Modelling of Tsunami Evacuation Behavior
Accounting for Dynamics of Heterogeneity
in Expected Utility
BinN International Research Seminar #07
Sep 25th, 2016

Kobe University
Junji URATA

Choice of Evacuation Start Time
Safetyshelterevacuation

LegendDecision-making
Decision-making in future

Distancefrom seaPrevious activity
Future activity (Alternatives)
Expected utility

stayDangerInformation

trip

Information

Disaster Risk (Low High)time t• The reason for evacuation is to avoid a future risk of their place.
• People choose an evacuation with an expected utility.

2

Dynamics of Heterogeneity in Expected Utility
Safetyshelter
evacuation

LegendDecision-making
Decision-making in future

Distancefrom seaPrevious activity
Future activity (Alternatives)
Expected utility

stayDangerInformation

trip

Information

Disaster Risk (Low High)time t• People can’t know their correct conditions under an extraordinary situation.
• However they have to decide to evacuate or not, they recognize their own
expected utility and decide.
• This recognized expected utility is different from the correct one.
• The difference is defined as “Dynamics of Heterogeneity”.
3Difference of Recognition in Space and Time••

Dynamics of heterogeneity is influenced from space and time.
People who stayed near a sea may recognized a low expected utility.
0-10min10-20min

20-35min35-45min4

Importance of Evacuation Start Time
Pel et al.(2010) evaluate
by DTA simulator
on Rotterdam metropolitan area

Departure
Arrival

Early start

Slow start• People can arrive safety places if they start to evacuate earlier and theeffects will be amplified on network.
• A purpose of many disaster mitigation policies, emergency warnings andrisk education, is to evacuate earlier.
• Evacuation choice model can evaluate these policies.
5Purpose• Propose a formulation of a tsunami evacuation behavior modelaccounting for dynamics of heterogeneity in expected utility• Construct an algorithm to estimate parameters of the proposed
dynamic model• Validation (parameter estimation)

6

Outline• Background and Purpose• Formulation of dynamics of heterogeneity• Algorithm for parameter estimation• Validation

7

Formulation 1: Sequential Choice
Sequential choice model Fu & Wilmot (2004)
evacuatet0
choicestayevact1choice

Probability ofevacuation at each time t’
Probability ofevacuation at time t
Log-likelihood

Evacuation Timewindow T

evact2
choicestay= (( )== log: choice(evacuation or not),

=( )

|, )=

(1 −( )

t3
choice ・・・

stayexp(∑∀' exp(),)

, ′)

(1)

(2)

(3)

: observed state variable, : parameter, : utility8Formulation 2: Expected Utility
Evacuation Timewindow T

Expected utility EV
evacuatet0
choiceevac

stay

Expected utility EV,<state;t1
choicestaystate;

evacchoicestay

Probability ofevacuation at each time t’

9:evacB−>? C(=|=t2 ;choice;t3
choice ・・・stay・・・, )=exp(,∑∀' exp(, ′+ 89:( ,))+ 89:( , ′ ))(4)transition probability to next states ?

log >

∀'exp

?

@

utility at t+1,+ 89:?

@,×

expected value function at t+1=?

@,=0
(5)
9

Formulation 3: Dynamics of Heterogeneity
Evacuation Timewindow T

Expected utility EV
evacuatet0
choiceevacevac<statestay;t1
choice=t2 ;choicestay;t3
choice ・・・

stay

Recognized expected utility EV’state;

evacchoicestay・・・

Recognized Expected utility EV

9:,B−>? Clog >transition probability to next states ?
∀'exp?

@utility at t+1,+ 89:?

@,×

expected value function at t+1=?

@,≠0(5)’

10

Formulation 4: Maximum Likelihood• Su and Judd (2010) propose an estimation method for structural model using
constrained optimization approach.
• This method regards EV as parameter in finite period problem:
maxF, GHF,GH

subject toI F, GH = 9:,B−>

? C

(6)
log >

∀'exp?

@,+ 89:?

@,×=?

