1. Dong-Yun Kim, Chao Han, and Evan Brooks Virginia Tech Department of Statistics

2.  Introduction  Data  SPI  Application  Fitting the Models  Finding Excess Life  Summary  References 2

3.  There is great interest in modeling droughts, with two key objectives being:  Prediction of next drought occurrence  Estimating the length of the drought 3

4.  We have monthly precipitation measurements from 47 weather stations in Eastern Africa.  Each station has data for a different range of years. 4

5.  We can use the Standardized Precipitation Index (SPI) to measure the amount of monthly precipitation.  To get SPI from at least 30 years’ worth of raw precipitation data,  Fit a Gamma distribution to a moving average window within the data and calculate the cumulative probabilities,  Apply the inverse standard normal cdf to these values  The resulting SPI measures the number of SD’s above or below the long-term mean precipitation. 5

6.  SPI is particularly useful, since  The window in the gamma fit can be adjusted to capture different aspects of weather patterns  6 months for seasonal and mid-term weather  12 months for long-term weather and effects on water supply  Being standardized by site, SPI allows for comparison of relative precipitations over different regions. 6

7.  There is no unique definition of drought.  G. Tsakiris and H. Vangelis (2004) define an event a drought when the SPI is continuously negative and reaches an intensity where the SPI is –1.0 or less. The event ends when the SPI becomes positive.  We will use this definition of drought for the study. 7

8.  Our basic approach is to model the SPI time series according to a renewal process:  Events occur independently of each other with a common interarrival distribution  We are actually using two processes to model two aspects of drought:  N(t) models number of drought occurrences, with interarrival times {T i }  D(t) models drought durations and drought-free durations with an alternating renewal process with interarrival times {(X i,Y i )}  We are interested in both the mean interarrival times and the excess life for each variable, denoted by γ. 8

9. 9 st T3T3 Y1Y1 X1X1 γ X,t γ T,t γ Y,s Drought Offset Drought Onset X: Drought Period Y: No-drought Period T: Time Between Drought Onsets

10.  The first step is to estimate the interarrival distributions.  Natural choices include  Exponential (would imply a Poisson Process)  Weibull  Gamma  In our case, some of the stations’ series could be fitted by an exponential curve, but all of them were best fitted by a Weibull curve. 10

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12.  The Weibull distribution, like the Gamma, has both a shape and scale parameter, denoted k and β, respectively. 12

13.  Once the parameters are estimated (unique for each station), we can calculate the mean and SD for each of T, X, and Y.  In an alternating renewal process, the long-term expected time in state X is given by  Multiplying this by 12 months gives the overall expected number of months per year in a drought state. 13

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15.  Using the estimated Weibull parameters, it is also fairly easy to compute the distribution function for the excess life of each variable, conditional on the current time in that variable.  For a Weibull random variable, we have 15

16.  Also, we can calculate the mean and SD for the excess, conditional on the current life, by using the first and second moments. 16

17.  Since the Weibull distribution is typically right- skewed, we also calculate the p th percentile of the excess life distribution via the formula  By doing so, we can produce a table of probabilities and moments for given values of time spent in the current period. 17

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19.  The alternating renewal process model answers the question of “how much time out of the year will I be in drought?”  The Weibull model answers the question of “if I have been in drought t months, how much longer should I be in drought?” 19

20.  Are there any questions? 20

21.  Edwards, D. C., and T. B. McKee, 1997: “Characteristics of 20th century drought in the United States at multiple time scales.” Climatology Rep. 97-2, Department of Atmospheric Science, Colorado State University, Fort Collins, CO, 155 pp.  Guttman, N.B. (1998) “Comparing the Palmer Drought Index and the Standardized Precipitation Index.” Journal of the American Water Resources Association, Vol. 34, No. 1, pp.113-121  Hayes, M. J.: 1999, “Drought Indices”, National Drought Mitigation Center – Drought Happens, Drought Indices.  Hayes, M., Wilhite, D. A., Svoboda, M., and Vanyarkho, O., 1999: “Monitoring the 1996 drought using the Standardized Precipitation Index.” Bulletin of the American Meteorological Society, 80, 429–438.  McKee, T. B., N. J. Doesken, and J. Kleist, 1993: “The relationship of drought frequency and duration to time scales.” Preprints, Eighth Conf. on Applied Climatology, Anaheim, CA, Amer. Meteor. Soc., 179–184.  McKee, T. B., N. J. Doesken, and J. Kleist, 1995: “Drought monitoring with multiple time scales.” Preprints, Ninth Conf. on Applied Climatology, Dallas, TX, Amer. Meteor. Soc., 233–236.  Rossi, G., Benedini, M., Tsakiris, G. and Giakoumakis, S.,1992: “On regional drought estimation and analysis.” Water Resources Management, 6(4), 249–277.  Taylor, H. M., and Karlin, S. (1998) An Introduction to Stochastic Modeling, 3 rd ed. Academic Press, CA, 419- 457.  Tsakiris G., Vangelis H., 2004: “Towards a drought watch system based on spatial SPI.” Water Resources Management, 18(1), 1-12.  Weibull, W. (1951) "A statistical distribution function of wide applicability." J. Appl. Mech.-Trans. ASME, Vol. 18, No. 3. 293–297. 21