intensity δ, ρ and DC decay. 4. Correlation in the quadratic map To understand the role of the distance correlation in the context of nonlinear systems, in this Section we compare the correlation between one quadratic map and a Gaussian noise and between two quadratic maps, one is the usual map and for the other one we add a Gaussian noise. We start describing the basic properties of both maps which are relevant for later discussions. 4.1. The usual map The quadratic map is a one-dimensional dynamical system given by the relation [24, 25] xn+1 = r − x2n ,
(19)
where r is a parameter, n = 0, 1, 2, 3, ..., N an integer which represents the number of iterates and xn ∈ [−2, 2] is the state of the system at time n. For reference we plot the well known bifurcation diagram in Fig.2(a) (see black line). Periodic orbits with period-1, 2, 4 and 8 are clearly visible. For r smaller than −0.25, all initial conditions diverge to −∞. A tangent bifurcation occurs at r = −0.25, where a periodic orbit-1 is created. It persists from r = −0.25 to r = 0.75. At r = 0.75, the period-1 orbit loses stability and is replaced by periodic-2 orbits. Increasing the values of r, a sequence of period doubling bifurcations occurs until the chaotic motion is reached with some periodic windows. For r > 2.0 all initial conditions diverge. Figure 2(b) the Lyapunov exponent (LE) λδ is presented. For positive Lyapunov exponents the dynamics 5
Figure 2: (a) Bifurcation diagram of the quadratic map for the noiseless case in black. Red dots is the bifurcation diagram including noise (see text). For 2250 equally spaced r values we have computed 1050 iterates with initial condition x0 = 0.1. For each r only the values of the iterates from x1001 to x1050 were plotted. (b) The Lyapunov exponent λδ in black for the map (19) and in red for the noise map (20).
is chaotic. It is easy to observe that all periodic orbits are regular (negative LE) and that LE is zero at the bifurcation points. 4.2. The map with noise The quadratic map with noise is given by yn+1 = r − yn2 + δξn ,
(20)
where ξn is the Gaussian noise with zero mean and variance 1, and δ gives the intensity of the noise. This map was studied many years ago [19] in the context of noise-induced escape from attractors. The author used Monte-Carlos simulations and analytical results to show that the escape rate follows R ≈ τ10 exp(−E0 /δ), PN −1 where E0 , proportional to n=1 ξn2 , is the minimum escape energy and τ0 , the time between attempts is proportional to the number of paths which are close to the path with minimum escape energy. The paths begin on the attractor and end on the boundary of the basin of attraction. Red points in Fig. 2(a) exemplify the bifurcation diagram obtained using the map (20) with δ = 0.05. There are no periodic orbits anymore but some relation (correlation) to the usual map is expected. In the simulations we discard 103 iterations (transient) before the noise is added. For increasing values of δ many ICs may diverge. Here we consider that an IC diverges when |yn | ≥ 103 . It is also possible to consider divergency when orbits are outside the corresponding basin of attraction from the noiseless case, as done in [19]. We observed that both methods provide similar results and in our case we do not need to determine the boundaries of the basins of attraction. In Fig. 2(b) the Lyapunov exponent λδ is plotted in red for the noisy map. Figure 3 displays the number of ICs which diverge after N = 10, 20, 30 and 40 iterations as a function of δ. Panel (a) shows r = 1, which is the regular period-2 dynamics from the noiseless case and (b) the chaotic case with r = 2. Clearly there is a critical value δcrit , above which many ICs diverge after N iterations and a noise-induced escape from the attractors occurs. Analysing the curves from Fig. 3 for the other values of r (not shown), we estimate that δcrit ≈ 0.2 represents well all periodic cases. This critical value decreases as r increases and for the chaotic case it approaches δcrit → 0. Similar behaviour was 6
10
10
10
10
(b)
(a) 8
10
6
10
Nº ICs - Diverge
10
10
10
10
4
10
N = 10 20 30 40
2
10
8
6
4
2
0
0
100.0
0.2
0.4
0.6
0.8
1.0
10 0.0
0.2
0.4
0.8
0.6
1.0
δ
δ
Figure 3: Number of ICs which diverge after N iterations as a function of δ for (a) r = 1 and (b) r = 2.
