Statistical Models for Rainfall with Applications to Index Insura

Transcription

1 Statistical Models for Rainfall with Applications to April 21, 2008

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3 Overview The idea: Insure farmers against the risk of crop failure, like drought, instead of crop failure itself. It reduces moral hazard and adverse selection.

4 Overview The idea: Insure farmers against the risk of crop failure, like drought, instead of crop failure itself. It reduces moral hazard and adverse selection. (Rainfall, Other Factors) Crop Yield Income.

5 Overview The idea: Insure farmers against the risk of crop failure, like drought, instead of crop failure itself. It reduces moral hazard and adverse selection. (Rainfall, Other Factors) Crop Yield Income. Statistically speaking: Crop Yield i = f (Rainfall i ) + ɛ i, Income i = f (Crop Yield i )

6 Stabilizing Annual Income Without Insurance: Net Income = Yield ($) With Insurance: Net Income = Yield ($) + Insurance Payout - Cost of Insurance Without Insurance With Insurance Income ($) Income ($) Year Year

7 Contract Design Net Income i = Yield i + Insurance Payout i Cost of Insurance = f (R i ) + ɛ i + g(r i ) c Payout = a(rainfall) + b, a < 0 Minimizing Variance Payout ($) SD of Net Income ($) Rainfall (mm) Slope = a

8 The Data 70 Daily Rainfall mm Year Figure: Daily Rainfall, Lilongwe, Malawi,

9 For most data sets, we have between 5 and 45 years of daily rainfall data. Wet Day Indicator X = {X mtd } = 1{Y mtd > 0}, the indicator of a wet day on day d in year t of month m. Rainfall Y = {Y mtd } denote the amount of rainfall on day d in year t of month m. We fit a model whose likelihood can be factored into two parts: the likelihood of the frequency of wet days, X, and the likelihood of the intensity of rainfall on wet days, Y : P(X, Y θ) = P(X θ 1 ) P(Y θ 2 ).

10 The model for rainfall intensity (Y ) Y mtd X mtd = 1 Gamma(α mt, β mt ),

11 The model for rainfall intensity (Y ) Y mtd X mtd = 1 Gamma(α mt, β mt ), ( log(αmt ) log(β mt ) ) ) N (µ m, Σ m,

12 The model for rainfall intensity (Y ) Y mtd X mtd = 1 Gamma(α mt, β mt ), ( log(αmt ) log(β mt ) ) ) N (µ m, Σ m, p(µ m, Σ m ) Σ m 3 2, for t = 1,..., 45, and d = 1,..., n d.

13 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

14 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

15 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

16 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

17 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

18 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

19 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

20 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

21 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

22 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

23 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

24 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

25 Illustration Rainfall Intensity by Month mm JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Month

26 The model for rainfall frequency (X ) ( ) p1mt 1 p X mtd X mt(d 1) P mt = 1mt 1 p 2mt p 2mt

27 The model for rainfall frequency (X ) ( ) p1mt 1 p X mtd X mt(d 1) P mt = 1mt 1 p 2mt p 2mt p jmt Beta(µ jm, s jm )

28 The model for rainfall frequency (X ) ( ) p1mt 1 p X mtd X mt(d 1) P mt = 1mt 1 p 2mt p 2mt p jmt Beta(µ jm, s jm ) µ jm U(0, 1) s jm U(0, ) for d = 2,.., n d, m = 1,..., 12, t = 1,..., 45, and j = 1, 2.

29 Fit the model

30 Fit the model

31 Fit the model

32 Fit the model

33 Fit the model

34 Fit the model

35 Fit the model

36 Fit the model

37 Model Checking - Posterior Predictive Checks 1. 1 data set = 45 years of daily rainfall. Measure ˆθ data. ex) ˆθ data = SD of annual rainfall 2. Simulate 500 copies of the data (45 years of daily rainfall each). Measure ˆθ rep for each copy from rep = 1,..., Plot the distribution of ˆθ rep (a histogram), and then plot the observed value ˆθ data and see if it falls within the distribution of simulated statistics.

38 Posterior Predictive Check *Could this model have produced this data? Interannual Variability mm

39 Interpreting Posterior Predictive Checks Interpret it like a p-value. 1 out of 20 values of ˆθ data will lie in the tail of the distribution of ˆθ rep just by random chance.

40 When a bad result is good: (ˆθ data = Y mt1 Y (m 1)t(31) ) This model exaggerates the difference in rainfall between the last day of one month and the first day of the next month. AUG SEP OCT NOV Frequency Frequency Frequency Frequency mean.last.sim[, i] mean.last.sim[, i] mean.last.sim[, i] mean.last.sim[, i] DEC JAN FEB MAR Frequency Frequency Frequency Frequency mean.last.sim[, i] mean.last.sim[, i] mean.last.sim[, i] mean.last.sim[, i] APR MAY JUN JUL Frequency Frequency Frequency Frequency mean.last.sim[, i] mean.last.sim[, i] mean.last.sim[, i] mean.last.sim[, i]

41 The Frequency Domain Rainfall-generating process is smooth through time: Y t gamma(α t, β t ), for t = 1,..., (45 365). α t N(X t η α + a sin (2π t) + b cos (2π 365 t), σ2 α), β t N(X t η β + c sin (2π t) + d cos (2π 365 t), σ2 β ). Non-informative priors on η α, η β, a, b, c, d.

42 Thanks : Dan Osgood (IRI) & colleagues Andy Robertson (IRI) Thanks. shirley