Modeling Building Classes Using Moment Matching

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Modeling Building Classes Using Moment Matching

In Ho Cho,a)

M.EERI, and Keith Porter,b)

M.EERI

Large-scale earthquake risk assessment necessitates models of the seismic

performance of building classes. This work addresses how to depict a class

with only a few (index) buildings whose designs span the attributes that most

impact the seismic behavior of the class. We propose a general numerical moment

matching (MM) technique to represent those seismic attributes of index

buildings, which can then be individually analyzed by second-generation

performance-based earthquake engineering methods (PBEE-2). The results prob-

abilistically combined to model the class behavior as a whole. Thereby, the modelhonors the joint distribution of variable features within the class and propagates

all other uncertainties that PBEE-2 already recognizes. Importantly, we can

reflect and propagate with rigor the uncertain attributes of buildings within a

class, notably without resorting to standard distributions. The MM enables

PBEE-2 to rigorously capture seismic risk assessment of a building class.

[DOI: 10.1193/071712EQS239M]

INTRODUCTION

Probabilistic risk assessments of building portfolios subjected to natural hazards employ

models of the performance of asset classes, models such as vulnerability functions, fragility

functions, etc. (Asset classes are also called structure types, building types, facility types, and

perhaps others). Asset class definitions follow no uniform rule. For example, Wesson et al.

(2004) address the seismic performance of all single-family dwellings in Southern California

in the 1994 Northridge earthquake. Depending on how one counts them, HAZUS-MH uses

between 190 and 5,300 classes ( NIBS and FEMA 2003). The Global Earthquake Model

offers an innovative genetic taxonomy (Scawthorn 2013) that can classify buildings in vir-

tually unlimited ways. Each of these depicts buildings as undifferentiated specimens of

(potentially very small) classes.

Some authors create models of the fragility or vulnerability of asset classes from

analytical models of individual specimens. Depicting and propagating uncertainty in the

asset-class definition can be an important issue, potentially introducing as much performance

uncertainty as record-to-record variability or component damageability. A probabilistic

model of the asset class should reflect the variability of the seismically important features

of the class (i.e., those features that vary between assets in the class and whose variability

contributes significantly to the overall performance uncertainty). Some features are

a) Iowa State University, 326 Town, Ames, IA 50011 b) University of Colorado, 428 UCB, Boulder, CO 80309

Earthquake Spectra, Volume 32, No. 1, pages 285 – 301, February 2016; © 2016, Earthquake Engineering Research Institute

285



convenient to simulate by Monte Carlo methods (e.g., material properties) and some are not

(e.g., number and configuration of bays and stories in a frame).

We present a procedure to select important features, perform a survey, and (the bulk of the paper) design a small sample — a few buildings — whose variability matches the survey.

To limit the number of required index buildings, we propose a sampling procedure called

moment matching. It is related to quadrature and requires designing fairly few buildings:

2n þ 1, where n is the number of the most important seismic attributes, such as degree

of vertical and plan irregularity.

PAST EFFORTS TO MODEL THE CHARACTERISTICS OF A BUILDING CLASS

Previous approaches to characterize class performance often resort to expert opinion (e.g.

ATC-13 1985) or statistical analyses of earthquake experience data (e.g., Wesson et al.

2004). In analytical performance-evaluation approaches (Czarnecki 1973, Kustu et al.

1982, Kircher et al. 1997, Porter et al. 2001 and 2002, Rossetto and Elnashai 2005), indi-

vidual assets are analyzed. Where the procedure is supposed to model a class, the model must

somehow quantify and propagate variability of important building features within the class.

For example, if the class comprises some subset of low-rise buildings (one to three stories),

then the variability in height within the class must be reflected in the uncertainty in the per-

formance of the class.

One can use census data (e.g., HAZUS-MH) or remote sensing data (e.g., Prasad et al.