2 observed in [19], i. e. an almost zero minimum escape energy E0 (proportional to δcrit ) is needed to obtain the noise-induced escape from the chaotic attractor.
4.3. Correlations decay between the map (19) and noise For reference we start analysing correlation coefficients ρ and DC between the states xn generated by the quadratic map (19) with the Gaussian noise. For this, the sample {X = (xn+1 ) : n = 0, 1, 2, ..., N } is obtained from (19) and the sample {Y = (δξn+1 ) : n = 0, 1, 2, ..., N } from the Gaussian generator. We use 2 × 104 ICs randomly selected and equally spaced in the interval [−2, 2]. In this way we obtain the average 0.35
1.00
10
-3
0
-10
2
0.30
0.28
-3
0.01
0.33
〈DC20 〉
0.10
〈ρn 〉
〈DCn 〉
r = 1.0 1.3 1.38 2.0
2
1000
500
10
100
(a)
(b) 1000
n
0.25 0.0
0.2
0.4
0.6
0.8
1.0
δ
Figure 4: Plot showing the (a) time dependence of the average distance correlation and the average Pearson’s (inset) for δ = 1 and (b) noise dependence of the average distance correlation for n = 20.
Pearson’s correlation hρn (X, Y)i and the average distance correlation hDCn (X, Y)i. For simplicity we write hρn i ≡ hρn (X, Y)i and hDCn i ≡ hDCn (X, Y)i. The mean quantities hρn i and hDCn i are plotted in Fig.4(a) as a function of the time for four distinct dynamical regimes from the map (19), namely r = 1.0 for which the state xn has period-2, r = 1.3 with period-4, r = 1.38 with period-8 and r = 2.0 with a chaotic attractor. We use red colours for period-2, green for period-4, blue for period-8 and magenta for the chaotic motion. Results show in Fig.4(a) that while hρn i oscillates around zero, hDCn i decays with ∝ n−0.5 . Curiously, for very small times (n . 5), hDCn i decays faster for smaller values of r, so that for n = 20, for example, the value of hDC20 i is different for each r and remains independent of δ, as shown in Fig.4(b). The fact that the distance correlation is constant is a consequence of the scale invariance in {Y} mentioned in Sec. 2.2. The interesting point is that the distance correlation between the chaotic motion and the noise is larger than between the regular motion and the noise. In addition, for smaller periods hDC20 i decreases. This shows that hDC20 i 7
between the noise and two regular points (period-2), is smaller than hDC20 i between the noise and four regular points (period-4), and so on. In other words, hDC20 (2) i < hDC20 (4) i < hDC20 (8) i < . . . < hDC20 (20) i, where the superscript refers to the period-p. In the chaotic case the period is infinity, but since we are just iterating until N = 20, we say the period is 20. Therefore, for a given n & 5, hDCn i assumes values depending on the number of points from the attractors, and could be used to compare and quantify the size of distinct attractors. These results show the first distinction between both correlations and the relevance of the distance correlation in the context of dynamical systems. It is worth to mention that DC decays to zero asymptotically, as expected. However, from the physical and dynamical point of view, it is very interesting to know the qualitative behaviour of such decay. It can elucidate the underline dynamics, as demonstrated next. 4.4. Correlations decay between maps (19) and (20) Now we wish to analyse the correlation coefficients ρ and DC between the states xn generated by the quadratic map (19) with states yn generated by the map (20) with noise. For this, the sample {X = (xn+1 ) : n = 0, 1, 2, ..., N } is obtained from (19) and the sample {Y = (yn+1 ) : n = 0, 1, 2, ..., N } is generated from (20). In other words, we determine the correlations between the noiseless case (black lines in Fig. 2(a)) and the noisy case (red points in Fig. 2(a) for δ = 0.05) for specific values of r and increasing values of δ. After eliminating a transient of 103 iterations, for each IC we determine the coefficients ρN (X, Y) and DCN (X, Y) between both time series until N = 20. Due to noise, the states yn may change attractors and diverge. In such cases we choose another IC until we have a total of 2 × 104 ICs which did not diverge for N = 20. Also here the ICs are randomly selected and equally spaced in the interval [−2, 2] and we look at the mean quantities hρ20 i and hDC20 i. 0.50
1.00
〈ρ20 〉
(a)
(b) 0.10
0.10
0.01
r = 1.00 1.30 1.38 2.00
0.01
0.01 0.10
1.00
0.20
δ 0.65
1.00
(c) 〈DC20 〉
1.00
0.50
δ
(d)
0.50
0.50
0.25
0.20 0.01
0.10 δ
1.00
0.20
0.50
1.00
δ
Figure 5: Log-log plots showing the noise dependence of the (a) average Pearson’s correlation, (b) adjusted fitted curves from (a) inside the δ interval [0.2, 1], (c) average distance correlation and (d) adjusted fitted curves from (c) inside the same δ interval from (b). Functions for the fits are presented in Table 1.
