Comparison of Time-Dependent Sequential Logit and Nested Logit for ...

Comparison of Time-Dependent Sequential Logit and Nested Logit for Modeling Hurricane Evacuation Demand Ravindra Gudishala and Chester Wilmot to effectively capture the effect of these dynamic conditions and thereby improve evacuation management (4). Thus, when demand models properly incorporate dynamic policy variables the models become more useful tools. For example, if an official wants to evaluate the influence of issuing an evacuating order at a certain time, that evaluation can be done by changing the appropriate input fed into the dynamic model. Recent application of the TDSLM has resulted in counterintuitive results. Normally one would expect the predicted number of evacuees, summed over the entire analysis period, to add up to the observed number of evacuees. However, the authors’ recent experience with the application of the TDSLM on survey data from Hurricane Gustav proved otherwise (5). This problem may arise from failure of the TDSLM to account for the dependency among interdependent decisions and also for treating error terms as independent over time periods. To remedy this condition, a time-dependent nested logit has been developed that takes interdependency and correlation of error terms of a household over a sequence of time periods into account. In this paper the sequential choice paradigm proposed by Fu and Wilmot is briefly discussed and used to derive two different specifications by using two different assumptions in modeling hurricane evacuation demand. The two choice models on hurricane evacuation data collected by Gudishala on Hurricane Gustav are then applied, and the results from the two models are compared (5). It is expected that relaxing the assumption that sequential decisions of a household are independent will result in a model capable of producing more accurate hurricane evacuation behavior.

Models that predict hurricane evacuation demand can play a crucial role in developing and evaluating alternative evacuation policies and plans. However, to evaluate alternative policies effectively, evacuation demand models should be sensitive to time varying characteristics of a storm and the contextual conditions surrounding an evacuee. The timedependent sequential logit is one such model, but it makes use of restrictive assumptions about the dynamic choices made by evacuees. A new model, a time-dependent nested logit model, relaxes those assumptions. It was formulated and derived in this study, and its performance was then compared with that of the time-dependent sequential logit model by applying both models to data from Hurricane Gustav. The results indicated that the time-dependent nested logit model has better predictive capability than the time-dependent sequential logit model.

Hurricane evacuation demand models are used to forecast evacuation demand as well as determine the value that evacuees place on various factors that affect their evacuation choices. Demand models can be either static or dynamic. Dynamic demand models relate an evacuee’s choice to the dynamic characteristics of an approaching hurricane (e.g., hurricane strength, direction, and location), actions of officials (e.g., evacuation orders, contraflow), and evacuees’ socioeconomic characteristics and the conditions in which they reside (e.g., income, type of house, vehicle ownership, house in flood-prone area, time of day). One example of a dynamic hurricane evacuation model is the time-dependent sequential logit model (TDSLM) developed by Fu and Wilmot, which relates timedependent factors to a sequence of choices made by an evacuee in multiple time periods (1). However, static demand models consider only static characteristics and disregard the dynamic nature of evacuation. Examples of static hurricane evacuation demand models are the logit model developed by Wilmot and Mei (2) and the mixed logit model developed by Hasan et al. (3). Static models estimate the total demand for an analysis period but do not describe how demand varies during the period. Dynamic models such as TDSLM allow analysis of dynamic conditions such as a developing storm, changing local conditions, and actions taken by emergency officials. It is often in the interest of emergency management and transportation officials to be able

Literature Review Static evacuation demand models predict the number of households that will evacuate but not when households will evacuate. In contrast, dynamic evacuation models are capable of predicting the number of evacuation households as well as their time of departure. Traditional evacuation demand modeling uses a two-step approach: the total evacuation demand is estimated in the first step by using logistic regression or other variants of regression-based models, and in the second step, the estimated demand is spread across time by applying an evacuation response curve or departure response curve (1). The response curves are assumed to follow a Weibull, uniform, sigmoid, or Poisson distribution. Examples of studies that used an evacuation response curve to model departure times include Liu et al. (6), Yuan et al. (7), Cova and Johnson (8), Jonkman (9), Lindell (10), Kalafatas and Peeta (11), and Xie et al. (12). PBS&J, Inc., developed a crossclassification type of evacuation demand model that relates evacuation demand to hurricane and household characteristics (13, 14).

