Analysis of Short Term and Long Term Dependence of Stream Flow Phenomenon in Seonath River Basin, Chhattisgarh

In this paper to investigate long rang phenomena ( Hurst effect) of river flows which characterizes hydrological time series is studied, especially in connection with various climate-related factors, is important to improve stochastic models for long-range phenomena and in order to understand the deterministic and stochastic variability in long-range dependence of stream flow. Long rang dependence represented by the Hurst coefficient H is estimated for 5 mean monthly discharge time series of Chhattisgarh state for a period of 32 years from 1980-2012.long memory analyzed for both monthly and seasonally stream flow time series of the Seonath River Basin at Chhattisgarh State by using Hurst exponent and testing specifically the null hypothesis of short-term memory in the monthly and seasonal time series by (Von Neumann ratio test, Kendall's rank correlation test, Median crossing test, Run above and below the median for general randomness, Turning point test, Rank difference test).


INTRODUCTION
The number of time scale studies have been analyzing for the long-term behaviour of streamflow has increase adequately in the accomplished duration with exceptional quality and data availability with increasing interest of influence of climate change and climate-related factors on stream flow processes (Bloschl and Montanari, 2010), [1] the extent and complexity of such a consideration have increased. The necessity of such research lies in the need for incorporate long-range dependence and to developed speculative models, which can be used for illustration in the management of water resources or reservoir action. Another property characterizing time series from a longrange perspective is the long-term dependence (Hurst phenomenon (Hurst, 1951). [2] The phenomena of longrange persistence have a long history and have been authenticated appropriately in hydrology, meteorology and geophysics. Present day studies have led to reawakening and to add analyze long-term persistence in temporal time series of hydrologic data and also to developed applicable methods for estimating and modelling the intensity of long-term persistence in time series, as well as providing the reason for the Hurst phenomena. Based on the consideration of long-term persistence, a stationary process xt processes long memory if there be present in a real number H ε (0.5, 1), called the Hurst exponent (Montanari et al., 2000). [3] The exponent H, in a hydrological time series, is called the devotion of long-term persistence and it can be numerically denoted by the Hurst coefficient H. when H>0.5 higher the intensity of long-term or long-range persistence in the data and when H <0.5 be identical to short-term negative time persistence, which is almost never encountered in the analysis of hydrological data (Montanari et al., 2000). [3] To test for significant statistically long-term memory on a hydrologic time series, a significant difference between short term and long term persistence must be accomplished (Rao and Bhattacharya,1999). [4] The phenomena of short-term persistence are based on the concept of strong mixing (Rosenblatt,1956) [5] which measure correlation sequentially among two cases distinct by increasing time lags. Against this background, Towards this end Hurst exponent used for ascertain the appearance of long term persistence in data series and testing specifically the null hypothesis of short term Long range dependence also called long memory or long-range persistence is a phenomenon that may arise in the analysis of spatial and time series data usually considered to have long range dependence if the dependence decays more slowly than an exponential decay and Short range dependence also called short memory or short range persistence a process is said to be short range dependence if the dependence among the observations diminishes fast. The phenomena of short-term dependence are based on the notion of strong mixing (Rosenblatt, 1956). [5] Which measure correlation successively between two events separated by increasing time lags. Against this background, the primary objective of this study is, (1) to investigate the streamflow time series of Seonath River To investigate the long term and short term dependence phenomena in the flow series, the average daily flows are aggregated to mean monthly stream flows by summing the average daily flow over the total number of days in the month. thus, for long-term dependence analysis, the seasonality must be removed. To remove the seasonality in the monthly flow series are log-transformed to normalise the data then deseasonalized; the deseasonalized is done as follows.

Test for independence:
The serial correlation coefficient (SCC) was performed to verify the dependency. It is the correlation between adjacent observations in time series data. According to Box and Jenkins the 1 serial correlation coefficient, 1 is computed as follows, for 5% significance level, if 1 < 0.5 then the station is considered as independent, 1 is given as below.

