The present work proposes a simulation-based Bayesian method for parameter
estimation and fragility model selection for mutually exclusive and
collectively exhaustive (MECE) damage states. This method uses an adaptive
Markov chain Monte Carlo simulation (MCMC) based on likelihood estimation
using point-wise intensity values. It identifies the simplest model that
fits the data best, among the set of viable fragility models considered. The
proposed methodology is demonstrated for empirical fragility assessments for
two different tsunami events and different classes of buildings with varying
numbers of observed damage and flow depth data pairs. As case studies,
observed pairs of data for flow depth and the corresponding damage level from
the South Pacific tsunami on 29 September 2009 and the Sulawesi–Palu
tsunami on 28 September 2018 are used. Damage data related to a total of five
different building classes are analysed. It is shown that the proposed
methodology is stable and efficient for data sets with a very low number of
damage versus intensity data pairs and cases in which observed data are
missing for some of the damage levels.
Introduction
Fragility models express the probability of exceeding certain damage
thresholds for a given level of intensity for a specific class of buildings
or infrastructure. Empirical fragility curves are models derived from
observed pairs of damage and intensity data for buildings and
infrastructure usually collected, acquired and even partially simulated in
the aftermath of disastrous events. Some examples of empirical fragility
models are seismic fragility (Rota et al., 2009; Rosti et al., 2021), tsunami fragility (Koshimura et al., 2009a; Reese et al., 2011; a comprehensive review
can be found in Charvet et al., 2017), flooding fragility (Wing et al., 2020)
and debris flow fragility curves (Eidsvig et al., 2014). Empirical fragility
modelling is greatly affected by how the damage and intensity parameters are
defined. Mutually exclusive and collectively exhaustive (MECE, see next
section for the definition) damage states are quite common in the literature
as discrete physical damage states. The MECE condition is necessary for
damage states in most probabilistic risk formulations, leading to the mean
rate of exceeding loss (e.g. Behrens et al., 2021).
Tsunami fragility curves usually employ the tsunami flow depth as the
measure of intensity; although different studies also use other measures
like current velocity (e.g. De Risi et al., 2017b; Charvet et al., 2015).
Charvet et al. (2015) demonstrate that the flow depth may cease to be an
appropriate measure of intensity for higher damage states, and other
parameters such as the current velocity, debris impact and scour can become
increasingly more important. De Risi et al. (2017b) developed bivariate
tsunami fragilities, which account for the interaction between the two
intensity measures of tsunami flow depth and current velocity.
Early procedures for empirical tsunami fragility curves used data binning
to represent the intensity. For example, Koshimura et al. (2009b) binned
the observations by the intensity measure, i.e. the flow depth; however, the
latest procedures have mostly used point-wise intensity estimates instead.
Fragility curves for MECE damage states are distinguished by their nicely
“laminar” shape; in other words, the curves should not intersect. When
fitting empirical fragility curves to observed damage data, this condition
is not satisfied automatically. For example, fragility curves are usually
fitted for individual damage states separately, and they are filtered
afterwards to remove the crossing fragility curves (e.g. Miano et al., 2020),
or ordered (“parallel”) fragility models are used from the start (Charvet
et al., 2014; Lahcene et al., 2021). Charvet et al. (2014) and De Risi (2017a)
also used partially ordered models to derive fragility curves for MECE
damage states. They used the multinomial probability distribution to model
the probability of being in any of MECE damage states based on binned
intensity representation. De Risi et al. (2017a) used Bayesian inference to
derive the model parameters for an ensemble of fragility curves.
Empirical tsunami fragility curves are usually constructed using generalised
linear models based on probit, logit or the complementary loglog link
functions (Charvet et al., 2014; Lahcene et al., 2021). As far as the
assessment of the goodness of fit, model comparison and selection are
concerned, approaches based on the likelihood ratio and Akaike information
criterion, (e.g. Charvet et al., 2014; Lahcene et al., 2021) and on k-fold
cross validation have also been used (Chua et al., 2021). For estimating
confidence intervals for empirical tsunami fragility curves, bootstrap
resampling has been commonly used (Charvet et al., 2014; Lahcene et al., 2021;
Chua et al., 2021).
The present paper presents a simulation-based Bayesian method for inference
and model class selection for the ensemble modelling of the tsunami
fragility curves for MECE damage states for a given class of buildings. By
fitting the (positive definite) fragility link function to the conditional
probability of being in a certain damage state, given that building is not
in any of the preceding states, the method ensures that the fragility curves
do not cross (i.e. they are “hierarchical” as in De Risi et al., 2017a).
The method uses adaptive Markov chain Monte Carlo simulation (MCMC, Beck and
Au, 2002), based on likelihood estimation using point-wise intensity values,
to infer the ensemble of the fragility model parameters. Alternative link
functions are compared based on log evidence, which considers both the
average goodness of fit (based on log likelihood) and the model parsimony
(based on relative entropy). This way, among the set of viable models
considered, it identifies the simplest model that fits the data best. By
“simplest model”, we mean the model having maximum relative entropy
(measured using the Kullback–Leibler distance; Kullback and Leibler, 1951) with respect to the data. This usually means the model has a small number of
parameters.
MethodologyDefinitions of intensity and damage parameters
The intensity measure, IM, (or simply “intensity”, e.g. the tsunami flow
depth) refers to a parameter used to convey information about an event from
the hazard level to the fragility level – it is an intermediate variable.
The damage parameter, D, is a discrete random variable, and
the vector of
damage levels is expressed as Dj,j=0:NDS, where Dj as the jth damage level (threshold) and NDS as
the total number of damage levels considered (depending on the damage scale
being used and on the type of hazard, e.g. earthquake, tsunami, debris
flow). Normally, D0 denotes the no-damage threshold, while DNDS defines
the total collapse or being totally washed away. Let us assume that DSj is the jth damage state defined by the
logical statement that the damage D is between the two damage thresholds
Dj and Dj+1; i.e. D is equal to or greater than Dj and
smaller than Dj+1 as follows (see also Fig. 1 for a graphical
representation of the above expressions):
DSj≡(D≥Dj)⋅(D<Dj+1),
where (⋅) denotes the logical product and is read as “AND”.
Obviously, for the last damage state, we have DSNDS≡D≥DNDS.
Graphical representation of damage levels Dj and
damage states DSj, where j=0:NDS.
Damage states DS0,DS1,…,DSNDS are mutually exclusive and collectively exhaustive (MECE) if and
only if P(DSi⋅DSj|IM)=0 (if i≠j, j=0:NDS) and ∑j=0NDSP(DSj|IM)=1; (⋅) denotes the logical product and is read as “AND”. In simple words, the damage states are MECE if being in one damage state excludes all others and if all the damage states together cover the entire range of possibilities in terms of damage. The ensemble of MECE damage states DSj, j=0:NDS is usually referred to
as the damage scale (e.g. the EMS98; Grünthal, 1998).
The proposed methodology herein is also applicable to fragility assessment
in cases where observed damage data are not available for some of the damage
levels. Let index be the vector of j values (j=0:NDS),
indicating damage levels Nj for which observed data are available (j
values are in ascending order). The new damage scale formed as DSindex(1),DSindex(2),…,DSindex(N) , where N is the length of
vector index, is also MECE. It is noteworthy that the number of fragility curves derived in this case is going to be equal to N-1. In the following, for simplicity and without loss of generality, we have assumed
that observed data are available for all damage levels, i.e.
index=0:NDS, that is,
NDS=N-1. However, the proposed methodology is also applicable to the modified damage scale formed by damage level indices in vector
index(1:N). We will later see examples of such applications in the
case studies.
Fragility modelling using generalised regression models
The generalised regression models (GLMs) are more suitable for empirical
fragility assessment with respect to the standard regression models. This is
mainly because the dependent variable in the case of the generalised
regression models is a Bernoulli binary variable (i.e. only two possible
values: 0 or 1). Bernoulli variables are particularly useful in order to
detect whether a specific damage level is exceeded or not (only two
possibilities). In the following, fragility assessment based on GLMs is briefly described.
