Let us move to a pseudo experimental approach. It is assumed that we only have a single sequence of measurements obtained at different times. We seek a hidden determinism in our experimental data. Here, the “experimental results” are given by a numerical integration of Equation (1). In this section, we just try the reconstruction method provided by the tool box CDA22 [1] .

It is assumed that only the x-components of the vector, which gives the state of the system, is measured or calculated. Then, , where G is a scalar function of the state vector. We define what is called delay coordinate vectors such as [3] [4] [6] [7] [8]

where is here a simple parameter and T is the time delay.

Packard et al. [4] [6] have shown that starting from the time series (5), one may reconstruct the trajectory of the attractor in a -dimensional embedding space by means of vectors

with, where j is an integer and is the minimum sampling time. Time is the sampling interval between the first components of successive vectors. Based on (6) and assuming the time window spanned by the p embedded points is included in the time window spanned by the N values of the initial time series, it holds that. If where n is an integer, it may be written that is:. Thereafter, we will set j = 1 and we shall have

The space reconstruction requires to select values of the reconstructed space dimension and the time delay. The embedding theorem [6] [9] [10] [28] tells us that the following sufficient but not necessary condition must be verified

where is the box-counting dimension of the attractor. Considering that [2] , the condition (8) is satisfied when

that is

To verify the relevance of this condition, the reconstruction was achieved considering several values of and using the CDA22 tool box [1] . Let x(t) be a one-dimensional data set evaluated at equal increments of the variable t. The values of x(t) were obtained by solving numerically Equation (1) using a fourth order Runge-Kutta scheme. These values of x play the role of experimental measurements. Using CDA22 tool box the correlation dimension of the Rössler attractor was calculated for. It is reminded here that where is the sampling rate, we chose and found . Then, was calculated for different values of. First the same values of and n were considered. The results are shown in Figure 4 Very different values of and n were also considered, we chose and. Figure 4 shows that the two curves merge.

Figure 4 shows the curves versus obtained with two set of parameters. The full line is obtained with and, squares are ob-

tained with and. When, squares are very closed to the full line curve and the plateau seen in corroborates condition (10) which means that, for the long limited noise free data sequence considered here, Taken’s theorem is satisfied. The correlation dimension becomes an invariant on the attractor when the embedding dimension used for the computation increases. Then, the optimal embedding dimension is reached [11] . Still, Figure 4 shows that is a sufficient but not necessary condition, would be quite suitable. The typical problem with this way to determine the embedding dimension is that it is very time-consuming for computation.

Figure 4 also shows that the results obtained for are not, at least in this range of values, sensitive to n and consequently to the time delay T. Because the initial data set contains about 32,000 values and is noise-free, results obtained are in good agreement with Takens [6] [7] [9] . In this case, the existence of a diffeomorphism between the original attractor and the reconstructed image exists for almost any choice of time delay and a sufficiently high embedding dimension.

It is known that [1] , then, as, one must have [29] which in good agreement with our numerical results.

The optimal embedding dimension has also been calculated by using Cao’s algorithm which is much less time-consuming [9] [14] . According to the IIIa paragraph notations and (7), Cao defines as a function of (with) the i^th reconstructed vector and the embedding dimension, is written as

where denotes the sup-norm, i.e.,

and is the i^th reconstructed vector in the dimensional reconstructed space. Subscript refers to the which is the nearest neighbor of in the dimensional reconstructed space. Integer depends on i and. If is qualified as an embedding dimension by the embedding theorem [1] [2] [3] [6] [27] , then any two points which stay close in a dimensional reconstructed space will be still close in a dimensional reconstructed space. Such a pair of points are called true neighbors, otherwise, they are called false neighbors. To qualify two points to be false neighbors, must be larger than a threshold value which depends on the i state point chosen. To avoid this problem, the quantity is defined as the mean value of all’s

with T =, so that is only dependent on the embedding dimension and the time delay. To investigate the variation of E between and, we define as. If stops changing when, then is the minimum embedding dimension we are looking for. When meaningful predictions from chaotic time sequence cannot be made, data appears to come from a random system. Considering that provided by a random set of numbers will never attain a saturation value as increases, it is necessary to distinguish deterministic chaotic from random data sequences. In most cases, it is difficult to resolve whether is slowly increasing or has stop changing if is sufficiently large. In fact, since available observed data samples are limited in number, it may happen that stops changing at some value although the time series is random. To solve this problem Cao [14] has suggested to consider the quantity which is useful to make distinction between deterministic signals from stochastic ones. Let us consider the following quantity

where the meaning of is the same as above. As for, to study the variations, an quantity is defined as . For random data, since the future values are independent of the past values, will equal unity for any. In the case of deterministic data is certainly related to and cannot be a constant for all. In other words, there must exist some values such that. Cao recommends to calculate both and for determining the minimum embedding dimension of a scalar time series. Figure 5 shows results obtained with Cao’s method applied to an about 32,000-data sequence by using CDA22 tool box.

Figure 5 shows that is related to and is not a constant for all

. This is in good agreement with the fact that the data used are deterministic. When n = 15, the minimum embedding dimension is. When n = 1, the same value is found for. This means that, when enough points are considered and when no noise is considered, the minimum value for is almost independent of T.