Test for randomness:
Test for randomness is performed to identify whether there is any recognised pattern. If the data is non-random it shows that process then generates the event is following a trend. The data should be random for any time series analysis; the run test is carried out for this purpose, for 5% significance level, if Z > 0.05 then the station is considered as random. As per test when randomness is more in given time series it means there is more probability to become trendless in such a time series.
Where R is observed number of runs, 1 is the expected number of run, n is the number of observation, A is the number of observation above k, B is the number of observations above k, and k is the mean of the observations 2.3 Test for consistency/Homogeneity: Consistency test is performed to identify that the behaviour mechanism that generates a part of time series data is considered with the segment of the time series data. For this purpose, standard normal homogeneity test (SNHT) is done with the help of XLSTAT plug in a package for MS-Excel is used. For 5% significance level, if p > 0.05 then the station is considered as consistent. HURST EXPONENT: Long range dependence is numerically expressed by the Hurst coefficient H (0 -1) in general holds H = 0.5, the time series is random noise for H < 0.5, the time series is said to be anti-persistence.
Where n is the no. of data set where, Where, σ is the standard deviation, Long rang dependence can be numerically by the Hurst coefficient this is a coefficient ranging between 0 and 1, where H > 0.5 indicates long-range dependence in the data.

VON NEUMANN RATIO TEST:
The null hypothesis test for short-term dependence is done by using the von Neumann ratio test, (Madansky, 1988). [6] The null hypothesis of no long-term dependence, the following test statistics is computed for both monthly and seasonal streamflow time series. The null hypothesis in this test is that the time series variable is independently and identically distributed (random). The alternate hypothesis is that the series is not random, The von Neumann ratio (N) is the most widely used test for testing a time series for the absence or presence of homogeneity and also identified the presence of shortterm dependence and the null hypothesis of no short-term dependence in given time series.
Where =hydrologic variable constituting the sequence in time, n = total number of hydrologic records, and x = average of , If data are independent, NR is approximately normally distributed with E (V) = 2 under the null hypothesis, E (NR) = 2. The mean of NR tends to be smaller than 2 for a non-homogeneous series and Var

KENDALL'S RANK CORRELATION TEST:
Rank correlation (Kendall, 1948;Abdi, 2007). [7,8] Can be used to establish whether an apparent trend in a series is significant or not. The number of times p in all pairs of observations , ; j > i that > is determined (i.e., for i = 1, N -1 how many times > is for j = i + 1, i + 2, N).

International Journal of Advanced Engineering Research and Science (IJAERS)
[ The test is carried out using the statistic Ԏ (known as Kendall's Ԏ and which varies between ±1) defined as Ԏ = 4 * ( −1) −1 for a random series, E (Ԏ) = 0, and its variance is given as Var (Ԏ) = 2 * (2 +5) 9 * ( −1) As N increases, converges to a standard normal distribution. It may also be possible to carry out a test using E (Ԏ) that takes values of -1 and 1, leading to the inference that there is a rising or falling trend. In this test we have to correlate the two adjacent variables in between -1 to 1 and after adding all the variables we get to chances of no trend in series if the value of 'z' lies within the limits ±1.96 at the 5% significance level, the null hypothesis of no trend cannot be rejected.

MEDIAN CROSSING TEST
: (Fisz, 1963). [9] x is replaced by zero if x~ < x (median), and X is replaced by one if > x. If the original sequence of x~ has been generated by a purely random process. In this test we have to compare all variables from a median of the series and after comparing us gets to a number of crossed or not crossed in given time series.

RUN ABOVE OR BELOW THE MEDIAN TEST FOR GENERAL RANDOMNESS:
(Shiau and Condie, 1980). [10] The necessary condition for applying this test is that the observations in the sample are obtained under similar conditions. Null hypothesis (H0) is made that the observations in a time series are independent of the order in the sequence, which is tested by the run test on successive differences. From the sequence of observations (t = 1, 2… n), a sequence of successive differences ( +1 -) is formed (i.e., each observation has the preceding one subtracted from it). In this test, we have to compare all variables from a median of the series and after comparing us get to a number of run above or run below in given time series. The teststatistic (K) is defined as the number of runs of '+' and '-' signs in the sequence of differences. If represents the total number of runs above and below the median of length s, then for a random process. , where s' is the maximum run length in the sequence.

