Jin, Bingzi; Xu, Xiaojie
A icle
P edic ing wholesale edible oil p ices h ough Gaussian
p ocess eg essions uned wi h Bayesian op imiza ion and
c oss- alida ion
Asian Jou nal o Economics and Banking (AJEB)
P o ided in Coope a ion wi h:
Ho Chi Minh Uni e si y o Banking (HUB), Ho Chi Minh Ci y
Sugges ed Ci a ion: Jin, Bingzi; Xu, Xiaojie (2025) : P edic ing wholesale edible oil p ices h ough
Gaussian p ocess eg essions uned wi h Bayesian op imiza ion and c oss- alida ion, Asian Jou nal
o Economics and Banking (AJEB), ISSN 2633-7991, Eme ald, Leeds, Vol. 9, Iss. 1, pp. 64-82,
h ps://doi.o g/10.1108/AJEB-06-2024-0070
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P edic ing wholesale edible oil p ices
h ough Gaussian p ocess eg essions
uned wi h Bayesian op imiza ion and
c oss- alida ion
Bingzi Jin
Ad anced Mic o De ices (China) Co., L d, Shanghai, China, and
Xiaojie Xu
No h Ca olina S a e Uni e si y a Raleigh, Raleigh, No h Ca olina, USA
Abs ac
Pu pose –De eloping p ice o ecas s o a ious ag icul u al commodi ies has long been a signi ican
unde aking o a a ie y o ag icul u al ma ke playe s. The weekly wholesale p ice o edible oil in he
Chinese ma ke o e a en-yea pe iod, om Janua y 1, 2010 o Janua y 3, 2020, is he o ecas ing issue we
explo e.
Design/me hodology/app oach –Using Bayesian op imisa ions and c oss- alida ion, we s udy Gaussian
p ocess (GP) eg essions o ou o ecas ing needs.
Findings –The p oduced models deli e ed p ecise p ice p edic ions o he one-yea pe iod be ween Janua y 4,
2019 and Janua y 3, 2020, wi h an ou -o -sample ela i e oo mean squa e e o o 5.0812%, a oo mean squa e
e o (RMSEA) o 4.7324 and a mean absolu e e o (MAE) o 2.9382.
O iginali y/ alue –The p ojec ion’s ou pu may be u ilised as s and-alone echnical p edic ions o in
combina ion wi h o he p ojec ions o policy esea ch ha in ol es making assessmen .
Keywo ds Wholesale edible oil, P ice o ecas ing, Gaussian p ocess eg ession, Bayesian op imiza ion,
C oss- alida ion, Chinese ma ke
Pape ype Resea ch pape
1. In oduc ion
Food p ice p ojec ions om he ag icul u e sec o a e c ucial o a ange o ma ke pa icipan s,
such as p ocesso s, specula o s, hedge s and policymake s (Raihan e al., 2023). P oduce s, o
example, o en need p ice o ecas da a o se sales p ices be o e p oduc ion begins, expo e s
and p ocesso s o ul il hei con ac ual du ies, specula o s o p o i , hedge s o con ol isks,
and policymake s o de elop, ack and e alua e s a egic plans and policies (Dacha e al.,
2021). China’s signi ican ag icul u al ma ke sha e (Ji e al., 2022), close linkages o he
ene gy sec o (Ranguwal e al., 2023), and he in luence o inancial ma ke s and
mac oeconomic ac o s (Raihan, 2023) make edible oil p ice o ecas ing he e no
excep ion. Because o he physical limi a ions on he ood supply—such as land,
ag icul u al echnology, en i onmen al sus ainabili y and clima e change—policymake s
see ood p ices as s a egic issues. This is especially ue, gi en China’s massi e popula ion
and expanding economy. A numbe o mac oeconomic and inancial ac o s, such as in e es
AJEB
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© Bingzi Jin and Xiaojie Xu. Published in Asian Jou nal o Economics and Banking. Published by
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Recei ed 13 June 2024
Re ised 16 Sep embe 2024
16 Oc obe 2024
2 No embe 2024
Accep ed 4 No embe 2024
Asian Jou nal o Economics and Banking
Vol. 9 No. 1, 2025
pp. 64-82
Eme ald Publishing Limi ed
e-ISSN: 2633-7991
p-ISSN: 2615-9821
DOI 10.1108/AJEB-06-2024-0070
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a es, s ock p ices, exchange a es and inancialisa ion le els, as well as changes in he ene gy
ma ke s, such as he p ice o oil, e hanol and he demand o bio uels, could pu ood p ice
s abili y a isk. P ice o ecas ing may no need o be mo i a ed a lo since ag icul u al
commodi y p ices o en show e a ic ola ili y pa e ns (Yeasin e al., 2020), ha e a signi ican
impac on ma ke pa icipan s’ decisions (Raho eanu e al., 2018), and ul ima ely a ec
esou ce alloca ions and o e all economic wellbeing (Wulanda i e al., 2021).