Then, Cao’s algorithm has been applied on a much smaller data sequence of about 4000 values, and were calculated. It was shown that can be different from zero in all the cases. It confirms the deterministic character of the data. It also shows that when n = 50 or when n = 10, the minimum embedding dimension is close to. A saturation value as increases is more difficult to discern when n = 1. Still, we can conclude that for this relatively small number of data, the minimum embedding dimension is still almost independent of n.

Cao’s method has been applied then to the 4000 values data-sequence with noise added. Figure 6 show results obtained when a white Gaussian noise with variance one is added to the 4000 values data-sequence.

In Figure 6, when n = 50 and n = 10, can be clearly different from unity, reaches a constant value for about the same value of:. For n = 1, remains close to unity, our data do not appear to be deterministic.

Then, a white Gaussian noise with variance five was added. Figure 7 show that a high value of n is necessary to determine a minimum embedding dimension. The following values for and were found

In this case is clearly different from a constant only for n = 100.

In summary, it was shown that for noise-free data of very long length, the reconstruction is valid for any time delay as far as the embedding dimension is high enough. When going to small number noisy data samples, the time delay used to determine the minimum embedding dimension cannot have any value.

The Cao’s method [14] , used to determine the optimal embedding dimension , has shown that for a sufficiently long noise free data set, the time delay T and the minimum embedding dimension are almost independent and the delay time T can be set arbitrarily. However, when white Gaussian noise is added, T varies with and an irrelevant partnership between T and will directly impact the equivalence between the original state space and the reconstructed one. Moreover, in the real world, measurements data set are finite and noisy, and in this case, a more useful quantity to estimate the embedding dimension is the delay time window which is the entire time spanned by the vectors. Martinerie et al. [17] have shown that is an essential factor for estimating the correlation dimension. In their paper, they have demonstrated that first, determines the correlation integral characteristics and second, the correlation integral is very sensitive to the values. H.S. Kim et al. [7] and Wei-Dong Cai et al. [8] have developed a method which uses the correlation integral and is based on a statistic similar to the BDS statistic that Brock et al. [12] [13] used for their development for distinguishing random time series from chaotic or nonlinear stochastic time series. By using the notations of Equations ((6) and (7)), the BDS statistic applied to the time series writes [7] [8]

where is the correlation integral

r is a search radius and H is the Heaviside function: H(a) = 0 if a < 0 and H(a)=1 if a ³0. N is still the size of the data set, n is the index lag, is the number of embedded points in the dimensional space and still denotes the sup-norm. measures the fraction of pair of points whose sup-norm distance is not greater than r. Brock et al. have proved that if the stochastic process is independent and identically distributed (iid) then, for all and r. The density of points in a hypersphere of radius r scales like. It means that the correlation integral of the process behaves like the one of an independent random variables product which is the product of the correlation integral of each random variable. This leads to interpret the statistic S as a dimensionless quantity which highlights determinism. A significant nonzero value of S is evidence for determinism in the time series. The technique developed by Kim et al. called the C-C method consists in subdividing the time series into n disjoint time series each one of N/n values. One has

The average of the statistical quantity given by Equation (14) is defined as

when, can be rewritten in the following way

The locally optimal times may be either the zero crossing of for all r or the times at which shows the least variation with r, since this indicates a nearly uniform distribution of points. From several representative values, we define the quantity

which measures the variations of with r.

For data set with finite sample sizes N, appropriate choices for, r and n should be in agreement with the BDS statistic. For example, when applied to a data set with a sequence of about 4000 values, varies in the range [2, 7], n varies in the range [1, 200] and varies in the range with k = 1, 2, 3, 4 and is the standard deviation of the time series. We then define the average of quantities given by Equations ((17) and (19)).

and

As locally optimal times are either zero crossing of or times at which shows the least variation with r, we look for the first zero crossing of or the first local minimum of to find the optimal times for data independence which will gives T. The optimal time is the time for which and are both closest to zero. As the two quantities and may not be minimum at the same time (see Figure 8), we may look at the minimum of the quantity [7] [8]

which gives the delay time window. T is in a sense the minimum value of, it is determined as the minimum of the curve versus n running from 1 to 200. The C-C method has been programmed in R by using the packages “nonlinearTseries and tseriesChaos”. An organigram of the C-C method used in this work is presented in Appendix 1 and the results obtained on low size data sequence of about 4000 values are presented in Figure 8 & Figure 9.

The first local minimum of (Figure 8) occurs when n = 12 and represents the optimal delay time T.

The minimum of (Figure 9) occurs when n = 157 which is the optimal time embedding window. Then, the embedding dimension is given by, where the function int() represents the integer part.

White Gaussian noises with different variances σ (σ = 1, σ = 5, σ = 10) has been added to the noise free original signal of 4000 data sequence and the C-C method has been applied to the time series where x is the

noise free original signal, is a white Gaussian noise with zero mean and a variance σ, α is the strength of the noise and represents the level of noise in percentage (20%, 30%, 50%, 70%, 100%). The C-C method is performed for each σ and α values and the variations of T and compared to the reference values obtained from the original 4000 noise free data set, are shown in Table 1. The σ and α values ensuring the stability of the C-C method when applied to noisy data set are those for which.

