توزیع تقریبی از برآورد اینترمیتنسی سری زمانی: برنامه های کاربردی برای داده های اقتصادی و بالینی

The intermittency of a time series can be defined as its normalized difference in scaling parameters. We establish the central limit theorem for the estimates of intermittency under the null hypothesis of a random walk. Simulations of random walks indicate that the distribution of intermittency estimates is slightly negatively skewed and positively biased, but that the skewness and bias approach zero as the length n of the random walks increases. We provide a formula by which the sample variance of the intermittency estimates of these simulations can be used to approximate the standard error of the intermittency for any large n. These results can be used to test whether the intermittency estimate of an observed long time series is significantly greater than zero, the intermittency of a random walk. This test reveals that the intermittency estimates of the S&P 500 index and of the heart rate of a human adult are significantly positive. The hypothesis testing proposed in this paper can also be applied to other observed time series to determine whether their intermittency estimates are sufficiently high for the series to be considered intermittent, or whether their estimates are small enough to be consistent with a random walk.

Many observed time series are intermittent in the sense that observations differ dramatically from previous observations from time to time. The intermittency of a stationary time series {Δxi}i=1n−1 is informally defined as the tendency of its absolute values, {|Δxi|}i=1n−1, to be much greater than the most probable absolute values. Thus, intermittency increases with the skewness of |Δxi| for single-modal distributions. The intermittency of a nonstationary time series {xi}i=1n with stationary increments can be quantified by the intermittency of the first difference, {Δxi≡xi+1−xi}i=1n−1 ( Davis et al., 1994). Although intermittency can easily be confused with corruption by outliers, intermittency is an essential property of the system studied, whereas corrupt data is contaminated by factors external to the system of interest. For example, unpredictable, fast increases in the heart rate resulting from physiological activity exhibit intermittency, but isolated jumps in heart rate data due to equipment failure are considered to be outliers. Thus, intermittency is estimated to characterize data, but outliers can impede the estimation of parameters of interest. Contamination by outliers can falsely elevate estimates of intermittency, requiring robust methods of intermittency estimation (Bickel, 2001b). Since intermittency generally does not depend on the mean or variance of a time series, it can be used as another summary statistic to describe data and compare data to models ( Davis et al., 1994; Bickel, 1999b). Several models of intermittency are available, such as multiplicative-rate point processes, fractal renewal point processes ( Bickel, 1999a), and multiplicative cascades ( Davis et al., 1994), including the multifractal α-model ( Schertzer and Lovejoy, 1987). There are multiple formal definitions of intermittency, some of which are based on a particular multifractal model. For example, Holley and Waymire (1992) defined an intermittency parameter to characterize a multifractal cascade. Herein we instead study the asymptotic distribution of an estimator using a more general definition of intermittency, based on the theory of multifractal scaling ( Bickel, 1999b). Rényi (1959) introduced the concept of multifractal scaling by generalizing the concept of dimension in the context of probability theory. Multifractal statistical analysis has since successfully described various data sets ( Falconer, 1994), including those of turbulence and the spatial distribution of rainfall ( Holley and Waymire, 1992). We assume that x(t) is a scaling process in the sense that it satisfies the relation s(q;θT)=θH(q)s(q;T), the solution of which is equation(1) s(q;T)∝TH(q), where the structure function, s(q;T), is equation(2) s(q;T)≡{E[|x(t+T)−x(t)|q]}1/q. Here View the MathML source, and E[•] denotes the expectation of its argument. Note that s2(2;T) is the variogram of x(t), and that s(2;T) is the standard deviation of x(t+T)−x(t) and s(q;T) is the qth-power deviation of x(t+T)−x(t) when E[x(t+T)−x(t)]=0. Bickel and Lehmann (1976) define the qth-power deviation and give the requirements for a measure of dispersion, which are satisfied by s(q;T). The class of scaling processes is broadened by including all x(t) for which Eq. (1) applies over the range of scales 0<Tmin⩽T⩽Tmax. The generalized Hurst exponents, H(q), obey View the MathML source, and View the MathML source ( Davis et al., 1994). For slowly varying processes, such as fractional Brownian motion (fBm), View the MathML source, where the constant H is called the Hurst exponent. Beran 1992 and Beran 1994 reviews several results on statistical inference for such processes. These processes are called unifractals or monofractals since they have a single scaling exponent; processes for which H(q) depends on q are called multifractals if qH(q) is nonlinear; such processes are said to be multiscaling. If x(t) is ordinary Brownian motion (Bm), then H=1/2. Bickel (1999b) defined