Bayesian analysis of a parametric semi-markov process applied to seismic data

Transcription

1 Bayesian analysis of a parametric semi-markov process applied to seismic data Ilenia Epifani, Politecnico di Milano Joint work with Lucia Ladelli, Politecnico di Milano and Antonio Pievatolo, (IMATI-CNR) July 8, 2013

2 Seismic Region Classification of earthquakes Exploratory Data Analysis 6 E 8 E 10 E 12 E 14 E 16 E 18 E 46 N 46 N MR3 44 N 44 N 42 N 42 N 40 N 40 N 38 N 38 N 36 N 36 N 6 E 8 E 10 E 12 E 14 E 16 E 18 E Earthquakes from 1838 to 2002 with magnitude Mw 4.5 occurred in a tectonically homogeneous macroregion in the central Northern Apennines in Italy

3 Seismic Region Classification of earthquakes Exploratory Data Analysis Three types of earthquakes according to their severity Low Class 1: 4.5 apple magnitude < 4.9 Medium Class 2: 4.9 apple magnitude < 5.3 High Class 3: magnitude 5.3

4 Seismic Region Classification of earthquakes Exploratory Data Analysis Three types of earthquakes according to their severity Low Class 1: 4.5 apple magnitude < 4.9 Medium Class 2: 4.9 apple magnitude < 5.3 High Class 3: magnitude 5.3 Data are collected by the CPTI04 (2004) catalogue, Subdivision in seismogenic Macroregion provided by DISS Working Group December 2002 is the closing date of the CPT104 (2004) (Rotondi, 2010 and Rotondi and Varini, 2012)

5 Some Statistics Seismic Region Classification of earthquakes Exploratory Data Analysis Table: Number of observed transitions in the whole dataset (271) 308 (278) 347 (791) (406) 373 (434) 433 (667) (301) 210 (287) 385 (385) Table: Mean inter-occurrence times and sd in days (rounded)

6 Seismic Region Classification of earthquakes Exploratory Data Analysis Weibull qq-plots of earthquake inter-occurrence times 1 to 1 1 to 2 1 to 3 sample quant sample quant sample quant theor. quant theor. quant theor. quant. 2 to 1 2 to 2 2 to 3 sample quant sample quant sample quant theor. quant theor. quant theor. quant. 3 to 1 3 to 2 3 to 3 sample quant sample quant sample quant theor. quant theor. quant theor. quant.

7 Trajectories of the process Semi-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 j 0, j 1,...,j n occur, with random inter-occurrence times x 1,...,x n The process starts at the time of an earthquake with magnitude of known class j 0 When the ending time T does not coincide with an earthquake then the time past from last earthquake to T is right-censored

8 Trajectories of the process Semi-Markov process We assume that 1 the states J 1,...,J n,...are a discrete MC(j 0, {1, 2, 3}, P) 2 the sojourn times X 1,...,X n,...conditionally on j 0, j 1,...,j n,... are independent with X n J n 1, J n F Jn 1 J n 3 F ij = Weibull( ij, ij )

9 Trajectories of the process Semi-Markov process We assume that 1 the states J 1,...,J n,...are a discrete MC(j 0, {1, 2, 3}, P) 2 the sojourn times X 1,...,X n,...conditionally on j 0, j 1,...,j n,... are independent with X n J n 1, J n F Jn 1 J n 3 F ij = Weibull( ij, ij ) (J n, X n ) n is a semi-markov process with Weibull intertimes

10 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion 1 the transition matrix P and the Weibull parameters ( ij, ij ) i,j=1,2,3 are independent

11 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS 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 vectors

12 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS 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 vectors 3 The 9 pairs of Weibull parameters ( 11, 11 ),...,( 33, 33 ) are independent with ij ij GIG(m ij, b ij ( ij ), ij ) ij 0 -shifted Gamma density (Berger and Sun, 1993 and Bousquet, 2010)

13 The prior Data Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion ij ij GIG(m ij, b ij ( ij ), ij ) ij 0 -shifted Gamma density can be seen as the output of the following mechanism

14 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion The prior ij ij GIG(m ij, b ij ( ij ), ij ) ij 0 -shifted Gamma density 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 (, ) / c 1( 0)1( 0 ), c 0, 0 > 0 Then Use the resulting posterior distributions as prior but conveniently modified