2009) to quantify variability in attributes, but the question remains how to relate that variability to uncertain structural parameters such as fundamental period. Rossetto and

Elnashai (2005) use engineering principles and data to set parameters, but for a practical

limit, their application is restricted to a few tractable parameters, such as material properties

rather than design diversity (e.g., number of bays, stories, or degree of vertical and plan

irregularity).

We seek to characterize the most important features of assets within the class as uncertain

and enable one to analyze specimens by second-generation performance-based earthquake

engineering (PBEE-2) [e.g., using FEMA P-58 (ATC 2012)]. PBEE-2 propagates uncertainty

in the seismic performance of a particular, unambiguous asset in each of the four major ana-lytical stages of PBEE-2 — hazard, structural, damage, and loss-analysis stage — but not in the

asset definition (Figure 1). How can one make the asset definition also uncertain?

One could randomly sample and analyze buildings within the class. Many have used

Monte Carlo simulation or Latin hypercubes (e.g., Borzi et al. 2008). Those methods

pose several challenges. First, creating PBEE-2 structural models is time consuming. Second,

analyzing each building can be computationally demanding. Also, the random sampling

method often resorts to simplified structural analyses, such as an equivalent oscillation

system with idealized macroscopic nonlinearity. Another problem is that underlying prob-

ability density functions (PDFs) of real-world seismic features may not resemble a conve-nient parametric distribution (see Figure 2 for an example data set that shall be discussed

later). How can one represent asset classes with arbitrarily distributed features in a robust

yet simple way?

286 I. H. CHO AND K. PORTER



PROPOSED APPLICATION OF MOMENT MATCHING

We suggest the following procedure to model uncertainty in the asset class: (1) select key

seismic features, (2) survey buildings in the class to quantify the joint distribution of those

features, (3) use MM to select sample points (attributes and weights of sample index

Height

Plan irregularity

Hazard

analysis

Structural

analysis

Damage

analysis

Loss

analysis

Asset

Figure 1. Illustration of PBEE-2 methodology augmented by a class-level asset definition. In the

top, four boxes represent key analytical stages. Street-level photos were obtained from authors’

field survey (aerial images from Google Earth).

Figure 2. Joint PDF plots of salient seismic features obtained from the field survey of 265 RCcommercial buildings in California. Both are far different from standard distributions.

MODELING BUILDING CLASSES USING MOMENT MATCHING 287



buildings) to match the first several moments of the survey data, and (4) select or design

index buildings with the desired attributes. Then, these can be analyzed by PBEE-2, although

the PBEE-2 analysis is not a part of the present work. The procedure is independent of the

type of underlying PDFs — one need not assume Gaussian distributions, for example.

APPLICATION TO REINFORCED CONCRETE COMMERCIAL

BUILDING CLASS IN CALIFORNIA

We applied the procedure to commercial reinforced concrete (RC) buildings in

California. The goal of this illustrative application is to better explain the general procedure.

SELECTION AND QUANTIFICATION OF KEY SEISMIC FEATURES

It would be difficult to objectively select the most important seismic features — the oneswith the largest effect on repair cost — that vary between buildings in a class. It might require

a large number of PBEE-2 analyses. Our approach is a meta-analysis: review a number of

well-received references (Table 1) and select the features that appear to be commonly con-

sidered important. From the references in Table 1, we infer that, conditioned on class (defined

here in terms of construction material and lateral force resisting system), the three features

commonly believed to have significant impact on seismic performance are vertical and plan

irregularities and building height. This particular selection does not limit the general

procedure. One could use more features (e.g., redundancy) or add features not listed here

(e.g., design base shear).

SURVEY BUILDINGS IN THE CLASS OF INTEREST

The second step is to survey assets and gather statistics for the important attributes. Rig-

orous sampling procedures were recently developed and tested for the Global Earthquake

Model that balance survey effort and accuracy (Porter 2013). Briefly, the sampling procedure

Table 1. Top three seismic features for commercial RC buildings along with other minor

factors

ASCE 7

(2010)

ConcreteCoalition

(2011)

ACI 318

(2005)

FEMA 154

(ATC 2002)

FEMA310

(1998)

FEMA273

(1997)

EQ Eng.Handbook

(2003)

Chopra &Goel

(2000)

Plan irreg.

