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SHORT-TERM SOLAR IRRADIATION FROM A SPARSE PYRANOMETER
NETWORK
ANNETTE ESCHENBACH
GRADO EN INGENIERÍA INFORMÁTICA, FACULTAD DE INFORMÁTICA,
UNIVERSIDAD COMPLUTENSE DE MADRID
T abajo Fin G ado en Ingenie ía
8 de junio de 2018
José Ignacio Gómez Pé ez
Ch is ian Tenllado an de Reijden
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Resumen
La p edicción p ecisa de la adiación sola es necesa ia pa a es ima co ec amen e
la p oducción de ene gía de los sis emas sola es o o ol aicos y su in eg ación en la ed
eléc ica. Es e abajo explo a has a qué pun o las écnicas de Machine Lea ning pueden se
u ilizadas pa a esol e es e p oblema. La me a es p edeci la adiación a co o plazo pa a un
obje i o con a ios ho izon es de p edicción. El obje o de las p edicciones es una de las 22
es aciones de edes de pi anóme os di usos con obse aciones de mues a de esolución 30’.
Se analizan las p es aciones y limi aciones de un modelo de Suppo Vec o Machine simple
y dos conjun os de mé odos de ap endizaje más so is icados – Random Fo es
Reg ession y G adien Boos ing. Se mues a que odos ellos uncionan bien en condiciones
climá icas cons an es pe o no ealizan p onós icos iables du an e días en que las condiciones
climá icas cambian ápidamen e. Una selección in eligen e de unciones es ú il pa a hace
que el modelo sea más e icien e y ápido sin necesa iamen e mejo a signi ica i amen e la
iabilidad de los esul ados. Con modelos ag egados pa a escena ios especí icos, se debe
p es a a ención a segui algunas eglas pa a no aumen a innecesa iamen e la complejidad
del modelo a expensas de la gene alización de nue os da os. Los modelos de en enamien o
en pequeñas can idades de da os p eseleccionados pueden causa sob eajus e o o e i ing.
Palab as cla e
P e isión a co o plazo de adiación sola , ap endizaje au omá ico, mine ía de da os,
á boles de decisiones, eg esiones de bosques alea o ios, máquina de ec o es de sopo e,
ed de pi anóme o di uso, Py hon, Pysola , Sciki Lea n, in eligencia a i icial, ap endizaje
de conjun o
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Abs ac
Accu a e o ecas ing o sola i adiance is necessa y o co ec es ima es o he
ene gy ou pu o sola pho o ol aic sys ems and hei in eg a ion in o he powe g id. This
pape explo es o which ex en Machine Lea ning echniques can be applied o sol e his
p oblem. The objec i e is o p edic sho - e m adia ion o a a ge wi h a ious o ecas
ho izons. The a ge is one o 22 s a ions o a spa se py anome e ne wo k wi h sample
obse a ions o 30’ esolu ion. The pe o mance and limi a ions o a simple Suppo Vec o
Machine model and wo mo e sophis ica ed ensemble lea ning me hods – Random Fo es
Reg ession and G adien Boos ing a e analyzed. I is shown ha all o hem pe o m well in
s eady wea he condi ions bu ail o make eliable p edic ions o days wi h apid wea he
changes. A sma ea u e selec ion p o es use ul o make he model mo e e icien and as e
wi hou signi ican ly imp o ing he eliabili y o he p edic ions. Wi h agg ega ed models
o speci ic scena ios one has o pay a en ion o ollow some ules in o de no o
unnecessa ily inc ease he complexi y o he model a he expense o gene aliza ion on new
da a. T aining models on small p eselec ed da a may cause o e i ing.
Keywo ds
Sho - e m sola adia ion o ecas ing, Machine Lea ning, da a mining, decision
ees, Random Fo es eg ession, Suppo Vec o machine, spa se py anome e ne wo k,
Py hon, Pysola , Sciki Lea n, a i icial in elligence, ensemble lea ning
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Table o Figu es
Figu e 1: Linea Reg ession model p edic ion (example), sou ce: [Ge17], p.109 .............. 16
Figu e 2: Polynomial Reg ession model p edic ion (example), sou ce: [Ge17], p.122 ...... 17
Figu e 3: High deg ee o polynomial eg ession shows o e i ing (example), sou ce: [Ge17],
S.123 .................................................................................................................................... 17
Figu e 4: SVM Reg ession wi h ma gins (dashed lines) and suppo ec o s (ci cles), sou ce:
[Ge17], p. 155 ...................................................................................................................... 18
Figu e 5: Decision T ee eg ession, sou ce: ........................................................................ 19
Figu e 6: AdaBoos Classi ie ha shows adap i e boos ing, sou ce: ................................ 19
Figu e 7: G adien Boos ing, sou ce: [Ge17], p.197 ........................................................... 20
Figu e 8: In o iego: selec ed s a ions and a ge TA ........................................................... 22
Figu e 9: example o nsamples- and o se -pa ame e s ...................................................... 23
Figu e 10: buil X, Y ma ixes (example) ........................................................................... 23
Figu e 11: MAE o he Pe sis ence model, SVM and Random Fo es o he whole es se
(agg ega ed by days) ............................................................................................................ 34
Figu e 12: Random Fo es model p edic s adia ion o a sunny day (easy scena io) ........ 35
Figu e 13: Random Fo es p edic s adia ion o a cloudy day (easy scena io) .................. 35
Figu e 14: Random Fo es p edic s adia ion o a day wi h uns able wea he condi ions
(di icul scena io) ............................................................................................................... 35
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Lis o Tables
Table 1: Resul s o C oss Valida ion o di e en algo i hms .............................................. 28
Table 2: Ex ac ion o he mos ele an ea u es wi h Random Fo es .............................. 29
Table 3: P edic ing wi h di e en o ecas ho izons ........................................................... 30
Table 4: Valida ion o di e en algo i hms on he es se .................................................. 34
Table 5: Agg ega ed model o “easy” and di icul p edic ions......................................... 37
Table 6: Agg ega ed model: N . o poo ly p edic ed samples ............................................ 37

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TABLE OF CONTENTS
RESUMEN ........................................................................................................................... 2
PALABRAS CLAVE ........................................................................................................... 2
ABSTRACT ......................................................................................................................... 3
KEYWORDS ....................................................................................................................... 3
TABLE OF FIGURES ........................................................................................................ 4
LIST OF TABLES ............................................................................................................... 5
LIST OF DEFINED TERMS ............................................................................................. 8
1. CAPÍTULO 1 - INTRODUCCIÓN ........................................................................ 9
