bayesian analysis of a parametric semi-markov process ......data probabilistic model bayesian...

DataProbabilistic ModelBayesian Analysis

Earthquakes in Central Northern AppenninesMaking decision on Building Maintenance

References

Bayesian analysis of a parametric semi-Markovprocess applied to seismic data

Ilenia Epifani, Politecnico di Milano

Joint work with Lucia Ladelli, Politecnico di Milano andAntonio Pievatolo, (IMATI-CNR)

July 8, 2013

I. Epifani GDRR 2013



References

Seismic RegionClassification of earthquakesExploratory Data Analysis

6˚E

6˚E

8˚E

8˚E

10˚E

10˚E

12˚E

12˚E

14˚E

14˚E

16˚E

16˚E

18˚E

18˚E

36˚N 36˚N

38˚N 38˚N

40˚N 40˚N

42˚N 42˚N

44˚N 44˚N

46˚N 46˚N

6˚E

6˚E

8˚E

8˚E

10˚E

10˚E

12˚E

12˚E

14˚E

14˚E

16˚E

16˚E

18˚E

18˚E

36˚N 36˚N

38˚N 38˚N

40˚N 40˚N

42˚N 42˚N

44˚N 44˚N

46˚N 46˚N

MR3

Earthquakes from 1838 to 2002 with magnitude Mw � 4.5 occurred ina tectonically homogeneous macroregion in the central NorthernApennines in Italy




References


Three types of earthquakes according to their severityLow Class 1: 4.5 magnitude < 4.9

Medium Class 2: 4.9 magnitude < 5.3High Class 3: magnitude � 5.3

Data are collected by the CPTI04 (2004) catalogue,Subdivision in seismogenic Macroregion provided by DISSWorking Group 200731 December 2002 is the closing date of the CPT104 (2004)

(Rotondi, 2010 and Rotondi and Varini, 2012)




References


Some Statistics

1 2 31 65 30 172 32 15 73 15 9 4

Table: Number of observed transitions in the whole dataset 1838-2002

1 2 31 274 (271) 308 (278) 347 (791)2 324 (406) 373 (434) 433 (667)3 246 (301) 210 (287) 385 (385)

Table: Mean inter-occurrence times and sd in days (rounded)




References


Weibull qq-plots of earthquake inter-occurrence times

●●

●●●●●

●●●●

●●●●●●●

●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●

●●●

−4 −3 −2 −1 0 1

−5−3

−11

theor. quant.

sample

quant

.

1 to 1

● ●

●●

●

●●●●●●

●●●●●●●●●●●●●●

●●

●●

●

−3 −2 −1 0 1

−3−2

−10

1

theor. quant.sam

ple qu

ant.

1 to 2

●

●

● ●● ●●

●●●●

●●●●●

●

−3 −2 −1 0 1

−3−1

01

2

theor. quant.

sample

quant

.

1 to 3

●

● ●

●●●●

●●●●●●●●●

●●●●●●

●●●●●●●●

●●

−3 −2 −1 0 1

−5−3

−11

theor. quant.

sample

quant

.

2 to 1

●

●

●●

● ●●

●●●

●●● ● ●

−2 −1 0 1

−3−2

−10

1

theor. quant.

sample

quant

.

2 to 2

●

● ●

●●

●

●

−2.0 −1.0 0.0 0.5

−2−1

01

theor. quant.

sample

quant

.

2 to 3

● ● ● ●

●

● ●

●●

●●●●

●

●

−2 −1 0 1

−3−2

−10

1

theor. quant.

sample

quant

.

3 to 1

●

●●

●●

●

●

●●

−2.0 −1.0 0.0

−4−2

0

theor. quant.

sample

quant

.

3 to 2

●

●

●

●

−1.5 −1.0 −0.5 0.0 0.5−1.

5−0.

50.5

theor. quant.

sample

quant

.

