Accounting for erroneous model structures in biokinetic process models

Accoun ing o e oneous model s uc u es in
biokine ic p ocess models
K is Villez
a,b
, Da io Del Giudice
a,c,d
, Ma c B. Neumann
e,
, Jö g
Riecke mann
a
a
Eawag: Swiss Fede al Ins i u e o Aqua ic Science and Technology, Übe lands asse 133,
8600 Dübendo , Swi ze land
b
ORNL: Oak Ridge Na ional Labo a o y, Oak Ridge, TN, USA
c
ETHZ: Swiss Fede al Ins i u e o Technology, S e ano-F anscini-Pla z 5, 8093 Zü ich,
Swi ze land
d
Depa men o Ci il, Cons uc ion & En i onmen al Enginee ing, NC S a e Uni e si y,
Mann Hall 311, 2501 S inson D i e, Raleigh, NC, 27695, USA
e
Basque Cen e o Clima e Change (BC3), Scien ic Campus o he Uni e si y o he
Basque Coun y, Sede Building 1, 1s oo , 48940 Leioa, Spain
IKERBASQUE, Basque Founda ion o Science, Ma ia Diaz de Ha o 3, 6 solai ua,
48013 Bilbao, Spain
Abs ac
In enginee ing p ac ice, model-based design equi es no only a good p ocess-
based model, bu also a good desc ip ion o s ochas ic dis u bances and
measu emen e o s o lea n c edible pa ame e alues om obse a ions.
Howe e , ypical me hods use Gaussian e o models, which o en canno de-
sc ibe he complex empo al pa e ns o esiduals. Consequen ly, his esul s
in o e condence in he iden ied pa ame e s and, in u n, op imis ic eac o
designs. In his wo k, we assess he s eng hs and weaknesses o a me hod o
s a is ically desc ibe hese pa e ns wi h au oco ela ed e o models. This
me hod p oduces inc eased wid hs o he c edible p edic ion in e als ol-
lowing he inclusion o he bias e m, in u n leading o mo e conse a i e
design choices. Howe e , we also show ha he augmen ed e o model is no
a uni e sal ool, as i s applica ion canno gua an ee he desi ed eliabili y o
he esul ing was ewa e eac o design.
Keywo ds:
bias desc ip ion; kine ic model; p ocess design; was ewa e
ea men ; unce ain y
Email add ess:
illezk@o nl.go
(K is Villez)
P ep in submi ed o Reliabili y Enginee ing and Sys em Sa e y July 21, 2020
This documen is he Accep ed Manusc ip e sion o a Published Wo k ha appea ed in inal o m in:
Villez K., Del Giudice D., Neumann M.B., Riecke mann J. 2020. Accoun ing o e oneous model s uc u es in
biokine ic p ocess models. RELIABILITY ENGINEERING & SYSTEM SAFETY. 203. DOI (10.1016/
j. ess.2020.107075).
© 2020 Else ie L d
This manusc ip e sion is made a ailable unde he CC-BY-NC-ND 3.0 license h p://c ea i ecommons.o g/licenses/
by-nc-nd/3.0/
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1. In oduc ion
1
In cu en en i onmen al enginee ing p ac ice, de e minis ic p ocess-based
2
modeling is a common ool o be e unde s and he unc ioning o complex
3
was ewa e collec ion and ea men sys ems. The gold s anda d is o im-
4
p o e p edic ion pe o mance o ou models by  ing hem o obse a ions.
5
Consequen ly, he ad en o ubiqui ous sensing leads o an unin ended ye
6
commonly obse ed si ua ion whe e senso s e eal mo e de ails han mech-
7
anis ic models can cap u e. When his is he case, unce ain y es ima es
8
ob ained om s a is ical in e ence wi h mechanis ic models a e almos ce -
9
ainly oo na ow as he applied model s uc u e is oo es ic i e ela i e
10
o he obse ed eali y. A long-s anding ques ion is he e o e whe he isk-
11
based design, based on unce ain y es ima es om s a is ical in e ence wi h
12
mechanis ic models, is ac ually easible. In his wo k, we es one me hod
13
designed o add ess his issue o a case o WWTP design and discuss i s
14
po en ial and limi a ions.
15
Accoun ing o model pa ame e unce ain y is c ucial o isk-based decision-
16
making, including in as uc u e design and ope a ions (e.g., Cagno e al.,
17
2011;Scheidegge e al.,2013;Kabi e al.,2015;Scheidegge e al.,2015;
18
Jensen and Je ez,2018). Con en ional me hods o unce ain y analysis a e
19
based on a wo-s ep app oach, consis ing o
(a)
quan ica ion o inpu un-
20
ce ain y, measu emen unce ain y, and subsequen unce ain y o model
21
pa ame e s ollowed by
(b)
p opaga ion o he quan ied unce ain y o he
22
sys em pe o mance measu e o in e es (e.g., Van G iens en and Meixne ,
23
2007;Sin e al.,2009;Guo and Mu phy,2012;Del Giudice e al.,2016).
24
2
Howe e , i has been demons a ed be o e how sys ema ic deciencies in
25
model s uc u e, nex o inpu and measu emen unce ain y, also lead o bi-
26
ased model pa ame e s and, consequen ly, inco ec design o in as uc u al
27
elemen s, such as biological eac o sys ems (Neumann and Guje ,2008).
28
Un o una ely, o ou knowledge, no one has a emp ed o p o ide a me hod
29
o sol e his pa icula p oblem, i.e. o iden i y sys ema ic disc epancies in
30
p ocess-based models so o accoun o hem du ing model-based design.
31
Recen ly, s a is icians ha e been sugges ing a p omising app oach o sol e
32
his dilemma. The unde lying idea is o no assume iden ically and indepen-
33
den ly dis ibu ed (i.i.d.) e o s o misma ches be ween models and obse -
34
a ions (Liu and Zacha a,2001), bu o explici ly accoun o misma ches by
35
adding a s ochas ic au o-co ela ed p ocess o he i.i.d. measu emen e o
36
model (C aig e al.,2001;Kennedy and O'Hagan,2001;Baya i e al.,2007).