the intermittency, χ(p,q), of x(t) by equation(3) View the MathML source The intermittency is thus the extent to which H(q) deviates from constancy. It thereby defines the intermittency of a nonstationary process x(t), with View the MathML source for unifractal, nonintermittent processes and View the MathML source for intermittent processes ( Bickel, 1999b). χ(p,q) can also quantify the intermittency of a stationary process View the MathML source through the cumulative process, View the MathML source; the intermittency of View the MathML source is then defined as the intermittency of x(t). For example, if View the MathML source is a stationary point process, then its intermittency is defined by χ(p,q) of the corresponding counting process, View the MathML source ( Bickel 1999b and Bickel 2001a). A stationary, nonnegative process View the MathML source has generalized codimensions C(q) equal to χ(1,q) of its cumulative process x(t) ( Bickel, 1999b). The codimensions are related to the dimensions D(q) of Rényi (1959) by C(q)=1−D(q). Holley and Waymire (1992) provide D(q) for a class of random processes. The correlation codimensions, C(2), of various uncorrelated and correlated point processes have been derived ( Bickel, 1999a) and those of fractal renewal point processes have been studied through simulation ( Bickel, 2001a). Several other methods of quantifying intermittency exist, but the one described herein is relatively simple, is computationally efficient, applies to both nonstationary and stationary processes, does not require the selection of scale or threshold nuisance parameters, is related to the parameters H(q) used to quantify nonstationarity, and is insensitive to the sampling time ( Bickel, 1999b). This technique has described the intermittency of human movement events ( Bickel 1999a, Bickel 1999b and Bickel 2000b), the human heart rate ( Bickel 1999b and Bickel 2001b), and viral DNA composition ( Bickel, 1999a), and has potential applications to several other intermittent data sets, such as those of microbiology, sociology ( Zeldovich et al., 1990), physics ( Wang and Wolynes, 1995; Feng et al., 1998), and geophysics ( Schertzer and Lovejoy, 1987; Davis et al., 1994). Section 4 of this paper describes applications to economic and biomedical data. Given {xi=x[(i−1)T0]}i=1n, a time series sampled n times at time-intervals of T0,χ(p,q) of x(t) can be efficiently estimated as follows. First, s(q;kT0) is estimated over time by the maximally overlapping estimator equation(4) View the MathML source Then H(q) is estimated by the slope, View the MathML source, of the least-squares regression of View the MathML source on View the MathML source using k=kmin,2kmin,4kmin,…,kmax/2,kmax, where kmin⩾Tmin/T0,kmax⩽Tmax/T0, and kmax<n. The parameters kmin and kmax are chosen such that a plot of View the MathML source versus View the MathML source is approximately linear. The scales are integer powers of two so that the values of View the MathML source are equally spaced; using k=kmin,kmin+1,kmin+2,…,kmax−1,kmax gives too much weight in the regression to the higher values of View the MathML source and is computationally inefficient. The intermittency estimates, View the MathML source, are obtained by substituting the estimates of H(p) and H(q) into Eq. (3) for p≠q. Since View the MathML source, the estimate View the MathML source is found by changing p and computing View the MathML source until p is sufficiently close to q, when View the MathML source no longer changes significantly. Typically, View the MathML source since View the MathML source for a time series realization for which View the MathML source is maximal: xi=X1 for i>n/2 and xi=X2≠X1 for i⩽n/2, with View the MathML source. Thus, View the MathML source describes the intermittency of a time series, with values closer to 0 for less intermittent series and values closer to 1 for more intermittent series ( Bickel, 1999b). Estimates of intermittency can be slightly greater than 0, giving the appearance of an intermittent process, when the actual intermittency of the process is 0; this finite-size effect is called residual or spurious multiscaling ( Davis et al., 1994). To address this problem in 2 and 3 we study the asymptotic and finite-size distributions of intermittency estimates under the null hypothesis that x(t) is Bm. If x(t) is Bm, then View the MathML source since View the MathML source, and the sample {xi}i=1n constitutes a discrete-time random walk, which is used in the next two sections to determine whether a given estimate View the MathML source from an observed time series is significantly greater than 0.

We have demonstrated that the distribution of intermittency estimates of random walks is asymptotically normal with zero mean and is slightly negatively skewed for random walks of finite length. These properties can be used with the standard error of the intermittency estimates to determine whether the intermittency of an observed time series is significantly greater than zero. If it is, then the process realized by the time series is considered intermittent; otherwise, the process is considered slowly varying. This methodology reveals intermittency in the stock market and the adult heart rate, and can similarly be applied to other time series.