3 to 3

Figure: Weibull qq-plots of earthquake inter-occurrence times (centralNorthern Appennines) classified by transition between magnitude classes.




References

Trajectories of the processSemi-Markov process

From a modelling point of view we observe over a period of time (0,T ]a process in which

different earthquakes with magnitude of class j0, j1, . . . , jn occur,withrandom inter-occurrence times x1, . . . , xn

The process starts at the time of an earthquake with magnitude ofknown class j0

When the ending time T does not coincide with an earthquakethen the time past from last earthquake to T is right-censored




References

Trajectories of the processSemi-Markov process

We assume that1 the states J1, . . . , Jn

, . . . are a discrete MC(j0, {1, 2, 3},P)2 the sojourn times X1, . . . ,Xn

, . . . conditionally on j0, j1, . . . , jn, . . .are independent with X

n

|Jn�1, Jn

⇠ F

J

n�1J

n

3F

ij

= Weibull(↵ij

, ✓ij

)

(Jn

,Xn

)n

is a semi-Markov process with Weibull intertimes




References

Prior DistributionsElicitation of the hyperparametersLikelihood of semi-Markov dataJAGS implementantion

1 the transition matrix P and the Weibull parameters (↵ij

, ✓ij

)i,j=1,2,3

are independent

2 The rows of P are independent Dirichlet vectors3 The 9 pairs of Weibull parameters (↵11, ✓11), . . . , (↵33, ✓33) are

independent with

✓ij

|↵ij

⇠ GIG(mij

, bij

(↵ij

),↵ij

)

↵ij

⇠ ↵0-shifted Gamma density

(Berger and Sun, 1993 and Bousquet, 2010)




References



, ✓ij

)i,j=1,2,3

are independent2 The rows of P are independent Dirichlet vectors

3 The 9 pairs of Weibull parameters (↵11, ✓11), . . . , (↵33, ✓33) areindependent with

✓ij

|↵ij

⇠ GIG(mij

, bij

(↵ij

),↵ij

)

↵ij






References



, ✓ij

)i,j=1,2,3

are independent2 The rows of P are independent Dirichlet vectors3 The 9 pairs of Weibull parameters (↵11, ✓11), . . . , (↵33, ✓33) are

independent with

✓ij

|↵ij

⇠ GIG(mij

, bij

(↵ij

),↵ij

)

↵ij






References


The prior✓

ij

|↵ij

⇠ GIG(mij

, bij

(↵ij

),↵ij

)

↵ij


can be seen as the output of the following mechanism

First Implement a Bayesian inference on some “historical data”x�m

, . . . , x�1 of size m with diffuse prior

⇡̃(↵, ✓) / ✓�1⇣

1 � ↵0↵

⌘c

1(✓ � 0)1(↵ � ↵0), c � 0, ↵0 > 0

Then Use the resulting posterior distributionsbut conveniently modified

as prior




References


x�m

, . . . , x�1 are m sojourn times in the string (i , j)

x̂

q

= its qth sample percentileb(↵) = x̂

↵q

[(1 � q)�1/m � 1]�1

l = m ln(x̂q

)�P

ln x�i

m ⇡(✓|↵) ⇡(↵)

2, 3, . . . GIG(m, b(↵),↵) ↵0-shifted �(m, l), ↵0 = 2m

1 GIG(1, x↵1 ,↵)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1), ↵0 = 2

3 , ↵1 = 10

0

(GIG(1,X↵,↵)

X ⇠ Unif (. . .)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1)




References


x�m


x̂

q


↵q

[(1 � q)�1/m � 1]�1

l = m ln(x̂q

)�P

ln x�i

m ⇡(✓|↵) ⇡(↵)

2, 3, . . . GIG(m, b(↵),↵) ↵0-shifted �(m, l),

↵0 = 2m

1 GIG(1, x↵1 ,↵)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1), ↵0 = 2

3 , ↵1 = 10

0

(GIG(1,X↵,↵)