1.1. Mo i ación y obje i os del es udio ................................................................................................... 9
1.2. Es uc u a de la esis ....................................................................................................................... 10
2. CHAPTER 1 - INTRODUCTION ....................................................................... 12
2.1. Mo i a ion and Goals o he P ojec .............................................................................................. 12
2.2. Thesis S uc u e ............................................................................................................................... 13
3. CHAPTER 2 - METHODS ................................................................................... 14
3.1. Machine Lea ning ........................................................................................................................... 14
3.1.1. Fundamen al Concep s o Machine Lea ning .................................................................................... 14
3.1.2. Ca ego ies o Machine Lea ning ....................................................................................................... 14
3.1.3. Mos Impo an and Es ablished Algo i hms o Reg ession in Supe ised Lea ning ....................... 16
(A) Linea and Polynomial Reg ession ................................................................................................. 16
(B) SVM Reg ession ............................................................................................................................. 17
(C) Decision T ees ................................................................................................................................ 18
(D) Ensemble Lea ning and Random Fo es ......................................................................................... 19
3.1.4. Py hon packages and lib a ies used o he implemen a ion .............................................................. 21
3.2. Da abase ........................................................................................................................................... 21
3.2.1. Resea ch A ea ................................................................................................................................... 21
3.2.2. Idea .................................................................................................................................................... 22
4. CHAPTER 3 - MODELLING APPROACHES .................................................. 24
4.1. Sola Model o Rela i e Radia ion ............................................................................................... 24
4.2. Fea u e Selec ion ............................................................................................................................. 24
4.3. Da a P epa a ion ............................................................................................................................. 26
4.3.1. De ec ion and Replacemen o Ou lie s ............................................................................................. 26
4.3.2. Replacemen o Missing Values ........................................................................................................ 26
4.4. Model Selec ion ................................................................................................................................ 27
4.5. T aining and E alua ing on he T aining Se ............................................................................... 27
4.5.1. Resul s o C oss Valida ion ............................................................................................................... 27
4.5.2. Ex ac ing ea u e impo ances wi h Random Fo es s ....................................................................... 28
4.5.3. T aining wi h di e en o ecas ho izons .......................................................................................... 30
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5. CHAPTER 4 - RESULTS ..................................................................................... 32
5.1. The Bias/Va iance T adeo ........................................................................................................... 32
5.2. Me ics o E alua ion o Model Accu acy o Indi idual Samples ........................................... 32
5.3. Me ics o measu ing he p edic ion quali y o days ................................................................ 33
5.4. Valida ion on he es se ................................................................................................................. 33
5.5. Re e ence Models ............................................................................................................................ 34
5.6. G aphic e alua ion .......................................................................................................................... 35
5.7. Agg ega ed model ............................................................................................................................ 36
6. CHAPTER 5 - CONCLUSIONS AND FUTURE WORK ................................. 38
6.1. Conclusions ...................................................................................................................................... 38
6.2. Fu u e Wo k .................................................................................................................................... 38
7. CAPÍTULO 5 - CONCLUSIONES Y TRABAJO FUTURO ............................ 39
7.1. Conclusiones .................................................................................................................................... 39
7.2. T abajo u u o ................................................................................................................................. 39
REFERENCES .................................................................................................................. 41
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Lis o de ined e ms
ANN
GHI
A i icial Neu al Ne wo ks
global ho izon al i adiance
NREL
Na ional Renewable Ene gy Labo a o y
ML
Machine Lea ning
PCA
RBF
P incipal Componen Analysis
Radial Basis Func ion
SPA
Sola Posi ion Algo i hm
SVM
Suppo Vec o Machine
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1. CAPÍTULO 1 - INTRODUCCIÓN
1.1. Mo i ación y obje i os del es udio
En los úl imos años, la ene gía sola se ha con e ido en una uen e de ene gía ba a a y limpia
y ha aumen ado signi ica i amen e su pa icipación en el suminis o de ene gía global. Sin
emba go, la i adiación sola depende de algunas condiciones climá icas incon olables,
como ene un cielo despejado. Como esul ado, la ince idumb e aún ep esen a un iesgo
eno me pa a la es abilidad de la ed eléc ica.
His ó icamen e, las in aes uc u as de las edes eléc icas gene almen e es aban diseñadas
pa a que u ie an ni eles de elec icidad ela i amen e cons an es, de modo que la demanda
y el suminis o de ene gía pudie an coincidi exac amen e. Sin emba go, la na u aleza de la
ene gía sola signi ica que a menudo hay caídas epen inas o picos en el suminis o de
elec icidad debido a los ápidos cambios de las condiciones climá icas, que c ean g andes
di icul ades a los ope ado es de la ed.