Many esea ch s udies ha e been conduc ed in he li e a u e on a a ie y o ime se ies
app oaches o p ice p edic ion (Jin and Xu, 2024a). In hese ea ly s udies desc ibed below,
models like ec o au o eg essi e (VAR) models, au o eg essi e in eg a ed mo ing a e age
models, ec o e o co ec ion models and many a ian s o hese models a e o en
men ioned. The au o eg essi e in eg a ed mo ing a e age (ARIMA), o example, has been
shown in p e ious s udies o be a highly p e e ed choice o a a ie y o ime se ies o ecas ing
applica ions. I was shown ha ARIMA pe o ms no iceably be e han expe iews and
s uc u al model-based o ecas s o he US hog and ca le ma ke s. The accu acy o hog p ice
p ojec ions will only be sligh ly imp o ed by mo ing om he ARIMA o models ha include
mo e da a om he sow’s a owing p ice, acco ding o ano he esea ch. A numbe o
dis inc ions exis be ween his empi ical da a and he whea p ice da a, which demons a ed
ha he ARIMA model’s o ecas accu acy may be enhanced wi h adding exchange a e se ies
da a. P io s udies ha e shown ha combining he ARIMA wi h a ious model ypes may
inc ease p edic ion accu acy mo e han depending jus on one da a sou ce. One well-liked
econome ic me hod o p ice se ies o ecas s is he VAR app oach, which emphasises he
connec ions be ween a ious economic ac o s. The compa ison’s conclusions demons a e
ha he VAR p edic s US co on p ices be e han s uc u al models a imes o no mal p ice
ola ili y. In dis inguishing he p edic i e con en o a se o whea u u es p ices om a ious
coun ies and in di e en ia ing be ween he p ices o soybeans and soy in a ious US egions,
he VAR was shown o be use ul. Long- e m ela ionships be ween economic a iables a e also
aken in o conside a ion by he ec o e o co ec ion model (VECM), which is closely ela ed
o he VAR, ia coin eg a ion. I may be pa icula ly help ul o long- e m p ice o ecas s.
Acco ding o s udies, o example, he VECM o en pe o ms be e han he VAR in p edic ing
global whea p ices.
In p ice p edic ing s udies, he p e iously ou lined econome ic models ha e p o en use ul,
especially in edible oil s udies. Ka ia e al. (2016), o example, used he au o- eg essi e
ac ionally in eg a ed mo ing a e age (ARFIMA) and ARIMA o palm, apeseed, soybean,
linseed and sun lowe oil o s udy p ice o ecas s. Resol ing he o e -di e encing issue had
li le in luence on he p edic ion accu acy o any model, and hey ound con adic o y esul s
abou he e icacy o se e al models. Acco ding o P iyanga e al. (2019), he ARIMA migh be
use ul o o ecas ing he p ice o coconu oil in Ke ala. The ARIMA model was used by
Da eka and Reddy (2017) o o ecas he p icing o an oil seed, namely g oundnu s, in India.
They ound ha a me s, legisla o s and ma ke e s may bene i om he p ojec ions. The e a e
mino di e ences be ween he model’s p ojec ed and ac ual alues, acco ding o Meena e al.
(2014)’s analysis o oil and mus a d seed p ices in India using he ARIMA model.