X ⇠ Unif (. . .)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1)




References


x�m


x̂

q


↵q

[(1 � q)�1/m � 1]�1

l = m ln(x̂q

)�P

ln x�i

m ⇡(✓|↵) ⇡(↵)


1 GIG(1, x↵1 ,↵)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1), ↵0 = 2

3 , ↵1 = 10

0

(GIG(1,X↵,↵)

X ⇠ Unif (. . .)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1)




References


x�m


x̂

q


↵q

[(1 � q)�1/m � 1]�1

l = m ln(x̂q

)�P

ln x�i

m ⇡(✓|↵) ⇡(↵)


1 GIG(1, x↵1 ,↵)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1),

↵0 = 23 , ↵1 = 10

0

(GIG(1,X↵,↵)

X ⇠ Unif (. . .)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1)




References


x�m


x̂

q


↵q

[(1 � q)�1/m � 1]�1

l = m ln(x̂q

)�P

ln x�i

m ⇡(✓|↵) ⇡(↵)


1 GIG(1, x↵1 ,↵)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1), ↵0 = 2

3 , ↵1 = 10

0

(GIG(1,X↵,↵)

X ⇠ Unif (. . .)

⇣1 � ↵0

↵

⌘1(↵0<↵<↵1)




References


The Likelihood of semi-Markov data depends only on

1 the number of visits to each string (i , j):(N11,N12,N13), (N21,N22,N23), (N31,N32,N33)

2 the times x

(1)ij

, x (2)ij

, . . . , spent in state i at 1st, 2nd, . . . , visit to (i , j)(all independent and identically distributed for each string)

3 if time u from last earthquake is right-censored, introducing thefuture unosservable state j

⇤, the full likelihood factorizes

L

full

=Y

ij

p

N

ij

ij

⇥Y

ij

Y

⇢

f (x (⇢)ij

|↵ij

, ✓ij

,Nij

)⇥ S

urvival

(u|↵ij

, ✓ij

)⇥ p

j

n

j

⇤




References


The Likelihood of semi-Markov data depends only on1 the number of visits to each string (i , j):

(N11,N12,N13), (N21,N22,N23), (N31,N32,N33)

2 the times x

(1)ij

, x (2)ij

, . . . , spent in state i at 1st, 2nd, . . . , visit to (i , j)(all independent and identically distributed for each string)

3 if time u from last earthquake is right-censored, introducing thefuture unosservable state j

⇤, the full likelihood factorizes

L

full

=Y

ij

p

N

ij

ij

⇥Y

ij

Y

⇢

f (x (⇢)ij

|↵ij

, ✓ij

,Nij

)⇥ S

urvival

(u|↵ij

, ✓ij

)⇥ p

j

n

j

⇤




References


For this Bayesian Statistical model,

JAGS is able to run an exact Gibbs sampler (P,↵, ✓, j⇤)

J

⇤ has discrete prior (pj

n

,1, pj

n

,2, pj

n

,3)

(N11,N12,N13) is multinomial-distributed withprobability vector (p11, p12, p13) ⇠ Dirichlet

exact times x

1ij

, . . . , xn

ij

ij

are Weibulllast right-censored time u is handled by a special instructionprovided by the Jags languageif m = 0, 1 the non-standard prior of ↵ is coded using the“zero-trick”other prior distributions are routine




References

Historical and current datasetsModel fittingParameters EstimateForecastingTime or Slip Predictable models

1 2 31 27 14 42 14 8 43 4 3 3

Table: (1832-1916)

1 2 31 38 16 132 18 7 33 11 6 1

Table: (1916-2002)




References


Couple of states

Inte

rtim

es (d

ays)

pos

terio

r pre

dict

ive 2

.5%

and

97.