15 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion x m,...,x 1 are m sojourn times in the string (i, j) ˆx q = its qth sample percentile b( ) =ˆx q [(1 q) 1/m 1] 1 P l = m ln(ˆx q ) ln x i

16 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion x m,...,x 1 are m sojourn times in the string (i, j) ˆx q = its qth sample percentile b( ) =ˆx q [(1 q) 1/m 1] 1 P l = m ln(ˆx q ) ln x i m ( ) ( ) 2, 3,... GIG(m, b( ), ) 0 -shifted (m, l),

17 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion x m,...,x 1 are m sojourn times in the string (i, j) ˆx q = its qth sample percentile b( ) =ˆx q [(1 q) 1/m 1] 1 P l = m ln(ˆx q ) ln x i m ( ) ( ) 2, 3,... GIG(m, b( ), ) 0 -shifted (m, l), 0 = 2 m

18 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion x m,...,x 1 are m sojourn times in the string (i, j) ˆx q = its qth sample percentile b( ) =ˆx q [(1 q) 1/m 1] 1 P l = m ln(ˆx q ) ln x i m ( ) ( ) 2, 3,... GIG(m, b( ), ) 0 -shifted (m, l), 0 = 2 m 1 GIG(1, x 1, )

19 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion x m,...,x 1 are m sojourn times in the string (i, j) ˆx q = its qth sample percentile b( ) =ˆx q [(1 q) 1/m 1] 1 P l = m ln(ˆx q ) ln x i m ( ) ( ) 2, 3,... GIG(m, b( ), ) 0 -shifted (m, l), 0 = 2 m 1 GIG(1, x1 1, ) 1 ( 0 < < 1 ), 0

20 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion x m,...,x 1 are m sojourn times in the string (i, j) ˆx q = its qth sample percentile b( ) =ˆx q [(1 q) 1/m 1] 1 P l = m ln(ˆx q ) ln x i m ( ) ( ) 2, 3,... GIG(m, b( ), ) 0 -shifted (m, l), 0 = 2 m 1 GIG(1, x1 1, ) 1 ( 0 < < 1 ), 0 = 2 3, 1 = 10 0

21 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion x m,...,x 1 are m sojourn times in the string (i, j) ˆx q = its qth sample percentile b( ) =ˆx q [(1 q) 1/m 1] 1 P l = m ln(ˆx q ) ln x i m ( ) ( ) 2, 3,... GIG(m, b( ), ) 0 -shifted (m, l), 0 = 2 m 1 GIG(1, x1 1, ) 1 ( 0 < < 1 ), 0 = 2 3, 1 =

22 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion x m,...,x 1 are m sojourn times in the string (i, j) ˆx q = its qth sample percentile b( ) =ˆx q [(1 q) 1/m 1] 1 P l = m ln(ˆx q ) ln x i m ( ) ( ) 2, 3,... GIG(m, b( ), ) 0 -shifted (m, l), 0 = 2 m 1 GIG(1, x1 1, ) 0 1 ( 0 < < 1 ), 0 = 2 3, 1 = 10 ( GIG(1, X, ) 0 X Unif (...) ( 0 < < 1 )

23 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion The Likelihood of semi-markov data depends only on

24 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion The Likelihood of semi-markov data depends only on 1 the number of visits to each string (i, j): (N 11, N 12, N 13 ), (N 21, N 22, N 23 ), (N 31, N 32, N 33 ) 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)

25 Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion The Likelihood of semi-markov data depends only on 1 the number of visits to each string (i, j): (N 11, N 12, N 13 ), (N 21, N 22, N 23 ), (N 31, N 32, N 33 ) 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 the future unosservable state j, the full likelihood factorizes L full = Y p N ij ij Y Y f (x ( ) ij ij, ij, N ij ) S urvival (u ij, ij ) p jnj ij ij

26 For this Bayesian Statistical model, Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion JAGS is able to run an exact Gibbs sampler (P,,,j )

27 For this Bayesian Statistical model, Prior Distributions Elicitation of the hyperparameters Likelihood of semi-markov data JAGS implementantion JAGS is able to run an exact Gibbs sampler (P,,,j ) J has discrete prior (p jn,1, p jn,2, p jn,3) (N 11, N 12, N 13 ) is multinomial-distributed with probability vector (p 11, p 12, p 13 ) Dirichlet exact times xij 1,...,x n ij ij are Weibull last right-censored time u is handled by a special instruction provided by the Jags language if m = 0, 1 the non-standard prior of is coded using the zero-trick other prior distributions are routine