A mul i a ia e ARIMA, which combines he ARIMA wi h an econome ic equa ion
p ede e mined o Malaysian palm oil p ice es ima es, was p oposed by Shamsudin and
A shad (2000) o imp o e o ecas s based on he ARIMA. Khin e al. (2011) ound simila
empi ical e idence o back hei Malaysian palm oil p ice p edic ions. The ARIMA,
exponen ial gene alised au o- eg essi e condi ional he e oscedas ic (GARCH) model
(EGARCH) and GARCH model we e s udied by Lama e al. (2015) in o de o o ecas
edible oil p ices bo h domes ically and in e na ionally. Because i be e cap u es he ola ili y
pa e n, hey ound ha he EGARCH model ou pe o ms he o he wo. In o de o an icipa e
he p ices o soybean and apeseed oil, Wang e al. (2013) demons a ed he e ec i eness o
he seasonal VECM in China. In o de o demons a e he a ailabili y o possible p edic i e
in o ma ion om c ude oil p ices o hose o edible oil, Hasano e al. (2016) used he
GARCH-in-mean model and ola ili y impulse esponse unc ion analysis. Thei indings
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showed ha c ude oil p ices migh aid in o ecas ing edible oil p ices. This empi ical e idence
may a y depending on he his o ical pe iods in ques ion. Using he VECM and di ec ed
acyclic g aph app oach, Yu e al. (2006) examined he p ice co ela ions be ween c ude oil and
a ious ood oils and concluded ha he e was no disce nible impac o c ude oil p ices on
edible oil p ices.
Resea che s ha e la ely shown a g ea lo o in e es in examining he uses o machine
lea ning algo i hms o ag icul u al commodi y p ice o ecas s because o he ease wi h which
compu e esou ces and echnology a e now accessible (Alade e al., 2021;Jin and Xu, 2024b).
The e o e, a a ie y o commodi ies—such as soybeans, suga , co n, whea , soybean oil,
co ee, co on, g een beans, canola, edible oil and peanu oil—ha e been he subjec o
esea ch using neu al ne wo ks, gene ic p og amming, deep lea ning, suppo ec o
eg essions, andom o es s, K-nea es neighbou s, mul i a ia e adap i e eg ession splines,
decision ees, ensembles, and boos ing. Neu al ne wo ks may be he mos popula machine
lea ning model o p edic ing he p ice o ag icul u al commodi ies, acco ding o hese and
o he indings om ea lie s udies, howe e his is by no means an exhaus i e analysis.
Fu he mo e, p e ious empi ical s udies demons a ing he e ec i eness o machine lea ning
me hods o inancial and economic o ecas ing a e o en in ag eemen wi h hese e alua ions.
The s udy demons a es ha machine lea ning echniques a e being used mo e and mo e o
o ecas edible oil p ices. In hei in es iga ion o p ice p ojec ions o Malaysian palm,
soybean, coconu , oli e, apeseed and sun lowe oils, o example, Kanchymalay e al. (2017)
ound ha he sequen ial minimum app oach o suppo ec o eg ession (SVR) imp o es
o ecas accu acy. The andom o es may be use ul in o ecas ing Myanma ’s edible oil p ice,
claim Mya and Tun (2019). In key Indian ma ke s, Singh (2021) and Jha and Sinha (2014)
examined neu al ne wo ks wi h ARIMA o p ice o ecas s o mus a d, g oundnu , apeseed,
and soybean oil. They ound ha , on a e age, neu al ne wo ks a e mo e accu a e han ARIMA.
Mish a and Singh (2013) ocused on p edic ing g oundnu oil p ices in Delhi using neu al
ne wo ks and ARIMA; hey ound mixed indings on he wo models’ e icacy. Lama e al.