Las p edicciones p ecisas de la i adiación sola exac a en un momen o conc e o pueden
usa se pa a calcula la can idad exac a de elec icidad que se alimen a á a la ed. Luego, la
gene ación de ene gía pod ía se ajus ada o se pod ían ac i a una ese a de ene gía, según
el caso. Las p e isiones p ecisas pueden pe mi i la ges ión de la capacidad de ene gía ex a
con encional (nuclea , de gas, ca bón, e c.), sis emas de almacenamien o de ba e ía y ca gas
con olables y ambién pueden ayuda a ope a con elec icidad o o ol aica y con la ges ión
de plan as de ene gía.
La p edicción es, po lo an o, un ac o c ucial pa a in eg a la ene gía sola en el sis ema
de ene gía a bajo cos o. Sin emba go, las écnicas de p edicción iables siguen p esen ándose
como un g an desa ío.
Se han desa ollado modelos so is icados de p onós ico de i adiación sola y se ha log ado
una mejo a signi ica i a en la p ecisión de los p onós icos. Las dos g andes ca ego ías de
p edicción sola son la u ilización de sis emas de imágenes de nubes que as ean el
mo imien o de la nube y las simulaciones/op imizaciones numé icas. Sin emba go, oda ía
hay mucho ma gen de mejo a.
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3.1.3. Mos Impo an and Es ablished Algo i hms o Reg ession in Supe ised
Lea ning
The ollowing algo i hms all wo k equally well o classi ica ion asks. As his s udy ocuses
on a eg ession p oblem, we show he algo i hms in his con ex . I should be e iden o he
eade how o p edic a class ins ead o a alue.
(A) Linea and Polynomial Reg ession
Linea eg ession is he mos commonly used eg ession and simply looks o a linea
co ela ion be ween wo ea u es x and y, ha i s he aining da a o a s aigh eg ession
line. The Linea Reg ession algo i hm inds he op imal pa ame e alues o a linea
equa ion wi h bias θ0 and a slope θ1 such ha i minimizes a cos unc ion ha measu es he
dis ance be ween he linea model’s p edic ions and aining labels. E en hough his model
is limi ed and canno adap o non-linea ela ionships, he ad an age is ha i can ne e
“o e i ” he da a and gene alizes well, i.e. is no in luenced by noisy da a.
Figu e 1: Linea Reg ession model p edic ion (example), sou ce: [Ge17], p.109
Polynomial Reg ession is mo e powe ul and complex. The model c ea es addi ional ea u es
om he powe s o exis ing ea u es and hen ains a linea model on hem. This echnique
is capable o de ec ing non-linea ela ionships bu is p one o o e i ing. The e a e
egula iza ion me hods o de ec and a oid his isk.

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Figu e 2: Polynomial Reg ession model p edic ion (example), sou ce: [Ge17], p.122
Figu e 3: High deg ee o polynomial eg ession shows o e i ing (example), sou ce: [Ge17], S.123
(B) SVM Reg ession
The SVM Reg ession i s he da a on a kind o b oad s ee (la ge ma gin) and chooses he
line ha maximizes his ma gin (maximum ma gin es ima o ) while limi ing ma gin
iola ions. The hype -pa ame e ε con ols he wid h o he s ee . The bounda ies and
p edic ions a e no a ec ed by new samples as long as hey i on he s ee . Only he samples
on he edge o he s ee (suppo ec o s) ma e o he i , which p o ides high lexibili y
and as compu a ion.
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Figu e 4: SVM Reg ession wi h ma gins (dashed lines) and suppo ec o s (ci cles), sou ce: [Ge17], p. 155
The SVM is a ke nel-based model, i.e. he model can imp o e by ans o ming he inpu s,
which is done by p ojec ing he ea u es in o highe -dimensional space simila o a
polynomial eg ession.
Wi h complex polynomial o b (‘RBF’)’-ke nels he da a can be p ojec ed in o highe -
dimensional space de ined by polynomials and Gaussian basis unc ions ([Va16], p.411).
Thus, he SVM can pe o m linea and nonlinea eg ession and e en ou lie de ec ion.
The hype -pa ame e s C and gamma γ con ol he lexibili y o he ma gins, i.e. he ole ance
o ma gin iola ions by ou lie s. Bo h ac as egula iza ion pa ame e s and ha e a simila
in luence. Dec easing γ and C makes he model mo e gene al wi h mo e in luence o
indi idual samples. In case o o e i ing, γ and C should be educed [Ge152].
(C) Decision T ees
Decision T ees a e ypically known o pe o ming classi ica ion asks bu hey can equally
be used o eg ession. They wo k e y in ui i ely as in each node a ques ion is asked o
p edic he class o he sample o a alue, espec i ely ( eg ession).
The samples a e spli (gene ally bina y) such ha hei a e age alue comes as close as
possible o he a ge p edic ion alue o a node such ha he alue in he lea node inally
ep esen s he a e age a ge alue o he samples associa ed wi h his lea . Du ing his
p ocess, he algo i hm pe o ms a linea eg ession ha app oxima es a sine cu e.
O e i ing can be con olled by limi ing he pa ame e s max_dep h o he maximum dep h
o a ee and min_samples_lea o a minimum o samples equi ed in a lea . O he wise he
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cu e will ge e y dense and he algo i hm will badly o e i aining da a wi hou il e ing
ou he noise [Ge17], p.175-176.