37
This is known as he bias desc ip ion me hod and ocuses on he modeling
38
o he symp oms o a misma ch be ween model s uc u e and eali y. While
39
his does no iden i y o ackle he oo cause o hese symp oms, i has
40
been p o en o be a compu a ionally ecien ool o inc ease he eliabili y
41
o model-based p edic ions compa ed o s anda d eg ession app oaches in a
42
a ie y o sys ems om lakes o na u al ca chmen s o u ban hyd ology (Di-
43
e zel and Reiche ,2012;Reiche and Schuwi h,2012;Del Giudice e al.,
44
2015). The e o e, we expec ha he bias desc ip ion me hod also imp o es
45
he eliabili y o p edic ions wi h s uc u ally decien was ewa e ea men
46
models in iew o isk-based design. Specically, adding a s ochas ic mea-
47
su emen e o e m o a model gi en he same amoun o expe imen al
48
measu emen s is expec ed o educe he ela i e in o ma ion- ichness o he
49
expe imen al da a and lead o la ge c edibili y in e als o model pa ame-
50
e s, wide p edic ion in e als and, by a oiding o e conden p edic ions, a
51
mo e us wo hy design.
52
No e ha he bias desc ip ion me hod can be ega ded as a g ey box o
53
hyb id modelling s a egy. Indeed, he esul ing model consis s o a mech-
54
anis ic model o he s udied p ocess (whi e box) and a s ochas ic model
55
o au o-co ela ed measu emen e o s (black box). O he g ey box ap-
56
p oaches may be based on he inclusion o ime- a ian pa ame e s (Reiche
57
and Mielei ne ,2009;Lin and Beck,2012) o in eg a ion o non-pa ame ic
58
elemen s in o a model s uc u e ha is mechanis ic o he wise (Ma²i¢ e al.,
59
2017).
60
In his con ibu ion, we apply he bias desc ip ion me hod o in es iga e
61
he impac o model s uc u e deci s o p ocess design. We use Neumann
62
3
and Guje (2008) as a benchma k o e alua e he bene s and limi a ions
63
o he bias desc ip ion me hod o model-based design and e e o i as
he
64
e e ence s udy
. While his e e ence s udy conce ns a concep ually simple
65
case, using i in his s udy highligh s
(a)
ha he appa en simplici y o his
66
case is a he decep i e and
(b)
ha challenges associa ed wi h model- eali y
67
misma ch a e o be expec ed o bo h simple and complex sys ems.
68
2. Ma e ial and me hods
69
2.1. Applied e o models
70
In a as majo i y o en i onmen al modeling s udies, he measu emen
71
e o is assumed o be i.i.d. Fo example, Hauduc e al. (2015) compa es an
72
ex ensi e lis o model pe o mance c i e ia o was ewa e ea men mod-
73
elling ye does no lis any c i e ion which accoun s o au oco ela ed model
74
p edic ion e o s. One app oach conside ed in Cie kens e al. (2012), consis s
75
o downsampling ime se ies o a oid he appea ance o au oco ela ion. As
76
explained in he same s udy, his leads o an inecien use o he a ailable
77
da a and, mo e impo an ly, canno accoun a all o model s uc u e deci s
78
as a po en ial oo cause o au oco ela ed esiduals. Igno ing he p esence
79
o au oco ela ed esiduals was shown o lead o o e condence in he p o-
80
duced model and, subsequen ly, poo decision-making, as was shown also in
81
he e e ence s udy. Mos o en, a Gaussian dis ibu ion is assumed o he
82
measu emen e o s. Such a model o measu emen e o canno accoun o
83
sys ema ic de ia ions be ween he assumed model and he obse ed mea-
84
su emen s, i.e. bias. One way o accoun ing o bias is by adding e ms, such
85
as a s ochas ic au oco ela ed e o bias e m, o he measu emen equa ion
86
(C aig e al.,2001;Kennedy and O'Hagan,2001;Baya i e al.,2007). In
87
his wo k, we desc ibe he obse able ou pu ime-se ies (i.e., measu ed con-
88
cen a ion,
yo
) as a sum o a de e minis ic dynamic model ou pu (
y
, he
89
modeled concen a ion), a classical Gaussian measu emen e o (
e(ψ)
), and
90
an au o-co ela ed e o e m (
b(ψ)
):
91
yo(θ, γ, ψ) = y(θ) + γ+ b(ψ) + e(ψ)
(1)
whe e
θ
and
γ
a e pa ame e s o he de e minis ic pa s o he model and
92
ψ
a e hose o he s ochas ic pa s (e o s). The bias e m
b(ψ)
dec ibes an
93
au oco ela ed e o (
b(ψ)∼ N(0,Σb(σb, τ))
) and can be included o accoun
94
4
o ime-dependen de ia ions be ween model and obse a ions (Reiche and
95
Schuwi h,2012). No e ha his bias e m ep esen s a s ochas ic p ocess,
96
hus desc ibing alea o y unce ain y, al hough he de ia ions be ween model
97
and obse a ions may ac ually be sys ema ic, possibly e en de e minis ic.
98
These de ia ions a e expec ed o be sys ema ic when hey a e caused by a
99
lack o knowledge abou he ue da a-gene a ing p ocess. This lack o knowl-
100
edge is ypically cha ac e ized as a sou ce o epis emic unce ain y a he han
101
alea o y unce ain y.
102
The bias e m has wo pa ame e s, he s anda d de ia ion
σb
and he
103
co ela ion leng h
τ
:
104
Σb(i, j) := σb2·e−| i− j|2/τ .
