The Modeling and Prediction of Great Salt Lake Elevation Time Series Based on ARFIMA

1. The Modeling and Prediction of Great Salt Lake Elevation Time Series Based on ARFIMA Rongtao Sun*, YangQuan Chen+, Qianru Li# * Phase Dynamics, Inc. Richardson,TX75081 +Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical & Computer Engineering, Utah State University #Department of Economics, Utah State University 3RD Int. Symposium on Fractional Derivatives and Their Applications (FDTA07) ASME DETC/CIE 2007, Las Vegas, NV, USA. Sept. 4-7, 2007

2. Applications to the Elevation Records of Great Salt Lake • The Great Salt Lake, located in Utah, U.S.A, is the fourth largest terminal lake in the world with drainage area of 90,000 km2. • The United States Geological Survey (USGS) has been collecting water-surface-elevation data from Great Salt Lake since 1875. • The modern era record-breaking rise of GSL level between 1982 and 1986 resulted in severe economic impact. The lake levels rose to a new historic high level of 4211:85 ft in 1986, 12.2 ft of this increase occurring after 1982. • The rise in the lake since 1982 had caused 285 million U.S. dollars worth of damage to lakeside. • According to the research in recent years, traditional time series analysis methods and models were found to be insufficient to describe adequately this dramatic rise and fall of GSL levels. • This opened up the possibility of investigating whether there is long-range dependence in GSL water-surface-elevation data so that we can apply FOSP to it.

3. Study area of Great Salt Lake • Adapted from M. Kemblowsli and Asefa, \Dynamic reconstruction of a chaotic system: The Great Salt Lake case," Utah Water Research Laboratory, Utah State University, 2003, working paper, 16p.

4. Long-term water-surface elevation graphs of the Great Salt Lake

5. Initial processing of GSL data • Let Itrepresents the levels of Great Salt Lake. We define rIas the logarithmic difference of the adjacent measurements. • Then, we define vIas the logarithmic of squared rI. • While the data set seems to be approximately uncorrelated, it is commonly accepted that they are not independent because the volatility clustering. • Log transformation reduces the skewness and makes unconditional distributions closer to the normal distribution.

6. Power spectral density of the volatility of GSL water-surface elevation

7. R/S Analysis • The R/S statistic is the range of partial sums of deviations of a time-series from its mean, rescaled by its standard deviation. A log-log plot of the R/S statistic versus the number of points of the aggregated series should be a straight line with the slope being an estimation of the Hurst exponent.

8. Aggregated Variance Method • The method plots in log-log scale the sample variance versus the block size of each aggregation level. If the series is long-range dependent then the plot is a line with slope greater than -1. The estimation of H is given by

9. Absolute Value Method • The log-log plot of the aggregation level versus the absolute first moment of the aggregated series is a straight line with slope of H -1, if the time series is long-range dependent.

10. Variance of Residuals Method • The method uses the least-squares method to fit a line to the partial sum of each block. A log-log plot of the aggregation level versus the average of the variance of the residuals after the fitting for each level should be a straight line with slope of H=2.

11. Periodogram Method • This method plots the logarithm of the spectral density of a time series versus the logarithm of the frequencies. The slope provides an estimate of H = 0.547.

12. Wavelet Based Method

13. FrFT Based Estimator

14. Autoregressive fractional integral moving average (ARFIMA) • Using a fractional differencing operator which defined as an infinite binomial series expansion in powers of the backward-shift operator, we can generalize ARMA model to ARFIMA model. where L is the lag operator, are error terms which are generally assumed to be sampled from a normal distribution with zero mean:

15. ARFIMA Modeling and Forecasting • Standard deviations of the forecasting by ARFIMA with different samples for training

16. ARFIMA Modeling and Forecasting • Standard deviations of the forecasting by ARFIMA with different forecasted samples

17. ARFIMA Modeling and Forecasting • 2002-2007 GSL elevation differences forecasting

18. ARFIMA Modeling and Forecasting • 2002-2007 GSL elevation forecast by ARFIMA

19. Conclusions • Fractional order signal processing techniques are very useful in analyzing real world data and is developing incredibly fast. • Great Salt Lake data are found to have long-range dependence. • ARFIMA model has been successfully implemented for fitting the volatility of GSL water-surface-elevation data. It has been found to provide satisfactory forecast of the water-levels in the future. The experiments also show that more previous data samples result in better forecasting.

20. Thank you!