Figu e 5: Decision T ee eg ession, sou ce:
h p://sciki -lea n.o g/s able/au o_examples/ ee/plo _ ee_ eg ession.h ml
(D) Ensemble Lea ning and Random Fo es
Figu e 6: AdaBoos Classi ie ha shows adap i e boos ing, sou ce:
h p:// insol.com/blog/2016/06/28/compu e - ision- ace-de ec ion/
In gene al, ensemble lea ning means ha a ew simple p edic o s a e combined in o an e en
mo e powe ul p edic o . One echnique is o agg ega e e y di e se p edic o s, ha ha e
been ained wi h di e en algo i hms ha a e as independen om each o he as possible.
Ano he way is o use he same aining algo i hm o e e y p edic ion bu ain hem on
di e en andom subse s o he aining se . I hese samples a e d awn wi h eplacemen
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(boo -s apping), his is called bagging. Random o es s a e an example o a bagging
ensemble me hod buil on decision ees wi h he boo s ap-pa ame e se o ue by de aul .
Con a y o he simple decision ees his model is less in ui i e and mo e o a black box.
The accu acy o e e y indi idual ee is imp o ed by a e aging he o e o he indi idual
sub ees. Each o he mul iple decision ees is buil on a andom subse o he aining
samples. Du ing he aining p ocess a speci ic numbe o ea u es is selec ed a andom o
ind he bes spli o he da a. The model accu acy can be e alua ed on he OOB-samples,
i.e. he “ou -o -bag” samples ha ha e no been used o he ee g owing and a e unknown
o he algo i hm [Zh17].
The e a e se e al mo e examples o he applica ion o he abo e p inciple.
Ano he ensemble me hod is o combine a couple o weak lea ne s in o a s ong lea ne .
P edic o s a e ained sequen ially, each ying o imp o e he p e ious esul . The mos
popula algo i hms ha use his me hod a e AdaBoos and G adien Boos ing. AdaBoos
assigns highe weigh s acco ding o he p edic ion e o o he ins ances such ha he nex
p edic o ocuses mo e on ha d cases ([Ge192], S.192).
G adien Boos ing is ano he sequen ial lea ning echnique, whe e he p edic o s all co ec
hei p e ious p edic o . This in each i e a ion he new p edic o s a e being i o he
unexplained e o s o he eg ession line ([GE17], p.195-197).
Figu e 7: G adien Boos ing, sou ce: [Ge17], p.197
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3.1.4. Py hon packages and lib a ies used o he implemen a ion
• Jupy e : Package o compu a ional en i onmen .
• NumPy: E icien manipula ion and s o age o n dimensional homogenous da a
wi h nda ays objec s.
• Pandas: E icien manipula ion and s o age o he e ogeneous and labeled da a in
da a ame objec s.
• Ma plo lib: Da a isualiza ions and plo ing.
• Sciki Lea n: E icien and clean implemen a ions o he mos common ML
algo i hms.
3.2. Da abase
3.2.1. Resea ch A ea
The app oach o his s udy is o choose an exis ing py anome e ne wo k o ob ain g ound-
based obse a ions o downwa d sola i adia ion in a gi en egion. These eal- ime
measu emen s would ideally p o ide a high le el o accu acy wi h a high empo al
esolu ion. This is e y ele an o sho - e m i adia ion- o ecas ing wi h he ho izon o
wo hou s as he accu acy o he p edic ions depends on he quali y o he aining da a o
he machine lea ning algo i hms. The inal model can a e wa ds be applied o o he da ase s
o o he ne wo ks. An al e na i e me hod consis s o ge ing less accu a e me eo ological
da a om emo e sensing ia sa elli es, ha a e no es ic ed o a speci ic a ea which allows
o ecas s wi h longe o ecas ho izons ac oss space and ime [Zh17].
In o iego is a go e nmen ally- unded and publicly accessible pla o m in Cas illa y León, an
au onomous communi y in No h-wes e n Spain. They main ain a ne wo k o mo e han 50
s a ions ac oss he egion in he di e en p o inces ha p o ide a ious me eo ological da a
mainly o ag icul u al pu poses wi h a esolu ion o 30 minu es3 ia a FTP-se e and a Res
3 A leas he da a used in his s udy (2015 o 2017) consis s ully o 30’obse a ions (moni o ed pe iod
In o iego: 2001 - now).

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API (web se ice): p ecipi a ion (mm.), empe a u e (°C), humidi y (%), sola
i adia ion(W/m2), wind eloci y (m/s) and wind di ec ion (°).
Figu e 8: In o iego: selec ed s a ions and a ge TA
3.2.2. Idea
Ou a ge s a ion is in he No h o Valladolid p o ince (see Figu e 8, VA01) idea is o pick
a a ge s a ion in he cen e o a local egion and he 21 su ounding s a ions in highes
p oximi y (See Figu e 8). The objec i e is o p edic he sola i adiance a he a ge TA o
a speci ic ime, based on samples om he o he s a ions wi h he ollowing pa ame e s. Ou
in ui ion is ha choosing he closes s a ions may help o imp o e p edic ions.
• P edic ion ho izon o o se : de ines he empo al di e ence be ween he sample
closes o he p edic ed alue and ime o p edic ion, i.e. om 8 a.m. i we wan o
p edic wi h a p edic ion ho izon o wo hou s o 10 a.m. [see example in igu e].