28 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models Table: ( ) Table: ( )

29 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models Couple of states Intertimes (days) posterior predictive 2.5% and 97.5% quantiles 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) mean median

30 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models (0.102) (0.252) (0.103) (0.173) (0.232) (0.473) (0.179) (0.150) (0.618) Table: Posterior means followed by (sd) of the shape parameters ij

31 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models (0.102) (0.252) (0.103) (0.173) (0.232) (0.473) (0.179) (0.150) (0.618) Table: Posterior means followed by (sd) of the shape parameters ij Table: Posterior means of the transition matrix P

32 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models To predict what type of event and when it is most likely to occur, Cross state-probability P ij t 0 x that the next earthquake occurs within a time interval x and is of a given magnitude j, conditionally on the waiting time from last earthquake t 0 and on its magnitude i

33 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models To predict what type of event and when it is most likely to occur, Cross state-probability P ij t 0 x that the next earthquake occurs within a time interval x and is of a given magnitude j, conditionally on the waiting time from last earthquake t 0 and on its magnitude i In CPTI04 catalogue: last recorded event had been in class i = 2 and had occurred about t 0 = 32 months earlier x = 1 Month 2 Months 3 Months 4 Months 5 Months 6 Months Year 2 Years 3 Years 4 Years j = j = j =

34 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models To assess the predictive capability of the model

35 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models To assess the predictive capability of the model re-estimating the Cross state-probabilities, using only the data up to 31 December 2001, 31 December 2000, and so on backwards down to 1992.

36 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models To assess the predictive capability of the model re-estimating the Cross state-probabilities, using only the data up to 31 December 2001, 31 December 2000, and so on backwards down to Dec 31st CSP good performace for bad predictive performance for

37 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models Look at transition matrix & cross state-probability to state Which mechanism govern the earthquake generation: Time or Slip Predictable models?

38 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models In a time predictable model when a maximal energy threshold is reached, some fraction of it is released; therefore, the waiting time distribution depends on the current event type, but not on the next event type The strength of an event does not depend on the strength of the previous one:

39 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models In a time predictable model when a maximal energy threshold is reached, some fraction of it is released; therefore, the waiting time distribution depends on the current event type, but not on the next event type The strength of an event does not depend on the strength of the previous one: P has equal rows

40 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models In a time predictable model when a maximal energy threshold is reached, some fraction of it is released; therefore, the waiting time distribution depends on the current event type, but not on the next event type The strength of an event does not depend on the strength of the previous one: P has equal rows Given i, P ij t 0 x are proportional to each other as j = 1, 2, 3 8 x

41 Data Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models In a time predictable model when a maximal energy threshold is reached, some fraction of it is released; therefore, the waiting time distribution depends on the current event type, but not on the next event type The strength of an event does not depend on the strength of the previous one: P has equal rows Given i, P ij t 0 x are proportional to each other as j = 1, 2, 3 8 x Ratios of couples of CSPs from 1 Ratio of CSPs ratio of CSP (1,1) to (1,2) ratio of CSP (1,1) to (1,3) ratio of CSP (1,2) to (1,3) 1M 3M 5M 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y

42 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models In a slip predictable model after an earthquake, energy falls to a minimal threshold and increases until the next event, where it starts to increase again from the same threshold. Here, the magnitude of an event depends on the length of the waiting time, but not on the magnitude of the previous one:

43 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models In a slip predictable model after an earthquake, energy falls to a minimal threshold and increases until the next event, where it starts to increase again from the same threshold. Here, the magnitude of an event depends on the length of the waiting time, but not on the magnitude of the previous one: P has equal rows

44 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models In a slip predictable model after an earthquake, energy falls to a minimal threshold and increases until the next event, where it starts to increase again from the same threshold. Here, the magnitude of an event depends on the length of the waiting time, but not on the magnitude of the previous one: P has equal rows Given j, P ij t 0 x are equal to each other as i = 1, 2, 3 8 x

45 Data Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models In a slip predictable model after an earthquake, energy falls to a minimal threshold and increases until the next event, where it starts to increase again from the same threshold. Here, the magnitude of an event depends on the length of the waiting time, but not on the magnitude of the previous one: P has equal rows Given j, P ij t 0 x are equal to each other as i = 1, 2, 3 8 x Ratio of couples of CSPs to 2 Ratio of CSPs ratio of CSP (1,2) to (2,2) ratio of CSP (1,2) to (3,2) ratio of CSP (2,2) to (3,2) 1M 3M 5M 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y