The term P(DSj|IM) denotes the probability
of being in damage state DSj for a given intensity level IM. Based on NDS damage thresholds, this conditional probability P(DSj|IM) can be read (see Eq. 1) as the probability
that D≥Dj and D<Dj+1 and can be estimated as follows (see Appendix A for the derivation):
P(DSj|IM)=PD≥Dj⋅D<Dj+1|IM=P(D≥Dj|IM)-P(D≥Dj+1|IM)for0≤j<NDSP(D≥Dj|IM)forj=NDS,
where P(D≥Dj|IM) is the fragility function for damage level Dj.
For each damage threshold, fragility can be obtained for a desired building
class considering that the damage data provide Bernoulli variables (binary
values) of whether the considered damage level was exceeded or not for given IM levels. For damage threshold Dj, all buildings with an observed damage level D<Dj will have a probability equal to zero, while those with
D≥Dj will have an assigned probability equal to one. In other
words, for building i and damage state j, the Bernoulli variable Yij indicates whether building i is in damage state jYij=1if buildingiexceedsDjwith probabilityP(D≥Dj|IMi)0if buildingidoes not exceedDjwith probability1-P(D<Dj|IMi),
where IMi is the intensity evaluated at the location of building i. A
Bernoulli variable is defined by one parameter, which is P(D≥Dj|IMi) herein. This latter is usually linked to a linear logarithmic predictor in the form
lij=α0,j+α1,jlnIMi,
where α0,j and α1,j are regression constants for
damage level j. We have employed generalised linear regression (e.g. Agresti, 2012) with different link functions “logit”, “probit” and “cloglog” to define probability function πij as the following:
πij=πjIMi=1+exp(-lij)-1logitΦlijprobit1-exp-exp(lij)cloglog.
The logit link function is equivalent to presenting πjIM with a logistic regression function. The probit link function is equivalent to a
lognormal cumulative distribution function for πjIM. In the cloglog (complementary log–log) transformation, the link function at the
location of building i can be expressed as lij=ln-ln1-πij. It is noted that the
generalised linear regression based on maximum likelihood estimation (MLE) is available in many statistical software packages (e.g. MathWorks, Python, R).
In the following, we have referred to the general methodology of fitting
fragility model to data – one damage state at a time – the “Basic method”. In the Basic method, the probability of exceeding damage level j is equal to the
probability function defined in Eq. (5); that is, πij=P(D≥Dj|IMi). This method for empirical
fragility curve parameter estimation is addressed in detail in the section “Results”, under the “MLE-Basic” method. The fragility curves obtained under the
“MLE-Basic” method could potentially cross, leading to the ill condition that
P(DSj|IM)<0. To overcome this, a hierarchical fragility modelling approach has been adopted like that in De Risi et al. (2017a).
Hierarchical fragility modelling
Equation (2) for 0≤j<NDS, and given IMi, can also be written as follows using the product rule in probability:
P(DSj|IMi)=PD<Dj+1⋅D≥Dj|IMi=1-P(D≥Dj+1|D≥Dj,IMi)⋅P(D≥Dj|IMi).
The term P(D≥Dj+1|D≥Dj,IMi) embedded in Eq. (6) denotes the conditional probability that
the damage exceeds the damage threshold Dj+1 knowing that it has
already exceeded the previous damage level Dj given IMi. By making πij=P(D≥Dj+1|D≥Dj,IMi) (see Eq. 5, which is positive definite), we ensure that the fragility curve of a lower damage level will not fall below the fragility
curve of the subsequent damage threshold (the ill condition of P(DSj|IM)<0 does not take place). Hence, Eq. (6) can be expanded as follows (see Appendix B for derivation):
P(DSj|IMi)=1-πij⋅1-∑k=0j-1P(DSk|IMi)forj≥11-πi0≜P(D<D1|IMi)forj=0.
In this way, the fragility curves are constructed in a hierarchical manner
by first constructing the “fragility increments” P(DSj|IMi). Note that for the last damage state DSNDS, the probability P(DSNDS|IMi), which is also equal to the fragility of the ultimate damage threshold DNDS, i.e. P(D≥DNDS|IM) (see Eq, 2), can be estimated by
satisfying the collectively exhaustive (CE) condition
P(DSNDS|IMi)=P(D≥DNDS|IMi)=1-∑j=0NDS-1P(DSj|IMi).
Accordingly, the fragility of other damage levels P(D≥Dj|IMi), where 0<j<NDS, can be obtained from Eq. (2) by starting from the fragility of the higher threshold P(D≥Dj+1|IM) and adding successively P(DSj|IM) (see Eq. 7) as follows:
P(D≥Dj|IMi)=P(DSj|IMi)+P(D≥Dj+1|IMi)for0<j<NDS.
As a result, the set of hierarchical fragility models based on Eq. (9)
has 2×NDS model parameters with the vector
θ=α0,j,α1,j,j=0:NDS-1. Obviously, with reference to Eq. (8), no model parameter is required for the last damage level, which
is derived by satisfying the CE condition. The vector
θ of the proposed hierarchical fragility models
can be defined by two different approaches
MLE method. A generalised linear regression model (as explained in previous the section) is used for the conditional fragility term πij=P(D≥Dj+1|D≥Dj,IM) for the jth damage state DSj (see Eq. 7, 0≤j<NDS). Herein, we need to work with partial damage data so that all buildings in DSj (with an observed damage Dj≤D<Dj+1) will be assigned a probability equal to 0, while those in higher damage states (with D≥Dj+1) will be assigned a probability equal to 1 (i.e. in order to model the conditioning on D≥Dj, the domain of possible damage levels is reduced to D≥Dj).
Bayesian model class selection (BMCS). The Bayesian inference for model updating is employed to obtain the joint distribution of the model parameters.
Detailed discussion about these two approaches, namely “MLE” and “BMCS”, for parameter estimation of empirical fragility curves are provided in Sect. 3.
Bayesian model class selection (BMCS) and parameter inference using adaptive MCMC
We use the Bayesian model class selection (BMCS) herein to identify the best
link model to use in the generalised linear regression scheme. However, the
procedure is general and can be applied to a more diverse pool of candidate
fragility models. BMCS (or model comparison) is essentially Bayesian
updating at the model class level to make comparisons among candidate model
classes given the observed data (e.g. Beck and Yuen, 2004; Muto and Beck,
2008). Given a set of NM candidate model classes Mk,k=1:NM, and in the presence of the data D, the posterior probability of the kth model class, denoted as P(Mk|D) can be written as follows:
P(Mk|D)=pD|MkP(Mk)∑k=1NMpD|MkP(Mk).
In lieu of any initial preferences about the prior P(Mk), one can assign equal weights to each model; thus, P(Mk)=1NM. Hence, the probability of a model class is dominated by the likelihood of p(D|Mk) (a.k.a. evidence). It is important to note that p herein stands for the probability density function (PDF). Here, data vector D=(IM,DS)i,i=1:NCL defines the observed intensity and damage data for NCL buildings surveyed for class CL. Let us define the vector of model parameters θk for model class Mk as θk=α0,j,α1,jk,j=0:NDS-1. We use the Bayes theorem to write the “evidence” p(D|Mk) provided by data D for model Mk as follows:
pD|Mk=pD|θk,Mkpθk|Mkpθk|D,Mk.
It can be shown (see Appendix C; Muto and Beck, 2008) that the logarithm of the
evidence (called log-evidence) lnp(D|Mk) can be written as
lnpD|Mk=∫ΩθklnpD|θk,Mkpθk|D,Mkdθk︸Term1-∫Ωθklnpθk|D,Mkpθk|Mkpθk|D,Mkdθk︸Term2,
where Ωθk is the domain of θk, and p(D|θk,Mk) is the likelihood function conditioned on model class Mk. “Term 1” denotes the posterior mean of the log likelihood, which is a measure of the average data fit to model Mk. “Term 2” is the relative entropy (Kullback and Leibler, 1951; Cover and Thomas, 1991) between the prior p(θk|Mk) and the posterior p(θk|D,Mk) of θk given model Mk, which is a measure of the distance between the two PDFs. The latter Term 2 measures quantitatively the amount of information (on average) that is “gained” about θk from the observed data D. It is interesting that Term 2 in the log-evidence expression penalises for model complexity; i.e. if the model extracts more information from data (which is a sign of being a complex model with more model parameters), the log-evidence reduces. The exponential of the log-evidence, p(D|Mk), is going to be implemented directly in Eq. (10) to provide the probability attributed to the model class Mk. More details on how to estimate the two terms in Eq. (12) are provided in Sect. 3.