• nsamples: numbe o samples ha will be included om a s a ion, o example n
samples = 3 o a gi en s a ion, i we use da a om 7 a.m., 7.30 a.m. and 8.00
a.m. o p edic adia ion a 10.
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Figu e 9: example o nsamples- and o se -pa ame e s
In he aining X, Y ma ix wi h known alues e e y X, Y- ow ep esen s a sample. A ow
consis s o all he inpu ea u es X and he ou pu alue Y – he a ge alue. Y will hold he
alue measu ed a he p edic ion ime co esponding o X (See Figu e 10).
The adia ion is exp essed as ela i e global ho izon al i adiance (“GHI”), i.e. he a io o
measu ed GHI and he expec ed adia ion gi en op imal wea he condi ions (clea -sky
model). This a io se es he model o indi ec ly de i e day o yea and hou .
To suppo his idea o a ime-se ies, azimu h angle and zeni h angle will be addi ionally
included as ex a ea u es o he sample closes o p edic ion ime, i necessa y. All alues
will be no malized, i.e. con e ed o he ange [0, 1]. Especially SVMs a e sensi i e o ea u e
scales ([Ge17], p.146) whe eas Random Fo es s do no equi e ea u e scaling.
P ecipi a ion, empe a u e, humidi y, wind eloci y and wind di ec ion a e no u he
conside ed o be included as inpu pa ame e s o he model as hese measu es end o c ea e
noise in he compu a ion.
The idea o his s udy is o es i a machine-lea ning model can de ec a kind o signal
simila o he mo ion o clouds ha a els be ween he s a ions o e ime, o a leas
ecognize pa e ns ha indica e a d op o adia ion. These pa e ns will ha e o be disco e ed
and i is no known whe he a d op o adia ion may be due o a local s o m o a kind o mis .
Figu e 10: buil X, Y ma ixes (example)
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4. CHAPTER 3 - MODELLING APPROACHES
4.1. Sola Model o Rela i e Radia ion
To de e mine he exac GHI o all he s a ions gi en a clea sky, a model is necessa y ha
compu es he sola posi ion as accu a ely as possible. The Sola Posi ion Algo i hm (“SPA”)
by he Na ional Renewable Ene gy Labo a o y (“NREL”) o he Uni ed S a es is cu en ly
he mos common solu ion wi h he highes accu acy used o PV-applica ions in gene al,
including he calib a ion o py anome e s (See [Re08], p.1). The e exis se e al applica ions
based on his algo i hm, including a eely a ailable one ha implemen s i in Py hon:
Pysola . Pysola is a collec ion o Py hon lib a ies o simula ing he i adia ion o any poin
on ea h and chosen as he bes solu ion o his s udy as i is specially aimed a modeling
pho o ol aic sys ems. Sunpy o example is a simila applica ion bu ocused in sola physics
modelling.
Pysola expec s a imezone-awa e da e ime as inpu pa ame e oge he wi h longi ude and
la i ude o he loca ion o compu e azimu h angle and zeni h angle (al i ude). Wi h da e ime
and al i ude he GHI is ob ained. The a io o he measu ed adia ion and he ‘ideal’ GHI
ep esen s he ela i e adia ion o a speci ic loca ion a a speci ic ime.
4.2. Fea u e Selec ion
As desc ibed in sec ion [3.2.2] we c ea e he X, Y aining ma ix wi h he n alues om -
p edic ion ho izon o -1h-n*30’ o ela i e adia ion o all 22 s a ions as he samples all ha e
a pe iod o 30 minu es. The basic model is o ake nsamples = 3 wi h a p edic ion ho izon o
2 hou s. As we s a a 5 am he i s p edic ed alue is a 8 am, he las is a 10 pm. The
samples o he a ge s a ion i sel a e also included.
Fo his s udy di e en me hods ha e been used o selec ea u es:
• By ial and e o : jus build models, selec di e en se s o ea u es and check wi h
which ea u es pe o mance is imp o ed;
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• by looking a he s anda d co ela ion coe icien 4 o look o linea co ela ions
be ween a ea u e and he a ge alue; and
• by e alua ing he ea u e impo ance wi h he Random Fo es algo i hm.
The i s op ion is clea ly he mos in ui i e. I has ac ually been applied many imes, o
example o analyse o which s a ions i makes sense o include azimu h angle and al i ude.
The bes esul was o also include he azimu h angle o e e y adia ion sample om e e y
s a ion and he al i ude only once o he a ge s a ion a he a ge ime. Na u ally, no all
possible cons ella ions can be es ed wi h his app oach.
Ano he as and simple me hod o check o a co ela ion be ween a pai o con inuous
a iables o a eg ession p oblem is he s anda d co ela ion coe icien . I gi es a alue
om he in e al [-1,1] wi h -1 meaning a pe ec nega i e, 1 a pe ec posi i e and 0 no
co ela ion a all ([Ge17], p.56). Un o una ely, his me hod only gi es in o ma ion abou
linea co ela ion, while ea u es can be pe ec ly co ela ed in a non-linea way.
A less ob ious, bu e y elegan me hod is o ind da a co ela ion wi h Random Fo es s and
Decision T ees ([Ts10], p.11) ha will be used o de e mine he key ea u es o ou di e en
models (See [T aining and e alua ing on he T aining se ]). These algo i hms ind ou he
s a is ical usages o each ea u e which can be accessed wi h he ea u e_impo ances
a ibu e ha gi es he ela i e impo ance o each ea u e whe e he sum o all impo ances
is 1 ([Ge17] p.190).