5% q

uant

iles

95% posterior predictive intervals of the intertimes

(1,1) (2,1) (3,1) (1,2) (2,2) (3,2) (1,3) (2,3) (3,3)

25

1020

5010

020

050

020

00

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

meanmedian

Figure: Posterior predictive 95 percent credible intervals of the meaninter-occurrence times in days with actual times denoted by (blue) solid dots.Suspect outliers are denoted by (red)-pointing triangles. The (green) dottedline shows the posterior mean and the (violet) dashed line the posteriormedian.




References


1 2 31 1.182 (0.102) 1.981 (0.252) 0.828 (0.103)2 1.318 (0.173) 1.218 (0.232) 1.716 (0.473)3 1.026 (0.179) 0.996 (0.150) 2.098 (0.618)

Table: Posterior means followed by (sd) of the shape parameters ↵ij

1 2 31 0.57 0.27 0.162 0.58 0.28 0.143 0.52 0.32 0.16

Table: Posterior means of the transition matrix P




References


To predict what type of event and when it is most likely to occur,

Cross state-probability P

ij

t0|�x

that the next earthquake occurs within atime interval �x and is of a given magnitude j , conditionally on thewaiting time from last earthquake t0 and on its magnitude i

In CPTI04 catalogue: last recorded event had been in class i = 2 andhad occurred about t0 = 32 months earlier

�x = 1 Month 2 Months 3 Months 4 Months 5 Months 6 Months Year 2 Years 3 Years 4 Yearsj = 1 0.070 0.122 0.171 0.210 0.243 0.270 0.364 0.410 0.418 0.420j = 2 0.059 0.105 0.150 0.187 0.221 0.249 0.363 0.444 0.468 0.477j = 3 0.014 0.024 0.034 0.041 0.048 0.054 0.075 0.088 0.092 0.093




References


To assess the predictive capability of the model

re-estimating the Cross state-probabilities, using only the data up to 31December 2001, 31 December 2000, and so on backwards down to1992.

Dec 31st 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992CSP .373 .505 .151 .252 .021 .175 .007 .007 .007 .006

good performace for 2001-1996bad predictive performance for 1995-1992




References


To assess the predictive capability of the modelre-estimating the Cross state-probabilities, using only the data up to 31December 2001, 31 December 2000, and so on backwards down to1992.

Dec 31st 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992CSP .373 .505 .151 .252 .021 .175 .007 .007 .007 .006

good performace for 2001-1996bad predictive performance for 1995-1992




References


Look at transition matrix & cross state-probability to state

Which mechanism govern the earthquake generation:

Time or Slip Predictable models ?




References


In a time predictable model when a maximal energy threshold isreached, some fraction of it is released; therefore,

the waiting time distribution depends on the current event type, butnot on the next event typeThe strength of an event does not depend on the strength of theprevious one:

P has equal rowsGiven i , P

ij

t0|�x

are proportional to each other as j = 1, 2, 3 8�x




References



the waiting time distribution depends on the current event type, butnot on the next event typeThe strength of an event does not depend on the strength of theprevious one: P has equal rows

Given i , P

ij

t0|�x





References



the waiting time distribution depends on the current event type, butnot on the next event typeThe strength of an event does not depend on the strength of theprevious one: P has equal rows

Given i , P

ij

t0|�x


●

●

●

●

●

●

● ● ● ● ● ● ● ● ● ●

05

1015

20

Ratio

of CSP

s

1M 3M 5M 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y

Ratios of couples of CSPs from 1

●●

●●

● ●● ● ● ● ● ● ● ● ● ●

●

●

ratio of CSP (1,1) to (1,2)ratio of CSP (1,1) to (1,3)ratio of CSP (1,2) to (1,3)




References


In a slip predictable model after an earthquake, energy falls to aminimal threshold and increases until the next event, where it starts toincrease again from the same threshold. Here,

the magnitude of an event depends on the length of the waitingtime, butnot on the magnitude of the previous one:

P has equal rowsGiven j , P

ij

t0|�x

are equal to each other as i = 1, 2, 3 8�x




References



the magnitude of an event depends on the length of the waitingtime, butnot on the magnitude of the previous one: P has equal rows

Given j , P

ij

t0|�x





References



the magnitude of an event depends on the length of the waitingtime, butnot on the magnitude of the previous one: P has equal rows

Given j , P

ij

t0|�x


●

●

●

●

●

●

●

●

●● ● ● ● ● ● ●

0.20.4

0.60.8

1.0

Ratio

of CSP

s

1M 3M 5M 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y

Ratio of couples of CSPs to 2

●

●

●

●

●

●

●

●

● ● ● ● ● ● ● ●

●

●

ratio of CSP (1,2) to (2,2)ratio of CSP (1,2) to (3,2)ratio of CSP (2,2) to (3,2)




References


Neither a SPM nor a TPM seem to be supported by the posteriordistributions of the parameters, because of the behavior of the waitingtimes




References

An example cost function

When to start the jobs? How long should them last?

Two kind of maintenance works on buildingsRestoration of the strength the building had before the damageoccurred. It’s a short/medium term decisionstrengthening of the building (i.e. increase the strength of thebuilding). It’s a long term decision

Focus on Restoration.

The cost to be minimize is a function of time to next earthquake, giventhe earthquake type and the restoration starting and ending times




References


c

R

= restoration costc

E

(j) = collapse cost (due to next earthquake of type j)a1, a2 = restoration starting and ending times (decisionalparameters); a1 a2

⌧ = time of next eartquake

– All time parameters have as origin the time of occurrence of latestearthquake J

n

= i

– Assume c

R

< c

E

(j).

C(⌧, j ; a1, cR

, a2, cE

(j)) =

8><

>:

c

R

if ⌧ > a2⌧�a1a2�a1

c

R

+ a2�⌧a2�a1

c

E

(j) if a1 ⌧ a2

c

E

(j) if ⌧ < a1




References


0 20 40 60 80 100

0.00.2

0.40.6

0.81.0

time of next eartquake

cost

restoration start restoration end




References


starting time

ending

time

cost

Cost of collapse or restoration

as a function of restoration starting and ending times given next earthquake time and type




References


Average Surfaces pointwise using

p

J,⌧ |Jn

=i

(j , t)�t ' P

ij

0|�t

(Pij

0|�t

= P(J = j , ⌧ 2 (t , t +�t ] | J

n

= i))

Unfortunately The average surface preserves the same monotonicitypatterns of individual surfaces and the best decision would bea1 = a2 = 0: not feasible

But one has to choose (a1, a2) which meets external constraints at anacceptable nonzero cost (risk)

Or one can assume c

R

> c

E

(1), for earthquake type 1 only, so that notall surfaces have the same monotonicity pattern

BUT: everything about this decision problem remains to be done




References


Average Surfaces pointwise using

p

J,⌧ |Jn

=i

(j , t)�t ' P

ij

0|�t

(Pij

0|�t

= P(J = j , ⌧ 2 (t , t +�t ] | J

n

= i))

Unfortunately The average surface preserves the same monotonicitypatterns of individual surfaces and the best decision would bea1 = a2 = 0: not feasible

But one has to choose (a1, a2) which meets external constraints at anacceptable nonzero cost (risk)

Or one can assume c

R

> c

E

(1), for earthquake type 1 only, so that notall surfaces have the same monotonicity pattern

BUT: everything about this decision problem remains to be doneI. Epifani GDRR 2013



References

I. Epifani, L. Ladelli, A. Pievatolo (2013). Bayesian estimation for a

parametric Markov Renewal model applied to seismic data,http://arxiv.org/abs/1301.6494

Thank you for your attention!


http://arxiv.org/abs/1301.6494

bayesian analysis of a parametric semi-markov process ......data probabilistic model bayesian...

Documents