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An inherent advantage of years Hipel and McLeod, There are the SARIMA family of models is that only few two types of structural models — deseasonalized model parameters are required for describing and periodic. In deseasonalized modeling, the time series which exhibit non-stationarity seasonal component of the time series to be both within and across seasons. Some useful modelled is removed by first subtracting each applications of these models in seasonal river seasonal mean from the corresponding seasonal flow forecasting are reported in McKerchar observations, and then dividing by the respective and Delleur , Panu et al.

In Mondal et al. Kavvas and Delleur have can account for variability in seasonal standard shown, both from analytical and empirical deviations and correlations that a SARIMA Journal of Hydrology and Meteorology, Vol. After transformation, the seasonal mean is To find out the required number of harmonics, removed from the series y r ,s by subtracting the and corresponding Fourier coefficients, seasonal mean m s from each observation and necessary for a good fit in equation 1 , a then dividing the result by the corresponding cumulative periodogram test, which is a seasonal standard deviation s s. Seasonal graphical test, is usually conducted.

This test is means and standard deviations can be obtained the most accurate for selecting the number of by parametric or non-parametric analysis. The The non-parametric method requires many test is carried out by computing first the mean parameters to remove seasonality, particularly squared deviation MSD of m s around m : 2 when a time series is weekly or decadal. Chowdhury 35 model parameter estimation, and iii diagnostic 1 2 checking of the model residuals.

Pi , which is the ratio of the variable z t. In the estimation stage, of the sum of the first i MSDs to the MSD m , maximum likelihood estimates of different is then computed from: model parameters are obtained. There are some additional statistical parts: a faster increasing periodic part and a tools, such as Akaike Information Criterion slower increasing sampling part. The two parts AIC, Akaike, and Bayes Information are approximated by two smooth curves, the Criterion BIC, Schwartz, , which can intersecting point of which provides the number be used to select the best model from several of significant harmonics. An ARMA model is then fitted to determine whether they satisfy the to the deseasonalized time series z t.

The assumptions of independence, normality, and general equation of the ARMA model Box homoscedasticity constant variance. The et al. When the model residuals pass each partitioned matrix are indicated. The where mt , j is the t , j th entry in the matrix M. The notation k indicates a time lag. Chowdhury 37 Steps 5 through 8 are repeated for generation The catchment area of the river is about 0. However, this 3. A gauge station of the river is located river and contributes about two-thirds of the at Bahadurabad, which is at 10 km downstream total dry season flows in Bangladesh.

It plays of the off-take of the Old Brahmaputra. The an important role in overall socio-economic distance between Bahadurabad and Aricha is development of the country. An application about km. The width of the river varies of the above deseasonalized ARMA model spatially and temporally, and the overall width was made to the decadal flow of this river ranges from 6 to 14 km FAP 24, b. Before proceeding to the model application, a general description of the Brahmaputra River and its flow characteristics 3. River Discharge data of the Brahmaputra are available 3. The data are missing for 18 months October The Brahmaputra River is one of the largest to March , and April to March rivers in the world. It originates in the Jima It has a long course for months August to March in the about km through the dry and flat region of FAP 24 a report.

The data for this period southern Tibet. Throughout this upper course, have been replaced with the data derived from the river is generally known as the Tsang-Po the three rating equations suggested in FAP 24 FAP 24, a. At its easternmost point, the a. As the river enters Arunachal state of India, The Brahmaputra flow, on an average, reaches it is called Siang.

The Brahmaputra appears its peak during the second decade of July and in the Assam valley as the Dihang River. It trough during the last decade of February. From flows for about km through Arunachal the second decade of June to the first decade of state and km through Assam. The Dihang October, flows are much higher compared to is joined by the Dibang and the Lohit from the rest of the year. There is a strong seasonal the east near Sadiya in northeast Assam. From pattern in the Brahmaputra flow, so the flow is this point of confluence, the river is called the intra-year non-stationary.

As Brahmaputra, the river flows To check whether or not the flow is inter-year through the entire stretch of Assam and sweeps stationary, a total of 36 time series, one for each round the Garo Hills and enters Bangladesh. In decade of each month of the year, was plotted. A Bangladesh, the Brahmaputra flows southward linear regression line was superimposed on each for nearly km before joining the Ganges at of these plots. The slopes of the least-squares Goalanda FAP 24, b. Thus Journal of Hydrology and Meteorology, Vol. Chowdhury August the flow has in general increasing trends except is also justified from the fact that the skewness some decades in the pre-monsoon and monsoon of all months except June and July decreases periods.

However, the slopes of the increasing due to the natural logarithmic transformation trend lines were generally low. The per year as reported in Mondal et al. A common increase was found to vary between 0. The values of the coefficient of with it is to plot the values of spread and level determination R2 of the trend lines were also for each period. If there is no relationship, the low, 0 to Therefore, the small increasing points would cluster around a horizontal line. To determine an the Brahmaputra River exhibits a trend in the appropriate power for transforming the data, annual peak or trough, the highest and lowest we can plot, for each period, the logarithm of flows of each year were found out from the the median against the logarithm of the inter- or daily values.