Finally, wi h he P incipal Componen Analysis (“PCA”) algo i hm i is possible educe he
dimensionali y o he inpu ea u es (see unsupe ised lea ning algo i hms in [Ca ego ies o
Machine Lea ning]). PCA indeed can be an e icien way o educe complexi y and pe o m
a ea u e selec ion. Fo ou s udy we did no use PCA.
4 Also called Pea son’s , he co a iance o wo a iables X and Y, s anda dized by he p oduc o hei s anda d
de ia ions.
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5. CHAPTER 4 - RESULTS
5.1. The Bias/Va iance T adeo
In supe ised ML, op imizing one pa ame e o en means ha ano he ge s wo se – as in
desc ibed in sec ion [4.5.1] - when he (R)MSE is minimized du ing aining and he s anda d
de ia ion augmen s.
One has o unde s and ha he gene aliza ion e o o a model is composed o h ee pa s:
Bias, Va iance and he I educible E o . The las one desc ibes he noisiness o he da a and
we al eady ied o minimize i by da a cleaning and in elligen ea u e selec ion. T ying o
minimize Bias and Va iance a he same ime is impossible and will p obably esul in
o e i ing he da a.
Ins ead one should make he igh assump ions abou he da a in o de o de ec co ela ions
while a oiding an excessi e sensi i i y o small luc ua ions in he aining da a ([Ge17],
p.127). To p e en o e i ing he algo i hms can be egula ized by special pa ame e s (See
[Mos impo an and es ablished algo i hms o Reg ession in supe ised lea ning]) o
inc easing he aining da a also helps gene alize mo e.
5.2. Me ics o E alua ion o Model Accu acy o Indi idual Samples
I is ela i ely simple o e alua e i a classi ie wo ks well by jus quan i ying co ec and
alse p edic ions. Fo eg ession models i is qui e di icul o ind app op ia e pe o mance
measu es o e alua e he accu acy o he models o o compa e hei pe o mances. We will
use he s anda d pe o mance measu es o ML and also de ine a couple o own me ics.
(i) MAE (Mean Absolu e E o ): also called A e age Absolu e De ia ion - measu es
he Manha an dis ance be ween he a ge - and he p edic ion- ec o , whe e you can
only mo e along o hogonal pa hs wi hin a g id. This measu e is less sensi i e o
ou lie s han he RMSE.
𝑀𝑀𝑀𝑀𝑀𝑀(𝑿𝑿,ℎ)=1
𝑚𝑚∑|ℎ(𝑥𝑥(𝑖𝑖))−𝑦𝑦(𝑖𝑖)|
𝑚𝑚
𝑖𝑖=1 ([Ge17], p.39)
Whe e m is he numbe o samples, x(i) is he ec o o all he ea u e alues o he i h
ins ance in he da ase , y(i) he label o ha ins ance and h he sys em’s p edic ion
unc ion (hypo hesis) ha compu es a p edic ion h(x(i)) = ŷ(i)

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(ii) RMSE (Roo -Mean-Squa ed-E o ): a ypical pe o mance measu e o eg ession
p oblems ha gi es an idea o how much e o he sys em ypically makes in i s
p edic ions wi h a highe weigh o la ge e o s:
𝑅𝑅𝑀𝑀𝑅𝑅𝑀𝑀(𝑿𝑿,ℎ)=�1
𝑚𝑚∑(ℎ�𝑥𝑥(𝑖𝑖)�−𝑦𝑦(𝑖𝑖))2
𝑚𝑚
𝑖𝑖=1 ([Ge17], p.37)
(iii) MSE(X,h): he pa ame e ha is usually minimized by he aining algo i hm (cos
unc ion). The sco ing unc ion in Sciki Lea n usually is he opposi e o his (as
absolu e o ela i e alue) ([Ge17], p.70).
5.3. Me ics o measu ing he p edic ion quali y o days
(i) s(day): e o ec o (s) gene a ing a ec o pe day like: 𝒔𝒔(𝑑𝑑𝑑𝑑𝑦𝑦) =
� �(𝑟𝑟𝑟𝑟𝑑𝑑𝑟𝑟(𝑑𝑑𝑑𝑑𝑦𝑦,ℎ)−𝑝𝑝𝑟𝑟𝑟𝑟𝑑𝑑𝑝𝑝𝑝𝑝𝑝𝑝𝑟𝑟𝑑𝑑(𝑑𝑑𝑑𝑑𝑦𝑦,ℎ))2 �,∀ℎ,ℎ≥10:00 , ℎ≤22:00
(ii) Simila i y measu e: o compa e simila i y be ween he signals ( eal and p edic ed)
I consis s on he scala p oduc o he wo signals, di ided by he p oduc o i s
no ms:
∑
𝑖𝑖=22:00
𝑖𝑖=10:00 (𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝑖𝑖)∗𝑝𝑝𝑟𝑟𝑟𝑟𝑝𝑝𝑖𝑖𝑝𝑝𝑝𝑝𝑟𝑟𝑝𝑝(𝑖𝑖)
�∑
𝑖𝑖=22:00
𝑖𝑖=10:00 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝑖𝑖) 2 ∗ �∑
𝑖𝑖=22:00
𝑖𝑖=10:00 𝑝𝑝𝑟𝑟𝑟𝑟𝑝𝑝𝑖𝑖𝑝𝑝𝑝𝑝𝑟𝑟𝑝𝑝(𝑖𝑖)2
(iii) Sco e: in ege numbe o each day ha measu es he quali y o i s sample
p edic ions. We de ine a h eshold pa ame e ha de e mines i he di e ence
be ween a p edic ed and a eal alue is oo high. We e alua e he di e ence o
each hal -hou -sample and add i o he sco e i i exceeds he h eshold. The highe
he sco e o a day he wo se a e he p edic ions o his day. We use he numbe o
days wi h high sco es (depending on you h eshold) as a inal “o e all” me ic o
he es se - i.e. numbe o poo ly p edic ed days.