The likelihood p(D|θk,Mk) can be derived, based on point-wise intensity information, as the likelihood of nCL,j buildings being in damage state DSj (considering that ∑j=0NDSnCL,j=NCL), according to data D defined before
p(D|θk,Mk)=∏j=0NDS∏i=1nCL,jP(DSj|IMi).
The posterior distribution pθk|D,Mk can be found based on Bayesian inference
pθk|D,Mk︸posterior=pD|θk,Mkpθk|Mk∫ΩθkpD|θk,Mkpθk|Mkdθk=C-1pD|θk,Mk︸likelihoodpθk|Mk︸prior,
where C-1 is a normalising constant. In lieu of additional information
(or preferences), the prior distribution, p(θk|Mk), can be estimated as the product of marginal normal PDFs for each model parameter, i.e. a multivariate normal distribution with zero correlation between the pairs of model parameters θk (see Appendix D). More detail about an efficient prior joint PDF is provided in Sect. 3. To sample from the posterior distribution p(θk|D,Mk) in Eq. (14), an
adaptive MCMC simulation routine (see Appendix E) is employed. MCMC is particularly useful for drawing samples from the target posterior, while it is known up to a scaling constant C-1 (see Beck and Au, 2002); thus, in Eq. (14), we only need un-normalised PDFs to feed the MCMC procedure. The MCMC routine herein employs the Metropolis–Hastings (MH) algorithm (Metropolis et al., 1953; Hasting, 1970) to generate samples from the target joint posterior PDF p(θk|D,Mk).
Calculating the hierarchical fragilities and the corresponding confidence intervals based on the vector of model parameters <inline-formula><mml:math id="M158" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">θ</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>
For each realisation of the vector of model parameters θk, the corresponding set of hierarchical fragility curves can be derived based on the procedure described in the previous sections. Since we have realisations of the model parameters drawn from the joint PDF p(θk|D,Mk) (based on samples drawn from adaptive MCMC procedure, see also Appendix E), we can use the concept of robust fragility (RF) proposed in Jalayer et al. (2017) (see also Jalayer et al., 2015; Jalayer and Erahimian, 2020) to derive confidence intervals for the fragility curves. RF is defined as the expected value for a prescribed fragility model considering the joint probability distribution for the fragility model parameters θk. The RF herein can be expressed as
P(D≥Dj|IM,D,Mk)=∫ΩθkP(D≥Dj|IM,θk)p(θk|D,Mk)dθk=Eθk|D,MkP(D≥Dj|IM,θk),
where P(D≥Dj|IM,θk) is the fragility given the model parameters θk associated with the model Mk (it has been assumed that once conditioned on fragility model parameters θk, the fragility becomes independent of data D); Eθk|D,Mk is the
expected value over the vector of fragility parameters θk for model Mk. The integral in Eq. (15) can be solved numerically by employing Monte Carlo simulation with Nd generated samples from the vector θk as follows:
PD≥Dj|IM,D,Mk≅1Nd∑l=1NdPD≥Dj|IM,θk,l,
where PD≥Dj|IM,θk,l is the fragility given the lth realisation (l=1:Nd) of the model parameters θk for model Mk. Based on the definition represented in Eqs. (15) and (16), the variance σθk|D,Mk2, which can be used to estimate a
confidence interval for the fragility considering the uncertainty in the
estimation of θk, is calculated as follows:
σθk|D,Mk2P(D≥Dj|IM,θk)=Eθk|D,MkP(D≥Dj|IM,θk)2︸≅1/Nd∑i=1NdP(D≥Dj|IM,θk,l)2-Eθk|D,MkP(D≥Dj|IM,θk)︸=P(D≥Dj|IM,D,Mk)2(Eq.16).
The empirical fragilities derived through the hierarchical fragility
procedure are not necessarily attributed to a lognormal distribution. Hence,
we have derived equivalent lognormal statistics (i.e. the median and
dispersion) for the resulting fragility curves. The median intensity, ηIMC, for a given damage level is calculated as the IM corresponding to 50 % probability on the fragility curve. The logarithmic standard deviation (dispersion) of the equivalent lognormal fragility curve at the onset of damage threshold, βIMC, is estimated as half of the logarithmic distance between the IMs corresponding to the probabilities of 16 % (IMC16) and the 84 % (IMC84) on the fragility curve; thus, the dispersion can be estimated as βIMC=0.50×lnIMC84IMC16. The overall effect of epistemic uncertainties (due to the uncertainty in the fragility model parameters and reflecting the effect of limited sample size) on the median of the empirical fragility curve is considered through (logarithmic) intensity-based standard deviation denoted as βUF (see Jalayer et al., 2020). βUF can
be estimated as half of the (natural) logarithmic distance (along the IM axis) between the median intensities (i.e. 50 % probability) of the RF's
derived with 16 % (denoted as IM84) and 84 % (IM16) confidence levels, respectively; i.e. βUF=0.50×lnIM84IM16. The RF and its confidence band, the sample fragilities θk,l (where l=1:Nd), the equivalent lognormal parameters ηIMC and βIMC of the RF, the epistemic uncertainty βUF, and finally the intensities IM16 and IM84 are shown in Figs. 2d to 6d in the following Sect. 3.
ResultsCase study 1: the 2009 South Pacific tsunami
The central South Pacific region-wide tsunami was triggered by an
unprecedented earthquake doublet (Mw 8.1 and Mw 8.0) on 29 September 2009, between about 17:48 and 17:50 UTC (Goff and Dominey-Howes, 2011). The tsunami seriously impacted numerous locations in the central South Pacific. Herein, the damage data related to the brick masonry residential (1 storey) and Timber residential buildings associated with the reconnaissance survey sites of American Samoa and the Samoan Islands were utilised. Based on the observed damage regarding different indicators (see Reese et al., 2011 for more details on damage observation), each structure was assigned a damage state between (DS0 and DS5). The original data documented in Reese et al. (2011) reporting the tsunami flow depth and the attributed damage state to each surveyed building can be found on the site of the European Tsunami Risk Service (https://eurotsunamirisk.org/datasets/, last access: 30 November 2022). The five damage levels (NDS=5) and a description of the indicators leading to
the classification of the damage states are given in Table 1 based on Reese
et al. (2011).
The classification of damage thresholds (the damage scale)
used for the 2009 South Pacific tsunami (from Reese et al., 2011).
Case study 2: the 2018 Sulawesi–Palu tsunami
On Friday 28 September 2018, at 18:02 local time, a shallow strike-slip
earthquake of moment magnitude Mw 7.5 occurred near Palu City, Central
Sulawesi, Indonesia followed by submarine landslides, a tsunami, and massive
liquefaction caused substantial damage (Muhari et al., 2018; Rafliana et al., 2022). In Sulawesi, more than 3300 fatalities and missing people, 4400 serious injuries and 170 000 people were displaced by earthquake, tsunami, landslides, liquefaction or building collapse, or combinations of these hazards (Paulik et al., 2019; Mas et al., 2020). Herein, the damage data related to the non-engineered unreinforced clay brick masonry (1 and 2 storey) and non-engineered light timber buildings located in Palu City were
utilised. Based on the observed damage (see Paulik et al., 2018 for
details on damage observation), each structure was assigned a damage state
between (DS0 and DS3). The original data reporting the tsunami flow depth and the attributed damage state to each surveyed building can be found as the Supplement to Paulik et al. (2018). The three damage levels (NDS=3) and a description of the indicators leading to the classification of the damage state are given in Table 2.
The classification of damage thresholds (the damage scale)
used for the 2018 Sulawesi–Palu tsunami (from Paulik et al., 2019).
The building classes
Table 3 illustrates the building classes, for which fragility curves are
obtained based on the proposed procedure and based on the databases related
to the two tsunami events described above. The taxonomy used for describing
the building class matches the original description used in the raw
databases. The number of data points available for different building
classes showcases both classes with large number of data available, e.g.
brick masonry 1 storey (South Pacific) and non-engineered brick masonry 1 storey (Sulawesi), and classes with few data points available, e.g. timber
residential (South Pacific) and non-engineered masonry 2 storeys and timber
(Sulawesi). The fifth column in the table shows the proportion of the
number of damage levels for which observed data are available N (see Sect. 2.1) to the total number of damage levels in the corresponding damage
scales, namely NDS+1=6 for South Pacific and NDS+1=4 for Sulawesi tsunami events (to include level 0). If the ratio is equal to
unity, it indicates that data are available for all the damage levels from 0
to NDS. Note that the number of fragility curves derived is going to be equal to N-1, that is, equal to the number of damage levels for which observed damage data are available minus one.