(2)
The andom measu emen e o is empo ally independen (
e∼ N(0,Σe(σe))
)
105
and is cha ac e ized by he pa ame e
σe
:
106
Σe(i, j) := (σe2, i =j
0, i 6=j.
(3)
Toge he , hese e o e ms wi h pa ame e s
ψ={τ, σb, σe}
accoun o
107
he ac ha he de e minis ic model may no ep oduce he modeled da a
108
se exac ly. No e ha he symbols
σb
and
σe
a e chosen o con ey he idea
109
ha hey bo h desc ibe he magni ude o a ia ion o a s ochas ic e m in he
110
measu emen equa ion. The symbol o he co ela ion leng h,
τ
, is chosen
111
o highligh he ac ha i desc ibes a ime-scale.
112
The s a is ical o mula ion in Eq. 1na u ally leads o he likelihood
113
unc ion
L(yo|θ, ψ)
which desc ibes how likely he conside ed model wi h
114
pa ame e s
(θ, ψ)
gene a ed he eco ded da a,
yo
. The likelihood o he
115
measu emen s condi ional o he model pa ame e s is:
116
L(yo|θ, ψ) = (2π)−n
2
qde Σ
exp −1
2hyo−yiT
(Σ)−1hyo−yi
(4)
whe e
Σ
is he a iance-co a iance ma ix o he s ochas ic de ia ions
117
be ween model and obse a ions:
118
5

Σ := Σe+ Σb,
(5)
wi h
Σb
and
Σe
dened in 2and 3, and wi h
n
equal o he numbe o
119
measu emen s. In o de o in e p e he esul s o pa ame e es ima ion, we
120
dene he pa ame e s
α
and
σ
such ha
σ2
e:= (1 −α)σ2
and
σ2
b:= α σ2
.
121
This means we can exp ess he a iance-co a iance ma ix abo e equi alen ly
122
as:
123
Σ := σ2·h(1 −α)In×n+α Ki
(6)
K(i, j) := e−| i− j|2/τ
(7)
In his o m,
σ
is a measu e o he o e all sp ead o he de ia ions be-
124
ween he model and he measu emen s and
α
is a pa ame e ha denes
125
he ela i e impo ance o he bias in he o e all a iance-co a iance ma ix.
126
Meaning ul alues o
α
a e be ween 0 and 1, wi h
α= 1
leading o he omis-
127
sion o he independen measu emen noise (
σe= 0
) and
α= 0
exp essing
128
ha he e is no bias (
σb= 0
). No e ha se ing
α= 0
ep oduces he model
129
wi hou a bias e m. Pu o he wise, he model wi h bias e m includes he
130
model wi hou bias e m as special case.
131
The addi ion o a bias e m accoun s o unde es ima ion o pa ame e
132
unce ain y when a con en ional ye un ealis ic dis ibu ion o he model
133
e o is assumed (e.g., unco ela ed). This is expec ed o p oduce a wide
134
p edic i e dis ibu ion, possibly leading o a be e quan ica ion o and a
135
educ ion o he isk o unde -design o o e -design. Howe e , special a en-
136
ion mus be gi en o he pa ame e es ima ion me hod as inc eased model
137
exibili y can lead o uniden iabili y (see e.g., Rena d e al.,2010).
138
2.1.1. Model pa ame e es ima ion
139
We apply a Bayesian app oach o wo easons. Fi s , we a ou a Bayesian
140
amewo k as a way o make p io belie s explici . Second, wi hou any o m
141
o p io , some o he pa ame e s o he a iance-co a iance ma ix
Σ
can be
142
s uc u ally uniden iable ( o deni ions, see Dochain e al.,1995;Dochain
143
and Van olleghem,2001;Pe e sen e al.,2003). Mo e specically, when
τ= 0
144
he ma ix
K
equals he iden i y ma ix and likelihood
L(yo|θ, ψ)
becomes
145
insensi i e o he alue o
α
. As a esul , no unique alue o
α
can be
146
6
iden ied unde any ci cums ances as long as
τ= 0
, i.e.
α
is s uc u ally
147
uniden iable. In he o mula ion wi h
σe
and
σb
, any inc ease o
σe
can
148
be compensa ed exac ly by an equi alen dec ease o
σb
when
τ= 0
. Fo
149
small alues o
τ
, e.g. close o he measu emen in e al o smalle , his is
150
expec ed o lead o a lack o p ac ical iden iabili y, e en i s uc u al iden i-
151
abili y could be gua an eed in p inciple. In ea ly expe imen s wi h uni o m
152
p io s o
τ
, we obse ed ha his can induce a lack o con e gence and poo
153
mixing condi ions o he applied sampling me hods, simila o obse a ions
154
desc ibed in Rena d e al. (2010). Applying an in o ma i e p io sol es his
155
iden iabili y p oblem and can he e o e also be in e p e ed as a o m o
156
egula iza ion (e.g., Scales and Teno io,2001;Mu phy,2012;Has ie e al.,
157
2015).
158
Bayesian calib a ion aims a cha ac e izing he dis ibu ion desc ibed by
159
he pos e io likelihood
L(θ, ψ|yo)∝ L(yo|θ, ψ)· L(θ, ψ)
, whe e he p io
160
likelihood
L(θ, ψ)
exp esses he p io belie s abou he pa ame e s. In his
161
wo k, he pos e io dis ibu ion is app oxima ed wi h a sample o
L(θ, ψ|yo)
162
d awn wi h a Ma ko Chain Mon e Ca lo (MCMC) sample (see Nume ical
163
implemen a ion).