Vertical irreg.

Building height

Redundancy,

# of bays

Adjacent

buildings

Opening on

side walls

Captive columns




calls for the analyst to randomly select sample city blocks in several neighborhoods with

common use and record the relevant features of the buildings on those blocks.

In our sample application, we conducted a field survey of commercial RC buildings inseveral California population centers: Los Angeles (population of 3.82 million as of 2011),

San Jose (967,000), and San Francisco (813,000). In addition, Pasadena (near Los Angeles,

2011 population of 138,000) was investigated as a proxy of smaller cities near a large one.

We focused on the downtown areas of the selected cities where old and new RC structures

can be found, thus (we hope) providing an unbiased distribution of the key features. We

emphasized all types of RC buildings lower than ∼40 stories in the chosen cities and obtained

265 samples with distinguishable key features. The full data set is presented in Cho and

Porter (2013a). With this real-world data we can now construct a few representative buildings

(denoted as index buildings hereafter), for which a robust and general mathematical tool, the

moment matching (MM) technique, will be adopted. In the following sections we expound onthe advantageous nature of MM and how to exploit the MM for arbitrary PDFs, without

resorting to standard distributions.

RECAP MOMENT MATCHING TECHNIQUE

Rosenblueth (1975) appears to be the first to outline an approach that samples over the

space of the joint distribution for the purpose of seismic risk management. He proposed two

samples per dimension in every combination, so the same size increases exponentially with

the number of features (i.e., 2n combinations of feature values for n features). Julier and

Uhlmann (2002) show how one can select a sample whose size increases linearly rather

than exponentially, and notably their method is not restricted to Gaussian distributions.The intuitive term “moment matching” was proposed by Ching et al. (2009), who applied

the reduced sigma point filter to seismic risk problems.

We depart from the work of Ching et al. (2009), but the differences and contributions of

our work should be noted. First, we suggest a general numerical tool that can tackle arbitrary

PDFs whereas they limited discussion to standard distributions. Second, our goal is to repre-

sent the diversity of a building class while theirs is efficient uncertainty propagation within

PBEE-2.

In MM, the continuous PDF of a random variable X (called X PDF hereafter) is replaced

by a discrete equivalent probability mass function (PMF) consisting of weighted delta func-

tions (here, delta function means the Dirac delta). The weights and positions of PMF are

selected such that the first few moments (mean, variance, skewness, and so on) of both

the discrete PMF and the original joint PDF are identical. In essence, the MM produces

point estimates to replace the continuous joint distribution (as done by Rosenblueth

1975). To better understand strengths of MM, it is instructive to review the underlying mathe-

matics (for details, see Ching et al. 2009).

The following discussion applies to a wide variety of problems, but the reader who wants

to think of it in the context of a loss-estimation problem can imagine a number of independent

variables X that describe the features of the assets in the class: their height, degree of planirregularity, degree of vertical irregularity, etc. We want to know Y, a vector of uncertain

measures of performance of the asset class, say repair cost, collapse probability, etc., at some

specific level of seismic excitation. The function Y ¼ gðXÞ gives that relationship; it might




be evaluated for a particular building specimen by PBEE-2. Now let us turn to the general

MM approach, which again is not limited to this particular application of X, Y, and gðXÞ.