The building classes.
Building class Tsunami eventNumberN/(NDS+1),indexof data1Brick masonry residential, 1 storeySouth Pacific 20091206/6, index=0,1,2,3,4,52Timber residential234/6, index=2,3,4,51Non-engineered masonry, unreinforced with clay brick, 1 storeySulawesi–Palu 20182793/4, index=0,1,22Non-engineered masonry, unreinforced with clay brick, 2 storeys373/4, index=0,1,23Non-engineered light timber143/4, index=1,2,3The different model classes
For each building class considered, we have considered the set of candidate
models consisting of the fragility models resulting from the three
alternative link functions used in the generalised linear regression in
Eq. (5). That is, M1 refers to hierarchical fragility modelling
based on “logit”, M2 refers to hierarchical fragility modelling based on “probit”, and M3 refers to hierarchical fragility modelling based on “cloglog”. For each model, both the MLE method using the MATLAB generalised regression toolbox and the BMCS using the procedure described in the previous section are implemented.
Fragility modelling using MLE
The first step towards fragility assessment by employing the MLE method (see
Sect. 2.3) is to define the vector of model parameters θ=α0,j,α1,j, where j=index(1:(N-1)) and where vector index is defined in Sect. 2.1 as the vector of damage level indices (in ascending order) for which observed damage data are available and N is the length of index. To accomplish this, the model parameters α0,j,α1,j are obtained by fitting the link functions in Eq. (5) to conditional fragility PD≥Dj+1D≥Dj,IM according to Eq. (6).
Herein, we have used MATLAB as a statistical software package (developed by
MathWorks) to estimate the maximum likelihood of model parameters α0,j,α1,j using the following MATLAB command:
glmfit(log(xj),yj,`binomial',`link',model'). The `model' will be either `logit', `probit', or
`comploglog'. For each damage level Dj+1,
index(1)≤j<index(N), the vector xj is the IM's for which the condition D≥Dj is satisfied; yj is the column vector containing one-to-one probability assignment to the IM data in xj, with zero (=0.0) assigned to those data corresponding to DSj (Dj≤D<Dj+1) and one (=1.0) to those related to higher damage states (with D≥Dj+1).
The vectors defining the MLE of the model parameters, θMLE, are presented in Table 4 for each of the building classes listed in Table 3 and for each of the three models M1, M2, and M3 defined in Sect. 3.4. Given the model parameter θMLE, the damage state probability PDSjIM can be estimated based on the recursive Eqs. (7) and (8). Then, the fragility
for the ultimate damage level is calculated first based on Eq. (8). For
the lower damage thresholds, the empirical fragility is derived based on Eq. (9). The resulting hierarchical fragility curves by employing the
direct fragility assessment given θMLE, i.e. P(D≥DjIM,θMLE) for
j=index2:N, are shown later in the next
section by comparison with those obtained from the BMCS method.
The model parameters θMLE.
BuildingModelα0,0α1,0α0,1α1,1α0,2α1,2α0,3α1,3α0,4α1,4classclassSouth Pacifictsunami 20091M15.2424.1903.9004.255-1.1754.805-1.3452.887-1.9942.917M22.7422.1902.0072.221-0.6702.804-0.8031.745-1.1571.733M32.0792.0111.3221.850-1.2683.057-1.3661.961-1.9812.2182M11.1271.5122.4840.771-2.8467.708M20.6570.9091.3900.426-1.5754.316M30.2510.8620.8830.355-2.1414.648Sulawesi–Palu tsunami 20181M16.0594.355-0.6302.909M23.2642.340-0.3711.709M32.4982.088-0.9072.0562M13.5563.672-2.4865.126M22.0772.144-1.4643.036M31.6642.239-2.4514.2173M10.4661.3750.4741.195M20.2950.8470.2960.774M3-0.0411.068-0.0760.835Fragility modelling using BMCS
In the first step, the model parameters are estimated for each model class
separately. For each model class Mk, the model parameters
θk are estimated through the adaptive MCMC method described in detail in Appendix E, which yields the posterior distribution in Eq. (14). With reference to Eq. (14), the prior joint PDF p(θk|Mk) is a multivariate normal PDF with zero correlation between the pairs of model parameters (see Appendix D). The vector of the mean values, μθ is set to be the MLE tabulated in Table 4 (=θMLE related to Mk). We have attributed a high value for the coefficient of variation (more than 3.20 herein) for each of the model parameters. Appendix F illustrates the histograms representing the drawn samples from the joint posterior PDF p(θk|D,Mk) for a selected building class.
The RF curves derived from the hierarchical fragility curves (see Sect. 2.5) and the corresponding plus/minus 1 standard deviation (±1σ) intervals from Eqs. (16) and (17) are also plotted in Figs. 2a to 6a corresponding to Classes 1–2 South Pacific tsunami and Classes 1–3 Sulawesi–Palu for one of the model classes Mk(k∈1,2,3) and for the damage thresholds Dj, j=index2:N. The colours of the hierarchical robust fragility curves, labelled as Dj-BMCS, match closely with those shown in Tables 1 and 2. The corresponding ±1σ confidence interval curve, which reflects the uncertainty in the model parameters, is shown as a light grey area with different colour intensities. Figures 2b to 6b compare the hierarchical robust fragility and its confidence interval with the result of hierarchical fragility assessment based on maximum likelihood estimation (see previous Sect. 3.5) labelled as Dj-MLE. The fragility curves are shown with similar colours (and darker intensity) and with the same line type (and half of the thickness) of the corresponding robust fragility curves. The first observation is that the results of MLE-based fragilities and the BMCS-based fragilities are quite close in all damage thresholds (as expected, see Jalayer and Erahimian, 2020). Moreover, the BMCS provides also the confidence bands for the
fragility curves, which cannot be directly provided by the MLE method. To
showcase an individual fragility curve, Figs. 2c to 6c illustrate the empirical fragility curves associated with the lth realisation of the vector of model parameters θk,l for model class Mk (where l is defined on each figure separately), i.e. P(D≥DjIM,θk,l) (see Sect. 2.5). Figures 2d to 6d illustrate the robust fragility curve
associated with the ultimate damage threshold Dindex(N), together with all the sample fragilities. The intensity values for which the damage level is not exceeded are shown with blue circles having the probability equal to zero. Other IMs that lead to the exceedance of the damage level are shown with red circles with a probability equal to one. Figures 2d to 6d also illustrate all the fragility parameters described in Sect. 2.5 including the equivalent lognormal parameters ηIMC and βIMC; the epistemic uncertainty in the empirical fragility assessment βUF; and also the intensities IMC16, IMC84, IM84, and
IM16 (the latter two are IMs at the median, i.e. 50 % probability, from the RF ±1 standard deviation, respectively. For all five building classes considered (see Table 3), ηIMC, βIMC, and βUF are tabulated in Table 5 for all damage thresholds associated to model classes M1 to M3.
Building class 1 (brick masonry residential, 1 storey) of the South Pacific 2009 tsunami considering fragility model class M3(a) hierarchical robust fragility curves and their ±1 standard
deviation confidence intervals; (b) comparison between hierarchical robust fragility curves and their confidence band (based on BMCS method) and
fragility assessment based on MLE method; (c) the fragility curves PD≥DjIM,θ3,1000, where 1≤j≤5 associated with the 1000th realisation of the model parameters, θ3,1000 (k=3 associated to model M3, l=1000); and (d) RF associated with the damage threshold D5, together with all the sample fragilities and the equivalent lognormal fragility parameters.
Building class 2 (timber residential) of the South Pacific 2009 tsunami considering fragility model class M3(a) hierarchical
robust fragility curves and their ±1 standard deviation confidence
intervals; (b) comparison between hierarchical robust fragility curves and their confidence band (based on BMCS method) and fragility assessment (based on MLE method); (c) the fragility curves P(D≥DjIM,θ3,800) where 3≤j≤5 associated with the 800th realisation of the model parameters, θ3,800 (k=3 associated to model M3, l=800); and (d) robust fragility associated with the damage threshold D5, together with all the sample fragilities and the equivalent lognormal fragility parameters.