164
2.2. Biokine ic model pa ame e iden ica ion wi h ba ch expe imen s
165
To s udy he eec s o model s uc u e e o and he u ili y o he bias de-
166
sc ip ion me hod, we execu e simula ions wi h he dynamic biokine ic model
167
used in he e e ence s udy. Conc e ely, a se ies o ba ch expe imen s is sim-
168
ula ed in which a subs a e, wi h concen a ion
s( )
, is consumed by a cell
169
cul u e wi h a xed concen a ion. The con e sion a e
( )
depends on he
170
subs a e by means o ime-in a ian Tessie kine ics so ha one can w i e:
171
ds( )
d =− ( )
(8)
s( = 0) = s0
(9)
( ) = Tessie
max ·1−exp −s( )
KTessie
S·x( )
(10)
Du ing he expe imen , noisy measu emen s o he ue subs a e con-
172
cen a ion,
yo( )
, a e simula ed by he ollowing measu emen e o model,
173
which is a ze o-mean Gaussian noise e m:
174
7
yo( ) = s( ) + e( )
(11)
e( )∼N(0, σe)
(12)
Fixed pa ame e s o each simula ion a e he same as in he e e ence
175
s udy:
s0
(ini ial subs a e concen a ion,
5g/m3
),
max
(maximum con-
176
e sion a e,
1g/m3.h
). The simula ed ime is
T= 8
hou s. The ani y
177
cons an (
KTessie
S
) and measu emen e o s anda d de ia ion (
σe
) a e a ied
178
ye cons an in e e y simula ed expe imen .
KTessie
S
is a ied om
0.1g/m3
179
o
1.5g/m3
in s eps o
0.2g/m3
. This allows simula ing a wide ange o
180
p ocess condi ions, including bo h low and high alues o
KTessie
S
ela i e
181
o he ini ial subs a e concen a ion. Two alues o he simula ed
σe
a e
182
conside ed, as in he e e ence s udy. In he low-noise case,
σe
akes he alue
183
0.01 g/m3
. In he high-noise case, i akes he alue
0.1g/m3
. The ec o
184
θ
equals
s0, max, KT essie
S
T
. The wo simula ed noisy ime se ies ob ained
185
wi h
KTessie
S= 0.7g/m3
a e shown in he supplemen a y in o ma ion (Fig.
186
S.1 and Fig. S.2).
187
Fo each simula ion expe imen , pa ame e iden ica ion is execu ed wi h
188
ou dis inc model s uc u es. The  s model ma ches he abo e model
189
s uc u e (Eq. 8-Eq. 12) exac ly. This ep esen s an idealized si ua ion whe e
190
he s uc u e o he calib a ed model ma ches eali y (g ound u h) exac ly.
191
The iden ied pa ame e s a e
S0
,
µTessie
max
,
KTessie
S
, and
σe
. A second model
192
is ob ained by eplacing he Tessie kine ics wi h he al e na i e and mo e
193
commonly used Monod kine ics. P ac ically, Eq. 10, is eplaced wi h he
194
ollowing equa ion:
195
( ) = Monod
max ·s( )
KMonod
s+s( )
(13)
The es ima ed pa ame e s a e now
s0
,
µMonod
max
,
KMonod
s
, and
σe
wi h
196
θ=s0, max, KMonod
S
T
. This case ep esen s he likely si ua ion ha a
197
modeling p ac i ione uses he common-place Monod model s uc u e and
198
does no obse e he bias ha esul s. This is e y likely in he high-noise
199
case (see e e ence s udy). Gi en his dicul y, he s ochas ic bias e m de-
200
sc ibed abo e is included o cap u e he sys ema ic de ia ions be ween he
201
model p edic ions and measu emen s. To achie e his, he p e iously applied
202
measu emen equa ion (Eq. 11) is eplaced wi h he ollowing equa ions:
203
8
yo( ) = s( ) + b( ) + e( )
(14)
e( )∼ N(0, σe)
(15)
b∼ N(0,Σb(σb, τ))
(16)
wi h he pa ame e s
τ
dened as abo e and
σb
and
σe
epa ame ized wi h
204
α
and
σ
. This esul s in a hi d model, whe e a Tessie model is combined
205
wi h he s a is ical bias desc ip ion and which equi es specica ion o he
206
pa ame e s
s0
,
µTessie
max
,
KTessie
S
,
σ
,
α
, and
τ
. As he Tessie model has
207
he same s uc u e as he da a-gene a ing model, one can expec a good
208
model  wi h
α
close o 0 and es ima es o
s0
,
µTessie
max
,
KTessie
S
, and
σ
209
ha a e close o g ound u h alues. Finally, he ou h model combines
210
he p esumed Monod kine ics wi h he s a is ical bias desc ip ion and he
211
iden ied pa ame e s a e
s0
,
µMonod
max
,
KMonod
s
,
σ
,
α
, and
τ
. In his case we
212
can expec ha he p esen model s uc u e bias is accommoda ed by means
213
o he s a is ical bias desc ip ion. I so, his should inc ease he wid h o he
214
p edic ion in e als and he eby imp o e he eliabili y o he model (Reiche
215
and Schuwi h,2012).
216
2.3. Nume ical implemen a ion
217
The biokine ic model, pa ame e es ima ion, and unce ain y p opaga-
218
ion we e implemen ed in Ma lab (R2019a). The p io p obabili ies o he
219
pa ame e s we e se based on he au ho s' expe ience. They a e all indepen-
220
den o each o he . All p io s a e uni o m, excep o
σ
and
τ
. The p io
221
likelihood o
σ
is p opo ional o i s in e se and is equi alen o he Je eys
222
p io condi ional o xed alues o all o he pa ame e s (see Box and Tiao,
223
1973). The p io likelihood o
τ
is he sine unc ion suppo ed be ween
0
and
224
2T
. This p io equals ze o a
τ= 0
and
τ= 2 T
and one a
τ=T
. This ex-
225
p esses he subjec i e belie ha he au oco ela ion leng h o he de ia ions
226
due o model s uc u e e o is expec ed o be simila o he du a ion o he
227
expe imen . The p io s a e specied comple ely in Table 1. We  s un an
228
adap i e MCMC algo i hm (Vihola,2012) o nd a good guess o he maxi-
229
mum a pos e io i es ima es and a good p oposal a iance-co a iance ma ix.