4.4.1. Concordances of hρ20 i and hDC20 i In Fig.5(a) the log-log plot of hρ20 i is displayed and is to be compared with Fig.5(c), where hDC20 i is plotted, also in a log-log scale. For δ = 0 both correlations are maximal (not shown) and they are 8
independent of the dynamics, even in the chaotic regime (for identical ICs in both maps). For the periodic motion we observe that both correlations are equal to 1 for values of δ smaller than ∼ 0.03. In the chaotic regime the mean Pearson’s correlation is ∼ 0.33 and the mean distance correlation ∼ 0.46. This already shows that the distance correlation is more sensitive to existing nonlinear correlations between the chaotic data. For a noise intensity δ = 0.1 we have for period-2: hDC20 (2) i = 0.88 and hρ20 (2) i = 0.82, for period-4: hDC20 (4) i = 0.76 and hρ20 (4) i = 0.63, period-8: hDC20 (8) i = 0.67. and hρ20 (8) i = 0.50 and for the chaotic case: hDC20 (20) i = 0.35 hρ20 (20) i = 0.17. Thus, in all cases hDC20 i > hρ20 i showing that the distance correlations is more resistant to noise effects, independent of the dynamical regime. In addition, both correlation are Table 1: For distinct values of r, the Table summarizes the associated period, Lyapunov exponent λδ and the functions used to adjust the curves of the mean quantities hDC20 i and hρ20 i. In the chaotic case the period is infinity, but since we are just iterating until N = 20, we say the period is 20.
r 1 1.3 1.38 2.0
Period-p 2 4 8 20
λδ=0 -∞ -0.42 -0.33 0.69
hDC20 (p) i ∼ α eβ/δ ∼ 0.2178 e0.169/δ ∼ 0.2551 e0.133/δ ∼ 0.2757 e0.110/δ ∼ 0.3239 e0.006/δ
hρ20 (p) i ∼ 0.0158 δ −1.78 ∼ 0.0263 δ −1.46 ∼ 0.0285 δ −1.36 ∼ 0.1680 e−1.73 δ
more resistant to noise for lower periods, since they start to decrease from 1 at distinct values of δ. In other words, hDC20 (2) i > hDC20 (4) i > hDC20 (8) i > . . . > hDC20 (20) i. This is directly related to the Lyapunov stability exponent λδ=0 (see Table 1), lower periods have smaller negative Lyapunov exponent are also more stable under noise. Chaotic attractors with λδ=0 > 0 are less stable under noise. For values δ > δcrit ≈ 0.2, both correlations cross around ∼ 0.24 and for δ > 0.24 their values interchange for regular and chaotic regimes when compared to the δ = 0.1 case. In other words, for δ > 0.24 we have hDC20 (2) i < hDC20 (4) i < hDC20 (8) i < . . . < hDC20 (20) i which is the inequality observed in Sec. 4.3, and the numerical values of hDC20 (p) i at δ = 1 are very similar to those from Fig. 4(b). This means that for large values of δ the distinct dynamical regimes of the noisy map (20) are not detected anymore in the distance correlation. Just the dynamical regimes from map (19) are still relevant. From the adjusted curves in Table 1 we conclude that for δ → ∞, while Pearson’s correlation approaches zero, the distance correlation approaches α, which clearly agrees with the values of hDC20 (p) i from Fig. 4(b). 4.4.2. Discordances of hρ20 i and hDC20 i After discussing some similarities of both correlations, we analyse now the differences, which are mainly related to their qualitative behaviour as a function of δ. Figures 5(b) and (d) display the best adjusted curves related to the data. Table 1 presents a summary. The curves for the mean Pearson correlation can be well adjusted by an inverse power-law in the periodic regime and an exponential decay in the chaotic regime. On the other hand, the mean distance correlation