Type: Controversial Essays
Sample donated: Marsha Bridges
Last updated: July 18, 2019
Abstract In this paper, the fractalityand stationarity of a usual wireless network has been investigated by exposingthe scaling pattern and nature of frequency fluctuation of the two crucialparameters,the daily peak hour call arrival number and daily call drop number, allied witha wireless network. The time series of these parameters between 3rd March,2005 to 31stOctober, 2015, of a sub-urban local mobile switching centre have been consideredfor revealing the nature of scaling (fractality) and stationary behaviour usingstatistical methodologies. Having the knowledge about the fractality, HurstExponent for the time series have been considered using the methodologies like GeneralHurst Estimation (GHE) and R/S. It has been observed that both the time series show Short Range Dependent (SRD) anti-persistentbehaviour.
Continuous Wavelet Transform (CWT) method has been used to find out thestationarity/non-stationarity of the data-series where both the time series exhibitthe nonstationarity. Theseobservations conclude that the both the time series are not a random phenomenonbut complex. However both the series found to have non-linearity and stability.
1. INTRODUCTION Withthe rapid growth in wireless technology different applications are vividlyapplied in smartphone. Now a day’s smartphones are widely used as the simpleand most common devices for communication.
The multi-featured attributes of smartphonedevices are widely acceptable across the world for various ways ofcommunications like data services and voice. With the repeated use of theseservices the demand for wireless networks increases rigorously. It becomes a trickytask for the service providers to maintain the Quality of Service (QoS) andcost effectiveness by upgrading the technical and infrastructural features ofthe wireless network system. So various issues consisting of system design,congestion control, and admission control should be addressed more efficientlyto provide multi-class services through desired wireless networks. To upgrade the service quality and to achievethe optimum performance there is a need to understand the nature of thefluctuation and underneath pattern (particularly the scaling, self-similarityproperty and stationarity) of the wireless network traffic data.
But with thegrowth of different factors like call drop rate and call arrival rate, theperformance of network traffic in mobile is highly affected. So it has become anecessity to understand the nature of fluctuations of these two parameters. Inthis paper an initiative has been taken to uncover the nature of the scalingbehaviour and time dependency of the frequency (stationarity or non-stationarity)of occurrence of the two parameters, daily busy hour call arrivals and droppedcalls, of a local mobile switching centre during 3rd March, 2005 to31st October, 2015 as shown in Figure 1 which can be treated as thesignatory representative of any wireless network traffic.The maximum number ofcall attempts in the peak hour of a day is defined as busy hour callinitiation. The resource of a network can be limited to or can be upgraded asper requirements depending on the maximum call arrival and the call drop causeddue to congestion. A concurrent study of busy hour call initiation and dailydropped call time series may give a feasible nature of the incoming trafficpattern, the call congestion, grade of service and blocking probability.In this workHurst exponent has been calculated for revealing the scaling behaviour of the time series, daily busy hour arrival call andcall drop. Two different methods like Rescaled Range analysis (R/S) method andGeneral Hurst Estimation (GHE) method (Hurst, Black, & Sinaika, 1965)have been used to calculate the Hurst Exponents to understand the nature of thesignals with respect to different scales to identify the signals as fractionalBrownian motion i.
e. whether they are stationary or non-stationary. There aremany limitations of calculating Hurst exponent using other methods.
So to get anon-controversial conclusion about the scaling property of the time series, itwill be useful to apply more than one method to estimate the Hurst Exponent.Hence two methods (mentioned above) have been chosen to calculate the HurstExponent. Thus confirming the authenticity of the conclusions taken out of theresults.Stationary ornon-stationary behaviour of the data series could be completed by analysing thefluctuating nature of the busy hour call initiation rate and call drop rate. Anon-stationary signal has changing frequency whereas stationary signal hasconstant frequency. The signals are checked with respect to time. The analysisfor non-stationary behaviour is necessary due to: 1) asymptotic analysis whichwill not be applicable for the regression model with non-stationary variables.Usually “t-ratios” does not follow a t-distribution, and hence valid testsabout the regression parameters cannot be undertaken.
2) The properties of thesignal are highly affected by the stationary or non-stationary behaviour.Different methods can be used to check the stationary/ non-stationary behaviourof the signals. Continuous Wavelet Transform (CWT) based method has beenimplanted in this paper to determine the nature of frequency dependency of thewireless network traffic. The advantagesof using CWT are: a) simultaneous localization in time and frequency domain andis computationally fast. ii) Wavelets have the great advantage of being able toseparate the fine details in a signal. Very small wavelets can be used toisolate very fine details in a signal, while very large wavelets can identifycoarse details. It decomposes a signal into component wavelets.