230
Wi h hese esul s, we execu e a (non-adap i e) MCMC algo i hm o ob ain
231
20,000
samples om
L(θ, ψ|yo)
. The  s
10,000
samples a e conside ed o
232
co espond o he bu n-in phase o he sample , du ing which eec s o he
233
ini ial sample may s ill be appa en . These samples a e he e o e disca ded,
234
as is common in p ac ice (Gilks e al.,1996).
235
9
Figu e 4: Dis ibu ions o he a io o p edic ed s eady s a e concen a ions o he g ound
u h concen a ion as a unc ion o he ani y cons an (
KS
)  High noise case (
σe=
0.1g/m3
). Red ho izon al whiske s indica e he wo-sided 99% c edible in e als.
Le
side beans:
wi hou bias desc ip ion;
Righ side beans:
wi h bias desc ip ion.
Top:
Tessie model - All c edible in e als include he ideal a io (equal o 1), excep o he
simula ion wi h
KS= 0.1g/m3
wi hou bias e m. The unce ain y inc eases when a bias
e m is added o he model.
Bo om:
Monod model  Including he bias e m in he
model inc eases he eliabili y o he c edible in e als. These in e als include he ideal
a io o wo cases (
KS= 0.5
and
0.7g/m3
).
Fo
KS= 0.3g/m3
his al eady amoun s o 8.6%. Since he model s uc u e
354
e o p ima ily ela es o he cu a u e o he con e sion a e in his egion,
355
i ollows ha model s uc u e e o will always be dicul o de ec when
356
his ime ac ion is low.
357
In he supplemen a y in o ma ion, we p o ide esul s ob ained wi h he
358
modied me hod. We omi he in o ma ion ob ained du ing he washou ex-
359
pe imen du ing p edic ion. In his case, he unce ain y in he p edic ions
360
16

Figu e 5: Dis ibu ions o he pa ame e
α
o bo h models wi h a bias e m in all high-
noise cases (
σe= 0.1g/m3
). When he Tessie model is selec ed (no model s uc u e
e o ), he pos e io dis ibu ion o
α
is shi ed o he le o he p io , hus sugges ing he
kine ic model s uc u e is adequa e. In con as , he pos e io o
α
is simila o o loca ed
a he igh o he p io when he Monod model is used in all bu one case (
KS= 0.1g/m3
),
hus p o iding a use ul indica ion o model s uc u e e o .
is educed signican ly o he poin ha none o he 99% c edible in e als
361
include he g ound u h (see Fig.S.3). This is explained by he ac ha
362
he es ima es o
max
and
KS
exhibi s ong co ela ion (see supplemen-
363
a y in o ma ion o de ails). Howe e , since he modica ion ela es o he
364
p edic ion s ep only, one can s ill use he pos e io s o
α
as a de ec ion
365
mechanism o bias.
366
17
4. Discussion
367
4.1. Summa y and limi a ions o he expe imen al simula ion s udy
368
Summa y.
The nume ical esul s desc ibed abo e sugges ha he inclusion
369
o an addi i e au o-co ela ed e o p ocess in o a measu emen e o model
370
can imp o e he eliabili y o model-based designs. This is ue e en when
371
only a subse o he iden ied pa ame e s a e used du ing p edic ion (he e we
372
only used he es ima es o
KS
) and e en when he expe imen al se ing o
373
p edic ion (s eady s a e) is die en om he expe imen al condi ions used
374
o model iden ica ion (ba ch expe imen ). In ou case, he bias desc ip ion
375
me hod imp o es he eliabili y in all cases. Despi e his imp o emen , he
376
compu ed c edible in e als include he g ound u h alue only in a lim-
377
i ed numbe o cases wi h model s uc u e e o , meaning ha gua an eed
378
eliabili y canno be ob ained wi h he s udied me hod. Thus, he inclusion
379
o a bias desc ip ion e m o he pu pose o p edic ion can be ad ised as
380
a ela i ely as and easy way o accoun o e o s in he p oposed model
381
s uc u e, howe e only when one is unable o modi y he model s uc u e
382
i sel . This is especially ele an in enginee ing applica ions whe e one is e-
383
s ic ed o specic p ocess ep esen a ions (e.g., Monod kine ics) o so wa e
384
wi h limi ed exibili y. While he bias desc ip ion me hod imp o es he e-
385
liabili y o he model p edici ions only in a limi ed way, i is e y use ul as
386
a ool o de ec he p esence o bias du ing model iden ica ion, especially
387
when e o mula ed wi h he
α
pa ame e .
388
Limi a ions.
In his s udy, a simple case was chosen delibe a ely o wo
389
easons. Fi s , his enabled an objec i e compa ison o he bias desc ip-
390
i e me hod wi h he his o ical esul s in he e e ence s udy (Neumann and
391
Guje ,2008). Second, he appa en simplici y o he case also highligh s he
392
challenge o gene a ing eliable p edic ions wi h mechanis ic models, induced
393
by he ypical lack o exibili y o such models. The chosen scope also means
394
ha ou s udy comes wi h some limi a ions, which a e:
395

The gene al applicabili y o he bias desc ip ion me hod is no demon-
396
s a ed. Howe e , he bias desc ip ion me hod could easily be adap ed
397
o mo e complex sys ems. One could inco po a e a bias e m o ex-
398
p ess co ela ion be ween mul iple measu emen s, o he same o dis-
399
inc a iables measu ed in he same loca ion o die en loca ions. In
400
his case, he co a iance be ween wo measu emen s, as exp essed by
Σ
,
401
would no only be a unc ion o
(a)
he ime die ence (
i− j
, see (6)),
402
18
as in ou s udy, bu also o
(b)
spa ial dis ance in one o mo e dimen-
403
sions and
(c)
eec s o measu emen e o co ela ion be ween dis inc
404
senso s measu ing he same o dis inc a iables. This gene aliza ion
405
o he p esen model is likely mos con enien when he bias e o e m
406
is modelled as spa io- empo al Gaussian p ocess (e.g., De Cesa e e al.,
407
2001;Gnei ing,2002;S ein,2005).