(2016) sugges ha combining he neu al ne wo k wi h GARCH migh imp o e he o ecas s
o he indi idual model o edible oil p ices in bo h domes ic and in e na ional economies. The
p ice o ecas ing challenges o maize and palm oil we e examined by Jaiswal e al. (2022)
using he ARIMA, deep long sho - e m memo y neu al ne wo k and ime-delay neu al
ne wo k. They disco e ed ha he mos accu a e p edic ions we e made by he deep long sho -
e m memo y neu al ne wo k. Acco ding o Jaiswal e al. (2023), a nonlinea au o eg essi e
neu al ne wo k con aining exogenous a iables could be a help ul ool o soybean oil p ice
o ecas ing. Silalahi (2013) disco e ed ha he neu al ne wo k model ha was op imised by
he gene ic algo i hm could p o ide a easonable le el o accu acy in p ice o ecas ing o
soybean and palm oil. Salman e al. (2018) p oposed uning he backp opaga ion neu al
ne wo k o palm oil p ice o ecas ing using pa icle swa m op imisa ion, which inc eases
p edic ion accu acy compa ed o he con en ional backp opaga ion neu al ne wo k. Amal
(2021) disco e ed ha he long sho - e m memo y neu al ne wo k model could be uned using
adap i e momen es ima e op imisa ion o p o ide a palm oil p ice p edic ion wi h a high
deg ee o accu acy.
Fo ime-se ies da a ga he ed on edible oil p ices, howe e , he p edic ions gene a ed by he
Gaussian p ocess (GP) eg ession ha e no go en much a en ion. A no el eg ession
echnique is pu o wa d, d awing on Neal’s wo k on Bayesian lea ning o neu al ne wo ks
(Neal, 2012). Since he echnique elies on p io s o e unc ions o Gaussian p ocesses,
simula ing noisy da a makes sense (Jin and Xu, 2024c). The con e gence o se e al neu al
ne wo k-based Bayesian eg ession models o Gaussian p ocesses close o he edge o an
in ini e ne wo k was demons a ed (Neal, 2012). Resea ch has shown he e ec i eness o GP
eg essions in eplica ing da a ha is ei he noisy (Williams and Rasmussen, 1995) o noise-
ee (Neal, 1997). In hei s udy o Gaussian p ocesses using adial basis unc ion neu al
ne wo ks o o ecas ing p oblems in ol ing s a iona y ime-se ies da a, B ahim-Belhoua i
and Vesin disco e ed ha Bayesian lea ning yields supe io p edic ion ou comes
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(B ahim-Belhoua i and Vesin, 2001). Acco ding o he esul s o B ahim-Belhoua i and
Be mak (2004), his s udy looked a a wide ange o co a iance unc ions. GP p edic ion
echniques a e also help ul o o ecas ing p oblems wi h non-s a iona y ime-se ies da a.
Based on he wo k o B ahim-Belhoua i and Be mak, GP eg essions a e mo e e ec i e han
adial basis unc ion neu al ne wo ks o e all (B ahim-Belhoua i and Be mak, 2004). The
exac ma ix ope a ions equi ed o in eg a e p io and noisy models a e also he eason o he
GP o mula ion’s e ec i eness and success (B ahim-Belhoua i and Be mak, 2004).
Addi ionally, B ahim-Belhoua i and Be mak p oposed he use o GP p edic o s in he
mul i-model o ecas ing echnique (B ahim-Belhoua i and Be mak, 2004), which unc ion
simila ly o how we would use model a e aging.
O e a en-yea pe iod, om Janua y 1, 2010 o Janua y 3, 2020, we use he weekly
wholesale edible oil p ice o conduc ou inqui y. The o ecas model we use is GP
eg ession. The edible oil p ice index o he Chinese ma ke has a signi ican unde lying
economic ele ance as i is mean o e lec he wholesale ma ke end o all ypes o edible
oil h oughou he coun y. This p ice index may p o ide use ul p ojec ions o
policymake s and o he ma ke pa icipan s. To he bes o ou knowledge, none o he
ew ea lie s udies—including he ones men ioned abo e—ha e examined he pe inen
p edic ion issue o his p ice index. Ou p edic ion exe cise is based on his p ice index, and
we ollow he li e a u e on commodi y p ice o ecas s in o de o ill his esea ch gap.
