CIplus
Band 5/2018
Modelling Ze o-ina 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 ’soce
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