408

The me hods applied in bo h he e e ence s udy and ou s a e based on
409
me hods ha accoun o alea o y unce ain y only. Howe e , he lack
410
o knowledge abou he model s uc u e is ypically epis emic in na u e
411
and may he e o e be dicul o accoun o in his way. Epis emic
412
unce ain y may howe e be educed by using mo e exible models
413
(Ma²i¢ e al.,2017) while inc easing pa ame ic unce ain y, which can
414
be handled as an alea o y sou ce o unce ain y wi h cu en ly a ailable
415
me hods. S ill, he adop ion o al e na i e amewo ks o unce ain y
416
analysis (Pa sons,2001;Rao e al.,2008) may be sui ed o handle
417
epis emic unce ain y di ec ly. In summa y, he handling o epis emic
418
unce ain y dese es mo e a en ion.
419
4.2. Gene al consequences o p ac ical unce ain y and eliabili y analysis
420
U ili y o he bias desc ip ion me hod.
In ou opinion, he de ec ion o sys-
421
ema ic de ia ions be ween he assumed model s uc u e and he da a-gene a ing
422
p ocess is he mos use ul ea u e o he bias desc ip ion me hod. Fo his
423
eason, we ecommend ha a model is inspec ed o bias by
(a)
adding a bias
424
e m in he assumed model, speci ying a p io o alpha concen a ed a ound
425
a s ic ly posi i e alue, as sugges ed he e, and
(b)
inspec ing he pos e io
426
o
α
whene e an inapp op ia e model s uc u e is suspec ed. Re o mula ion
427
o he e o model (bias + measu emen e o ) wi h
α
,
σ
, and
τ
p o ed e y
428
help ul as i enables in e p e ing
α
as an indica o o he ela i e impo -
429
ance o model s uc u e e o . In cases whe e he pos e io p obabili y mass
430
is no shi ed owa ds ze o, ela i e o he p io , he modele should suspec
431
he p esence o bias. When his is de ec ed, po en ial model imp o emen s
432
may include he use o ime-dependen pa ame e s (Reiche and Mielei ne ,
433
2009;Lin and Beck,2012) and/o inpu e o s (Del Giudice e al.,2016) o
434
a change in model s uc u e (Del Giudice e al.,2015;Ma²i¢ e al.,2017).
435
While he me hod inc eases he eliabili y o he ob ained s eady-s a e pol-
436
lu an concen a ion p edic ions, i is impo an o no e ha he obse a ion
437
o his bene depends s ongly on he oo cause o he obse ed bias. Fo
438
19
his eason, de ec ion o bias should be ollowed by explo a o y analysis o
439
he esiduals and de elopmen o a be e model s uc u e (e.g., Reiche
440
and Mielei ne ,2009;Del Giudice e al.,2013). We do no ecommend ex-
441
ploi ing he bias e m o p edic ion wi hou sea ch o he unde lying causes
442
o model s uc u e deci s, especially conside ing ha he g ound u h is
443
a ely included in he p oduced c edible in e als. Ul ima ely, he u ili y
444
o any app oach depends on whe he i can success ully desc ibe he ele-
445
an sou ces o he de ia ions be ween model p edic ions and he measu ed
446
a iables (B ynja sdó i and O'Hagan,2014;Wani e al.,2019).
447
Pa ame e in e p e a ion and ans e abili y.
The mechanis ic in e p e a ion
448
o iden ied alues o he pa ame e s in he de e minis ic pa o he model
449
is nea ly impossible when bias is p esen . Adding an au o-co ela ed addi i e
450
e o e m con ibu es o a be e eliabili y o he model p edic ions bu can-
451
no p o ide a clea e in e p e a ion o he pa ame e alues o a di ec ion o
452
a mo e app op ia e model s uc u e. Indeed, he pa ame e es ima es emain
453
biased. Impo an ly, his is a likely scena io in was ewa e enginee ing due o
454
he ex emely simplied ep esen a ion o biological p ocesses du ing model
455
cons uc ion. Fu he mo e, ob aining p oo s o a lack o bias is ex emely
456
dicul o achie e so ha a s aigh o wa d in e p e a ion o pa ame e al-
457
ues is unlikely, e en when he model s uc u e may be app op ia e. Howe e ,
458
g ey-box o hyb id models may oe in ep e abili y and anspa ency a he
459
cos o compu a ional eo s (see in oduc ion abo e).
460
Da a quali y.
The quali y o he simula ed measu emen s in he s udied case
461
is ai ly high ela i e o cu en expe ience in he was ewa e sec o . Howe e ,
462
senso ha dwa e has become inc easingly obus in he las h ee decades
463
(Olsson,2012) and he e is no ob ious eason why his end should s op
464
now. I is he e o e easonable o expec ha he p esence o bias can be
465
de ec ed easily in he u u e, ei he by s a is ical es s o au o-co ela ion
466
o he esiduals, as in he e e ence s udy, o wi h desc ip i e me hods, as in
467
his s udy. This will also acili a e he modica ion o he model s uc u e
468
in acco dance o he en isioned high-quali y da a.
469
4.3. Fu u e wo k
470
Th ough his wo k, we iden ied se e al a enues o u he esea ch.
471
These include:
472
20

De elop and s udy me hods o pa ame e es ima ion and pa ame e
473
in e p e a ion unde p esence o model s uc u e e o .
474

De elop a sys ema ic app oach o he o mula ion o p io dis ibu ions,
475
especially when exibili y is a odds wi h model s uc u e o pa ame e
476
iden iabili y.
477

E alua ion o expe imen al design me hods o imp o e he chances o
478
de ec ion o model s uc u e e o s.