Department of Civil and Environmental Engineering, Louisiana State University, 3418 Patrick F. Taylor Hall, Baton Rouge, LA 70803. Corresponding author: R. Gudishala, [email protected]. Transportation Research Record: Journal of the Transportation Research Board, No. 2312, Transportation Research Board of the National Academies, Washington, D.C., 2012, pp. 134–140. DOI: 10.3141/2312-14 134

Gudishala and Wilmot

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The sequential choice paradigm as applied by Fu and Wilmot postulates that the decision to evacuate or stay in the face of an approaching hurricane can be described as a sequence of independent binary decisions over time (1). Each binary decision consists of the decision on whether to stay or to evacuate. Although these decisions are probably made almost continuously in reality, the theory proposes discretizing time into intervals for computational feasibility. Thus, the assumption is that the evacuation decision of each household can be described as a set of independent sequential binary choices made at discrete points in time. Specification and Derivation of Choice Probabilities for the TDSLM

n =1

t =1

(3)

The log likelihood function shown here should be custom written into an estimation package. The alternative method, in the absence of custom building a function, is expounded in Fu and Wilmot (1). In deriving the probability of a sequence of choices, it was assumed that the choice made by a household in time period t is independent of choices in other time periods. However, one would expect that unobserved factors that affect the choice in one period would persist into the next time period, introducing dependence between choices over time. The issue of choice dependency over time is taken into consideration in deriving the time-dependent nested logit model (TDNLM) in the next section. Specification and Derivation of the TDNLM The sequential choice paradigm described earlier is used in this section with a slight modification to derive a TDNLM. The decision to evacuate or stay in the face of an approaching hurricane is treated as a sequence of interdependent binary decisions over time. Each binary decision consists of the decision on whether to stay or to evacuate. Thus, the assumption that the evacuation decision of each household described as a set of independent sequential binary decisions by the TDSLM model is relaxed and binary decisions are treated as interdependent over time. The derivation of the time-dependent nested logit is described, for convenience, in the context of three sequential time periods: t1, t2, and t3. Assume that the decision-making process of a household proceeds in a sequential fashion as shown in Figure 1. In the first time period, the household chooses either to evacuate or stay. However, it is reasonable to assume that in making the decision, the household will take into account the conditions existing in the first time period as well as anticipated conditions in the next time periods, t2 and t3. It is suggested that the household will repeat this decisionmaking process until the household either evacuates or reaches the

Stay

iod

t2

Evacuate

pe r Evacuate

Stay d

t3

The probability of not evacuating (i.e., staying) in time period t, Pnst, is (1 − Pnet) because there are only two choices in each time period. If choices are independent over time, then the joint event that household n chooses to evacuate in time period t* after choosing to not evacuate in all t* − 1 previous periods, can be written as

Ti m e

(1)

e

expβn xnet + expβn xnst exp β n x net

Ti m

Pnet =

t = t *−1

pe

The TDSLM is derived by applying the stated choice paradigm as follows: a household, n, threatened by an impending hurricane chooses either to evacuate or stay in each time period t, and these decisions occur sequentially over time until the household either evacuates in one of the time periods or fails to evacuate entirely. By applying utility theory, a household is assumed to acquire a certain level of utility from each alternative choice in each time period. The utility that household n obtains from choosing the alternative to evacuate, e, in time period t is Unet = βn xnet + εnet and from the alternative choice to stay, s, in time period t is Unst = βnxnst + εnst where t = 1, 2, 3, . . . , T; xnet and xnst are vectors of observed variables; β is a vector of parameters; and εnet and εnst are unobserved random error terms that are independently and identically Gumbel distributed. Thus, in keeping with discrete choice theory, the probability that household n will choose to evacuate in time period t, Pnet, can be described by a standard binary logit model as shown in Equation 1 because it is assumed that choices in one time period are unaffected by conditions in others:

n= N

L (β ) = ∏ ( Pnet ) ∏ (1 − Pnet )

t1

Sequential Choice Paradigm to Model Hurricane Evacuation Demand

tive choices, then the parameter vector β can be estimated from a sample of N households by using the following likelihood equation:

rio d

A new approach in evacuation demand modeling is to model evacuation demand and departure time simultaneously. Fu and Wilmot developed a model that predicts evacuation demand and time of departure simultaneously (1, 15, 16). Fu and Wilmot were the first to pioneer the idea of modeling time-dependent evacuation demand by using a sequential logit behavior model (1). Surprisingly, very few studies exist that follow that approach.

where Pnet is given in Equation 1. If it is further assumed that households make evacuation decisions independent of each other and that households display the same “taste” or value system in evaluating the attributes of alterna-

pe e m Ti

(2)

t =1

rio

t =1*−1

Pnet* = Pns1Pns 2 Pns 3 i Pnst*−1Pnet* = Pnet* ∏ (1 − Pnet )

Evacuate FIGURE 1 Conceptual representation of sequential decision-making process.

