# 3. and is slower compared to standard Brownian

3. Hurst Exponent Estimation

One of the statistical measures used in

to classify the time series is Hurst exponent. Random series is recognised by

H=0.5 while H>0.5 indicated reinforcing series in trends. When two

consecutive data intervals are very high then the consistance of the signal is

negative. The value of H=0 denotes that the time series

is a white noise whose autocorrelation function (ACF) decreases rapidly with

delay.. For this, the upcoming values have a

tendency to return to a long-term mean. Hence it becomes slower than

standard Brownian motion. With an increase in the tendency in the time series,

the value of H will tend to 0. The signal contains short-range

dependent (SRD) memory that exhibits fractal behaviour. The ACF decreases

exponentially with lag and is relatively slower than that of the white noise,

and H=0.5 denotes that the time series will show Standard Brownian motion

through Markov chain feature. The ACF decay is slow compared to the

anti-persistent time series. Arbitrary fluctuations are seen in the signal. Irregularity in

behaviour will appear with the difference in the various data points of the

time series. When the value of H lies within the range of 0.5-1.0

then it shows that with an increase in the successive data intervals the

persistency of the signal shows positive behaviour.

The Hurst value will tend towards 1. The signal shows long-range

dependence (LRD) and non-periodical cycle. LRD unlike the SRD series exhibits

similar statistical properties at different scales (lower or higher). The ACF

decays hyperbolically and is slower compared to standard Brownian motion. The

consistency of the signal is smooth.When the value of H is equal to 1.0 then the

time series appears to be perfectly smooth and the ACF comes to a constant

level.

Different estimators

for the estimation of the Hurst Exponent of any signal or data are available.

In this paper, two Hurst estimation methods have been used. The very recent

method, Rescaled Range (R/S) analysis has been used along with traditional Generalized

Hurst Exponent (GHE) estimation method. The Rescaled Range method is used for

statistical measurement of a time series. Its aim is to provide an estimation

of how the variability of a series changes with the length of the time-period. GHE

provides the best finite sample behaviour among all the methods in respect of

the bias and lowest variance. GHE is suitable for any data series/signal

irrespective of the size of its distribution tail.

3.1. R/S Analysis:

R/S analysis (Rescaled Range analysis) was initially coined by Harold

Edwin Hurst in the year 1951. This method can be implemented in a program by

providing a direct estimation of the Hurst Exponent. The Hurst Exponent is a

precious indicator of the state of randomness of a time-series.

Given a time-series with n elements

X

, X

,…,X

, the R/S

statistic is defined as:

=

Where

,

is the

arithmetic mean and

is the standard deviation from the mean.

With this R/S value, Hurst found a generalization of a result in the

following formula:

E

= C

as n

Where H is the Hurst exponent.From there, it is clear that an estimation

of the Hurst exponent can be obtained from an R/s analysis.

3.2. Generalized Hurst Exponent (GHE) method:

This method was

coined by (Hurst,

Black, & Sinaika, 1965) defines a function

as

Where

is the time series.pis

the order of the moment of distribution and

is

the lag which ranges between

and

.

Generalised Hurst Exponent (GHE), is related to

through a power law:

Depending upon

whether it is

independent of p or not, a time series can be

judged as uni-fractal or multi-fractal (Matteo, 2007)

respectively. The GHE h

yields the value of original Hurst Exponent

for

,

i.e.

.

3. Test for Stationarity of Non-Stationarity:

3.1.Kwiatkowski–Phillips–Schmidt–Shin

(KPSS) tests:

Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests

(Kwiatkowski, Phillips, Schmidt, & Shin, 1992)are used for testing a null hypothesis to check

whether the observable time series is stationary

or termed stationary or is non-stationary.This

test is used as a complement to the standard tests in analyzing time series

properties.

The KPSS test is based on linear

regression. The time series is broken down into three parts: a deterministic trend

(?t), a random walk

(rt), and a stationary error (?t), with the regressio

equation:

xt = rt + ?t + ?1

If the data is stationary, it will have a fixed

element for an intercept or the series will be stationary around a fixed level (W.Wang, 2006).

The test uses OLS to find

the equation, which differs slightly depending on whether you want to test for

level stationarity or trend stationarity. A simplified version, without the

time trend component, is used to test level stationarity.

3.2.

Continuous Wavelet Transform (CWT) test:

Realword data or signals are frequently

exhibit slowly changing trend or oscillations punctuated with transient. Though

Fourier Transform (FT) is a powerful tool for data analysis, however it does

not represent abrupt changes efficiently. FT represents data as sum of sine

waves which are not localized in time or space. These sine waves oscillate

forever, therefore to accurately analyse signals that have abrupt changes, need

to use new class of functions that are well localized with time and frequency.