The e o e, ou esul s se e o p o ide use ul o ecas in o ma ion o he impo an p ice
index o di e en o ecas consume s. We examine how well models ained using c oss-
alida ion and Bayesian op imisa ions p o ide o ecas s. As i u ns ou , ou models a e
a he s aigh o wa d and p o ide accu a e and eliable o ecas s. As a as we a e awa e,
and conside ing he a o emen ioned s udies, his is he i s s udy o p edic China’s
wholesale edible oil p icing using he machine lea ning echnique o GP eg ession. Model
p edic ion pe o mance may be imp o ed by using Bayesian op imisa ion o p o ide GP
eg essions mo e lexibili y, especially when dealing wi h unde lying da a ha exhibi s
nonlinea p ope ies. On he one hand, since i doesn’ ake a lo o compu ing ime o
implemen , his p edic ion amewo k is a he e icien . Howe e , his app oach p oduces
o ecas s ha a e qui e accu a e. The e is no doub ing he impo ance o accu a e and imely
ag icul u al commodi y p ice p ojec ions o bo h policymake s and ma ke playe s. Risk
managemen , ma ke e alua ions and po olio adjus men s may bene i om such
o ecas s. This esea ch assis s decision-make s in making imely judgemen s by looking a
di icul ies ela ed o p ojec ing ag icul u al commodi y p ices and u ilising weekly da a,
which a e a he high- equency o he wholesale ma ke . Ou esul s may be u ilised as
independen echnical p ice p ojec ions, on he one hand. None heless, hey migh be used
in combina ion wi h o he (basic) p edic ion indings o policy esea ch and he c ea ion o
hypo heses abou p icing ends. This me hod may be help ul o bo h ma ke pa icipan s
and policymake s as i acili a es he ex ension o he amewo k o po en ial commodi y
p ice p ojec ions in o he business sec o s.
2. Da a
O e he cou se o en yea s, om Janua y 1, 2010, o Janua y 3, 2020, we examine weekly
wholesale edible oil cos s o he Chinese ma ke . These p ices a e aken om China’s
Na ional Wholesale P ice In o ma ion Sys em. Figu e 1’s op panel displays he p ice se ies
plo on he le , he quan ile-quan ile plo agains he s anda d no mal dis ibu ion on he igh ,
and he 50-bin his og am wi h ke nel es ima es in he cen e. Fo he p ice se ies’ di e ences,
he ele an da a isualisa ion is shown in he bo om panel o Figu e 1. Fo he p icing da a,
Table 1 o e s c ucial summa y in o ma ion. The p ice se ies de ini ely de ia es om no mal
dis ibu ions, as shown by he esul s o he Ja que-Be a and Ande son–Da ling es s. This may
no come as a su p ise conside ing he na u e o economic da a (Jin and Xu, 2024d).
In pa icula , he p ice se ies exhibi s pla yku ic beha iou and igh skew. As can be seen
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Figu e 1. Visualiza ion o weekly wholesale edible oil p ices and hei i s di e ences du ing he pe iod o Janua y 1, 2010–Janua y 3, 2020, oge he wi h associa ed 50-bin
his og ams wi h ke nel es ima es and quan ile-quan ile plo s agains he s anda d no mal dis ibu ion
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Table 1. Da a summa y o weekly wholesale edible oil p ices du ing he pe iod o Janua y 1, 2010–Janua y 3, 2020
Se ies Minimum
1s
pe cen ile
5 h
pe cen ile Mean Median
S anda d
de ia ion
95 h
pe cen ile
99 h
pe cen ile Maximum Skewness Ku osis
Ja que-
Be a
Ande son-
Da ling
P ice 65.4700 87.6898 90.1130 107.4965 102.3900 16.1876 133.6985 138.1588 140.1600 0.4619 1.7983 <0.001 <0.0005
Fi s
di e ence �36.2600 �6.7388 �3.4080 �0.02600 �0.1000 3.6374 3.8600 7.3028 34.9400 �0.0096 46.2291 <0.001 <0.0005
Sou ce(s): Table by au ho s
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om Figu e 1, he p ice se ies has ou no able su ges. Keep in mind ha he base pe iod p ice,
which is 100, is de i ed om he a e age weekly p ice o June 1994.
P e ious s udies on he eme gence o nonlinea ea u es a highe momen s o e a b oad
ange o ime-se ies da a ha e been published in signi ican numbe s by he inancial and
economic domains (Yang e al., 2008). To ind any possible nonlinea ends, he p ice se ies is
pu h ough he B ock–Deche –Scheinkman (BDS) es (B ock e al., 1996). As can be seen,
he es yielded almos ze o p- alues. These esul s imply he p esence o nonlinea i ies in he
p ice se ies. The pu pose o his s udy is o p edic he p ice se ies using GP eg essions while
aking hese pa icula s in o conside a ion.