Building class 1 (non-engineered masonry, unreinforced
with clay brick, 1 storey) of the Sulawesi–Palu 2018 tsunami considering
fragility model class M1(a) robust fragility curves (RF) and their ±1 standard deviation confidence intervals; (b) comparison between hierarchical robust fragility and its confidence band (based on BMCS method) and fragility assessment based on MLE method; (c) the fragility curves P(D≥DjIM,θ1,10), where 1≤j≤2 associated with the 10th realisation of the model parameters, θ1,10 (k=1 associated to model M1, l=10); and (d) robust fragility curve associated with the damage threshold D2, together with all the sample fragilities and the equivalent lognormal fragility parameters.
Building class 2 (non-engineered masonry, unreinforced with clay brick, 2 storey) of the Sulawesi–Palu 2018 tsunami considering fragility model class M3(a) hierarchical robust fragility curves and
their ±1 standard deviation confidence intervals; (b) comparison between robust fragility and its confidence band (based on BMCS method) and fragility assessment based on MLE method; (c) the fragility curves P(D≥DjIM,θ3,1850), where 1≤j≤2 associated with the 1850th realisation of the model parameters, θ3,1850 (k=3 associated to model M3, l=1850); and (d) robust fragility curve associated with the damage threshold D2, together with all the sample fragilities and the equivalent lognormal fragility parameters.
Building class 3 (non-engineered light timber) of the Sulawesi–Palu 2018 tsunami considering fragility model class M2(a) hierarchical robust fragility curves and their ±1 standard
deviation confidence intervals; (b) comparison between robust fragility and its confidence band (based on BMCS method) and fragility assessment based on MLE method; (c) the fragility curves P(D≥DjIM,θ2,120), where 2≤j≤3 associated with the 120th realisation of the model parameters, θ2,120 (k=2 associated to model M2, l=120); and (d) robust fragility curve associated with the damage threshold D3, together with all the sample fragilities and the equivalent lognormal fragility parameters.
The equivalent lognormal parameters and the epistemic
uncertainty in the RF assessment for all the building classes, damage
thresholds, and model classes M1 to M3.
Building classDamage levelModel 1 (M1) Model 2 (M2) Model 3 (M3) ηIMCβIMCβUFηIMCβIMCβUFηIMCβIMCβUF(m)(m)(m)South Pacific tsunami20091D10.290.400.200.300.460.220.330.510.22D20.430.350.150.460.370.150.500.400.15D31.280.350.081.280.350.071.370.370.07D41.800.450.071.810.420.071.890.370.06D52.490.470.072.480.470.072.500.350.062D30.641.080.580.631.200.640.631.260.53D40.731.010.480.751.010.460.840.980.35D51.520.250.081.540.260.081.610.230.07Sulawesi–Palutsunami 20181D10.250.390.150.260.430.150.270.560.18D21.240.570.061.240.590.071.310.580.062D10.390.440.200.380.450.190.430.430.17D21.600.310.111.590.320.121.650.280.113D20.741.010.340.711.030.380.810.850.29D31.160.980.391.201.100.431.290.950.31Model selection
With reference to Eq. (12), the log-evidencelnp(D|Mk) can be estimated by subtracting Term 2 from Term 1. Term 1 denotes the posterior mean of the log likelihood, and Term 2 is the relative entropy between the prior and the posterior. Within the BMCS method, these two terms are readily computable.
Given the samples generated from the joint posterior PDFs θk, Term 1 (the average data fit) can be seen as the expected value of the log likelihood over the vector of fragility parameters θk given the model Mk, i.e.
EθkD,Mklnp(D|Mk). Term 2
(the relative entropy) is calculated as the expected value of information gain or entropy between the two PDFs posterior and prior over the vector θ given the model Mk, i.e. EθkD,Mklnp(θk|D,Mk)p(θk|Mk). It is noted that based on Jensen's inequality, the mean information gain (relative entropy) of the posterior compared to the prior is always non-negative (see e.g. Jalayer et al., 2012; Ebrahimian and Jalayer, 2021). Hence, Term 2 should always be positive. Herein, p(θk|D,Mk) is constructed by an adaptive kernel density function (see Eq. E5, Appendix E) as the weighted sum (average) of Gaussian PDFs centred among the samples θk given model Mk (k=1:3). The prior pθk|Mk is a multivariate normal PDF,
respectively, with the mean and covariance described previously for each
model (see Eq. D1 in Appendix D). Table 6 shows the results for model
class selection for all five building classes considered. The last column
illustrates the posterior probability (weight) of the model P(Mk|D) according to Eq. (10) assuming that the prior P(Mk)=13 (where
k=1:3). The best model for each building class is shown in bold.
For instance, for Class 1 (masonry residential) for the South Pacific tsunami,
model class M3 (using a complementary log–log “cloglog”
transformation of πij to the linear logarithmic space, see Eq. 5) is preferred, since it has an overall larger difference between data fit and mean information gain, which leads to a higher log-evidence. The
posterior weights (last column of Table 6, see also Eq. 10) of 6 %,
11 % and 83 % are stabilised through different runs of the BMCS method. It should be noted that in Figs. 2 to 6 we reported directly the fragility results for the “best” fragility model
class (i.e. the one that maximises log evidence) identified based on the
procedure described here.
Bayesian model class selection results for empirical
fragility models. The best model for each building class is shown in bold.
BuildingModelTerm 1: AverageTerm 2: InformationLog-evidencePosterior probabilityclassclassdata fitgainof the modelSouth Pacifictsunami 20091M1-124.456123.6853-148.14140.055M2-123.465923.9566-147.42240.113M3-120.645424.7810-145.42640.8322M1-20.47919.7549-30.23400.315M2-19.910610.4660-30.37660.273M3-19.756510.2117-29.96820.411Sulawesi–Palu tsunami20181M1-161.95659.5690-171.52550.445M2-161.232010.5660-171.79790.339M3-161.882110.3673-172.24940.2162M1-23.36966.8292-30.19870.213M2-22.74297.2697-30.01260.257M3-22.43076.8551-29.28580.5313M1-15.80344.2741-20.07750.210M2-15.15753.9226-19.08020.570M3-14.62945.4015-20.03090.220The “basic” (MLE-<italic>Basic</italic>) method: fitting data to one damage state at a time
In the basic method (see Sect. 2.2), the fragility P(D≥DjIM) is obtained by using a generalised linear
regression model according to Eq. (5) with logit, probit or
cloglog link function fitted to the damage data (Mk where k=1:3). With reference to the MLE method described previously, the vector
xj herein is the IM associated with all damage data (and not partial, as in the hierarchical fragility method described in Sect. 3.3), and yj is the column vector of one-to-one probability assignment to the IM data in xj, with zero (=0) assigned to those data with an observed damage threshold D<Dj and one (=1) to those with D≥Dj. Thus, for the empirical fragility associated with the damage threshold Dj, and based on the model Mk, there are two model parameters to be defined, namely θMLE-Basic=α0,α1k. As noted previously, there might be conditions (depending on the quantity of the observed damage data) where a part of the fragility of damage threshold Dj lies below the fragility of the higher damage level Dj+1, indicating that P(DSjIM)<0. This is due to the fact that in the traditional method there is no explicit requirement to satisfy P(DSjIM)>0 as compared to the proposed method. The MLE of model parameters α0,α1 for the damage levels Dj, j=index2:N associated with the building classes in Table 3 are presented in Table 7.
The Model parameters θMLE-Basic.