479

Adop and e alua e me hods o handle epis emic unce ain y in model-
480
based p ocess design and ope a ion.
481
5. Conclusions
482
In his pape , we in es iga ed he challenge o s uc u al model deci s in
483
isk-based eac o design. This is a ele an p oblem, because digi aliza ion
484
will imp o e senso esolu ion and spa ial co e age o eac o s, which will
485
e eal misma ches (i.e, bias) in ou common enginee ing models (which ha e
486
been de eloped in he da a-sca ce pas , o en by g ab sampling). Au o-
487
co ela ed ma hema ical o mula ions ha e been sugges ed o imp o e he
488
desc ip ion o such biases.
489
In summa y, ou s udy shows ha
490

Adding au o-co ela ion e ms in he measu emen e o model as a
491
way o accoun o model s uc u e deci s signican ly imp o es he
492
eliabili y o biokine ic models.
493

Bias desc ip ion enables accoun ing o p edic i e unce ain y du ing
494
p ocess design o a la ge deg ee. This does no p oduce a gua an eed
495
eliabili y o he esul ing design howe e . I is he e o e no a bulle -
496
p oo solu ion o he p esence o model- eali y misma ch.
497

The s udied bias desc ip ion me hod is an adequa e ool o iden i y he
498
p esence o model s uc u e deci s in p esence o noisy expe imen al
499
da a.
500
21

Acknowledgmen s
501
We hank Pe e Reiche and Sanda Dejanic o hei help ul insigh in o
502
he s udied p oblem. Ma c B. Neumann acknowledges nancial suppo p o-
503
ided by he Spanish Go e nmen h ough he BC3 Ma ía de Maez u ex-
504
cellence acc edi a ion 2018-2022 (MDM-2017-0714) and he Ramón y Cajal
505
g an (RYC-2013-13628); and by he Basque Go e nmen h ough he BERC
506
2018-2021 p og am.
507
6. Au ho con ibu ions
508
Da io del Giudice: Me hodology, So wa e, Da a analysis, W i ing - Re-
509
iew & Edi ing. Ma c B. Neumann: Me hodology, So wa e, Da a analysis,
510
W i ing - Re iew & Edi ing. Jö g Riecke mann: Concep ualiza ion, Me hod-
511
ology, W i ing - Re iew & Edi ing. K is Villez: Me hodology, So wa e,
512
Fo mal analysis, Da a analysis, Visualiza ion, W i ing - O iginal D a .
513
Re e ences
514
Baya i, M.J., Be ge , J.O., Paulo, R., Sacks, J., Ca eo, J.A., Ca endish, J.,
515
Lin, C.H., Tu, J., 2007. A amewo k o alida ion o compu e models.
516
Technome ics 49, 138154. doi:
10.1198/004017007000000092
.
517
Box, G.E.P., Tiao, G.C., 1973. Bayesian In e ence in S a is ical Analysis.
518
Addison-Wesley, Reading, MA, USA.
519
B ynja sdó i , J., O'Hagan, A., 2014. Lea ning abou physical pa ame-
520
e s: The impo ance o model disc epancy. In e se P oblems 30, 114007.
521
doi:
10.1088/0266-5611/30/11/114007
.
522
Cagno, E., De Amb oggi, M., G ande, O., T ucco, P., 2011. Risk analysis
523
o unde g ound in as uc u es in u ban a eas. Reliabili y Enginee ing &
524
Sys em Sa e y 96, 139148. doi:
10.1016/j. ess.2010.07.011
.
525
Cie kens, K., Plano, S., Benede i, L., Weije s, S., de Jonge, J., Nopens, I.,
526
2012. Impac o inuen da a equency and model s uc u e on he quali y
527
o WWTP model calib a ion and unce ain y. Wa e Science & Technology
528
65, 233242. doi:
10.2166/ws .2012.081
.
529
22
C aig, P.S., Golds ein, M., Rougie , J.C., Seheul , A.H., 2001. Bayesian
530
o ecas ing o complex sys ems using compu e simula o s. Jou nal
531
o he Ame ican S a is ical Associa ion 96, 717729. doi:
10.1198/
532
016214501753168370
.
533
De Cesa e, L., Mye s, D.E., Posa, D., 2001. Es ima ing and modeling space-
534
ime co ela ion s uc u es. S a is ics & P obabili y Le e s 51, 914.
535
doi:
10.1016/S0167-7152(00)00131-0
.
536
Del Giudice, D., Albe , C., Riecke mann, J., Reiche , P., 2016. Desc ibing
537
he ca chmen -a e aged p ecipi a ion as a s ochas ic p ocess imp o es pa-
538
ame e and inpu es ima ion. Wa e Resou ces Resea ch 52, 31623186.
539
doi:
10.1002/2015WR017871
.
540
Del Giudice, D., Hon i, M., Scheidegge , A., Albe , C., Reiche , P., Rieck-
541
e mann, J., 2013. Imp o ing unce ain y es ima ion in u ban hyd ological
542
modeling by s a is ically desc ibing bias. Hyd ology and Ea h Sys em
543
Sciences 17, 42094225. doi:
10.5194/hess-17-4209-2013
.
544
Del Giudice, D., Reiche , P., Ba es, V., Albe , C., Riecke mann, J., 2015.
545
Model bias and complexi y - Unde s anding he eec s o s uc u al deci s
546
and inpu e o s on uno p edic ions. En i onmen al Modelling & So -
547
wa e 64, 205214. doi:
10.1016/j.en so .2014.11.006
.
548
Die zel, A., Reiche , P., 2012. Calib a ion o compu a ionally demand-
549
ing and s uc u ally unce ain models wi h an applica ion o a lake wa-
550
e quali y model. En i onmen al Modelling & So wa e 38, 129146.
551
doi:
h p://dx.doi.o g/10.1016/j.en so .2012.05.007
.
