An R based rainfall generator using a stochastic seasonal switching model

Transcription

1 An R based rainfall generator using a stochastic seasonal switching model Trevor Carey-Smith a, Peter Thomson b and John Sansom a a National Institute of Water and Atmospheric Research, New Zealand b Statistics Research Associates Ltd, New Zealand

2 WxGenerator?!? a library for the R statistical programming language. Package Objectives: 1. initially implement: a multi-site rainfall generation model a seasonal switching model 2. simple to use 3. inbuilt model validation/adequacy 4. simple and informative output 5. be extendable 2

3 The talk/package outline 1. The dataset importing, preparation 2. Initialisation a. rainfall model b. seasonal model c. reduced models 3. Parameter estimation 4. Simulation 5. Validation 3

4 Southland rainfall time series 12 locations 55 years ( ) Daily rainfall totals 4

5 Rainfall data within R package Held in a wg_data object: dat <- as.wg_data( rain=cbind(1:5,4:8), date=seq(sys.date(),,'days',5) ) wg_data sl_data <- read.wg_data( southland.csv ) 5

6 wg_data methods For example: subset.wg_data(), plot.wg_data(), etc. plot( subset(sl_data,select=1:4,year=1950,mon=1:10), cum=true ) 6

7 The rainfall model Based on Wilks(1998)and Thompson et al. (2007) This is a (partially) hidden three state model: dry 0 light rain 1 heavy rain 2 Dry 0 Light 1 Wet Heavy 2 Wilks, D.S., Multisite generalization of a daily stochastic precipitation generation model. J. Hydrol. 210, Thompson, C.S., P.J. Thomson and X. Zheng, Fitting a multisite daily rainfall model to New Zealand data. J. Hydrol.340,

8 The rainfall model The rainfall amounts depend solely on the current rainfall state: 0,1,2 0 where: and are unit exponential random variables. 8

9 Model parameters Dynamic parameters: 0 1 0, 0,1,2 0,1,2 Static parameters: : mean light rainfall : mean heavy rainfall Spatial parameters: : covariance matrix for rainfall states : covariance matrix for rainfall amounts 9

10 Model initialisation Model parameters held in wg_theta object: Dimensions: 8 sl_theta <- wg_theta(sl_data) 10

11 plot(sl_theta) 11

12 Interpretable parameters stationary probabilities: P(dry), P(light rain), P(heavy rain). expected durations: E(dry duration) E(light duration) etc. expected rainfall amounts: E(light rain) E(heavy rain) E(total rain) P(light rain) P(heavy rain) 12

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14 plot(sl_theta,param= OmegaPsi ) 14

15 Seasonal switching model 1. Season anchor points. 2. An onset (or hazard) function for season change. e.g. standard deterministic seasons: sl_seas <- wg_seas( c(15,105,196,288), function(t) as.numeric(1:t > T/2), ) Equivalently: sl_seas <- wg_seas( seasons, fixed ) 15

16 Deterministic seasons plot(wg_seas( seasons, fixed )) Anchor points Onset times Season change interval 16

17 Stochastic seasons uniform onset probability function(t) rep(1/t, T) 17

18 Stochastic seasons beta onset probability sl_theta <- wg_theta(sl_data, seas= seasons,fun= betaonset ) function(t,q0,q1) -diff(1-pbeta(0:t/t,q0,q1)) 18

19 Reduced models The rainfall model can be simplified. For instance, the original Wilks (1998) model has: & wilks <- wg_reduce(theta=sl_theta,p=c(0, 12 ),a= 012 ) Our favoured model keeps and fixed across seasons: forces seasonal variability into the dynamic parameters 19

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22 Parameter estimation This is a multi-step process: 1. Dynamic and static parameters univariate EM algorithm 2. Season change parameters (if necessary) estimated between each EM iteration sum loglik over all sites quasi-likelihood 3. Multi-site covariance matrices are estimated pairwise the site specific univariate parameters are kept fixed uses maximum likelihood (non-linear minimisation) 22

23 Example estimation Estimation can be done in stages: # define seasons sl_seas <- wg_seas( seasons,fun= unifonset ) # initialise theta sl_theta <- wg_theta(sl_data,seas=sl_seas) #univariate estimation sl_est <- wg_estimate(sl_data,sl_theta,method= em ) # bivariate estimation sl_est <- wg_estimate(sl_data,sl_est,method= ml, ml_stage= bivar ) or equivalently: sl_est <- wg_estimate( sl_data,method= auto, seas= seasons,fun= unifonset ) 23

24 Goodness of fit Improvements in BIC come from: 1. not partitioning into seasonal datasets 2. reducing parameters 12 to 4 seasons; non-seasonal betas 3. using stochastic seasons Change in AIC BIC 24

25 Estimated parameters 25

26 Simulation sl_sim <- wg_simulate(sl_est,years=100) wg_data 26

27 Recovery of season change points 0, 1R 1 0, 1R 27

28 Validation 1. General properties: a. rainfall distribution b. auto and cross correlations c. wet/dry durations 2. Annual/seasonal statistics a. total rainfall b. maximum daily rainfall c. number of wet days d. longest dry period 28

29 Rainfall distribution wg_properties(sl_data,sl_sim,,property= dist ) 29

30 Durations Wet Dry 30

31 Autocorrelation Occurrence Amount 31

32 Cross-correlation Occurrence Amount 32

33 Monthly summaries 33

34 Beta distribution onset probability 34

35 Annual rainfall amount 35

36 Maximum daily rainfall 36

37 Number of wet days 37

38 Length of longest dry 38

39 Conclusion 1. R package Multi-site rainfall model Seasonal switching model Reasonably fast and straightforward to use Additional models?? 2. Seasonal switching model Improved model fitting Enhanced seasonality in rainfall parameters More model flexibility Tool to examine seasonality of a region Much still to be explored 39