The level crossing and inverse statistics analysis of DAX and oil price time series are given. We determine the average frequency of positive-slope crossings, View the MathML sourceνα+, where View the MathML sourceTα=1/να+ is the average waiting time for observing the level αα again. We estimate the probability P(K,α)P(K,α), which provides us the probability of observing KK times of the level αα with positive slope, in time scale TαTα. For analyzed time series, we found that maximum KK is about ≈6≈6. We show that by using the level crossing analysis one can estimate how the DAX and oil time series will develop. We carry out the same analysis for the increments of DAX and oil price log-returns (which is known as inverse statistics), and provide the distribution of waiting times to observe some level for the increments.
Stochastic processes occur in many natural and man-made phenomena, ranging from various indicators of economic activities in the stock market, velocity fluctuations in turbulent flows and heartbeat dynamics, etc. [1]. The level crossing analysis of stochastic processes has been introduced by (Rice, 1944, 1945) [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] and [26], and used to describe the turbulence [16], rough surfaces [27], stock markets [28], Burgers turbulence and Kardar–Parisi–Zhang equation [29] and [30]. The level crossing analysis of the data set has the advantage that it gives important global properties of the time series and do not need the scaling feature. The almost of the methods in time series analysis are using the scaling features of time series, and their applications are restricted to the time series with scaling properties. Our goal with the level crossing analysis is to characterize the statistical properties of the data set with the hope to better understand the underlying stochastic dynamics and provide a possible tool to estimate its dynamics. The level crossing and inverse statistics analysis can be viewed as the complementary method to the other well-known methods such as, detrended fluctuation analysis (DFA) [31], detrended moving average (DMA) [32], wavelet transform modulus maxima (WTMM) [33], rescaled range analysis (R/S) [34], scaled windowed variance (SWV) [35], Langevin dynamics [36], detrended cross-correlation analysis [37], multifactor analysis of multiscaling [38], etc.
We start with formalism of the level crossing analysis. Consider a time series of length nn given by x(t1),x(t2),…,x(tn)x(t1),x(t2),…,x(tn) (here x(ti)x(ti) is the log-return of DAX and oil prices). The log-return x(ti)x(ti) is defined as x(ti)=ln(yi/yi−1)x(ti)=ln(yi/yi−1), where yiyi is the price at time titi. Let View the MathML sourceNα+ denote the averaged number of positive slope crossing of x(t)=αx(t)=α in time scale T=nΔtT=nΔt with Δt=1Δt=1 (we set also the average 〈x〉〈x〉 to be zero). The averaged View the MathML sourceNα+ can be written as View the MathML sourceNα+(T)=να+T, where View the MathML sourceνα+ is the average frequency of positive slope crossing of the level αα. The positive level crossing has specific importance that it gives the next average time scale that the price yiyi will be greater than the yi−1yi−1 again up to specific level. For narrow band processes, it has been shown that the frequency View the MathML sourceνα+ can be deduced from the underlying joint probability distribution function (PDF) for xx and View the MathML sourcedx/dt=ẋ. Rice proved that [2]
equation(1)
View the MathML sourceνα+=∫0∞ẋp(x=α,ẋ)dẋ,
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where View the MathML sourcep(x,ẋ) is the joint PDF of xx and View the MathML sourceẋ. For discrete time series (of course all of real data are discrete), the frequency View the MathML sourceνα+ can be written in terms of joint cumulative probability distribution, P(xi>α,xi−1<α)P(xi>α,xi−1<α) as [39],
equation(2)
View the MathML sourceνα+=P(xi>α,xi−1<α)=∫−∞α∫α∞p(xi,xi−1)dxidxi−1,
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where p(xi,xi−1)p(xi,xi−1) is the joint PDF of xixi and xi−1xi−1. The inverse of frequency View the MathML sourceνα+ gives the average time scale TαTα that one should wait to observe the given level αα again.
The rest of this paper is organized as follows. Section 2 is devoted to summary of level cross analysis of DAX and daily oil price log-returns. The inverse statistics of DAX and Oil price time series are given in Section 3. Section 4 closes with a discussion and conclusion of the present results.
In summary, we analyzed the DAX and oil daily price log-return time series using the level crossing method and find the average waiting time TαTα for observing the level αα again. This is a similar analysis as what has been done in Refs. [53], [54] and [40]. They have been carried out the level crossing of the volatility time series, instead of the time series itself. We define and estimate the probability of observing KK times of the level αα, P(K,α)P(K,α) in time scale TαTα. We show that by using the level crossing analysis one can estimate the future of the daily DAX and oil time series with good precision for the levels in the interval −0.5<α<0.5−0.5<α<0.5. Also, using the inverse statistics we estimate the waiting time probability distribution for two financial markets, i.e. oil and DAX time series.