They can be found in Mondal quartile range. Figure 1 shows such a plot for the et al. The analysis of the two extreme Brahmaputra flow data. We can see that there is a value series did not indicate the presence of any fairly strong linear relationship between spread linear trend in either series. The been any temporal change in the annual peak slope of the least-squares line is about 1. After highest and lowest water levels were determined applying this power transformation, a spread- for each year. It is found that the median date versus-level plot was again obtained.

No further of occurrence of peak flow is 30 July with a relationship was evident from such a plot. Dividing the peak and low flow time series into two halves each half with a year length , it is found that there is no significant difference in the time of occurrence of either the peak discharge or the low discharge between the two halves of the available periods. Furthermore, no trend is found in the two time series of the dates of occurrences of the highest and lowest flows. Decadal means and standard deviations of the Brahmaputra flow were found to be approximately proportional see Mondal et al.

The existence of such proportionality Figure 1: Spread versus level plot of the Brahmaputra flow at Bahadurabad indicates that a power or logarithmic transformation should be applied to the raw Normality checks of the negative power data before model construction. This conclusion transformed data were made with normal Journal of Hydrology and Meteorology, Vol. Chowdhury 39 probability plots as well as with tests of After removal of the seasonal component, normality. It is seen Shapira and Wilk, indicated that the power transformation improved the normality of the data significantly.

Therefore, the negative power transformed data were used for model building in the following sections. The result was then divided by the corresponding decadal standard deviation to obtain the deseasonalized series. The 36 decadal means and 36 decadal standard deviations for the transformed decadal flows were obtained by the parametric method. The first 5 and 13 harmonics were found to be significant for the decadal means and standard deviations, respectively.

To identify the significant harmonics, the graphical criterion of separating the harmonics into the periodic and sampling variation parts from the plot of the cumulative periodogram, as outlined earlier, was followed. These patterns indicate that the model can be a mixed model having both AR and MA parameters. After a few iterations, the model that appeared to be suitable was ARMA 1, 3.

The estimated parameters of the fitted Figure 2: Separation of the cumulative periodogram of model are given in Table 1. It is seen from the decadal means into two parts: periodic significant last column of the table that three parameters and sampling variation insignificant Journal of Hydrology and Meteorology, Vol. Chowdhury August Table 1: Estimated values, standard errors, t-statistics, etc. The cumulative periodogram of the residuals is shown in Figure 5. This figure does not show any periodic pattern in the model residuals. Brahmaputra River. The total number of 3. The procedures of forecast generation from the model are described in Mondal et al. The parameters of the model were estimated with the data up to February and the model was validated with the data from March to February using one-step- ahead validation forecasts.

Chowdhury 41 Figure 6: One-step-ahead forecasted flows along with the observed flows from the first decade of March to the last decade of February flows are given in Figure 6. It is seen from of that year. The influence of the disturbance the figure that the fitted model captures the on future flows reduces with the increase in observed decadal pattern of the Brahmaputra lead time.

This is also understandable from a flow reasonably well. The model performs very physical point of view. For example, if there is well during the dry season for which synthetic some rain in a time period, this rain will have flow would basically be required. The effect of rain on river flow will period affects the current and future flows, the decrease gradually as time passes away. This deseasonalized ARMA model was written in physical explanation of the behavior of the random shock form and the shock coefficients deseasonalized ARMA model gives it a strong were estimated. A plot of the coefficients against basis for use in river hydrology. It is evident from MAE of the one-step-ahead forecasts are given the figure that a disturbance in a decade of a year in Table 2.

The general algorithm Stedinger and Taylor suggested two described earlier for exact simulation with an diagnostics — model verification and validation ARMA model was used for the data generation. According to these authors, model using the SPSS package, with different verification is the demonstration that the random number seeds for different sequences. For this test, the theoretical variance the developed model. The 7 autocorrelations were computed. Chowdhury 43 Table 3: Observed, theoretical and generated variances, and lag-1 to lag-7 autocorrelations, for the deseasonalized decadal flows Parameter Observed Theoretical Generated Variance 0. For this, the Hurst coefficient and expected values. These are reported given in Table 3.

It is evident from the table in Table 4. It is seen from the table that the that the theoretical, as well as the observed, historical sequence has a Hurst coefficient of values of all the parameters are well inside the 0. It an expected value of 0. The sequences exhibit short-term characteristics, observed sequence has a RAR of The probability of Stedinger and Taylor described model exceedence of the historical Hurst coefficient validation as the demonstration that the and RAR was found to be Therefore, the developed deseasonalized ARMA model can be considered to have the capability of preserving the important long- term statistics of the Brahmaputra flow. Sugar plays the biggest role in the epidemic we see of chronic diseases, including diabetes, hypertension, obesity, and heart disease, to name a few.

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