5.4. Valida ion on he es se
The es se included he whole yea 2016 (10 days we e missing).
Model RMSE S anda d
de ia ion
N . ea u es N . o
samples
Suppo Vec o eg ession 0.13227 0.04588 133 10308
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G adien Boos ing eg ession 0.11623 0.03569 133
Random Fo es eg ession 0.1185 0.036436 133
Random Fo es eg ession
wi h bes ea u es
0.11965 0.03708 32
Table 4: Valida ion o di e en algo i hms on he es se
5.5. Re e ence Models
Ano he me hod ha can p o ide a ough idea o how well you model wo ks is o compa e
i wi h o he models. The mos ‘ i ial’ o so-called ‘nai e’ o ecas ing model is o assume
‘ hings s ay he same’. I.e. i jus akes he las alid alue om he sample ec o o he
a ge s a ion as a p edic ion. This ‘pe sis ence’ model basically demons a e, i ou
o ecas ing has any e ec a all ([Vo17], p. 21).
Figu e 11: MAE o he Pe sis ence model, SVM and Random Fo es o he whole es se (agg ega ed by days)
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5.6. G aphic e alua ion
Figu e 12: Random Fo es model p edic s adia ion o a sunny day (easy scena io)
Figu e
14
:
Random Fo es p edic s adia ion o a day wi h uns able wea he condi ions (di icul scena io)
Figu e 13: Random Fo es p edic s adia ion o a cloudy day (easy scena io)
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5.7. Agg ega ed model
To imp o e he quali y o ou p edic ions we wan ed o ain wo di e en models o
“di icul ”, i.e. ha d o p edic days and o ‘easy’ o p edic days. To iden i y hose days we
used he me ics o indi idual days, see pa ag aph [4.3]. Fo his expe imen aining and
es se we e exchanged. F om he las alida ion se (yea 2016) wi h he Random Fo es
model he samples wi h a sco e > 0.3 and a sco e <=0.3 espec i ely we e selec ed as he
wo new aining se s o he wo sepa a e models o p edic di icul and easy days. Thei
ea u e impo ances we e he same and bo h we e e ained wi h 42 key ea u es.
Fo he alida ion he old ainse (yea 2015) was spli in o wo ca ego ies: ‘easy’ and
‘di icul ’ days. This ime he GHI om Pysola wi h absolu e adia ion alues was used
oge he wi h he absolu e adia ion labels o compu e a sco e. To b eak he es se in o
simila p opo ions as he aining se , he h eshold o he sco e was 7500. The models we e
hen indi idually es ed on he wo es se s and a e wa ds he esul s we e combined.
Model RMSE S anda d
de ia ion
N . ea u es
N . o
samples
Random Fo es
o “easy” days
0.08250 0.02800 133 5229
Random Fo es
o “di icul ” days
0.15913 0.01150 133 5079
Random Fo es
o “easy” days
wi h op imized ea u es
0.08101 0.02888 42 5229
Random Fo es
o “di icul ” days
wi h op imized ea u es
0.15992 0.01020 42 5079
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Agg ega ed Model o
“di icul ” and “easy” days
(on es se )
0.09963
0.03196
42 10178
Random Fo es o
“easy” days (indi idual)
(on es se )
0.13032
0.04672
42 4743
Random Fo es o
“di icul ” days (indi idual)
(on es se )
0.13454
0.04218
42 5435
Table 5: Agg ega ed model o “easy” and di icul p edic ions
Model N . o poo ly p edic ed
samples
N . o o al
days
no agg ega ed
T aining se
5079
359
Agg ega ed
Tes se
3653
349
Table 6: Agg ega ed model: N . o poo ly p edic ed samples
The numbe o mis-p edic ed samples dec eased signi ican ly wi h he agg ega ed model and
he model may p edic e y well o some speci ic days. Howe e , he a iance o he da a
inc eases eno mously. An explana ion o his could be ha he combined model o e i s he
da a and does no gene alize well. The educed numbe o aining ins ances o each
indi idual model may also be esponsible ha he algo i hm is mo e sensi i e o indi idual
da a.
Un o una ely, aining and es se we e exchanged o his expe imen ha made a mo e
de ailed compa ison wi h he non-agg ega ed Random Fo es model impossible.

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6. CHAPTER 5 - CONCLUSIONS AND FUTURE WORK
6.1. Conclusions
The empo al acking o signals did no eally succeed as olde samples did no p o e use ul
o he p edic ions and only he newes ones we e ele an . None heless, he selec ed models
all pe o m well in clea -sky si ua ions o on comple ely cloudy days bu do no p edic well
o pa ly cloudy days wi h quick wea he changes. The Random Fo es model acili a es he
ea u e selec ion by gi ing di ec access o hei impo ances. This leads o he idea o a
highly complex model whe e o e e y poin in he da ase he bes combina ion o ea u es
could be used o build he ideal model. Ye he e o e, an ins ance is needed ha ells us wi h
100% accu acy be o ehand he bes ea u e selec ion o he gi en scena io. The app oach
o de eloping agg ega ed models o di e en scena ios comes wi h a high isk o o e i ing
he da a and may no gene alize well on unknown da a.