TURNING POINTS TEST:
Let's assume that a turning point occurs in the series (t = 1, 2… n) at any time t (t = 2, 3… n-1) if is larger than each of −1 and +1 or is smaller than −1 and +1 this situation has four chances of occurrence in six different possibilities of the occurrence of −1 and +1 , assuming that all three elements have different values. In this test, we have to identify the number of turning points in given time series, when number turning point is more it means more chances to randomness or trendless in the dataset. Accordingly, the chance of having a turning point in a sequence of three values is 4/6 or 2/3, for all the values of' except for t = 1 and t = n. In other words, the expected number of turning points (p ̅ ) in the given random series can be expressed as (Kendall and Stuart, 1976 for the same random series, variance is given by (Kendall, 1973) The test-statistic is represented by the standard normal variate (z), and is given as: where p is observed number a of turning points It is a very easy to test to apply to a series of randomness observation involves the counting of the number of local maxima and minima, the interval between two turning points is called phase Turning point test reasonable against cyclicity but poor as a test against the trend.

RANK DIFFERENCE TEST:
(Meacham, 1968).  Where the bold value represents no long-term dependence for the given time series and remaining value shows term persistence. Where, Table -2 represent The Null hypothesis of no short-term dependence in the series is accepted and thus the given series can be assumed to be random at 10%, 5% and 1% significance level. Where,  The Bold value represents null hypothesis of No short-term dependence in the series is accepted and thus the given series can be assumed to be having No trend at 1% significance level.  ____ Marks represents null hypothesis of No short-term dependence in the series is accepted and thus the given series can be assumed to be having No trend at 5% and 1% significance level.  1.8811 represent represents null hypothesis of No short-term dependence in the series is accepted and thus the given series can be assumed to be having No trend at 10% significance level.  (*) marks represents null hypothesis of No short-term dependence in the series is accepted and thus the given series can be assumed to be having No trend  The remaining value represents null hypothesis of No short-term dependence in the series is accepted and thus the given series can be assumed to be having No trend at 10%, 5% and 1% significance level.  The Bold value represents short-term dependence is observed in the given series thus the data cannot be random at 5% significance level.  The remaining value represents No short-term dependence is observed in the given series thus the data can be random at 5% significance level. Where The Bold value represents "No short-term dependence" in the given series and remaining has No dependence at 5% significance level in the given series.  (*) Marks represents null hypothesis of no short-term dependence in the series is accepted and thus the given series can be assumed to be random at 10%, 5% and 1% significance level.  The Bold value represents null hypothesis of no short-term dependence in the series is accepted and thus the given series can be assumed to be random at 5% and 1% significance level.  ____ Sign represents null hypothesis of no short-term dependence in the series is accepted and thus the given series can be assumed to be random at 1% significance level. Where,  At 5% significance level, the value of standard normal variate is "1.95996" below this level show "No short-term dependence observed" thus given series can be random at 5% significance level.  The Bold value represents Null hypothesis of no short-term dependence in the given series, at 5% significance level.

V. CONCLUSION
The main objective of this study was to detect short-term and long-term dependence of streamflow time series. As a first step, Hurst coefficient was estimated at 5 Gauge and Discharge stations of daily river discharge time series for Seonath River Basin, Chhattisgarh State. For Hurst phenomena, the Hurst exponent was greater than 0.5. And this Statistical analysis H is estimated greater than 0.7 in the majority of different time scales and also observed that the null hypothesis of no dependence at 10%, 5% and 1% significance level for all the estimators. The finding of this study has more important implications for hydrological modelling especially in reservoir operation and water resource management for example, in order to estimate the risk of supply from a reservoir the long-term dependence primary incorporated into the model, this study suggests that to identify the main factors associated with the climate variability and storage that affects the long-term dependence of streamflow at a regional scale. Change in climate could have directly and indirectly affected by the various environmental variables including discharge in many countries of the world. Change in discharge regime directly affects the management of water resources, agriculture, hydrology and ecosystems. Hence it is important to identify the changes in the magnitude of the temporal and spatial behaviour of discharge is imperative for suggesting the suitable strategies for sustainable management of water resources, agriculture, environment and ecosystems.