We seek to estimate E ðYÞ, where Y ¼ gðXÞ, X and Y are n-dimensional random vari-ables, and E ½: is the expectation. Notably, only several moments of X are to be used for the

estimation, and the PDF of X is arbitrary for generality (not limited to Gaussian, etc.). Taking

a multivariate Taylor series expansion of gðXÞ around E ½X and then considering their expec-

tations lead to

EQ -TARGET;t em p:int r a lin k -;e1;41;548

E ½Y ¼ gð E ½XÞ |fflfflfflffl{zfflfflfflffl} 0th

þ 1

2! E

Xn

j ¼1

∂

∂ X j

ð X j E ½ X j Þ

gðXÞjX¼ E ðXÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 1st

þ 13! E Xn

j ¼1

∂∂ X j

ð X j E ½ X j Þ2gðXÞjX¼ E ðXÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

2nd

þ · · · ð1Þ

Consider the third term (i.e., second-order term) on the right-hand side (when n ¼ 2):

EQ-TARGET;t em p:int r al ink -;e2;41;429

1

3! E

ð X 1 E ½ X 1Þ2

∂2g

∂ X 21

þ ð X 2 E ½ X 2Þ2

∂2g

∂ X 22

þ 2ð X 1 E ½ X 1Þð X 2 E ½ X 2Þ ∂2g

∂ X 1∂ X 2X¼ E ðXÞ

(2)

As seen above, if we know E ½ X 2i (i.e., the moments of X PDF up to the second order),

then we can exactly estimate the E ½Y with the same second-order accuracy, and no higher

term is needed. Hence, with the first pth moments of X PDF we can obtain E ½Y of pth-order

accuracy and in the similar fashion the variance of Y (denoted by V ar ½Y) of b p∕2cth-order

accuracy (b:c means the integer part of the number). Also, in terms of computational effi-

ciency and accuracy, the MM exhibits superiority over the first-order second-moment

(FOSM), which holds emphasis on the mean and variance of the final loss ( Baker and Cornell

2003) as addressed by Ching et al. (2009).MM further manifests its power in collaboration with a point-estimation process (i.e.,

substituting weighted delta functions — point masses — for the continuous PDF of X).

With the originally continuous PDF of X, considering variability often necessitates compli-

cated numerical integration. With the equivalent discrete PMF (the weighted masses), we

merely need a weighted summation of the function of the point masses, which can be suc-

cinctly stated by

EQ-T ARGET;t em p:int r a link -;e 3;41;172 E ðYÞ ¼

ð PDF ðXÞgðXÞd X ≈

|{z} byMM Xq

i¼1 wigðχiÞ (3)

where i indicates ith delta function and q is the total number of delta functions, each of which

is represented by the location χi and weight wi.




Naturally, the next question would be how to select the delta functions that exactly match

the first several moments. As mentioned by Ching et al., for general cases we need to solve

the nonlinear system of equations that can be rewritten for an arbitrary variable X

in a suc-cinct form:

EQ-TARGET;t em p:int ra link -;e4;62;591

2666664

χ 11 χ

12 … χ

1q

χ 21 χ

22 … χ

2q

..

. . .. ..

.

χ p1 χ

p2 … χ

pq

3777775

ð pqÞ

8>>><>>>:

w1

w2

..

.

wq

9>>>=>>>;

¼

8>>><>>>:

E ½ X 1 E ½ X 2

..

.

E ½ X p

9>>>=>>>;

andXqi¼1

wi ¼ 1 (4)

Thus, we have ( p þ 1) equations and 2q unknowns (i.e., q delta functions each represented by

its location χ i and weight wi). Hence, using the q delta functions, the first p ¼ ð2q 1Þmoments of the X PDF can be exactly matched. When X is an n-dimensional “independent ”

uncertain variable (the variables in X are independent of each other), with each uncertain

entity X i (i.e., X i ∈ X; i ∈ ½1; n) we need to perform the aforementioned procedure to obtain

the delta functions in the n-dimensional X space (leading to n q delta functions).

As it turns out from our case study — California commercial RC buildings — the cor-

relation coefficients among the salient seismic features are small enough that we can safely

assume that the seismic features considered herein are virtually uncorrelated and appear to be

independent (Figure 3). As pointed out by Ching et al. (2009), a special remedy exists when

the parameters in X are not independent. We will not further discuss the correlated X here; theinterested reader is referred to Ching et al. (2009) or Julier and Uhlmann (2002) for further

detail.