6.2. Fu u e Wo k
A way o imp o e ou model could be he enginee ing o new ea u es. An in e es ing
app oach would be o example o c ea e a new ea u e ec o wi h he di e ences be ween
consecu i e samples, ha kind o app oxima es o he de i a i e. As inc easing he
complexi y o ou models comes wi h a high isk o o e i ing we should a he y o
inc ease he quali y and a ie y o ou da ase . Though di icul o ob ain a ne wo k wi h
highe esolu ion and maybe a mo e dense g id s uc u e may be key o ealize he idea o a
model ha is mo e esponsi e o he signals o su ounding s a ions and sensi i e o quick
wea he changes. We could y o buil agg ega ed models wi h e y di e se p edic o s ha
could wo k on he same da ase and implemen a o ing sys em among hem. We could also
apply he SVM on di e en andomly selec ed subse s o he aining se and implemen a
so o bagging me hod simila o he Random Fo es algo i hm. Also ea u e selec ion could
be done andomized and hen imp o ed by he algo i hm.
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7. CAPÍTULO 5 - CONCLUSIONES Y TRABAJO FUTURO
7.1. Conclusiones
El seguimien o empo al de las señales no u o ealmen e éxi o, ya que las mues as más
an iguas no esul a on ú iles pa a las p edicciones y solo las más ecien es ue on pe inen es.
No obs an e, los modelos seleccionados uncionan bien en si uaciones de cielo despejado o
en días comple amen e nublados, pe o ienen un endimien o de icien e pa a los días
pa cialmen e nublados con cambios ápidos de clima. El modelo Random Fo es acili a la
selección de ca ac e ís icas al da acceso di ec o a las impo ancias. Es o lle a a la idea de
un modelo al amen e complejo donde pa a cada pun o del conjun o de da os pod ía usa se la
mejo combinación de ca ac e ís icas pa a cons ui el modelo ideal. Sin emba go, se
necesi a una ins ancia que nos indique con 100% de p ecisión de an emano la mejo
selección de ca ac e ís icas pa a un escena io o ci cuns ancias conc e as. El en oque de
desa olla modelos ag egados pa a dis in os escena ios iene apa ejado con un g an iesgo
de o e i ing de los da os y puede que no se gene alice bien con da os desconocidos.
7.2. T abajo u u o
Una o ma de mejo a nues o modelo pod ía se la ingenie ía de nue as ca ac e ís icas. Un
en oque in e esan e se ía, po ejemplo, c ea un nue o ec o de ca ac e ís icas con las
di e encias en e mues as consecu i as, que de algún modo se ap oxima a la de i ada. Como
aumen a la complejidad de nues os modelos implica un al o iesgo de sob eajus e o
o e i ing, debe íamos in en a aumen a la calidad y la a iedad del conjun o de nues os
da os. Aunque es di ícil ob ene , la u ilización de una ed con mayo esolución y al ez una
es uc u a de ed más densa puede se cla e pa a ealiza la idea de un modelo que esponda
mejo a las señales de las es aciones ci cundan es y sea sensible a los ápidos cambios
climá icos. Pod íamos a a de cons ui modelos ag egados con p edic o es muy di e sos
que pod ían unciona en el mismo conjun o de da os e implemen a un sis ema de o ación
en e ellos. También pod íamos aplica el SVM en di e en es subconjun os seleccionados al
aza del conjun o de en enamien o e implemen a una especie de mé odo de ensacado
(bagging me hod) simila al algo i mo de Random Fo es . La selección de ca ac e ís icas
ambién pod ía hace se de o ma alea o ia y luego mejo ada a a és del algo i mo.
40/41
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Re e ences
[Ge17] Gé on, Au élien: Hands-On Machine Lea ning wi h Sciki -Lea n& Tenso Flow –
Concep s, Tools and Techniques o build in elligen Sys ems, i s edi ion, O’Reilly
Media, 2017
[Mc12] McKinney Wes, Py hon o da a analysis, i s edi ion, O’Reilly Media, 2012
[Re08] Reda, I.; And eas, A.: Sola Posi ion Algo i hm o Sola Radia ion Applica ions.
55 pp.; NREL Repo No. TP-560-34302, 2003, e ised Janua y 2008,
h ps://www.n el.go /docs/ y08os i/34302.pd
[Ts10] Tsanas, A hanasios: A Simple Fil e Benchma k o Fea u e Selec ion, in Jou nal
o Machine Lea ning Resea ch, 2010,
h p://www.maxli le.ne /publica ions/ sanas10a.pd
[Va16] Vande Plas, Jake: Py hon Da a Science Handbook: Essen ial Tools o Wo king
wi h Da a, i s edi ion, O’Reilly Media, 2016
[Vo17] Voyan , Cy il; No on, Gilles; e .al.: Machine lea ning me hods o sola adia ion
o ecas ing: A e iew, Uni e si y o Co sica, 2017
[Zh17] Zhou, Qing ao: A machine lea ning app oach o es ima ion o downwa d sola
adia ion om sa elli e-de i ed da a p oduc s: An applica ion o e a semi-a id
ecosys em in he U.S., in PLOS jou nal, Boise S a e Uni e si y, Boise, Idaho,
USA, h ps://www.ncbi.nlm.nih.go /pmc/a icles/PMC5544233/, published online
04.08.2017
The code o he p ojec can be ound he e:
h ps://d i e.google.com/d i e/u/1/ olde s/1o eOP5 Si29k57Uo_i_ScRB0e u0pQ0