Bayesian Hierarchical Modelling of Rainfall IFD Curves · 3 | Bayesian Hierarchical Modelling of Rainfall IFD Curves IFD curve example •Intensity (or depth) How much rain falls

Bayesian Hierarchical Modelling of Rainfall IFD Curves Eric A. Lehmann | Aloke Phatak, Rex Lau, Joanne Chia, Mark Palmer

CSIRO Computational Informatics, Perth, Australia

HCSS Team Meeting, May 2014

2 |

Presentation overview

• Some background information • Intensity—frequency—duration (IFD) curves

• Bayesian modelling

• Etc.

• My experiences with BHM implementation • Computational issues

• Coding issues

• Etc.

• Discussion …

Bayesian Hierarchical Modelling of Rainfall IFD Curves

Bayesian Hierarchical Modelling of Rainfall IFD Curves 3 |

IFD curve example • Intensity (or depth)

How much rain falls (per unit of time).

• Duration Period over which the rain falls.

• Frequency How often we expect to observe such an event, e.g., level exceeded once in 100 years.

Background • Climate extremes (not just

means), under climate change

• Civil engineers, design of hydraulic infrastructures, ...

• Australian Rainfall & Runoff project (Engineers Australia)

Source: Australian Rainfall & Runoff (R87) (http://www.bom.gov.au)

http://www.bom.gov.au/


Typical dataset

• 252 stations around the Greater Sydney region.

• Annual maxima extracted between 1960 and 2000.

• Pluviometer data registered at 5 min “resolution”.

• 12 different accumulation durations: 5 min to 72 h.

• Under certain conditions, the maximum value of a sequence of random variables has a generalized extreme value (GEV) distribution.

• If z is the annual max. rainfall:

with GEV parameters: location , scale , and shape .

• Calculate IFD curves based on estimated GEV parameters (distribution quantiles).


Modelling rainfall extremes

0,,,)(

1exp)(GEV

1

zz


Spatial modelling of extremes (BHM) Data model: Y GEV(, , )

Process model:

MVN{ XT , (, ) } log() MVN{ XT , (, ) } MVN{ XT , (, ) } where: [(,)]ij = exp( xi - xj / )

Priors: for {, , } IG(, ) Gamma(, ) MVN(,

)

Inference: model fitting via MCMC simulations (Metropolis-Hastings within Gibbs), 150’000 iterations; prototyping in R code (~0.9s per iteration).

parameters vary smoothly according to a spatial stochastic process

Data model: Y GEV(, , )

Process model:



)



Spatial modelling of extremes (BHM)


Spatial modelling of extremes (BHM) Data model: Y GEV(, , )

Process model:



)


parameters vary smoothly according to a spatial stochastic process


Bayesian IFD curves

2 different return periods gauged vs. ungauged

Example: 100-year return level/intensity of precipitation (mean and 95% credible/confidence intervals).


Spatial return levels

Example: 100-year return level/intensity of precipitation (mean and 95% credible interval).


BHM summary • Provide estimates of uncertainty • BHM allows us to borrow strength spatially and over durations • Provide estimates of extremes at gauged and ungauged locations • Unified and flexible framework ... Features / extensions to this framework

– Koutsoyiannis parameterisation: * /

– duration dependence for GEV parameter: d d / ( d + )

– r-largest order statistics (use top r maxima)

– combination of pluvio and daily data

– integration of different covariates

– influence of climate change (RCM/GCM outputs)

– use of max-stable approach ...

For more info, please request copy of MODSIM’2013 and HWRS’2014 papers...


BHM implementation issues

MCMC simulation • Coding relatively straightforward (once you’ve worked out the

maths...), though many bells and whistles to account for (BHM)

• Computationally intensive and memory hungry!

• Which software?... • BUGS / WinBUGS / openBUGS / JAGS / WBDev

• PyMC

• many functions in R: BRugs, R2WinBUGS, rbugs, rjags, R2jags, runjags, etc.

• R packages: mcmc, MCMCpack, coda, BOA, etc.

• LibBi, spTimer, etc.

Need for full flexibility (modelling & coding), previous BHM code available, avoid new SW learning curve, etc. own implementation “from scratch” in R...


BHM implementation in R

First tests: ~40 s. per iteration!... (~46 days per run, 100’000 iterations)

not great for prototyping!...

Profiling R code With the help of:

• Rprof: enable profiling of the execution of R code (built-in)

• R packages: profr, proftools, ggplot2, Rgraphviz

Identify “slow” sections of code, then consider optimisation, Rcpp, parallelisation, etc.







BHM implementation in R: latest code







Speeding up your code without altering it: super-computers and optimised libraries...

Benchmark cross-product matrix

(b = a’ * a)

20 iterations of BHM code

Self time of %*% operations

Bragg-l 1.233s 2m 49s 65%

BANNISTER-FL 12.95s 4m 46s 65%

CMIS-08 16.48s 7m 26s 76%

CMIS1-PER 4.95s 10m 47s 74%

CSIRO Computational Informatics Dr. Eric A. Lehmann

T +61 8 9333 6123 E [email protected] W www.csiro.au

CSIRO COMPUTATIONAL INFORMATICS

Thank you – feedback welcome!...

Documents

Bayesian Hierarchical Modelling of Rainfall IFD Curves · 3 | Bayesian Hierarchical Modelling of Rainfall IFD Curves IFD curve example •Intensity (or depth) How much rain falls