Ching et al. (2009) suggested three ways to attain the delta functions, which completely

matches the first p ¼ ð2q 1Þ moments: (1) solving the nonlinear system of equations;

(2) adopting the Gauss-quadrature integration points and weights, when the X PDF is in

the similar form of a standard one in reference to Hildebrand (1956) or Abramowitz and

Stegun (1972); (3) practical shortcut by assuming the locations a priori (referring to the

0

10

20

30

40

50

1.0 3.0 9.0

N u m b e r o f S t o r i e s

Log(Plan Irregularity)

1.0

1.5

2.0

2.5

3.0

3.5

1.0 2.0 4.0 8.0

V e r t i c a l I r r e g u l a r i t y

Log(Plan Irregularity)

0

10

20

30

40

50

1 2 3

N u m b e r o f S t o r i e s

Vertical Irregularity

correlation

coefficient: 0.163

correlation

coefficient: 0.033

correlation

coefficient: 0.013

Figure 3. Scatter plots of salient seismic features and correlation coefficients indicating mutual

independence. The logarithm of plan irregularity is used only for the visualization purpose.




most similar standard PDF) and then finding the weights by solving the now-linear system of

equations. Ching et al. (2009) focused on some well-behaved cases where standard Gauss-

quadrature can be exploited; we examine a more general case here.

FINDING DELTA FUNCTIONS USING MM

In a larger work, we present three methods to employ MM: (1) assume parametric dis-

tributions for the important attributes and use Gauss-quadrature; (2) use a numerical method

to find the MM points for each of dimension (each attribute) separately and then combine

(1-D MM); and (3) use a numerical method to find the MM points for all of the dimensions

simultaneously (multidimensional MM). Methods 2 and 3 involve direct nonlinear system

solving. In the present briefer work we focus on method 3 because of its generality, and 1-D

MM is essentially the same as the multidimensional MM.

MULTIDIMENSIONAL MM

In a larger work (Cho and Porter 2013a), we show a simple yet universal numerical

approach to obtaining three delta functions that exactly match the first five moments of

one attribute with an arbitrary distribution (i.e., not necessarily Gaussian, lognormal,

beta, etc.). We here formulate the extension to multiple dimensions. To keep the derivation

as easy to follow as possible, we focus on a three-dimensional (3-D) case with independent

variables. The case for four or another number of attributes is very similar, and, as previously

noted, one can also use MM for correlated attributes with the aid of specific mathematical

remedies.If X is a 3-D random variable, we need nine delta functions (i.e., three delta functions for

the PDF in each dimension). Then, we have 18 unknowns, χ ¼ f χ i;wig, i ∈ ½1; 9 along with

18 conditions: (condition 1) the normality of the weights (they sum to 1.0); (conditions 2

through 16) the first five moments of the weighted samples match those of the population;

and (conditions 17 and 18) the two equality conditions among three weights, which makes

the three independent axes intersect at the central sample for each attribute. In this formula-

tion we arrange the variables as follows: χ 1, χ 2, χ 3 are the low, medium, and high values of

the first attribute, χ 4, χ 5, χ 6 are the low, medium, and high values of the second attribute, etc.

(see Figure 4). Weights are similarly arranged. For example, w1, w2, w3 are the weights for

the low, medium, and high values of the first attribute.

Now the problem becomes “root finding” of the multivariate function FðχÞ ¼fF 1;F 2;…F 18g:

EQ-TARGET;t em p:int r a link -;e5;41;206F 1 ¼

X9

i¼1

wi

1.0 (5)

EQ-T ARGET;t em p:int r a link -; e6;41;154F ð3k þi2Þ ¼ w3i2 χ

k 3i2 þ w3i1 χ

k 3i1 þ w3i χ

k 3i E ð X k i Þ k ∈ ½1;5; i ∈ ½1;3 (6)

EQ-TARGET ;t em p:int r a link -;e7;41;12 6F 17 ¼ w2 w5 (7)

EQ-TARGET ;t em p:int r a link -;e8;41;99F 18 ¼ w5 w8 (8)






convergence rate of the multivariate NR iteration is apparent. However, considering the fifth

power of the location terms in Equation 6, a slight change to them appears to severely disturb

the stability of the equilibrium, satisfied at the fifth iteration. Once the instability emerges, the

error grows exponentially. Thus, given intuitively reasonable initial values, prudent abortion

of the iteration appears necessary to obtain an acceptably accurate solution. That is, the ana-

lyst should carefully monitor the calculation and stop it when FðχÞ becomes small.

