Modelling Zero-inflated Rainfall Data through the Use of Gaussian Process and Bayesian Regression
Abstract
Rainfall is a key parameter for understanding the water cycle. An accurate rainfall measurement is vital in the development of hydrological models. By means of indirect measurement, satellites can nowadays estimate the rainfall around the world. However, these measurements are not always accurate. As a first approach to generate a bias-corrected rainfall estimate using satellite data, the performance of Gaussian process and Bayesian regression is studied. The results show Gaussian process as the better option for this dataset but leave place to improvements on both modelling strategies.
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Modelling Ze o-ina ed Rain all Da a
h ough he Use o Gaussian P ocess and
Bayesian Reg ession
Ma ga i a Alejand a Rebolledo Coy and Thomas Ba z-Beiels ein
Modelling Ze o-in la ed Rain all Da a h ough he Use o
Gaussian P ocess and Bayesian Reg ession
Ma ga i a Alejand a Rebolledo Coy and Thomas Ba z-Beiels ein
Ins i u e o Da a Science, Enginee ing, and Analy ics, TH-K¨oln
No embe 29, 2018
1 In oduc ion
Rain all is a key pa ame e o unde s anding he wa e cycle. An accu a e ain all
measu emen helps in he de elopmen o mo e accu a e hyd ological models. These
hyd ological models can be used la e in he design o be e managemen plans o he
a ailable wa e esou ces o in he implemen a ion o lood o d ough wa ning sys ems
o egions a isk. In he ecen decades, ain all es ima ion done by sa elli e p oduc s
ha e been made a ailable, p o iding a wo ldwide high spa io- empo al es ima ion o
p ecipi a ion. Howe e , as hese sa elli e ain all es ima es (SRE) a e done using indi ec
measu emen s om he sa elli es’ senso s a alida ion p ocess needs o be ca ied ou in
o de o a oid hei inco ec use [2]. Following [1] we aim o gene a e a bias-co ec ed
es ima e o ain all using sa elli e da a and ain gauge da a. Rain gauges a e ain all
senso s loca ed in a ne wo k in some gi en a ea. One o he main obs acles in using
hese senso s as a eliably sou ce o p ecipi a ion measu emen is he lack o co e age
in la ge a eas. Using he a ailable ain gauges we wan o calib a e he SREs o he
poin in which he ain gauge is loca ed and i s adjacen a ea. Fo his we use Guassian
p ocess eg ession and Bayesian linea eg ession on a ain all da a se .
2 Da a Desc ip ion
The selec ed ain all da a se comes om he Impe ial basin loca ed in Chile. This is
a ela i ely small a ea une enly co e ed wi h 13 ain gauges. One o he impo an
cha ac e is ic his a ea p esen s is i s plu ial hyd ological egime, meaning mos o i s
wa e comes om ain all. The collec ed da a co e s a ange o 13 yea s, om 2003 o
2015. The ain all measu emen s a e o ganised in 13 ables each wi h 4748 da a poin s.
All ables con ain he ollowing in o ma ion:
•The da e on which he measu emen was aken.
1
•The p ecipi a ion alue in millime es (mm) measu ed by he ain gauge (obse ed
alues).
•The SRE p ecipi a ion alue in mm agg ega ed yea ly eco ded o he speci ic
s a ion a ea (SRE annual).
•The SRE p ecipi a ion alue in mm agg ega ed seasonally eco ded o he speci ic
s a ion a ea (SRE seasonal).
The da a in all o he 13 ables show e y simila cha ac e is ics, wi h high dispe sion
and a lo o da a poin s in o a ound ze o, as illus a ed in ig. 1.