Building classModel classD≥D1D≥D2D≥D3D≥D4D≥D5α0α1α0α1α0α1α0α1α0α1South Pacifictsunami 20091M15.2424.1903.6554.556-1.2214.884-2.6664.213-4.2714.652M22.7422.1901.9462.486-0.6952.846-1.5062.425-2.2932.515M32.0792.0111.3472.361-1.3193.139-2.3893.009-3.9193.8062M11.1271.5120.8131.392-2.9316.055M20.6570.9090.4720.873-1.7473.597M30.2510.8620.0510.914-2.4163.922South Pacific tsunami 20091M16.0594.355-0.6382.948M23.2642.340-0.3751.728M32.4982.088-0.9182.0882M13.5563.672-2.5005.160M22.0772.144-1.4653.039M31.6642.239-2.4734.2683M10.4661.375-0.5011.843M20.2950.847-0.2861.141M3-0.0411.068-0.7871.565
Figure 7 compares the fragility assessment obtained based on MLE-based
hierarchical fragility modelling (see also the MLE-based curves in Figs. 2b
to 6b) with the result of the fragility assessment by employing the
MLE-Basic method for the “best” model class Mk(k∈1,2,3) identified according to the procedure outlined in the previous section. It is noted that the fragility functional form is
different between the two methods. MLE-based fragility assessment given
Mk uses Eqs. (7) to (9) to construct a hierarchical fragility curve given that the conditional fragility term πij=P(D≥Dj+1D≥Dj,IMi), j=index1:N-1 has one of the functional forms in Eq. (5). However, the fragility assessment using the
MLE-Basic method employs directly one of the expressions in Eq. (5)
(corresponding to Mk, k=1:3) to derive the fragility curve πij=P(D≥DjIMi) (based on the whole damage data) and j=index(2:N). This difference manifests itself in Fig. 7a (for brick masonry residential, Class 1, South Pacific tsunami) as the deviation between the two fragility models for higher damage thresholds D4 and D5. The deviations between the fragility curves are particularly noticeable at higher IM values (with exceedance probability >50 %); however, their medians are quite similar. In Fig. 7b for (timber residential buildings, Class 2, South Pacific tsunami) and 7e (light informal timber buildings, Class 3, Sulawesi–Palu tsunami), we can observe that fragility curves intersect in the case of the MLE-Basic fragility assessment. However, they do not intersect for hierarchical fragility curves. The intersection points of the consequent damage states Dj+1 with Dj when using the MLE-Basic fragility estimation method are shown with colour stars on each figure.
Comparison between the fragility assessment by
MLE-based hierarchical fragility modelling and MLE-Basic fragility assessment given (a) South Pacific 2009 tsunami, Class 1, M3; (b) South Pacific 2009 tsunami, Class 2, M3; (c) Sulawesi–Palu 2018 tsunami, Class 1,
M1; (d) Sulawesi–Palu 2018 tsunami Class 2, M3; and (e) Sulawesi–Palu 2018 tsunami Class 3, M2.
Table 8 reports the fragility assessment parameters of the MLE and
MLE-Basic methods for the damage thresholds Dindex(2) to
Dindex(N) with the equivalent lognormal parameters ηIMC and βIMC (explained in Sect. 2.5) for M1 to M3 for all five classes considered. The medians are almost identical among the four models while there are higher dispersion estimates for the MLE method derived by hierarchical fragility modelling.
Comparison between fragility assessment based on MLE method (by hierarchical fragility modelling) and the MLE-Basic method for damage
thresholds Dindex(2) to Dindex(N).
The results outlined in this section show a fragility assessment for two different data sets corresponding to observed damaged in the aftermath of South Pacific and Sulawesi–Palu tsunami events. We have demonstrated the versatility of the proposed workflow and tool for hierarchical fragility assessment both for cases in which a large number of data points are available (e.g. Class 1, brick masonry residential, South
Pacific tsunami; Class 1, one-storey non-engineered masonry, Sulawesi–Palu
tsunami) and cases where very few data points are available (e.g. Class 2,
timber residential, South Pacific tsunami; Class 3, non-engineered light
timber, Sulawesi–Palu tsunami). Moreover, we demonstrated how the proposed
workflow avoids crossing fragility curves (e.g. Class 2, timber residential, South Pacific tsunami; Class 3, non-engineered light timber, Sulawesi–Palu tsunami). The results illustrated for the five building classes demonstrate that the proposed workflow for hierarchical fragility assessment can be applied in cases in which data points are not available for all the damage levels within the damage scale.
Conclusion
An integrated procedure based on Bayesian model class selection (BMCS) for
empirical hierarchical fragility modelling for a class of buildings or
infrastructure is presented. This procedure is applicable to fragility
modelling for any type of hazard as long as the damage scale consists of
mutually exclusive and collectively exhaustive (MECE) damage states, and the
observed damage data points are independent. This simulation-based procedure
can (1) perform hierarchical fragility modelling for MECE damage states, (2) estimate the confidence interval for the resulting fragility curves, and (3) select the simplest model that fits the data best (i.e. maximises log
evidence) amongst a suite of candidate fragility models (herein, alternative
link functions for generalised linear regression are considered). The
proposed procedure is demonstrated for empirical fragility assessment based
on observed damage data to masonry residential (1 storey) and timber
residential buildings due to the 2009 South Pacific tsunami in American
Samoa and the Samoan Islands and non-engineered masonry buildings (1 and 2 storeys) and non-engineered light timber buildings due to the 2018
Sulawesi–Palu tsunami. It is observed that
For each model class, the same set of simulation realisations is used to estimate the fragility parameters, the confidence band and the log evidence. The latter, which consists of two terms depicting the goodness of fit and the information gain between posterior distribution resulting from the observed data and the prior distribution, is used to compare the candidate fragility models to identify the model that maximises the evidence.
Hierarchical fragility assessment can be done also based the maximum likelihood estimation (MLE) and the available statistical toolboxes (e.g. MATLAB's generalised linear model). For each damage level, the reference domain should be the subset of data that exceeds the consecutive lower damage level, instead of taking the entire set of data points as reference domain. Note that the basic fragility estimation (“MLE-Basic”, non-hierarchical fragility model) fits the damage data for each damage level at a time. In other words, the reference domain is set to all damage data.
The procedure is also applicable to cases in which observed data are available only for a subset of the damage levels within the damage scale. The number of fragility curves is going to be equal to the total number of damage levels for which data are available minus one. This means, in order to have at least one fragility curve, one needs to have data available for at least two damage levels.
Although the resulting fragility curves are not lognormal (strictly speaking), equivalent statistics work quite well in showing the fragility curves (median and logarithmic dispersion) and the corresponding epistemic uncertainty (logarithmic dispersion).
The proposed BMCS method and the one based on MLE lead to essentially the same set of parameter estimates for hierarchical fragility estimation. However, the latter does not readily lead to the confidence band and log evidence.
Using the basic method for fragility estimation (MLE-Basic) leads to results that are slightly different from the hierarchical fragility curves. The difference grows for higher damage levels. It is important to note that following the basic method MLE-Basic led to ill-conditioned results (i.e. fragility curves crossing) in some of the cases (Class 2 for the South Pacific tsunami and Class 3 for the Sulawesi–Palu tsunami, both Timber constructions) studied in this work.
The major improvement offered by this method is in providing a tool that can
fit fragility curves to a set of hierarchical levels of damage or loss in an
ensemble manner. This method, which starts from prescribed fragility models
and explicitly ensures the hierarchical relation between the damage levels,
is very robust in cases where few data points are available and/or where
data are missing for some of the damage levels. This tool provides confidence
bands for the fragility curves and performs model selection among a set of
viable link functions for generalised regression. It is important to note that the
proposed method is in general applicable to hierarchical vulnerability
modelling for human or economic loss levels and to different types of
hazards if (1) the defined levels are mutually exclusive and collectively
exhaustive, and (2) a suitable intensity measure (IM) can be identified.
The derivation of Eq. (2)
The probability of being in damage state DSj for a given intensity measure IM can be estimated as follows:
PDSj|IM=P(D≥Dj)⋅(D<Dj+1)|IM=1-P(D≥Dj)⋅(D<Dj+1)‾|IM=1-P(D≥Dj)‾+(D<Dj+1)‾|IM=1-P(D<Dj)+(D≥Dj+1)|IM︸ME≜P(D<Dj|IM)+P(D≥Dj+1|IM)=1-P(D<Dj|IM)-P(D≥Dj+1|IM)=P(D≥Dj|IM)-P(D≥Dj+1|IM),
where the upper-bar sign stands for the logical negation and is read as
“NOT”, and (+) defines the logical sum and is read as “OR”. The above
derivation is based on the rule of sum in probability and considers the fact that the two statements D<Dj and D≥Dj+1 are mutually exclusive (ME); thus, the probability of their logical sum is the sum of their probabilities.