follows α exp (β/δ) for regular and chaotic regimes, where α and β are adjusting parameters. Astonishingly, this is exactly the qualitative behaviour of the escape time decay 1/R observed in [19] (see also above). By comparing the values of β from Table 1 with the values of E0 from Fig. 3 in [19], we can affirm that β ≈ E0 . In addition, by comparing α with Fig. 5 from [19], we observe roughly that log10 (p/α) ≈ log10 (τ0 ), with p being the period of the orbit. From this the dimensionless adjusting parameter α ≈ p/τ0 is the period of the orbit divided by the escape time. Thus, we conclude that the mean distance correlation properly describes the noise-induced escape times through τ02 hDCN i ≈ 1/R. p
(21)
In other words, the mean distance correlation is able to reproduce correctly the qualitative behaviour of noise-induced escape times, while the Pearson correlation is not. 9
5. Measuring mixing in chaotic regimes There are several models to evaluate mixing process in chaotic dynamics, such as, recurrence, transitivity, ergodicity and entropy. The purpose of the present Section is to check the ability of the distance correlation to describe mixing in chaotic regimes and compare it to other mixing measure, namely the intensity of segregation [22, 21]. We call to attention that while the distance correlation is determined comparing pairs of data X and Y, the intensity of segregation is obtained from just one set of data. 5.1. Distance correlation As described above, to reckon DC we need a pair of samples X and Y. They can be obtained by starting two distinct ICs, x01 = 0.1 and x02 = 0.2 in the chaotic regime and iterate them for N = 2 × 103 . Thus, we obtain the data sequence from (X, Y) = {(x01 , y01 = x02 ), (x2 , y2 ), . . . , (xn , yn ), . . . , (xN , yN )}, and can determine DC. Figure 6(a) displays the the log-log plot of DC as a function of time for distinct values of r = 1.6, 1.7, 1.8, 1.9 and 2.0, all in the chaotic regime. If DC tends to zero in time, it can be considered as a valid statistical measure of mixed states. The strong oscillations observed in the behaviour of DC disappear
DCn (x01,x02)
1.00
0.10
0.01 2
(b)
1
(a)
0.1
r = 1.6 1.7 1.8 1.9 2.0
10
0.01 1000 2
100
10
n
100
1000
n
Figure 6: (a) Log-log plot of the time evolution of DCn (x01 , x02 ) between two trajectories with initial conditions x01 = 0.1 and x02 = 0.2 in chaotic regimes and (b) hDCn (x01 , x02 )i for many ICs starting distinct values of x02 .
when more ICs are used. This is shown in the log-log plot in Fig.6(b) by starting ICs with x01 = 0.1 and x02 chosen randomly and equally spaced in the interval [−2, 2]. It clearly shows that hDCn (x01 , x02 )i decays as a power-law with ∝ n−0.5 . 5.2. Intensity of Segregation The first step to obtain the intensity of segregation [26] is to determine Dj =
measured quantity , q
(22)
where q is the total number of observations. Dividing the interval [−2, 2] of states from the one-dimensional quadratic map in S subintervals of length ∆xj (j = 1, ..., S), we calculate the probability that each subinterval is visited using Eq.(22). The mean of Dj by the sum over subintervals is hDi =
S 1X Dj , S j=1
(23)
and the mean of the square density Dj2 is given by hD2 i =
S 1X 2 D . S j=1 j
10
(24)
1.0000
S=103 S=104 S=105 n-1
I
0.1000
0.0100
0.0010
0.0001 0 10
101
102
n
103
104
Figure 7: Log-log plot of the intensity segregation I as a function of the iteration.