552
Dochain, D., Van olleghem, P.A., 2001. Dynamical modelling & es ima ion
553
in was ewa e ea men p ocesses. IWA Publishing, London, UK.
554
Dochain, D., Van olleghem, P.A., Van Daele, M., 1995. S uc u al iden ia-
555
bili y o biokine ic models o ac i a ed sludge espi a ion. Wa e Resea ch
556
29, 25712578. doi:
10.1016/0043-1354(95)00106-U
.
557
Gilks, W.R., Richa dson, S., Spiegelhal e , D.J., 1996. Ma ko Chain Mon e
558
Ca lo in p ac ice. Chapman and Hall, London, UK.
559
23
Gnei ing, T., 2002. Nonsepa able, s a iona y co a iance unc ions o space-
560
ime da a. Jou nal o he Ame ican S a is ical Associa ion 97, 590600.
561
doi:
10.1198/016214502760047113
.
562
Guo, M., Mu phy, R.J., 2012. LCA da a quali y: sensi i i y and unce ain y
563
analysis. Science o he To al En i onmen 435, 230243. doi:
10.1016/j.
564
sci o en .2012.07.006
.
565
Has ie, T., Tibshi ani, R., Wainw igh , M., 2015. S a is ical lea ning wi h
566
spa si y: he Lasso and gene aliza ions. Chapman and Hall/CRC, Boca
567
Ra on, FL, USA.
568
Hauduc, H., Neumann, M.B., Muschalla, D., Game i h, V., Gillo , S., Van-
569
olleghem, P.A., 2015. Eciency c i e ia o en i onmen al model quali y
570
assessmen : A e iew and i s applica ion o was ewa e ea men . En i-
571
onmen al Modelling & So wa e 68, 196204. doi:
10.1016/j.en so .
572
2015.02.004
.
573
Jensen, H.A., Je ez, D.J., 2018. A s ochas ic amewo k o eliabili y and
574
sensi i i y analysis o la ge scale wa e dis ibu ion ne wo ks. Reliabili y
575
Enginee ing & Sys em Sa e y 176, 8092. doi:
10.1016/j. ess.2018.04.
576
001
.
577
Kabi , G., Tes ama iam, S., Sadiq, R., 2015. P edic ing wa e main ailu es
578
using Bayesian model a e aging and su i al modelling app oach. Relia-
579
bili y Enginee ing & Sys em Sa e y 142, 498514. doi:
10.1016/j. ess.
580
2015.06.011
.
581
Kennedy, M.C., O'Hagan, A., 2001. Bayesian calib a ion o compu e models.
582
Jou nal o he Royal S a is ical Socie y: Se ies B (S a is ical Me hodology)
583
63, 425464. doi:
10.1111/1467-9868.00294
.
584
Lin, Z.L., Beck, M.B., 2012. Accoun ing o s uc u al e o and unce ain y
585
in a model: An app oach based on model pa ame e s as s ochas ic p o-
586
cesses. En i onmen al Modelling & So wa e 27-28, 97111. doi:
10.1016/
587
j.en so .2011.08.015
.
588
Liu, C., Zacha a, J.M., 2001. Unce ain ies o Monod kine ic pa ame e s
589
nonlinea ly es ima ed om ba ch expe imen s. En i onmen al Science &
590
Technology 35, 133141. doi:
10.1021/es001261b
.
591
24
Ma²i¢, A., S ini asan, S., Bille e , J., Bon in, D., Villez, K., 2017. Shape
592
cons ained splines as anspa en black-box models o biop ocess mod-
593
eling. Compu e s & Chemical Enginee ing 99, 96105. doi:
10.1016/j.
594
compchemeng.2016.12.017
.
595
Mu phy, K.P., 2012. Machine lea ning: A p obabilis ic pe spec i e. MIT
596
P ess, Camb idge, MA, USA.
597
Neumann, M.B., Guje , W., 2008. Unde es ima ion o unce ain y in s a-
598
is ical eg ession o en i onmen al models: Inuence o model s uc-
599
u e unce ain y. En i onmen al Science & Technology 42, 40374043.
600
doi:
10.1021/es702397q
.
601
Olsson, G., 2012. ICA and me - A subjec i e e iew. Wa e Resea ch 46,
602
15851624.
603
Pa sons, S., 2001. Quali a i e me hods o easoning unde unce ain y. MIT
604
P ess, Camb idge, MA, USA.
605
Pe e sen, B., Ge naey, K., De issche , M., Dochain, D., Van olleghem, P.A.,
606
2003. A simplied me hod o assess s uc u ally iden iable pa ame e s
607
in Monod-based ac i a ed sludge models. Wa e Resea ch 37, 28932904.
608
doi:
10.1016/S0043-1354(03)00114-3
.
609
Rao, K.D., Kushwaha, H.S., Ve ma, A.K., S i idya, A., 2008. Epis emic
610
unce ain y p opaga ion in eliabili y assessmen o complex sys ems. In-
611
e na ional Jou nal o Pe o mabili y Enginee ing 4, 7184.
612
Reiche , P., Mielei ne , J., 2009. Analyzing inpu and s uc u al unce ain y
613
o nonlinea dynamic models wi h s ochas ic, ime-dependen pa ame e s.
614
Wa e Resou ces Resea ch 45, W10402. doi:
10.1029/2009WR007814
.
615
Reiche , P., Schuwi h, N., 2012. Linking s a is ical bias desc ip ion o
616
mul iobjec i e model calib a ion. Wa e Resou ces Resea ch 48, W09543.
617
doi:
10.1029/2011WR011391
.
618
Rena d, B., Ka e ski, D., Kucze a, G., Thye , M., F anks, S.W., 2010. Un-
619
de s anding p edic i e unce ain y in hyd ologic modeling: The challenge
620
o iden i ying inpu and s uc u al e o s. Wa e Resou ces Resea ch 46.
621
doi:
10.1029/2009WR008328
.
622
25