2. Experimental dataset: First and foremost the real time data arerecorded in the Server positioned in the Mobile Switching Centre (MSC) of theISP. The recorded data sets collected from the ISP sited in our city for theperiod 3rd March, 2005 to 31st October, 2015used for exclusivelyresearch purpose. The data can not be exported commercially, it comprises ofcall initiation, call holding time, call drops and its causes, time and delayof hand-off etc.
From these dataset the call initiation and the call dropstatistics for each hour of a day have been considered such as the peak hourcall initiation and the call drop statistics have been taken for analysis. Thesummary statistics and plotting of original data set of signal are described intable1 and figure1 respectively: Table 1: Summary statistics for daily dropped call andcall initiation signal Scores Call Initiation Signal Call Drop Signal Mean 168.3746 0.2596 Median 171 0 Mode 171 0 Standard Deviation 14.7867 1.3505 Variance 218.6452 1.
8238 Maximum 197 23 Minimum 14 0 Skewness -3.9571 10.0416 Kurtosis 29.2970 133.6682 Figure1: plotting of theinitiated calls and dropped calls3.
Hurst Exponent EstimationOne of the statistical measures used into classify the time series is Hurst exponent. Random series is recognised byH=0.5 while H>0.5 indicated reinforcing series in trends. When twoconsecutive data intervals are very high then the consistance of the signal isnegative. The value of H=0 denotes that the time seriesis a white noise whose autocorrelation function (ACF) decreases rapidly withdelay.
. For this, the upcoming values have atendency to return to a long-term mean. Hence it becomes slower thanstandard Brownian motion. With an increase in the tendency in the time series,the value of H will tend to 0. The signal contains short-rangedependent (SRD) memory that exhibits fractal behaviour. The ACF decreasesexponentially 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 motionthrough Markov chain feature. The ACF decay is slow compared to theanti-persistent time series. Arbitrary fluctuations are seen in the signal. Irregularity inbehaviour will appear with the difference in the various data points of thetime series. When the value of H lies within the range of 0.5-1.
0then it shows that with an increase in the successive data intervals thepersistency of the signal shows positive behaviour.The Hurst value will tend towards 1. The signal shows long-rangedependence (LRD) and non-periodical cycle. LRD unlike the SRD series exhibitssimilar statistical properties at different scales (lower or higher). The ACFdecays hyperbolically and is slower compared to standard Brownian motion. Theconsistency of the signal is smooth.When the value of H is equal to 1.0 then thetime series appears to be perfectly smooth and the ACF comes to a constantlevel.
Different estimatorsfor 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 recentmethod, Rescaled Range (R/S) analysis has been used along with traditional GeneralizedHurst Exponent (GHE) estimation method. The Rescaled Range method is used forstatistical measurement of a time series. Its aim is to provide an estimationof how the variability of a series changes with the length of the time-period. GHEprovides the best finite sample behaviour among all the methods in respect ofthe bias and lowest variance. GHE is suitable for any data series/signalirrespective of the size of its distribution tail.
3.1. R/S Analysis:R/S analysis (Rescaled Range analysis) was initially coined by Edwin Hurstin the year 1951. This method can be implemented in a program by providing adirect estimation of the Hurst Exponent. The Hurst Exponent is a preciousindicator of the state of randomness of a time-series. Given a time-series with n elements X , X ,…,X , the R/Sstatistic is defined as: = Where , is thearithmetic mean and is the standard deviation from the mean.
With this R/S value, Hurst found a generalization of a result in thefollowing formula:E = C as n Where H is the Hurst exponent.From there, it is clear that an estimationof the Hurst exponent can be obtained from an R/s analysis. 3.2.
Generalized Hurst Exponent (GHE) method:This method wascoined by (Hurst,Black, & Sinaika, 1965) defines a function as Where is the time series.pisthe order of the moment of distribution and isthe lag which ranges between and .Generalised Hurst Exponent (GHE), is related to through a power law: Depending uponwhether it isindependent of p or not, a time series can bejudged 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 checkwhether the observable time series is stationaryor termed stationary or is non-stationary.Thistest is used as a complement to the standard tests in analyzing time seriesproperties.
The KPSS test is based on linearregression. The time series is broken down into three parts: a deterministic trend(?t), a random walk(rt), and a stationary error (?t), with the regressioequation:xt = rt + ?t + ?1 If the data is stationary, it will have a fixedelement for an intercept or the series will be stationary around a fixed level (W.Wang, 2006).The test uses OLS to findthe equation, which differs slightly depending on whether you want to test forlevel stationarity or trend stationarity.