Second, the 3-D MM method often produces nonpositive location values in positively

valued variables during iteration (negative stories, for example). This behavior is commonly

found in many tests using reliable tools such as Matlab or Mathematica. It is well known that

negative weights can emerge during the application of MM technique and are acceptable

(Ching et al. 2009). However, a nonpositive location (e.g., number of stories) has no practical

interpretation. Additional constraints can be imposed to obtain positive locations, but in

many tests of our sample application, such automatic remedies often caused divergence problems.

We recommend manually adjusting the negative locations and assigning them positive

values. In our sample application the associated weights were small compared to other

weights (e.g., see w4 ¼ 0.00056 and w7 ¼ 0.001 before modification in Table 2), so

this adjustment had minimal effect. Indeed, after modification of χ 4 (from -5.222 to

1.097) and χ 7 (from -10.183 to 4.487), the mean error increases by only 0.08%

(i.e., from 2.56% to 2.64%). The final nine points are shown in Figure 6.

COMPARATIVE STUDY

To quantify the efficiency of MM, we conducted a comparative study against another

practical sampling method. We adopted a simple method of sampling over points at the

μ σ (hereafter, denoted as μ and σ method) values of each attribute in each dimension

(i.e., sampling 3n points in the n-space of attributes and assigning weights to each point).

Table 2. Nine delta functions (see Figure 6) of 3-D MM and positive locations after

prudent modification (bold letters for the modified locations). Terms i ¼ 1 to 3

correspond to vertical irregularity, 4 to 6 are plan irregularity, and 7 to 9 are number of

stories. “Error ” refers to values of FðχÞ.

Before modification (mean error: 2.56%) After modification (mean error: 2.64%)

i χ i wi i χ i wi

1 0.197 0.329 1 0.197 0.329

2 1.375 0.163 2 1.375 0.163

3 2.324 0.134 3 2.324 0.134

4 5.222 0.00056 4 1:097 0.00056

5 1.438 0.163 5 1.438 0.163

6 11.910 0.004 6 11.910 0.004

7 10.183 0.001 7 4:

487 0.001

8 9.473 0.163 8 9.473 0.163

9 30.871 0.041 9 30.871 0.041




μ AND σ METHOD FOR SAMPLE BUILDINGSIervolino et al. (2004) suggested using 3n sample buildings to cover n salient seismic

features within the building class of interest. To capture variability along each feature,

they used three points at μ and μ σ . Later, this approach is further enhanced by denser

sampling (Iervolino et al. 2007) that uses a larger number of sample points. Thus, more gen-

eral and smooth capacity surfaces can be constructed for arbitrary cases. The downside of this

remedy is that because of the increased samples, computation (structural analysis per sample)

becomes expensive. Figure 7a illustrates the 3n sampling method and its variation of dense

samples. Contrarily, MM uses considerably fewer points (Figure 7b). For example for four

seismic features, MM needs 2 4 þ 1 ¼ 9 samples whereas the μ and σ method

requires 34 ¼ 81.

Let us consider just vertical irregularity and number of stories, applying the MM and μ

and σ methods to the survey data of California RC buildings. Figure 8a and Table 3 show the

five MM sample values and weights and the joint distribution of attributes of the surveyed

buildings. Figure 8b and Table 4 show the nine μ and σ positions and weights. Weights for

the latter method are calculated by integrating the volume under the PDF surface in Figure 8a,

dividing the X-space at μ ½σ . There is substantial difference in locations and weights of

the two methods.

