Hurst coefficient on constant series?

Question

I want to evaluate the Hurst coefficient in my batch of time series, but I have an issue with these two specific types of `time series, for zero and constant values.

To give you a rough idea of the time series, it's a system that is perturbed at t=0, and I let it free running run for 2000 time steps. I change some parameters of the system, and for each specific parameter, I want to associate its H value.

To compute the Hurst exponent, I tried two technics that I found online:

The R/S ratio from Wikipedia.

Here is the code that I applied:

n_min = 10
ns = np.arange(n_min, len(time_series), 5)


R = []
S = []
E = []
for n in ns:
    Xt = time_series[:n]
    m = np.mean(Xt)
    Yt = Xt - m
    Zt = [sum(Yt[:i]) for i in range(len(Yt))]
    # Compute the range
    R.append(max(Zt)-min(Zt))
    # Compute standard deviation
    S.append(np.std(Xt))
    E.append(np.mean(np.array(R)/np.array(S)))   

reg = np.polyfit(np.log(ns), np.log(E), 1)
H = reg[0]

And I used a second method that I found here, based on the first exponent:

def hurst(p):  
    tau = []; lagvec = []  
    #  Step through the different lags  
    for lag in range(2,20):  
        #  produce price difference with lag  
        pp = subtract(p[lag:],p[:-lag])  
        #  Write the different lags into a vector  
        lagvec.append(lag)  
        #  Calculate the variance of the differnce vector  
        tau.append(sqrt(std(pp)))  
    #  linear fit to double-log graph (gives power)  
    m = polyfit(log10(lagvec),log10(tau),1)  
    # calculate hurst  
    hurst = m[0]*2  
    # plot lag vs variance  
    #py.plot(lagvec,tau,'o'); show()  
    return hurst

Now both techniques give me entirely different results:

Zero time series
H (S/R ratio) = 0.4732894
H (1st exponent) = 0.001971641

Note that fix and zero time series give me almost the same results.

So as you can see, depending on the technique I am using, I have very different results, and now I am wondering if the analysis I am performing on the whole set of time series is even accurate.

So I have a bunch of questions related to this:

I have read that the Hurst coefficient can be applied on stationary distribution, so my guess is that this might not be possible to measure H for these time series. Do you know a way to compute it on non-stationary time series, maybe?
Initially, I removed the first half of these time series, but it can't evaluate H when the signals are constant, and I get errors! If I want to stick with the stationary time series, is there a way for me to evaluate H correctly for these constant series?
Lastly, how can I evaluate the statistical significance of these results, any python code? That could help me decide which implementation is correct or not.

Thank you!

Hurst coefficient on constant series?

0 Answers