Recurrence Plot Approach
Two main problems with the correlation integral approach are: (1) considerable amount of data necessary for reliable estimates and (2) a priori choice of embedding parameters for reconstructed phase space. Recently, a more elaborate technique proved to be useful for analysis of nonlinearity. This new approach uses recurrence plots as a tool for visualization of observed trajectories. Recurrence plots show similarity in dynamics over time without specifying the structure of underlying processes. For observed series xt it can be expressed as
R {U, tj) = 0 (є –  xtj D,
where є is a specified threshold parameter. Usually the system dimension must be 2 or 3 to allow visualization, otherwise its trajectories can be observed only through projection on two or three dimensional subspaces. A recurrence plot enables us to investigate mdimensional trajectories through a twodimensional representation of its recurrences. Figure 5 demonstrates RPs for white noise processes and for predictable periodic sine function (diagonal lines are marked red).
Continuous diagonal lines prevail for sine RP, which is expected for predictable systems. Base structures in the recurrence plot can be easily interpreted: diagonal lines parallel to the main diagonal mean predictability at some periods of time, line length measures period of predictable behavior; horizontal and vertical lines indicate stability of the system state over a period of time. Diagonal lines turn out to be the main characteristic for research of complex deterministic behavior. Unfortunately, real finance data series produce quite complicated RPs that cannot be analyzed visually and need quantitative measures for determinism. Figure 6 shows RPs for Lukoil stock return and price.
Fig. 5 Recurrence plot for white noise process (left) and sine function (right) 
Table 2 Recurrence quantification analysis measures

White spaces in a price’s RP mean abrupt changes in price dynamics, which is the consequence of nonstationarity of the variable. Stationarity of the input signal is one of the implied properties in many nonlinear analysis techniques. Correlation integral methods described the above produced results consistent with our expectations about price. However, as will be shown below for recurrence analysis, applying methods to nonstationary data can lead to counterintuitive results.
A number of measures were introduced with the aim of quantifying structures found in RP to go beyond visual classification. Table 2 shows some of the main characteristics.
Results of quantification analysis are shown in Table 3. A presence of deterministic behavior is shown for stock returns, relative spread and absolute value of price change. The situation is unclear for price change series and no determinism was detected for spread. As we can also see, price series has the best DET value and low entropy, which implies deterministic dynamics. The result is counterintuitive and not consistent with expectations from a simple visual examination of RP, phase trajectories or previous results. The observed effect originates due to the nonstationary
Variable 
RR (%) 
DET (%) 
ENTR 
LAM (%) 
TT 
Return 
6.23 
0.91 
0.19 
56 
16 
Price 
6.95 
3.7 
0.28 
78.02 
21 
Price change 
5.91 
0.17 
0.32 
58.92 
16 
Absolute value of price change 
7.91 
0.39 
0.26 
69.81 
17 
Spread 
8.19 
0.03 
0.68 
78.38 
21 
Relative spread 
6.43 
0.51 
0.23 
73.92 
18 
Table 3 Recurrence quantification analysis results for stock dynamics 
nature of price dynamics and, thus, an insufficient number of recurrence points. This leads to unreliable estimates of quantification measures.
Another use of recurrence plot approach was introduced by Thiel et al. (2004). Let P£(l) be the probability to find a diagonal line of at least length l. It can be shown that the following approximate equality holds:
Pe(7) ^ sve~lpK2,
Where p is embedding lag parameter for reconstructed space (introduced in this chapter), v is the correlation dimension and K2 is order2 Renyi entropy of the system. Based on this formula one can estimate Renyi entropy as a slope of Pe(l) in log scale and correlation dimension via simple formula:
Thiel et al. (2004) have shown that both estimates are independent of embedding parameters, which solves one of the correlation integral problems at least to some extent.
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