3. Me hod
The p ima y ocus o his wo k is on he o ecas ing app oach o GP eg essions, a kind o
p obabilis ic ke nel model ha has shown p edic i e abili y in p edic ing a ange o nonlinea
pa e ns ac oss se e al scien i ic ields (Jin and Xu, 2024e). The aining da a used o es ima e
model pa ame e s wi h an unce ain dis ibu ion is ep esen ed by he model as ollows:
xi;yi
ð Þ;i¼1;2;...;T g. These a e he exp essions o he d�dimension p edic o s: xi∈Rd,
and he e lec ion o he a ge occu s ia yi∈R. By use o wel e-lag p ices as p edic o s, he
p ice es ima e is p oduced. P ices om he p e ious wel e weeks, o ins ance, will be used as
p edic o s o de e mine he p ice o he hi een h week.
The app oach below may be used o exp ess a linea eg ession: y5x
T
βþε, wi h
ε∼N0;σ2
ð Þ epo ing he e o i em. Howe e , in GP eg essions, an explici basis and la en
a iables a e used o de ine he a ge a iable. l(x
i
) ep esen s he la en a iables om
Gaussian p ocesses ha sa is y he equi emen s o a join Gaussian dis ibu ion, whe eas b
ep esen s he basis unc ion. The basis unc ion’s pu pose is o p ojec di e en p edic o s
on o he ea u e space; he objec i e’s smoo hness will be shown by he la en a iables’
co a iance unc ion (Zhang and Xu, 2020;LI e al., 2015).
Usually, a Gaussian p ocess (GP) is desc ibed by wo me ics: mean and co a iance. We
would adop k x;x0
ð Þ ¼ Co lðxÞ;l x0
ð Þ½ � o epo he co a iance and m(x)5E(l(x)) o epo he
mean. I would be epo ed ha y5b(x)
T
βþl(x) exp ess he GP eg ession, whe e
lðxÞ∼GP 0;k x;x0
ð Þð Þand bðxÞ∈Rp. Via a hype -pa ame e named θ,k x;x0jθð Þwould be able
o be pa ame e ized. A GP eg ession is o en ained using he ollowing a iables, which a e
o en calcula ed using a pa icula app oach: σ
2
,θ, and β. Fu he mo e, we would p o ide he
basis unc ions (called b’s) and ke nels (called k’s) ha would be used h oughou he model’s
aining p ocesses. This pape would examine wo ca ego ies o ke nels: one is he
noniso opic ke nel (also known as au oma ic ele ance de e mina ion ke nel) and he o he is
he iso opic ke nel. To explo e bo h iso opic ke nels and noniso opic ke nels, i e dis inc
ke nels o each ca ego y would be applied. Equa ions (1) h ough (10) p o ide he de ails o
each ke nel unde discussion. To indica e he scale-mix u e pa ame e , we would use he
no a ion α> 0. We would u ilize σ
l
o showing he cha ac e is ic leng h scale associa ed wi h
iso opic ke nels. We would use σ
o showing he s anda d de ia ion associa ed wi h he
signal. ¼ i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i
xi−xj
� �0xi−xj
� �
q. Applying θ5(θ
1
,θ
2
)5(log σ
l
, log σ
) sugges s an app oach
o he pu pose o gua an eeing ha σ
l
and σ
a e abo e ze o. The leng h scale associa ed wi h
each p edic o ha co esponds o a noniso opic ke nel would be unique and would be
e lec ed ho ough he use o σ
m
(m51, 2, . . .,d). Consequen ly, θwould be e lec ed ia he
use o θ5(θ
1
,θ
2
,...,θ
d
,θ
dþ1
)5(log σ
1
, log σ
2
,..., log σ
d
, log σ
).