The derivation of Eq. (7)
The probability of being in damage state DSj (where j≥1) given the intensity measure evaluated at the location of building i, denoted as IMi, based on Eq. (6) can be expanded in a recursive format as follows:
P(DSj|IMi)=1-P(D≥Dj+1|D≥Dj,IMi)︸1-πij⋅1-P(D<Dj|IMi)=1-πij⋅[1-P((D<Dj)⋅D≥Dj-1+⋯+(D<D1)⋅(D≥D0)|IMi)]=(1-πij)⋅1-∑k=0j-1P((D<Dk+1)⋅(D≥Dk)|IMi)=(1-πij)⋅1-∑k=0j-1P(DSk|IM),
where (+) defines the logical sum and is read as “OR”. The above
derivation is based on the rule of sum in probability and considers the
fact that the recursive statements in the second term expressed generally as
(D<Dk+1)⋅(D≥Dk), where 0≤k≤j-1, are ME; hence, the probability of their logical sum is the sum of their probabilities. It is important to note that in case where j=0, the above equation can be written as
P(DS0|IMi)=1-πi0≜P(D<D1|IMi).
The derivation of log-evidence in Eq. (13)
From an information-based point of view, the logarithm of the evidence
(log-evidence), denoted as lnp(D|Mk), can provide a quantitative measure of the amount of information as evidence of model Mk. Moreover, the posterior PDF p(θk|D,Mk) (see Eq. 14) over the domain of the model parameters Ωθ given the
kth model is equal to unity. Thus, lnp(D|Mk) can be written as follows:
lnpD|Mk=lnpD|Mk⋅∫Ωθkpθk|D,Mkdθk︸=1.0.
Since the log-evidence is independent of θ, we can bring it inside the integral and do some simple manipulation (also using
the relation in Eq. 11) as follows:
lnpD|Mk=∫ΩθklnpD|Mk⋅pθk|D,Mkdθk=∫ΩθlnpD|θk,Mkpθk|Mkpθk|D,Mk⋅pθk|D,Mkdθk=∫ΩθlnpD|θk,Mkpθk|D,Mkpθk|Mk⋅pθk|D,Mkdθk=∫ΩθlnpD|θk,Mk⋅pθk|D,Mkdθk︸Term1-∫Ωθlnpθk|D,Mkpθk|Mk⋅pθkD,Mkdθk︸Term2.
Multivariate normal distribution and generating dependent
Gaussian variables
Let us assume that the vector of parameters for the kth model is set to
θ; i.e. θ=θk. A multivariate normal PDF can be expressed as follows:
p(θ)=1(2π)nSexp-12θ-μθTS-1θ-μθ,
where n is the number of components (uncertain parameters) of vector
θ=θi,i=1:n;
μθ is the vector of the mean value of θ; and S is the covariance matrix. The positive definite matrix Sn×n can be factorised based on Cholesky decomposition as S=LLT, where Ln×n is a lower triangular matrix (i.e. for all j>i, Lij=0 where Lij denotes the (i, j)-entry of the matrix L). A Gaussian vector θn×1 with mean μθ and covariance S can be generated as follows:
θ=μθ+LZ,
where Zn×1 is a vector of standard Gaussian i.i.d. random variables with zero mean 0n×n and a covariance equal to the identity matrix In×n. To verify the properties of θ, we know that with reference to Eq. (D2), it should have a mean equal to μθ and a covariance matrix equal to S. The expectation of θ, denoted as E(θ), can be estimated as
E(θ)=Eμθ+LZ=E(μθ)+LE(Z)︸=0n×1=μθ.
The covariance matrix of θ can be written as
Eθ-μθθ-μθT=ELZZTLT=LEZZT︸=In×nLT=LLT=S.
Thus, the vector θ can be written according to Eq. (D2).
Adaptive MCMC schemeMCMC procedure
The MCMC simulation scheme has a Markovian nature where the transition from the current state to a new state is done by using a conditional transition function that is conditioned on the current (last) state. Let us assume that the vector of parameters for the kth model is set to θ; i.e. θ=θk. To generate (i+1)th sample θi+1 from the current ith sample θi based on MH routine, the following procedure is adopted herein
Simulate a candidate sample θ* from a proposal distribution q(θ|θi). It is important to note that there are no specific restrictions about the choice of q(⋅) apart from the fact that it should be possible to calculate both q(θi+1|θi) and q(θi|θi+1).
Calculate the acceptance probability min(1,r), where r is defined as follows (it is to note that the following Eq. (E1) is written in the general format for brevity compared to Eq. (14) of the paper, and we have used θ instead of θk and dropped the conditioning on Mk; hence, when we write the ith sample θi it is actually the ith sample drawn from θk and “k” is dropped for brevity):r=pθ*|Dpθi|D⋅qθi|θ*qθ*|θi=pD|θ*pD|θi︸likelihoodratio⋅pθ*pθi︸priorratio⋅qθi|θ*qθ*|θi︸proposalratio.
Generate u from a uniform distribution between (0,1), u∼ uniform (0,1).
If u≤min(1,r)→ set θi+1=θ* (accept the candidate state to be taken as the next state of the Markov chain); else set θi+1=θi (the current state is taken as the next state).
Estimating the likelihood in the arithmetic scale based on Eq. (E1) may
encounter instability as p(D|θ) may become very small; thus, the likelihood ratio becomes indeterminate. To avoid this numerical instability, it is desirable to substitute the likelihood ratio in Eq. (E1) with explnpD|θ*-lnpD|θi if the ratio becomes indeterminate or zero.
With reference to Eq. (E1), samples from the posterior can be drawn
based on MH algorithm without any need to define the normalising C-1
coefficient according to Eq. (14). Equation (E1) always accepts a
candidate if the new proposal is more likely under the target posterior
distribution than the old state. Therefore, the sampler will move towards
the regions of the state space where the target posterior function has high
density.
The choice of the proposal distribution q is very important. The ratio
q(θi|θ*)/q(θ*|θi) corrects for any asymmetries in the proposal distribution. Intuitively, if q(θ*|θi)=p(θ*|D), the candidate state is always accepted (with r=1); thus, the closer q is to the target posterior PDF, the better the acceptance rate and the faster the convergence. This is not a trivial task as information about the important region p(θ|D) is not available. If the proposal distribution q is non-adaptive, it means that the information of the current sample θi is not used to explore the important region of the target posterior distribution p(θ|D); thus, we can say that q(θ*|θi)=q(θ*). Therefore, it is more desirable to choose an adaptive proposal distribution which depends on the current sample (Beck and Au, 2002). Having the proposal PDF q centred around
the current sample renders the MH algorithm like a local random walk that
adaptively leads to the generation of the target PDF. However, if the Markov
chain starts from a point that is not close to the region of the significant
probability density of p(θ|D), the chance of generating a candidate state θ* will become extremely small (and we will face high rejection of candidate samples). Therefore, most of the samples will be repeated. To solve this problem, Beck and Au (2002) introduce a sequence of PDFs that bridge the gap between the
prior PDF and the target posterior PDF. This issue will be further explored
hereafter under the adaptive MCMC. Finally, it can mathematically be shown
that (see Beck and Au, 2002) if the current sample θi
is distributed as p(⋅|D), the next sample θi+1 is also distributed as p(⋅|D).
The adaptive MH algorithm (Beck and Au, 2002) introduces a sequence of
intermediate candidate evolutionary PDF's that resemble more and more the
target PDF. Let p1,p2,…,pNchain be the sequence (chain) of PDFs leading to p(θ|D)=pNchain, where Nchain is the number
of chains and each chain contains Nd samples (as indicated
subsequently). The following adaptive simulation-based procedure is
employed:
Step 1. Simulate Nd samples θ1,θ2,…,θNd(1), where the superscript (1) denotes the first simulation level or the first chain (nc =1, where nc denotes the chain number/simulation level), with the target PDF p1 as the first sequence of samples. Instead of accepting or rejecting a proposal for θ involving all its components simultaneously (called block-wise updating scheme), it might be computationally simpler and more efficient for the first chain to make proposals for individual components of θ one at a time (called component-wise updating approach). In the block-wise updating, the proposal distribution has the same dimension as the target distribution. For instance, if the model parameters involve n uncertain parameters (e.g. the vector of model parameters θn×1 in this paper has n=2(N-1) variables for each of the three models M1, M2, and M3), we design an n-dimensional proposal
distribution and either accept or reject the candidate state (with all n
variables) as a block. The block-wise updating approach can be associated
with high rejection rates. This may cause problem when we want to generate
the first sequence of samples (first chain). Therefore, we have utilised the
more stable component-wise updating for the first chain. We start from the
first variable and generate a candidate state based on a proposal
distribution for this individual component and finally accept or reject it
based on MH algorithm. Note that in this stage we have varied the current
component and kept the other variables in vector θ constant. Then, we move to the next components one by one and do the same
procedure while taking into account the previous (updated) components.