@,

=0J ∈ ∀( , ; , )Proposed approach• The recognized expected utility will be similar to the correct expected utility• This study propose that c is not equal to zero vector and is included in a range of
constraint L:
max F, GH′F,GH

subject toM F, GH′ ∈ L(L includes 0 )

(7)

11

Formulation 5: Range of Constraint• Specialize the range of constraint for parameter estimation.
• The recognized expected utility of the respective states has a different divergence fromthe correct expected utility.
• This difference in the amount of divergence can be explained by this formulation :∑∀ |I (F, GH)|≤ QR ⇔ > I F, GHNO∀

≤ NO QR = Φ(8)NO : number of state i
QR: upper constraint for the average amount of divergence from the corrected expected utility• This setting allows the recognized expected utility to be distributed flexibly.
• The dispersion of the distribution of the recognized expected utility is more unformalized
if an amount of each divergence of state i is limited : I F, GH < QVWX , ∀J.
• The analysis of distributions of EV’ clarify a tendency of people to recognize the futurestates because this approach obtain EV’ like a non parametric method.
12Outline• Background and Purpose• Formulation of dynamics of heterogeneity• Algorithm for parameter estimation• Validation

13

Proposed algorithm for parameter estimation

Proposed problem

max

F,GHF, GH′

subject to >∀I F, GH≤Φ(9)

• However the inequality constraint is a non-linear function.
• Number of parameters is more than the number of constraints.
• Apply a heuristic algorithm to solve and obtain a local optimum.
• Proposed algorithm is based on SQP (sequential quadratic programming)
and .

14

Apply SQP method
Primal Problem : Reformulated by using an exact penalty function as:

Descent
direction];

Update: Z;@ = Z; + \; ];(10)

Z=(11)

k: number of iteration^: penalty parameterZ;

Sub problem for the optimal search direction, 9: , [ = −hessian matrix of the Lagrangianfunction defined in the equation (9)(12)

This equation (12) is reformulated the quadratic problem of sk with no constraints.
(13)

Steepest descent direction(14)

Line search

(15)
15

Avoid the convergence to a local solution• The problem with non-linear constraints have many local solutions• Proposed algorithm add other direction for avoiding the convergence to one local solution• This heuristic searching algorithm is iterated and obtain the best solution
Objective function of primal problem

Steepest direction

Other direction

p16

Example of Calculation Process

The sum of divergence (Σ|ci|)

Computer : Intel Core(TM) CPU i5-4200M @ 2.50GHz & RAM 8.00GBLanguage : C One iteration : 5~60second

Log likelihood
17

Outline• Background and Purpose• Formulation of dynamics of heterogeneity• Algorithm for parameter estimation• Validation

18

Damage of Rikuzentakata city
Damages of the city

City Feature

population (people)

24,246dead and missing (people)

1,732
13

Flooded area (km2)
Structural damage to houses (houses)• City has ria coast and 2km square plain area
• Tsunami reached the coast about 37 - 45
minutes at the earthquake

3,368

Maps of Flooded area and Damaged building
Mountain

SeaSea20Surveys and Behavioral Data
Evacuation behavior data in Rikuzantakata
1. Questionnaire by MLIT (Ministry of Land, Infrastructure and Transport of Japan)
Days: September – December 2011
Respondent: 10,603 people (510 people in Rikuzentakata)
Questions: Preparation of Tsunami before the day, Evacuation behavior of the day

2. Questionnaire by University of Tokyo
Days: September 2012
Respondent: 373 people in Rikuzenntakata (31 people by face-to-face survey)
Questions: Evacuation behavior of the day
Dairy travel behavior in after-quake

Evacuation behavior of the day (Contents of Survey):
all trips after the quake; start and end time of each trip; trip purpose;route; mobility; traveling companion.
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Behavior example from evacuation data
ID2: woman & elderly
14:46 (Earthquake occurred) at home (1)14:50 moved by walk
14:55 picked up a family member at community center (2)15:00 moved by walk
15:05 to call for refugee at junior high school (3)15:05 climbed up a mountain (4)15:31 Tsunami arrived

421
322

Setting: Utility function and State variables=?,Utility of Evacuation

VJ_+'`J]?,Elapsed time after the quake [min](Divide by four: 0-5; 5-15; 15-25; 25-45)

Non-evacuationab=?,cV d_? + OWe If^? +?, ,Non-evacuation

ablikelihood=?, ,==exp

exp?

?

m,c g dJ h? +gV h_?,With someone

Ride a carFemale

Probability of Evacuation

Distance from sea [m](Divide by four: 0-400;400-1000; 1000-1500; 1500-)

exp

?,

+ exp

ab

exp

ab+ 89:?, , lj?,

+ exp

ab+ 89:?

+ m ,ab?,?,bi' jk ? + W` f]?

Had stayed home elderly?,?,++ 89:

Had assisted someone?, , lj?, , ljab

Choice result on time t of individual i

Other Settings:•People can choose to evacuate or not in 4 period.

•The number of observed state i is 386.

•EV at last period are given exogenously : EV(t4) = -0.01.