Figure 6. Nine delta functions (three for each feature) attained from the one-step 3-D MM.

Figure 7. Comparison between practical sampling methods and the MM: (a) schematic descrip-tion of 3n sampling method (e.g., nine gray circles represent sample points of two seismic fea-tures) and dense sampling method that introduces additional points at intersections of dashed

lines; (b) the MM in which 2n þ 1 samples (triangles) are employed.




An attractive advantage of the μ and σ method is that the positions appear to be more

physical and plausible than the index buildings of MM since μ and σ positions have no nega-

tive weight value and are less scattered. But MM offers two competing advantages. First is

the smaller number of samples, an advantage that increases with the number of attributes

studied. Second is how well MM matches higher moments. Table 5 shows the resulting

moments of the samples under each method. It also shows the difference between the

moments of the samples and that of the surveyed population; we call this difference the

error. The μ and σ method leads to sample buildings that precisely capture the mean quan-

tities of the two seismic features: see E ½ x column of Table 5. But it fails to retain the higher

order moments of the features, with errors substantially increasing at higher moments (see

columns of E ½ x 2 through E ½ x 5 in Table 5 This implies that the sample buildings of the μ and

σ method can lead to reliable mean estimates of the building class only when the original data

have a symmetric or a regular form. When the original data distribution is considerably irre-

gular or higher-order moments matter, μ and σ will perform poorly and at greater expense

compared with MM.

NOTE ON PBEE-2 ANALYSIS AND SEISMIC VULNERABILITY FUNCTIONS

The present work is part of a larger effort to analyze the seismic vulnerability of a class of

buildings in part using PBEE-2 analysis. We have discussed here a method to select index

Figure 8. Index buildings generated from (a) MM and (b) μ and σ approaches.

Table 3. Sample points by MM

No. Vert. irr. Stories Weight

0 1.6924 13 0.08046

1 1.6924 4 0.52194

2 1.6924 35 0.02013

3 1.1389 13 0.48708

4 2.6951 13 0.05131




buildings whose important seismic attributes match the first five moments of those of a sur-

veyed population of buildings. Each index building is subsequently analyzed by PBEE-2

methods based on FEMA P-58 to produce a seismic vulnerability function, a relationship

between ground motion and loss such as repair cost as a fraction of replacement cost.

Let us denote the expected value of the seismic vulnerability function for index building

i as yið x Þ, where x denotes the intensity measure (IM) level, say the 5% damped spectral

acceleration response at one-second period or some other convenient IM. The mean seismicvulnerability function for the class can be denoted by yð x Þ and estimated as the weighted

average of the yið x Þ using the MM weights wi, as in Equation 10. Higher moments can

be similarly estimated as in Equation 11:

EQ -TARGET;t em p:int r a lin k -;e10;62;202 yð x Þ ¼X2n

i¼0

wi · yið x Þ (10)

EQ-T ARGET;t em p:int r a link -; e11;62;150 E ½ yk ð x Þ ¼ X2n

i¼0

wi · yk i ð x Þ (11)

The larger methodology also involves a procedure for the structural analysis that employs a

advanced structure-independent parallel multiscale finite-element (FE) analysis platform that