For the case of mixing in a trajectory, the final state of mixing is characterized by a minimum statistical square density h(Dj − hDi)2 i ≡ hD2 i − hDi2 → 0. (25)
The rate of decrease of hD2 i/hDi2 characterizes the mixing process and depends on time and also on the number of subinterval. From the above quantities it is possible to determine the the intensity of segregation (I), which is a statistical measure [26]. For comparison with DC, we use the modified intensity of segregation (normalized between 0 and 1) defined as [22, 21] h(Dj − hDi)2 i I= . (26) hDi(1 − hDi)
When I tends to zero in time, the result is a good mixing state. In Fig.7 we display the decrease of the intensity of segregation for one trajectory in the chaotic regime from the quadratic map (19) with r = 2.0 and S = 103 , 104 and 105 . For this we consider N = 104 iterations of the map and discard a transient of 4 × 103 . Until iterations around n = 400 we observe a linear decay in the log-log scale. For the case of S = 105 the linear decay exist for √ all interval of iterations. The regime with a power-law decay of I satisfies ∝ n−1.0 . Thus, the square-root I decays in time as the mean distance correlation shown in Fig. 6(b), showing that the later is capable of correctly describing the qualitative of chaotic mixing. 6. Conclusions
The distance correlation [1] is a well defined statistical quantity able to measure dependence between random vectors of multivariate data. The relevant property is that it is zero if and only if the random vectors are independent (see [1, 3]). In addition, the random vectors can have any dimension. As detailed in the Introduction, the distance correlation has recently being applied in distinct areas. This work analyses properties of the distance correlation to describe dynamical systems. More specifically, it is studied for the quadratic map in the periodic and chaotic regimes subjected to a Gaussian noise. In other words, we determine the distance correlation (i) between the usual quadratic map (19) and a Gaussian noisy signal, and (ii) between the map (19) and the quadratic map (20) with the Gaussian noise. Results are compared to the classical Pearson’s correlation which only detects linear dependences between sample data. In general, we found that the distance correlation value is always larger than the Pearson’s correlation. In case (i), while Pearson’s correlation is almost zero for all times, the distance correlation decays in time roughly with n−0.5 and it is able to detect distinct size of the attractors from the map (19). In case (ii) both correlations are more resistant to noise for trajectories with smaller Lyapunov exponents from the noiseless case. Thus, the local linear stability of trajectories affects both correlations for small noise values. For larger noise values, for which almost all ICs from the map (20) diverge very fast, both correlations are sensitive to the number 11
of points which contain the attractors, i. e. the size of attractors. Also in this case the distance correlation is better than Pearson’s correlation to detect nonlinearities. In distinction to Pearson’s correlation, we found that the qualitative decay of the distance correlation as a function of the noise intensity is remarkably equivalent to the noise-induced escape times decay from attractors [19]. This means that only the distance correlation correctly detects the complicated nonlinear dynamics involved in escape processes. In addition, for the noiseless case of the quadratic map, the distance correlation is able to quantify the degree of mixing between chaotic trajectories. This is confirmed by comparing the decay of the distance correlation with the intensity of segregation. Future investigations may focus on the relevance of the distance correlation in describing other properties in dynamical systems, like decays in leaking chaotic systems [27], in open billiards [28], in stickiness [29], in high-dimensional non-integrable systems [30] in coupled oscillators [31], in chimera states [32], in the evolution of scientific disciplines [33], to mention a few. acknowledgments CFOM thanks FAPEAM and MWB thanks CNPq for financial support. The authors also acknowledge computational support from Professor Carlos M. de Carvalho at LFTC-DFis-UFPR. 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] [25] [26] [27] [28] [29] [30] [31] [32] [33]
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