Dong-Yun Kim, Chao Han, and Evan BrooksVirginia Tech Department of Statistics

Introduction Data SPI Application

Fitting the Models Finding Excess Life

Summary References

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

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We have monthly precipitation measurements from 47 weather stations in Eastern Africa. Each station has

data for a different range of years.

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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.

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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.

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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.

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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 {Ti} D(t) models drought durations and drought-free

durations with an alternating renewal process with interarrival times {(Xi,Yi)}

We are interested in both the mean interarrival times and the excess life for each variable, denoted by γ.

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s t

T3Y1

X1

γX,t

γ T,t

γ Y,s

Drought Offset

Drought Onset

X: Drought PeriodY: No-drought PeriodT: Time Between Drought Onsets

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.

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

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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.

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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

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

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Since the Weibull distribution is typically right-skewed, we also calculate the pth 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.

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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?”

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Are there any questions?

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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, 3rd 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.

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