Iso opic Exponen ial:k xi;xj��θ
� �¼σ2
e−
σl(1)
Iso opic Squa ed Exponen ial:k xi;xj��θ
� �¼σ2
e
−1
2
xi�xj
ð ÞTxi−xj
ð Þ
σ2
l(2)
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Iso opic Ma e n 5=2: k xi;xj��θ
� �¼σ2
1þ i i i5
p
σlþ5 2
3σ2
l
!e− i i5
p
σl(3)
Iso opic Ra ional Quad a ic:k xi;xj��θ
� �¼σ2
1þ 2
2ασ2
l
� �−α
(4)
Iso opic Ma e n 3=2: k xi;xj��θ
� �¼σ2
1þ i i i3
p
σl
� �e− i i3
p
σl(5)
Noniso opic Exponen ial:k xi;xj��θ
� �¼σ2
e
− i i i i i i i i i i i i i i i i i i i i i i
X
d
m¼1
xim�xjm
ð Þ2
σ2
m
s(6)
Noniso opic Squa ed Exponen ial:k xi;xj��θ
� �¼σ2
e
−1
2X
d
m¼1
xim�xjm
ð Þ2
σ2
m(7)
Noniso opic Ma e n 5=2: k xi;xj��θ
� �¼σ2
1þ i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i
5X
d
m¼1
xim �xjm
� �2
σ2
m
u
u
þ5
3X
d
m¼1
xim �xjm
� �2
σ2
m
0
@1
Ae
− i i i i i i i i i i i i i i i i i i i i i i i i
5X
d
m¼1
xim�xjm
ð Þ2
σ2
m
s
(8)
Noniso opic Ra ional Quad a ic:k xi;xj��θ
� �¼σ2
1þ1
2αX
d
m¼1
xim �xjm
� �2
σ2
m
!−α
(9)
Noniso opic Ma e n 3=2: k xi;xj��θ
� �¼σ2
1þ i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i
3X
d
m¼1
xim �xjm
� �2
σ2
m
u
u
0
@1
Ae
− i i i i i i i i i i i i i i i i i i i i i i i i
3X
d
m¼1
xim�xjm
ð Þ2
σ2
m
s
(10)
Analyses ha a e close o hose o a ious kinds o ke nels o ou di e en possible basis
unc ions (indic ed in Equa ions (11)–(14)) would be ca ied ou in ou wo k as well. F om
Equa ions (11–14),X¼x1;x2;. . . ;xn
ð Þ0,X2¼
x2
11 x2
12 ��� x2
1d
x2
21 x2
22 ��� x2
2d
.
.
..
.
..
.
..
.
.
x2
T1x2
T2��� x2
Td
1
C
C
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
@
,B¼b x1
ð Þ;ð
b x2
ð Þ;... ;b xn
ð ÞÞ0, and an emp y ma ix is e e ing o a ma ix who has a leas one o i s
dimensions ha is ze o.
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p ecise es ima es o commodi y p ices. P ice p ojec ions a e equi ed o policymake s o
know in o de o conduc ou ma ke e alua ions, c ea e policies, pu such policies in o
ope a ion and make con inual changes. P ope o ecas ing is c ucial o a numbe o easons,
including main aining a posi i e co po a e en i onmen and a e ing ma ke collapse. To he
bes o hei knowledge, econome ic models—in pa icula , ime-se ies econome ic
models—a e he basis o he o ecas ing app oach ha he go e nmen and many in es o s
use when a signi ican p opo ion o commodi y p ices a e a isk. In he meanwhile, p ice
es ima es a e ne e heless o en based on expe opinions. This is suppo ed by he possibili y
ha de eloping, implemen ing, and main aining econome ic models and expe judgemen s
will be compa a i ely simple. And because many o hese models ha e been ex ensi ely used
o decades by a b oad ange o o ecas clien s, i is likely ha many o hem ha e a espec able
p edic ion accu acy. Gi en he g owing a o dabili y o compu e esou ces and he solid
ounda ion o expec ed nonlinea p ope ies in p ice se ies o many commodi ies, machine
lea ning models a e widely ecognised o ha e p omise and wo h mo e in es iga ion. I may
be di icul o some in es o s and policymake s o p ope ly examine hese models, howe e ,
since some decision-make s may s ill see hem as oo complica ed o ecas ing ools. Ac ually,
ad anced in es o s and some go e nmen s ha e been inc easingly in e es ed in machine
lea ning echnologies in ecen yea s. The s udy conduc ed he e is a componen o a b oade
app oach ha explo es he possible uses o GP eg ession as a machine lea ning echnique o
sol e he edible oil o ecas ing issue. The indings p esen ed sugges ha machine lea ning
models may be wo h looking in o, maybe o a wide ange o commodi ies, gi en he
sugges ed me hod o building such a model and he shown s ong o ecas accu acy and
s abili ies.