•

The number of EV which are assumed as parameter is 288.

•Transition probability p3(x’|xj) to next states is given as exogenously:•Time discount rate is given as 0.80 exogenously.
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Estimation Result
Dyanamic model &Heterogeneity(Φ = 300)
Attributes

Param.

t-StatDynamic model &No heterogeneity
Param.

t-StatStatic modelParam.

t-StatElapsed time

0.584

8.27*

0.687

10.84*

0.838

12.72*

Distance from sea-0.363-8.04*-0.369-7.53*-0.632-14.51*

Female

0.227

1.77-

-0.012-0.09-

0.426

4.98*

Ride a car-0.039-0.32*-0.087-0.71-

0.557

4.84*

With someone-0.737-4.74*-0.327-2.08*

0.185

1.43*

Had stayed home

0.253

1.92-

-0.026-0.19-

0.204

1.59*

Elderly-0.144-0.81*-0.260-1.52--0.341-1.97*

Had assisted

0.120

0.77*

0.437

2.99*

0.615

4.55*

Observations

1591

1591

1591

Likelihood at 0-1102.8-1102.8-1102.8

Final likelihood-643.0-732.0-885.8

ρ20.417

0.336

0.197

Adjusted ρ20.410

0.328

0.190
*: significant at 0.05
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Distribution of recognized expected utility• Decrease of average shows that people evaluate a low expected utility by time.

IJ
Average
Quantile(EV’ - EV)

Divergence ci between recognized EV’ and correct EV• Wider distribution in period 3 shows that people recognized the different future in
more urgent situation.

Period 1

Period 2

Period 3
state i25

Size of constraint range• “Φ = 400” is similar to a result of no constraint because the estimation result is far
from the border of “Φ = 400”.
• “Φ = 100” is a severe constraint because the s.d. of period 3 is small.
• Choose “Φ = 300” because the case is fitter.
Table Compare with the size of constraint range ΦValue of Φ
Final likelihood

0−732.0100−688.2200−657.5300−643.0400−630.9Divergence ci between recognized EV’ and correct EV (EV’-EV)
Period 1 Ave.
s.d.
Period 2 Ave.
s.d.
Period 3 Ave.
s.d.

-−0.17
0.52
−0.21
0.38
0.11
0.42

−0.10
0.88
−0.32
0.71
−0.24
1.35

−0.10
1.10
−0.18
1.08
−0.26
1.76

−1.18
1.68
0.12
1.65
−0.31
2.32

26

Flexibility for simulation• Simulated probabilities by proposed model are closer to observed probabilities

Simulated evacuation probability• Proposed model have a flexibility of evaluation because of its less-parametrically
O Proposed model(Φ=300)
O No heterogeneity (Φ=0)

Observed evacuation probability of each state i27

Transition of EV’ in Space
Near sea : People gradually had small expected utility ; had small s.d. in period 2
Far from sea : People gradually had big expected utility and had big s.d.

s.d. of EV’ - EVperiod 3

period 1

period 2

Distance from sea (km)period 1

Average of EV’ - EV28

Conclusion
Conclusions• Formulate a dynamic discrete choice mode with dynamics of heterogeneity.
• Algorithm for parameter estimation can avoid the convergence to a localoptimum.
• Proposed model provides a better goodness of fit and show the spatial andtemporal characteristics of dynamics of heterogeneity.
Future works• Need a sophisticated approach for exogenous variables :range of constraints,
time window, transition probability and line search vector.
• The EV’ in final period should be distributed, like a MXL model, to express
time windows which people recognized are distributed.
• The dataset has only behaviors of survived people.

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Thank you for your listening.
Mail: urata@person.kobe-u.ac.jp
References:Pel, A.J., Hoogendoorn, S.P., Bliemer, M.C.J.: Impact of variations in travel demand andnetwork supply factors for evacuation studies, Transportation Research Record, Vol.
2196, pp. 45-55, 2010.
Fu, H., Wilmot, C.G.: A sequential logit dynamic travel demand model for hurricane
evacuation, Transportation Research Record, Vol. 1882, pp. 19-26, 2004.
Su, C.J., Judd, K.L.: Constrained optimization approaches to estimation of structuralmodels, Econometrica, Vol. 80, pp. 2213-2230, 2012.
Urata, J., Hato, E., Yaginuma, H., Modelling of Tsunami Evacuation Behavior Accounting
for Dynamics of Heterogeneity in Expected Utility, Journal of Japan Society of Civil
Engineers, Ser. D3 (Infrastructure Planning and Management), accepted. (in Japanese)30