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0 10 20 30 40 50 60 70
0 20 40 60 80 100 120
SRE Annual Values
Rain Gauge Values
Figu e 1: Visualiza ion o one o he da a ables. The obse ed alues s SRE annual
show a high dispe sion wi h a lo o poin s concen a ed a ound ze o
3 Expe imen s
Two eg ession models a e i ed, he i s using he R package SPOT [3] o he Gaussian
eg ession. The second implemen s Bayesian eg ession using he s a is ical language
STAN [4]. We use Roo Mean Squa e E o (RMSE) and Kling-Gup a E iciency (KGE)
o e alua e he goodness-o - i ness (GOF) o he models. As a baseline we de ine he
RMSE and KGE ha he obse ed alues ha e agains he measu ed SRE. I a model
2
su passes he baseline GOF hen i is conside ed ha his model gi es a be e app oxi-
ma ion o he ain all han he aw SRE. To es o s abili y he eg ession models we e
un 10 imes wi h di↵e en s a ing poin s in each o he da a ables.
4 Resul s
Gi en he la ge amoun o da a poin s clus e K iging was implemen ed when execu ing
he Gaussian eg ession on he yea ly SRE da a. Acco ding o ou esul s nei he o he
models ga e a good app oxima ion o ain all when using he yea ly da a. On he o he
hand, Gaussian eg ession deli e ed a be e app oxima ion on he ain all eal alues
when using seasonal SRE da a. In his poin i was no ed ha Bayesian eg ession was
no able o cap u e medium o hea y ain all e en s.
5 Conclusion and Fu u e wo k
O e all Gaussian p ocess eg ession showed a be e pe o mance o his da a in com-
pa ison o Bayesian eg ession. Howe e wi h i s high ime complexi y i may be a
p oblem when applied o bigge da a se s. Clus e K iging was implemen ed as a solu-
ion o his p oblem howe e his inc eased he e o in he model. In he case o he
Bayesian eg ession i was no ed ha i s pos e io dis ibu ion was no able o escape
a e y educed a ea, losing in o ma ion o hea y ain all e en s. In u u e wo ks, we
would like o explo e a di↵e en app oach o clus e K iging ha can educe he amoun
o in oduced e o as well as di↵e en dis ibu ions and cons ain s o he Bayesian
eg ession.
Re e ences
[1] M. Zamb ano-Bigia ini, A. Naudi , C. Bi kel, K. Ve bis , L. Ribbe. “Tempo al and
spa ial e alua ion o sa elli e-based ain all es ima es ac oss he complex opog aph-
ical and clima ic g adien s o Chile”. In: Hyd ology and Ea h Sys em Sciences. 21.2.
2017.
[2] M. Geb emichael, E.N. Anagnos ou, M.M. Bi ew. “C i ical S eps o Con inuing
Ad ancemen o Sa elli e Rain all Applica ions o Su ace Hyd ology in he Nile
Ri e Basin”. In: jJAWRA Jou nal o The Ame ican Wa e Resou ces Assosia ion
46.2. 2010.
[3] T. Ba z-Beiels ein, C. Lasa czyk, M. P euss “Sequen ial Pa ame e Op imiza ion”.
In: IEEE Cong ess on e olu iona y compu a ion. 2005.
[4] S an De elopmen Team. “RS an: he in e ace o S an in R” Package e sion
2.16.2 h p://mc-s an.o g 2017
3
Kon ak /Imp essum
Diese Ve ö en lichungen e scheinen im Rahmen de Sch i en eihe "CIplus". Alle Ve ö -
en lichungen diese Reihe können un e
abge u en we den.
Die Ve an wo ung ü den Inhal diese Ve ö en lichung lieg beim Au o .
Da um de Ve ö en lichung: 07.11.2018
He ausgebe / Edi o ship
P o . D . Thomas Ba z-Beiels ein,
P o . D . Wol gang Konen,
P o . D . Bo is Naujoks,
P o . D . Ho s S enzel
Ins i u e o Compu e Science,
Facul y o Compu e Science and Enginee ing Science,
TH Köln,
S einmülle allee 1,
51643 Gumme sbach
u l:
Sch i lei ung und Ansp echpa ne / Con ac edi o ’soce
P o . D . Thomas Ba z-Beiels ein,
Ins i u e o Compu e Science,
Facul y o Compu e Science and Enginee ing Science,
TH Köln,
S einmülle allee 1, 51643 Gumme sbach
phone: +49 2261 8196 6391
u l:
eMail: homas.ba z-beiels ein@ h-koeln.de
ISSN (online) 2194-2870