Stay

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end of the time periods considered. In the example shown here, the number of periods is limited to three for illustration purposes. If a household is to base its evacuation decision on the utilities of each alternative in the present and the future, a method must be available to compute the utilities of future time intervals. Assume that a household in the first period, t1, decomposes its second-period utility into a known and unknown part, for example, Une2 = Vne2 + εne2, Uns2 = Vns2 + εns2, and assume that ε follows a Gumbel distribution. This unknown factor becomes known to the household in the second period, so that the second period choice involves maximization over known utilities for choices “evacuate” and “stay.” However, in the first period the factor is unknown. Because the Gumbel distribution is stable under maximization, the maximum of two Gumbel distributed parameters, max(Une2, Uns2), is again Gumbel distributed with parameters (1/µ ln(exp(µη1) + exp(µη2)), µ), where η is a location parameter and µ is a scale parameter defining the distribution (17). Thus, the expected value of the distribution of max(Une2, Uns2) is equal to (1/µ ln(exp(µη1) + exp(µη2))). This expected value is nothing but the logsum term used in the nested logit model that links upper and lower levels (18). This observation was first noted and used by Rust (19). (For exposition purposes, the logsum term is labeled LS in further discussion.) When applied in the context of the current discussion, this feature can be used to incorporate the utility of the best alternative from a future time period (i.e., the alternative with the maximum utility in the next time period) into the assessment of the utilities of the present. Therefore, the utility of household n evacuating in time period t1, Une1, becomes Vne1 + εne1 and Uns1 becomes Vns1 + εns1 + LS2. Here, LS2 is defined as a logsum term and is given by LS2 = λ ln ( exp (Vne 2 ) + exp (Vns 2 ))

(4)

where 1 − λ represents the correlation coefficient between time period t1 and time period t2. Logsum term LS2 takes into account the utility of staying in time period t2 and the utility of staying in time period t3. The implication is that a household when making a decision in time period t1 considers all the projected conditions of an impending hurricane from future time periods t2 and t3. This can be verified by the fact that LS2 = λ ln ( exp (Vne 2 ) + exp (Vns 2 )) Substituting the value of Vns2 in the above will result in the following expression: LS2 = λ ln ( exp (Vne 2 ) + exp ( λ ln ( exp (Vne 3 ) + exp (Vns 3 )))) Because Une1 and Uns1 are now known, the probability of choosing to evacuate or stay in time period t1 can be readily computed. That is, the probability that household n evacuates in time interval 1 can be written as Pne1 = prob (U ne1 ≥ U ns1 ) = prob (Vne1 + ε ne1 ≥ Vns1 + ε ns1 + LS2 )

(5)

Because εne1 and εns1 are assumed to be Gumbel distributed, the difference between them is logistic distributed, and the above equation can be expressed as a binary logit model of the following form: expVne1 Pne1 = Vne 1 exp + expVns1 + LS2

(6)

If the household did not evacuate in time interval 1, the process will be repeated in time interval 2, taking into account the expected utility from time period t3 in evaluating the choices available in time period t2. On the basis of the argument made in the previous paragraph, the marginal probability that the household evacuates in time interval 2 can be written as Pne 2 =

expVne 2 exp + expVns 2 + LS3 Vne 2

(7)

and the conditional probability that the household evacuates in time interval 2 is Pne 2 = (1 − Pne1 ) Pne 2 expVne1 expVne 2     Pne 2 =  1 −  Vne 1 Vns 1 + LS2  Vne 2    exp + expVns 2 + LS3  exp + exp

(8)

Similarly, the marginal probability that household n evacuates in time interval 3 is Pne 3 =

expVne 3 exp + expVns3 Vne3

(9)

There is no logsum term in the above equation because it is the final time period. The conditional probability that the household evacuates in time interval 3 is Pne 3 = (1 − Pne1 ) (1 − Pne 2 ) Pne 3

(10)