The parameter α is linked to the signal to noise ratio SNR which can be defined as where is the mean value of the noisy data sequence and is the standard deviation of the noisy part of that is. As the standard deviation of may be written with and, one has:

then, the C-C method should be stable against the noise for values of α such as.

In Table 1, we observe that for σ = 1, the C-C method is stable against the noise when. For σ = 5, it is stable when, and for σ = 10, it is stable when.

In summary, the C-C method is a relatively simple method easy to implement that can be used for relatively small data set to determine both the time delay T and the time delay window. This method is robust against low and intermediate noise level.

In their paper, W.D Cai et al. [8] pointed out some problems that limit to the C-C method. The first one is that there are local minimal points whose values are very close to the minimum of and they disturb the minimum estimation (see Figure 9). The second one is that shows high frequencies oscillations, increasing with n, that can affect the estimation of the first local minimum of (see Figure 8). Based on these remarks, W.D. Cai et al. have developed in their paper [8] a new method called C-C-1 different from the C-C method calculating with another average method.

Starting from the N values initial data set and according to (7), the number of the embedded points calculated from the delay time T in the dimensional reconstructed space is. A positive integer q independent of the delay time T is selected as a constant, to subdivide the embedded points series into q subseries, each with embedded points where the “int” function means integer part.

as each component is composed of components, Equation (24) can be rewritten as

Kim et al. have shown in their paper [7] that, when using the BDS statistic on time series, the sample data size N should be appropriate relatively to the values of, r and n. They have shown that for finite time series of size N ³ 500 the statistic represents the true correlation of the time series. The parameter q can be adjusted so that the size of the time subseries will not be too short.

The average of the statistical quantity given by Equation (14) is defined as follows:

The definitions of are given formally by Equations (20)-(22). The first local minimum of is the optimal delay time T.

If we define the mean orbital period P of a chaotic system as the mean period generated by the oscillations of the chaotic attractor in the phase space orbits, an optimal value for would coincide with the first period of the N values initial time series S. Cai has shown in his paper that with the new statistical quantity average he defines in (26), the peak values of corresponds to the orbital period P values of S and all the points that bring this values are the minima of.

Therefore, a new quantity is defined by

By looking for the minimum of, we estimate the optimal time window corresponding both to the minima of and to the first period P of the initial N values time series S.

The C-C-1 method has been programmed in R language by using the same packages as with the C-C method. An organigram of the C-C-1 method is presented in Appendix 2 and results obtained the 4000-values data sequence with q = 19 are shown in Figure 10.

High frequencies oscillations occurring when applying the C-C method (red dashed line) have disappeared in the C-C-1 method (black solid line) (see Figure 10). In Figure 10, the first local minimum occurs when n = 10 while it

was n = 12 for the C-C method. n = 10 represents the optimal delay time T.

Figure 10 shows that the estimate of the optimal delay time T in the C-C-1 method (n = 10) is quite the same as with the C-C method (n = 12). The first local minimum of coincide with the first period P of the chaotic time series S, and gives the optimal delay time window. The graph of in Figure 11 (black solid line) enables to distinguish clearly the first local minimum from the other local minima. The estimate of the optimal delay time window occurs for n = 33 when is minimum, or with the first peak value of [7] and is different from the optimal delay time window estimated in the C-C method. The optimal embedding dimension is given by and is closer to the estimation given by the Cao’s method (paragraph III c) and agrees with the results presented in paragraph IIIb.

An optimization of the C-C-1 method should be to define a criterion for the optimal selection of the q parameter value. Based on the results obtained from the C-C method, a criterion is suggested here to select optimally the q parameter value. We define first the quantity

where is given by Equation (19) and. The op-

timal choice of the q parameter value should coincide with the first value of n at which shows the least variation with. This requires to define the quantity

Figure 12 shows the evolution of Q(n) as a function of n.

As the value of the q parameter should be chosen so that the time subseries will not be too short. The optimal q value may be chosen as the first value of n for which shows a minimum variation with, that is q = 19. An organigram for obtaining the graph of the variable Q as a function of

n is presented in Appendix 3.

The C-C-1 method has been applied to the time series where is the noise free original signal obtained from the about 4000 values data set, is a white Gaussian noise with zero mean and a variance σ (σ = 1, 5, 10), α is the strength of the noise and represents the level of noise in percentage (20%, 30%, 50%, 70%, 100%). T and variations with the different values of σ and α are shown in Table 2.

We observe that the C-C-1 method gives stable results for σ = 1, for σ = 5 since, and for σ = 10 since.

In conclusion, the C-C-1 method is an improvement of the C-C method. The original time series is subdivided by setting a parameter q which is independent of the time delay T. Tests performed on this method show that it gives more reliable and stable estimates of the optimal delay time T and the optimal time delay window. Tests show also the robustness of this method in presence of noise as the embedding dimension remains equal when noise free data set is degraded with white Gaussian noises with variances respectively equal to 1, 5, 10.