Table 4. Sample points by μ and σ method

No. Location Vert. irr. Stories Weight

1 ( μ1, μ2) 1.47425 9 0.205

2 ( μ1, μ2 þ σ 2) 1.47425 15 0.100

3 ( μ1, μ2 σ 2) 1.47425 3 0.116

4 ( μ1 þ σ 1, μ2) 1.866 9 0.154

5 ( μ1 þ σ 1, μ2 þ σ 2) 1.866 15 0.112

6 ( μ1 þ σ 1, μ2 σ 2) 1.866 3 0.054

7 ( μ1 σ 1, μ2) 1.082 9 0.093

8 ( μ1 σ 1, μ2 þ σ 2) 1.082 15 0.062

9 ( μ1 σ 1, μ2 σ 2) 1.082 3 0.104

Table 5. Error comparison of the MM and the μ and σ method against original data

moments

Quantity Method E ½ x E ½ x 2 E ½ x 3 E ½ x 4 E ½ x 5

Vertical Irregularity MM 0.0002% 0.0006% 0.0018% 0.0034% 0.0055%

μ & σ 1.62% 0.27% 5.15% 14.85% 27.83%

Number of Stories MM 0.0066% 0.0047% 0.0109% 0.0169% 0.021%

μ & σ ∼0% 12.66% 37.68% 64.51% 83.07%




requires only the building geometry and two generic material parameters (see Cho and Porter

2013b). It can explicitly capture multidirectional cracks, crushing, progressive bar buckling,

nonlinear shear, and localized damage (Cho 2013). Owing to these multidisciplinaryadvances, in the future class-level loss estimations we will be able to reflect building-specific

characteristics such as construction era and so on, without resorting to idealized models. This

advance in structural analyses in conjunction with a few index buildings of MM is significant

since by doing so we can accurately consider the construction era, which substantially affects

reinforcement detailing as well as the largely fluctuating design base shear. It should be noted

that when the MM numerically generates an unrealistic index building, a simple intuitive

adjustment of the seismic feature is recommended as addressed before. Still, even in

such a rare case, we can rigorously control the error.

CONCLUSIONS

We show how one can characterize an asset class with a sample set of index buildings for

purposes of societal loss modeling. The sample set is selected to represent the probabilistic

variability in a few important seismic attributes that vary between buildings in the class. The

attributes can have arbitrary probability distributions (i.e., not necessarily Gaussian, lognor-

mal, etc.) and can be correlated. Doing so extends PBEE-2 by treating the asset definition as

uncertain, just as PBEE-2 currently quantifies and propagates uncertainties in hazard, struc-

tural response, damage, and loss.

The procedure begins by identifying the features that matter most. We selected three

top features by a meta-analysis (i.e., by seeking those features that many respected sources

commonly considered important). Here we chose vertical and plan irregularities and build-

ing height, although the use of other features would be straightforward since the meth-

odology is not limited by which features or how many are believed to contribute most to

the variability of the performance of the class. The analyst then performs a field survey to

quantify the joint distribution of the attributes in the class. We carried out a field survey

of 265 California commercial RC buildings and determined the joint distribution of the

selected features.

One then selects index buildings with particular values of each attribute and a weight for

each index buildings so that the weighted samples represent the population of the asset class.The method employs moment matching: a sampling strategy in which the sample values and

weights are selected so that the first several moments of the weighted samples match those of

the surveyed population. Even with correlated attributes, one can select 2n þ 1 samples to

match the first five moments of the population, where n is the number of attributes of interest.

Other methods that sample over the joint attribute space tend to require a number of samples

that increases exponentially with n.

In a larger work we offer three procedures to select sample positions and weights, includ-

ing one where the attributes have an approximately parametric distribution. Here we present

only one of these three methods. It uses a multivariate Newton –

Raphson iteration to select positions and weights for all the attributes simultaneously and allows the attributes to have an

arbitrary, nonparametric distribution. The method exhibits strong sensitivity to initial con-

ditions and has convergence problems, but we offer practical remedies.




As shown in the practical application and a comparative study against a practical sam-

pling method (using μ and μ σ points), a small number of 2n þ 1 index buildings proved

sufficient to rigorously mimic the statistics of the original data. Being capable of tackling

arbitrary PDFs and also achieving both accuracy and computational efficiency, we can now

proceed toward large-scale loss estimation with no restrictions of region, building classes,

and associated PDFs.

ACKNOWLEDGMENTS

This research is funded by the Willis Research Network and the Global Earthquake

Model, whose support is gratefully acknowledged.

APPENDIX

Please refer to the online version of this paper to access the supplemental material.

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(Received 27 September 2012; accepted 10 November 2013)




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Modeling Building Classes Using Moment Matching