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 m-dimensional trajectories through a two-dimensional 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




Recurrence rate (RR)

Percentage of black points in RP

Correlation integral

Determinism (DET)

Percentage of black points which are part of diagonal lines of at least length L

Measures predictability

Entropy (ENTR)

Shannon entropy of the distribution of diagonal lines P(L)

Quantifies the complexity of the deterministic structure

Laminarity (LAM)

Same as DET for vertical lines

Quantifies the occurrence of laminar states

Trapping Time (TT)

Mean length of vertical lines

Measures the mean time that the system sticks to a certain state

Подпись: Fig. 6 Recurrence plot for stock return (left) and price (right)

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


RR (%)

DET (%)


LAM (%)














Price change






Absolute value of price change












Relative spread






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,

Подпись: v = ln Подпись: РДР Pe+Ae(l) Подпись: ln Подпись: s + As Подпись: -1

Where p is embedding lag parameter for reconstructed space (introduced in this chapter), v is the correlation dimension and K2 is order-2 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:

Подпись: Conclusion A review of modern methods for identifying nonlinear dynamics was given; all algorithms were applied to real microstructure intraday MICEX data while describing difficulties of implementation in practice. It is worth mentioning that all the procedures are applicable without a priori knowledge of the underlying model or class of models. Results can be structured as follows: • According to all identification techniques, return, price changes and relative spread show signs of a complex nonlinear underlying structure. Thus a random walk model isn’t appropriate for them (such as Merton model for returns). The Scheinkman-LeBaron procedure shows that the history

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.


Подпись: of these variables helps to predict future values. Unfortunately, obtained results indicate but do not imply deterministic behavior of the variables. • Price and spread dynamics in the correlation integral approach show purely stochastic behavior, which can also be due to the large amount of noise in initial data. Future values are not fully predicted by information in history. • Recurrence quantification analysis shows the presence of determinism in returns, relative spread and absolute price change dynamics, but no determinism for spread. Results for price are clearly incorrect due to nonstationarity of the initial series. Nonstationarity also questions the reliability of obtained price results of other methods. This can explain the contradictory conclusion: return and price change appears to be deterministic in nature, while price is purely stochastic— though it is a deterministic function of return/price change. A simple explanation can be proposed: due to integral transformation of return/price change, the price series loses stationarity and becomes inappropriate for given methods. Dependency on time makes it impossible to recognize similar patterns in data, so the price series is identified as stochastic process.


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