The derivation shown in this section with three levels of nests can be extended to any number of time periods or levels. However, adding more levels adds to the complexity of the estimation process and might demand more resources and sample data. To give an example, the estimation of a three-level nested logit model typically involves estimation of coefficients of explanatory variables, estimation of alternative specific constants, estimation of two correlation coefficients, and finally estimation of two logsum terms. Given the highly nonlinear nature of nested logit models, and the added burden of many parameters, most estimation routines require an analytical second derivative for producing a reliable set of estimates. Adding many levels to an already complex problem compounds the estimation problem considerably. Most of the popular estimation packages, such as SAS, Stata, and LIMDEP, do not allow users to specify levels beyond four while estimating a nested logit model. Therefore, to estimate a nested logit model with more than four levels, special programs that have the capability of using sophisticated optimization routines are needed. Examples of such software products include TOMLAB, GAMS, and AMPL. For a sample of N households, the vector of parameters β can be estimated by applying maximum likelihood to the following log likelihood expression: n= N

L (β ) = ∏ ∏ Pnetyni n =1

(11)

t

where yni = 1 if household n chooses to evacuate in time interval t and 0 otherwise, and Pnet is the conditional probability of evacuating in time interval t.


Application The TDSLM and TDNLM were applied to evacuation data from Hurricane Gustav. The evacuation behavior during Hurricane Gustav was collected as part of a study aimed at developing a new data collection procedure and explained in detail in Gudishala and Wilmot (20). Part of the study involved collecting revealed preference data from 300 households in the New Orleans, Louisiana, area and surrounding parishes in 2009 by using a self-administered, mail-out mail-back survey. In the revealed preference survey, respondents were asked to report their evacuation behavior for Hurricane Gustav. Specifically, sampled respondents were asked about their evacuation decision in addition to their socioeconomic data and other information such as evacuation mode and date, time of evacuation, and type of refuge sought. The revealed preference data were enhanced with Hurricane Gustav’s storm-related information by retrieving information from the archives of the National Hurricane Center. Dynamic information, such as hurricane category at every time interval, actions taken by public officials, the predicted path of the storm, and the potential storm surge for the surveyed area, was appended to the collected data. Model Estimation Data collected from the revealed preference study were rearranged for the estimation process. Each row of observations from a single household was expanded into 22 rows. In the expanded data, each row represented a time period of 6 h, and the value of dynamic variables varied between these time periods. The 22 rows represented a total duration of 22 × 6 = 132 h, which was the total length of the analysis period considered. Explanatory variables appeared in columns in the data set. The intersection of each row and column was populated with the value taken by a particular explanatory variable in the corresponding time interval. Thus, the data presented time-dependent conditions experienced by the sampled households during Hurricane Gustav. The dependent variable was whether a household chose to evacuate in each time interval. When a household reported evacuating in a certain time period, no further rows of data were included in the data set for that household. For example, if a household reported evacuating in time period 13, then only 13 rows of data would appear for that household. If a household did not evacuate at all, all 22 rows of data were present in the data set. The variables considered for inclusion in the model were selected on the basis of previous experience and also on other research conducted by Baker (21) and Fu and Wilmot (1). The variables considered for both models are discussed below.

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scale parameters of 6 and 0.6, respectively. This transformation produces a skewed distribution with a mode at 249 mi, a fat tail to the right declining to a probability density value of 0.0001 at 700 mi, and reducing to a probability density value of zero at a distance of zero mi. Hurricane Category The hurricane category variable was entered as a variable with potentially five values corresponding to the five categories of hurricanes in the Saffir–Simpson scale. In Hurricane Gustav, the storm category ranged from a maximum of four to a minimum of two as the hurricane approached the coastline, and these dynamic values were entered into the data. Evacuation Order An evacuation order refers to the action taken by public officials specifying the type and timing of an evacuation order issued. This variable was entered as a dummy variable acquiring the value of 0 or 1. A mandatory or voluntary order was represented by 1, and no evacuation order was represented by 0. Time of Day The variable time-of-day (TOD) was represented with the use of three dummy variables, TOD1, TOD2, and TOD3. If the TOD was between midnight and 6 a.m., then TOD1 was coded as 1, and 0 otherwise. If the TOD was between 6 a.m. and noon, then TOD2 was coded as 1, and 0 otherwise. TOD3 represented the time between noon and 6 p.m. and was coded as 1 if the TOD fell in that category, and 0 otherwise. The time between 6 p.m. and midnight was used as the base and was represented in the data with zeros on TOD1, TOD2, and TOD3. Storm Surge The variable storm surge represents the threat of flooding a household may face and enters the models as a dummy variable. Whenever the storm surge from Hurricane Gustav resulted in an estimated inundation depth greater than 10 ft above ground level at a household’s geographical location, the variable storm surge was coded as 1, and 0 when it was less than 10 ft. The value of 10 ft was used because the home sites are often raised above mean ground level as a result of the construction of retention ponds or lakes, depressed roads, and raised foundations.