These bring the topic of wavelets.

The primary objective of the Continuous Wavelet Transform

(CWT) is to get the signal’s energy distribution in the time and frequency

domain simultaneously.The continuous wavelet

transform is a generalization of the Short-Time FourierTransform (STFT) that

allows for the analysis of non-stationary signals at multiple scales.Key

features of CWT are time frequency analysis and filtering of time localized

frequency components. The mathmetical equation for CWT is given below(Shoeb & Clifford, 2006):

C (a,

) =

(

) x(t) dt

Where C(a,

) is the function of the parameter a,

.

The a parameter is the dilation

of wavelet (scale) and

defines a

translation of the wavelet and indicates the time localization, ?(t)

is the wavelet. The coefficient

is an energy

normalized factor (the energy of the wavelet must be the same for different a

value of the scale).

4.

Results & Discussion

The values of

Hurst exponents for the two time series a) daily

dropped calls and b) daily busy hour call initiated has been calculated using

the three methods, VGA, HFD and GHE which are being tabulated below in Table 2.

Table

2: Hurst

parameter

values for daily

dropped calls and daily busy hour call initiation

Hurst exponent (H)

Methods

Daily dropped calls

Daily busy hour

Initiated calls

R/S

0.2707

0.2405

GHE

0.2461

0.1565

The Hurst

exponents for both the series are less than 0.5. The Hurst exponent for daily

busy hour initiated calls is lower than that of the daily dropped calls.

These results claim the anti-persistent behaviour of both of them i.e. their

future values have the tendency to revert to their long-term mean with the

daily busy hour initiated calls profile has more tendency to return to its mean

compared to the daily dropped calls profile. Since there are the tendencies for

both the profiles to return to their respective mean, it can be said that there

must be some driving forces which bring back the series towards their means

when the profiles deviate from the mean (the most stable position of any

fluctuation). This implies that some negative feedback system must be working

which continuously try to stabilise the profiles. Moreover these low values of

H signify that both the signals

have short-range dependent (SRD) memory. The self similar nature in short scale

for both the times series is evident from this SRD phenomenon of them.

The SPWVD based time-frequency

spectrum for the two time series are shown in Figure 2 and Figure 3

respectively.

Figure

2 CWT for daily call initiation

Figure

3 CWT for daily call drop

Figure 3

undoubtedly indicates that the daily dropped calls frequency is varying with

time.So, daily dropped calls data set is non-stationary in nature.

Figure

2 shows that this signal is nearly stationary as the frequency contents do not

change with time. So it can be concluded that busy hour initiated calls data

are stationary. In a non-stationary signal the frequency contents are the

functions of time i.e. they are not independent of time change. Frequency any

event signifies the number of events happen per unit time. So, it can be

inferred that the number call drops per unit time is not independent of time

but varies with time. In case of busy hour call initiation profile there are

nearly eight types of frequency contents as is evident

from figure 3 but all of them remains constant with respect to time. This can

be interpreted as the rates of busy hour call initiation is not varying with

time and hence proper modelling and forecasting of the busy hour call

initiation can be made easily.

5. Conclusion

One

of the statistical measures used in to classify the time series is Hurst

exponent. Using the value of H, the attributes within the time series can be

predicted: H=0: The time series is a white noise whose autocorrelation function

decreases rapidly with lag, a value of H in the range 0 – 0.5 indicates a

time series with long-term switching between high and low values in adjacent

pairs, meaning that a single high value will probably be followed by a low

value and that the value after that will tend to be high, with this tendency to

switch between high and low values lasting a long time into the future.

The persistency of the signal is negative (or anti) where the probability of

opposite trend between any two successive data intervals is very high. This

means that future values have a tendency to return to a long-term mean and

hence it is slower than classical/standard Brownian motion. If this

tendency is more in the time series, the value of H will be found to be closer

to 0. A value of H=0.5 can indicate a completely uncorrelated series, but

in fact it is the value applicable to series for which the autocorrelations at

small time lags can be positive or negative but where the absolute values of

the autocorrelations decay exponentially quickly to zero. Whereas H=1 denotes time

series ideally smooth and the autocorrelation function does not vary with lag

but settle to constant level signal has arbitrary fluctuation. If the

value of H is in this range 0.5–1, indicates a time series with long-term

positive autocorrelation, meaning both that a high value in the series will

probably be followed by another high value and that the values a long time into

the future will also tend to be high. The persistency of the signal is

positive where the probability of related trend between any two successive data

intervals is very high. The stronger the trend, the H value moves towards 1.