Therefore, what happens in the current step is conditional on the updated
parameters in the previous steps.
Step 2. Construct a kernel density function κ(1) as
the weighted sum (average) of n-dimensional Gaussian PDFs centred among the samples θ1,θ2,…,θNd(1), with the covariance matrix S(1) of the samples θi(1) and the weights associated to each sample as wi, where i=1:Nd as follows (see Ang et al., 1992; Au and Beck, 1999):
κ(1)(θ)=1Nseed∑i=1Nseed1win(2π)nS(1)⋅exp-12wi2θ-θi(1)TS(1)-1θ-θi(1).
The kernel density κ(1) constructed in Eq. (E2) approximates p1. The kernel function κ can be viewed as a PDF consisting of bumps at θi, where width wi controls the common size of the bumps. Therefore, a large value of wi tends to over-smooth the kernel density, while a small value may cause noise-shaped bumps. In view of this, wi can be assumed to have a fixed width (=w), or alternatively the adaptive kernel estimate can be employed (Ang et al., 1992; Au and
Beck, 1999) that is defined for each sample θi,
i=1:Nd. The adaptive kernel has better convergence and smoothing
properties over the fixed-width kernel estimate. The fixed width w is
estimated as follows (Epanechnikov, 1969):
w=4(n+2)Ndist1n+4,
where Ndist is the number of samples (Ndist≤Nd). For one-dimensional problems (n=1), this leads to the well-known fixed-width value of (4/3)Nseed1/5. In the adaptive kernel method, the
idea is to vary the shape of each bump so that a larger width (flatter bump)
is used in regions of lower probability density. Following the general
strategy used in the past (see Ang et al., 1992; Au and Beck, 1999), the
adaptive band width wi for the ith sample θi can be written as wi=wλi, where the local bandwidth factor λi can be estimated as follows:
λi=κ(θi)∏j=1Nseedκ(θj)1Nseed-ω,
where 0≤ω≤1.0 is the sensitivity factor, and κ(θi) is calculated based on Eq. (E2) where
θ=θi, with the choice of fixed-width w in Eq. (E3). The denominator in Eq. (E4) is a geometric mean of the kernel estimator at all Nd points. The value of ω=0.50 is employed herein as also suggested by other research endeavours (Abramson, 1982; Ang et al., 1992; Au and Beck, 1999). It is numerically more stable to estimate the denominator in Eq. (E4) as ∏j=1Nseedκ(θj)1/Nseed.
Step 3. Simulate Nd Markov chain samples θ1,θ2,…,θNd(2) with the target PDF p2 as the second simulation level (nc =2). We use κ(1) as the proposal distribution q(⋅) in Eq. (E1) in this stage to generate the second chain of samples. To generally simulate sample θ from the kernel κ(nc) (where nc =1:Nchain), we generate a discrete random index from the vector [1,2,…,Nd] with the corresponding weights w1,w2,⋯,wNseed using an inverse
transformation sampling; if index =j, then generate θ from
the Gaussian PDF κj, where
κj(θ)=1(wλj)n(2π)nS(nc)⋅exp-12(wλj)2(θ-θj)TS(nc)-1(θ-θj)=1(2π)nSj(nc)⋅exp-12(θ-θj)TSj(nc)-1θ-θj,
where Sj(nc)=wj2S(nc),
where S(nc) is the covariance matrix of the samples
θ1,θ2,…,θNd(nc). Appendix D shows how a sample θ can be drawn from the Gaussian PDF κi. From this sequence on, the MCMC updating is done in a block-wise manner as we generate a candidate θ and accept/reject it as a block. The second chain of samples θ1,θ2,…,θNd(2) are then used to construct the kernel density κ(2) based on Eq. (E2).
Step 4. In general, κ(nc) is used as the proposal distribution in order to move from the ncth simulation level
(which approximates pnc) into (nc+1)th chain (with target PDF pnc+1). This will continue until the Nchainth simulation level where Markov chain samples are simulated for the target updated p(θ|D)=pNchain.
MCMC samples for each model
The adaptive MCMC procedure for drawing samples from the model parameters
from the joint posterior PDF p(θk|DMk) is carried out by considering Nchain=6 chains (simulation levels) and Nd=2000 samples per each chain (see Appendix E). In the first simulation level (first chain, nc=1), for which a component-wise updating approach is employed (see
Appendix E, Step 1 for the description of component-wise and block-wise updating), the first 20 samples are not considered in order to reduce the initial transient effect of the Markov chain. The proposal distribution (see Eq. E1) for each component is assumed to be a normal distribution with a coefficient of variation COV =0.30 herein. In addition, the prior ratio according to Eq. (E1) will become the ratio of two normal distributions for each component one at a time. In the next simulation levels (i.e. nc=2:6), the adaptive kernel estimate (Eq. E2) is employed; i.e. the MCMC updating is performed in a block-wise manner. There will be Nd Markov chain samples generated within the each chain, denoted as θk,1,θk,2,⋯,θk,Nd(nc), where nc=2:Nchain(=6). The Nd samples of the last chain (nc=6) will be used as the fragility model parameters, as discussed in Sect. 3.6. It is important to note that the likelihood p(D|θk,Mk) (used in calculating the acceptance probability within the MCMC procedure in Eq. E1) is estimated according to Eq. (13).
Figure F1 illustrates the histograms representing the drawn samples from the
joint posterior PDFs corresponding to the sampled model parameters θk,1,θk,2,⋯,θk,Nd(6) corresponding to the brick masonry residential, Class 1 South Pacific tsunami classes Mk (k∈1,2,3) shown
in Fig. 2. The marginal normal prior PDFs are also shown with
dashed orange-coloured lines. The statistics of the samples, mean and
confidence interval (CI) between 2nd and 98th percentiles for the posterior of model parameters θk are shown on the figures associated to each parameter. It is expected to have the mean values of the marginal posterior samples close to and comparable with those obtained by the MLE in Table 4.
Distribution of the fragility model parameters θk=α0,j,α1,jk,j=0:4 based on model class Mk (where k=1:3) for Class 1, brick masonry residential, South Pacific tsunami, by employing an adaptive MCMC procedure including samples drawn from the joint posterior PDF with their statistics (mean and COV) and the marginal normal priors (subfigures show the posterior statistics).
Code and data availability
The code implementing the methodology and data used to produce the results in this article are available at the following URL:
https://github.com/eurotsunamirisk/computeFrag/tree/main/Code_ver2 (last access: 30 November 2022; Ebrahimian and Jalayer, 2022).
Author contributions
FJ designed and coordinated this research. HE performed the simulations and developed the fragility functions. KT cured the availability of codes and software on the European Tsunami Risk Service (ETRiS). BB provided precious insights on the damage data gathered for American Samoa and the Samoan Islands in the aftermath of the 2009 South Pacific tsunami (documented in Reese et al., 2011). All authors have contributed to the drafting of the paper. The first two authors contributed in an equal manner to the drafting of the paper.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Special issue statement
This article is part of the special issue “Tsunamis: from source processes to coastal hazard and warning”. It is not associated with a conference.
Acknowledgements
The authors would like to acknowledge partial support from Horizon Europe
Project Geo-INQUIRE. Geo-INQUIRE is funded by the European Commission under
project no. 101058518 within the HORIZON-INFRA-2021-SERV-01 call. The
authors are grateful to the anonymous reviewer and Carmine Galasso
for their insightful and constructive comments.
The authors gratefully acknowledge partial support by PRIN-2017 MATISSE (Methodologies for the AssessmenT of anthropogenic environmental hazard: Induced Seismicity by Sub-surface geo-resources Exploitation) project no. 20177EPPN2 funded by the Italian Ministry of Education and Research.
Financial support
This research has been partially supported by PRIN-2017 MATISSE (Methodologies for the AssessmenT of anthropogenic environmental hazard: Induced Seismicity by Sub-surface geo-resources Exploitation) project no. 20177EPPN2 funded by the Italian Ministry of Education and Research.
Review statement
This paper was edited by Animesh Gain and reviewed by Carmine Galasso and one anonymous referee.
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