Políticas de Capacitación

4.2.1 The DFA algorithm

A standard approach for the detection of long-range autocorrelation, via estimation of the scaling exponent 2.4 of a time-series is the so-called Detrended Fluctuation Analysis (DFA). The method, originally introduced in (Peng, S. V. Buldyrev, Havlin et al., 1994) is nowadays widely utilized due to its robustness to non-stationarity, e.g. trends. Earlier methods, such as the Hurst’s rescaled range analysis (Hurst, 1951) can lead to the false detection of long-range autocorrelation (Bryce et al., 2012). Although a number of extensions of the originally proposed DFA have been proposed (e.g. Bashan et al., 2008), the method as of (Peng, S. V. Buldyrev, Havlin et al., 1994) is widely utilized and constitutes the methodological approach of Publication III.

The following summarizes the DFA procedure for an arbitrary time-series of N

observations,{x_t}_t=1,...,N:

I Altought not obligatory, a common practice is to consider theprofileof the

6_{Providing here expressions forD}

BandDWwould leave the notation undefined. This requires a set

of mathematical definitions along with their respective notation, in the very exact way as addressed in Publication II

time-series, by considering the integrated sum: y(k) =k

t=1

(x_t−x¯).

Subtracting the mean ¯xset the mean of the integrated series to zero.

II By setting a lengths, the profile is divided inN/snon-overlapping windows (of

equal length). In each window, a first-degree7polynomial approximationy_{t r},

representing the local trend of the data, is estimated (by ordinary least-squares). III In each windowm, differences (residuals)y_m−y_{m,t r}, m=1, ...,N/s define

the m-th window detrended profile. By varying the widows of size s and

considering the square root of the average variance of the residuals across the N/swindows, the fluctuationF(s)is evaluated:

F(s) = 1 N/s N/s m=1 1 s s i=1 y_m(i)−y_{m,t r}(i) 2 .

IV In presence of power-law scaling,F(S)∼sα: the slope of the line approximating F(s)againstsin a log-log plot, estimates the scaling exponentα.

Figure 4.3 resembles the procedure.

0.5 < α <1.5 indicates long-range correlations, whereasα=0.5,α=1 andα=

1.5 respectively correspond to White-noise, Brownian noise and pink-noise signals. Exponentsα <0.5 correspond to anti-correlations (Bashan et al., 2008; Kantelhardt, 2009). In case of “crossovers” where different scaling exponents apply at different time-scales, i.e. the slope of F(s) against s in the log-log plot changes, the same

interpretation hold, but on a limited time-scale range. Refer to Chapter 6 for a

discussion on the hypotheses underlying the DFA methodology, its applicability, and a discussion on how it relates to time-series stationarity.

7_{This corresponds to 1}st_{order DFA, which removes linear trends. More generally an-degree}

0 500 1000 1500 2000 2500 3000 t 0 50 100 150 200 Durations x(t) 0 500 1000 1500 2000 2500 3000 k -1000 0 1000 2000 3000 y(k) 0 500 1000 1500 2000 2500 3000 k -1000 0 1000 2000 3000 y_m(k) ym,tr(k) 500 1000 1500 2000 2500 3000 k & m (s: 300) 103 104 105 106 r2 _Mean(r2_{) F}2_(s) 101 ₁₀2 Log₁₀ s 10-1 100 101 102 103 104 Log10 F(s) Log10 F(s) = Log10 s + q

Figure 4.3 Visual representation of the DFA procedure. Order-to-order duration series for DK0010268606 (Vestas Wind Systems) on June 01, 2010 (for orders submitted at the best level on the bid side).

4.2.2 Stationarity issues in DFA

An analysis on non-stationarities in time series when applied to DFA is provided in (Z. Chen et al., 2002; Bryce et al., 2012). This are perhaps the only analysis available in this regard. For a simulated long-range correlated signal, the effect of three types on non-stationarities is analyzed in (Z. Chen et al., 2002).

(a) Signals with segments removed, i.e. non-stationarities caused by discontinuities in the signal. This is relevant in financial applications, e.g. due to the fact that markets do not trade on weekends, holidays, and at night. Z. Chen et al., 2002 find that the scaling of correlated signals is not affected by the cutting procedure, independently on the size of the cutting segment and on the number of segments removed.

(b) Signals with random spikes. In the duration series used in Publication III these correspond to seldom andisolatedexceptionally long durations. Following Z. Chen et al., 2002, when uncorrelated spikes are added to the signal a change

in the cross-over of the scaling exponent at a characteristic scale is observed. For positively correlated signals, this is observed as small-time scales. Our data does not show any intra-day cross-over features, rather the log-log pot of fluctuation against window size is remarkably linear, suggesting that spikes- related non-stationarity are not relevant in our analyses.

(c) Signals with different local behavior, which include signals with (i) a number of segments of a certain length with different standard deviation and (ii) with different local correlation. (Z. Chen et al., 2002, Fig. 4d) shows that for correlated signals,α >0.5, with segments characterized by two different values of standard deviation no difference is found in the scaling exponent compared to the stationary correlated signal. Whereas the variance of durations may vary across the day, e.g. as a consequence of the U-shaped trading activity profile, DFA still constitutes a robust method.

For correlated signals, the presence of segments with different correlation can either lead to no differences in the scaling exponent wrt. to the stationary signal or to double cross-overs, with a characteristics plateau characterized by a flatter slope in the central part, which is not observed in our data. Based on the results of (Z. Chen et al., 2002) analyzing non-stationary sources of relevance also in financial time series, DFA is shown to be capable of detecting the correlation of the non-stationary signal in some circumstance, while producing crossovers in other, which are however not observed in our data. (Bryce et al., 2012) generally warn about the use of DFA in time series concluding that it does not provide any protection against non-stationarity, introduces biases, and suffers from small-sample effects. (Bryce et al., 2012) devises that explicit detrending followed by measurement of the diffusional spread of a signals’ associated random walk is preferable. Note that in (Hu et al., 2001) “stationariety” is intended as “presence of trends”, rather limited when compared to its broader meaning in econometrics.8_.

8_{At this point, (Z. Chen et al., 2002) and (Hu et al., 2001) provide sufficient reasons to question DFA’}