7. Conclusion
Commodi y p ice o ecas s a e impo an o a numbe o ag icul u al indus y s akeholde s. In
his esea ch, we use weekly wholesale edible oil p ices o e a en-yea pe iod, om Janua y 1,
2010, o Janua y 3, 2020, o p edic he Chinese ma ke . No enough a en ion has been paid o
he p ojec ions o his signi ican p ice se ies in he li e a u e. As a o ecas ing ool, he GP
eg ession is in es iga ed using c oss- alida ion and Bayesian op imisa ions, yielding accu a e
and dependable indings. Mo e p ecisely, o he pe iod spanning om Janua y 4, 2019 o
Janua y 3, 2020, he gene a ed models we e capable o gene a ing p edic ing esul s o he
p ices wi h an ou -o -sample RRMSE o 5.0812%, RMSE o 4.7324 and MAE o 2.9382. The
bene i o he gene a ed models is shown by ou benchma k s udy, which compa es he GP
eg ession models wi h a numbe o o he ime-se ies and machine lea ning models. This da a
may be used o independen echnical p edic ions o combined wi h o he es ima ions when
doing policy esea ch ha calls o p ice end iews. The modelling me hodology ha
unde lies hese p edic ions may also be used o o ecas p oblems o a simila kind in o he
economic a eas. The simplici y and con enience o use o his amewo k may be essen ial o
a numbe o decision-making p ocesses. The echnique p esen ed he e may be limi ed by he
absence o po en ially use ul p edic i e da a om o he economic aspec s, which migh
imp o e p edic ion pe o mance. Examining eg ession models using Gaussian p ocesses and
exogenous inpu s migh help esol e his po en ial limi a ion. This is an impo an a ea o
u he s udy i da a on o he economic ac o s could be ga he ed. Fu u e esea ch on addi ional
Bayesian op imisa ion p ocedu es may be wo hwhile, since he cu en s udy ocusses on he
EI pe second plus me hod o Bayesian op imisa ions. Gi en ha he ime pe iod unde
examina ion concludes in Janua y 2020, i may also be a aluable di ec ion o u u e esea ch
ha conside s mo e cu en ime pe iods o analysis. We use he Bayesian op imisa ion
echnique o guide ou model de elopmen p ocedu e. The me hod de e mines he bes GPR
model based on he aining sample by expe imen ing wi h di e en ke nels, basis unc ions,
and ma ching pa ame e s. Following he cons uc ion o he op imum model, ou -o -sample
o ecas ing is pe o med. A p omising di ec ion o u u e esea ch is he no ion o building
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se e al models and selec ing signi ican ones based on he model con idence se . The p ice
p ojec ion p oblem o China’s wholesale edible oil ma ke has been in es iga ed in his pape .
In es iga ing he possibili ies o his empi ical o ecas ing amewo k based on o he
commodi y p ices would be in e es ing.
Abb e ia ion
ARIMA – Au o eg essi e In eg a ed Mo ing A e age
VAR – Vec o Au o eg essi e
VECM – Vec o E o Co ec ion Model
BDS – B ock–Deche –Scheinkman
GP – Gaussian P ocess
EIPSP – Expec ed Imp o emen Pe Second Plus
EI – Expec ed Imp o emen
EIPS – Expec ed Imp o emen Pe Second
RRMSE – Rela i e Roo Mean Squa e E o
RMSE – Roo Mean Squa e E o
MAE – Mean Absolu e E o
CV – C oss Valida ion
AR – Au o eg essi e
GARCH – Gene alized Au o eg essi e Condi ional He e oskedas ici y
SVR – Suppo Vec o Reg ession
RT – Reg ession T ee
MSE – Mean Squa e E o
MDM – Modi ied Diebold-Ma iano
GPR – Gaussian P ocess Reg ession
CART – Classi ica ion Analysis and Reg ession T ee
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