Time-Dependent Distance The time-dependent distance from the center of a hurricane to a survey respondent’s home is a measure of the proximity of the hazard and is expected to play an influential role in evacuation behavior. However, the effect of proximity is unlikely to be linear because changes in proximity when the storm is distant will have less effect than when the storm is close. Paradoxically, when the storm is very close, the threat is greater, but evacuation as a means of mitigation is no longer appropriate because there is insufficient time to clear the area. To accommodate this nonlinear effect of the proximity of the storm on evacuation behavior, time-dependent distance was transformed by using a lognormal probability density function with location and

Income The variable income entered the models as a quantitative variable and was entered in thousands of dollars. Because all households reported their incomes by using a range criterion, the midpoint of the range was used for estimation purposes. Vehicles Owned The variable vehicles owned entered both models as a quantitative variable.

138


TABLE 1 Estimation Results for TDSLM

TABLE 2 Estimation Results for Nested Logit

Explanatory Variable Name

Estimate

Standard Error

Lognormal (time-dependent distance) Hurricane category Evacuation order TOD1 TOD2 TOD3 Storm surge Income Vehicles owned Constant

769.52

18.58

0.47 0.68 1.23 1.92 0.84 0.91 0.0028 0.0798 −6.29

0.07 0.22 0.29 0.29 0.30 0.377 0.0014 0.0682 0.36

t-Statistic


Estimate

Standard Error

t-Statistic

4.26

Lognormal (time-dependent distance) Hurricane category Evacuation order TOD1 TOD2 TOD3 Storm surge Income Vehicles owned Constant Lambda

1,402.96

400.56

3.56

6.58 3.08 4.20 6.62 2.72 2.43 2.05 1.16 −17.07

Note: Number of cases = 288; number of observations = 6,224; L(0) = −3,309; L(C) = −1,299; L(β) = −718.94; ρ2 = .44.

Results from Estimation The TDSLM was estimated by using Equation 4; the results of the estimation are shown in Table 1. All parameters have the correct sign, and with the exception of the parameter for the variable vehicles owned, all were found to be significant at the 95% significance level. The variable vehicles owned was retained in the model because it is considered a relevant variable in the decision to evacuate or not. Among the TOD explanatory variables, the variables TOD2 and TOD3 are positive and large, suggesting that a household is more likely to evacuate during the daytime as compared with evening and particularly in preference to the early hours of the morning as represented by the base case from midnight to 6 a.m. The coefficient of the transformed time-dependent distance is large as a result of the transformation used. Among the dummy variables, storm surge has the second largest parameter value after time of day. Its value is higher than for the variable evacuation order. The high value implies that households living in a high potential storm surge area are more sensitive to their vulnerability to flooding than they are even to evacuation orders from officials, although the time at which households will evacuate is still strongly affected by the time of day. The variable income is normally expected to have a direct relationship to the evacuation decision. That is, a household with higher income is expected to evacuate more readily than a household with a low income. And as expected, the parameter associated with the variable income has a positive value. Similarly, a household that owns a vehicle is more likely to evacuate than is a household with no vehicle. This finding is reflected in the sign of the parameter associated with the variable vehicles owned. The TDNLM was estimated with Equation 11. A full information maximum likelihood procedure programmed in Matlab’s custom environment, TOMLAB, was used to maximize Equation 11. The TDNLM applied on Hurricane Gustav data used 22 time intervals, or levels. Each level represented a 6-h time period. In estimating the model it was assumed that λ, the factor representing the correlation between time intervals, is constant across all time periods and only one constant was used in the representative portion of the utility for all levels. If a unique constant is used in each time period, the model is overspecified and the constants ensure that the observed evacuations are reproduced in each cell without any contribution from the