46 Historical and current datasets Model fitting Parameters Estimate Forecasting Time or Slip Predictable models Neither a SPM nor a TPM seem to be supported by the posterior distributions of the parameters, because of the behavior of the waiting times

47 An example cost function When to start the jobs? How long should them last?

48 An example cost function When to start the jobs? How long should them last? Two kind of maintenance works on buildings Restoration of the strength the building had before the damage occurred. It s a short/medium term decision strengthening of the building (i.e. increase the strength of the building). It s a long term decision Focus on Restoration.

49 An example cost function When to start the jobs? How long should them last? Two kind of maintenance works on buildings Restoration of the strength the building had before the damage occurred. It s a short/medium term decision strengthening of the building (i.e. increase the strength of the building). It s a long term decision Focus on Restoration. The cost to be minimize is a function of time to next earthquake, given the earthquake type and the restoration starting and ending times

50 An example cost function c R = restoration cost c E (j) =collapse cost (due to next earthquake of type j) a 1, a 2 = restoration starting and ending times (decisional parameters); a 1 apple a 2 = time of next eartquake All time parameters have as origin the time of occurrence of latest earthquake J n = i Assume c R < c E (j).

51 An example cost function c R = restoration cost c E (j) =collapse cost (due to next earthquake of type j) a 1, a 2 = restoration starting and ending times (decisional parameters); a 1 apple a 2 = time of next eartquake All time parameters have as origin the time of occurrence of latest earthquake J n = i Assume c R < c E (j). 8 >< c R if >a 2 a C(,j; a 1, c R, a 2, c E (j)) = 1 a >: 2 a 1 c R + a 2 a 2 a 1 c E (j) if a 1 apple apple a 2 c E (j) if <a 1

52 An example cost function cost restoration start restoration end time of next eartquake

53 An example cost function Cost of collapse or restoration cost ending time starting time as a function of restoration starting and ending times given next earthquake time and type

54 Average Surfaces pointwise using p J, Jn=i(j, t) An example cost function t ' P ij 0 t (P ij 0 t = P(J = j, 2 (t, t + t] J n = i))

55 Average Surfaces pointwise using p J, Jn=i(j, t) An example cost function t ' P ij 0 t (P ij 0 t = P(J = j, 2 (t, t + t] J n = i)) Unfortunately The average surface preserves the same monotonicity patterns of individual surfaces and the best decision would be a 1 = a 2 = 0: not feasible

56 Average Surfaces pointwise using p J, Jn=i(j, t) An example cost function t ' P ij 0 t (P ij 0 t = P(J = j, 2 (t, t + t] J n = i)) Unfortunately The average surface preserves the same monotonicity patterns of individual surfaces and the best decision would be a 1 = a 2 = 0: not feasible But one has to choose (a 1, a 2 ) which meets external constraints at an acceptable nonzero cost (risk)

57 Average Surfaces pointwise using p J, Jn=i(j, t) An example cost function t ' P ij 0 t (P ij 0 t = P(J = j, 2 (t, t + t] J n = i)) Unfortunately The average surface preserves the same monotonicity patterns of individual surfaces and the best decision would be a 1 = a 2 = 0: not feasible But one has to choose (a 1, a 2 ) which meets external constraints at an acceptable nonzero cost (risk) Or one can assume c R > c E (1), for earthquake type 1 only, so that not all surfaces have the same monotonicity pattern

58 Average Surfaces pointwise using p J, Jn=i(j, t) An example cost function t ' P ij 0 t (P ij 0 t = P(J = j, 2 (t, t + t] J n = i)) Unfortunately The average surface preserves the same monotonicity patterns of individual surfaces and the best decision would be a 1 = a 2 = 0: not feasible But one has to choose (a 1, a 2 ) which meets external constraints at an acceptable nonzero cost (risk) Or one can assume c R > c E (1), for earthquake type 1 only, so that not all surfaces have the same monotonicity pattern BUT: everything about this decision problem remains to be done

59 I. Epifani, L. Ladelli, A. Pievatolo (2013). Bayesian estimation for a parametric Markov Renewal model applied to seismic data,

60 I. Epifani, L. Ladelli, A. Pievatolo (2013). Bayesian estimation for a parametric Markov Renewal model applied to seismic data, Thank you for your attention!