A simplified version, without thetime trend component, is used to test level stationarity.3.2.Continuous Wavelet Transform (CWT) test:Realword data or signals are frequentlyexhibit slowly changing trend or oscillations punctuated with transient. ThoughFourier Transform (FT) is a powerful tool for data analysis, however it doesnot represent abrupt changes efficiently. FT represents data as sum of sinewaves which are not localized in time or space.
These sine waves oscillateforever, therefore to accurately analyse signals that have abrupt changes, needto 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 frequencydomain simultaneously.The continuous wavelettransform is a generalization of the Short-Time FourierTransform (STFT) thatallows for the analysis of non-stationary signals at multiple scales.Keyfeatures of CWT are time frequency analysis and filtering of time localizedfrequency 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 dilationof wavelet (scale) and defines atranslation of the wavelet and indicates the time localization, ?(t)is the wavelet. The coefficient is an energynormalized factor (the energy of the wavelet must be the same for different avalue of the scale).4.Results & DiscussionThe values ofHurst exponents for the two time series a) dailydropped calls and b) daily busy hour call initiated has been calculated usingthe three methods, VGA, HFD and GHE which are being tabulated below in Table 2.Table2: Hurstparametervalues for dailydropped 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 Hurstexponents for both the series are less than 0.5.
The Hurst exponent for dailybusy 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. theirfuture values have the tendency to revert to their long-term mean with thedaily busy hour initiated calls profile has more tendency to return to its meancompared to the daily dropped calls profile. Since there are the tendencies forboth the profiles to return to their respective mean, it can be said that theremust be some driving forces which bring back the series towards their meanswhen the profiles deviate from the mean (the most stable position of anyfluctuation). This implies that some negative feedback system must be workingwhich continuously try to stabilise the profiles. Moreover these low values ofH signify that both the signalshave short-range dependent (SRD) memory.
The self similar nature in short scalefor both the times series is evident from this SRD phenomenon of them. The SPWVD based time-frequencyspectrum for the two time series are shown in Figure 2 and Figure 3respectively.Figure2 CWT for daily call initiation Figure3 CWT for daily call drop Figure 3undoubtedly indicates that the daily dropped calls frequency is varying withtime.So, daily dropped calls data set is non-stationary in nature. Figure2 shows that this signal is nearly stationary as the frequency contents do notchange with time.
So it can be concluded that busy hour initiated calls dataare stationary. In a non-stationary signal the frequency contents are thefunctions of time i.e. they are not independent of time change.
Frequency anyevent signifies the number of events happen per unit time. So, it can beinferred that the number call drops per unit time is not independent of timebut varies with time. In case of busy hour call initiation profile there arenearly eight types of frequency contents as is evidentfrom figure 3 but all of them remains constant with respect to time. This canbe interpreted as the rates of busy hour call initiation is not varying withtime and hence proper modelling and forecasting of the busy hour callinitiation can be made easily. 5. ConclusionOneof the statistical measures used in to classify the time series is Hurstexponent. Using the value of H, the attributes within the time series can bepredicted: H=0: The time series is a white noise whose autocorrelation functiondecreases rapidly with lag, a value of H in the range 0 – 0.
5 indicates atime series with long-term switching between high and low values in adjacentpairs, meaning that a single high value will probably be followed by a lowvalue and that the value after that will tend to be high, with this tendency toswitch between high and low values lasting a long time into the future. The persistency of the signal is negative (or anti) where the probability ofopposite trend between any two successive data intervals is very high. Thismeans that future values have a tendency to return to a long-term mean andhence it is slower than classical/standard Brownian motion. If thistendency is more in the time series, the value of H will be found to be closerto 0. A value of H=0.5 can indicate a completely uncorrelated series, butin fact it is the value applicable to series for which the autocorrelations atsmall time lags can be positive or negative but where the absolute values ofthe autocorrelations decay exponentially quickly to zero. Whereas H=1 denotes timeseries ideally smooth and the autocorrelation function does not vary with lagbut settle to constant level signal has arbitrary fluctuation. If thevalue of H is in this range 0.
5–1, indicates a time series with long-termpositive autocorrelation, meaning both that a high value in the series willprobably be followed by another high value and that the values a long time intothe future will also tend to be high. The persistency of the signal ispositive where the probability of related trend between any two successive dataintervals is very high. The stronger the trend, the H value moves towards 1.