0.73 1.87 2.34 2.81 1.26 0.26 0.01 0.16 −9.39 0.644

0.17 0.43 1.06 1.16 0.93 0.34 0.00 0.15 2.46 0.12

4.24 4.36 2.21 2.41 1.34 0.766 1.89 1.07 −3.82 5.5

Note: Number of cases = 288; number of observations = 6,224; L(0) = −2,067; L(C) = −1,960; L(β) = −704; ρ2 = .64.

remaining variables. The results from the estimation of the model are shown in Table 2. All the parameter estimates have the right signs, and the majority of the parameters are significant. The model specification used for the TDSLM was retained in the TDNLM for comparison purposes (parameter values, parameter significance), but it is possible that a different specification in the TDNLM could produce better estimation results. The parameter values from corresponding explanatory variables in the two models are different. Some variables appear to be more influential in one model than in the other. For example, time-dependent distance, hurricane category, time of day, income, and vehicles owned appear to have a greater effect in the TDNLM than in the TDSLM. In contrast, storm surge seemed to have a greater effect in the TDSLM. The parameter that is not estimated in the TDSLM but estimated in the TDNLM is the parameter λ. As explained elsewhere, the parameter when subtracted from 1 indicates the correlation between two successive time periods. Because it was assumed that λ is constant across all time periods, 1 − λ = 1 − 0.64 = 0.36 is the estimated correlation coefficient between all time periods among all households. Prediction and Comparison The estimated model parameters from both models were used to predict the observed hurricane evacuation demand. Figure 2 shows the results from the comparison. The predictions from the two models were similar in almost all time intervals, except in time intervals 3, 10, and 21, in which the TDSLM overpredicted the number of evacuations. Both models overpredicted in time interval 14 and underpredicted in time interval 18. Two factors, the root mean square error (RMSE) and the likelihood ratio index (ρ2) were used for objective comparison purposes. The RMSE for the TDNLM was 3.16, and for the TDSLM the RMSE was 4.63, showing a 32% better predictive performance of the TDNLM over the TDSLM. The likelihood ratio index shown in Tables 1 and 2 uses the log likelihood of the models with constants as the base and suggests good fits for both models although a ρ2 value of .64 for the TDNLM is considerably better than the .44 value obtained with the TDSLM. The number of TDNLM-predicted evacuations when summed over all 22 time intervals matched the sum of observation evacuations,


139

50 45

Number of Evacuations

40 35 30 25

TDSLM

20

TDNLM

15

Observed

10 5 0 -5

0

5

10

15

20

25

Time Intervals FIGURE 2 Comparison of predictions from TDSLM and TDNLM with observed evacuations.

whereas the sum of TDSLM-predicted evacuations exceeded the observed evacuations by 40. Thus, it appears that the TDNLM is better than the TDSLM in predicting hurricane evacuation demand.

TABLE 3 Estimation Results for TDSLM Model for External Validation

External Validation


Estimate

Standard Error

Two new models were developed to validate externally the TDSLM and TDNLM models. The specification of the two models was similar to what is shown in Table 1 and Table 2 but without the variables “income” and “vehicles owned.” The two variables had to be excluded because the data set on which these models were tested did not have data on income and vehicle ownership. The results from the estimation are presented in Table 3 and Table 4. The estimated models were then applied to data from Hurricane Georges. Before the application of the models, the estimated constants (−5.91 and −8.46) were recalibrated to reflect the fact that unobserved factors are different in the predicted context than in the context in which the model was estimated. The recalibration procedure described by Train was used (18). The results from the application of the models on the Georges data are shown in Figure 3. Clearly, results demonstrate that the TDSLM and the TDNLM predict with equal accuracy. To objectively compare the predictions, RMSE values were computed and compared. RMSE values (19.24 for TDSLM and 19.64 TDNLM) indicate that the two models were equally accurate.

Lognormal (time-dependent distance) Hurricane category Evacuation order TOD1 TOD2 TOD3 Storm surge Constant

760.15

179.84

Conclusions The results from the study showed that the TDNLM model performed better than the TDSLM. The better performance could possibly be attributed to the capability of the TDNLM in accommodating dependence between sequential choices made by a household over time. The TDSLM is simple and easy to estimate and requires less computing time in estimation than does the TDNLM, which uses full information maximum likelihood in its estimation. The TDNLM estimated that the correlation coefficient between the probabilities of choices among adjoining time periods is 0.36, suggesting a relatively weak dependence between sequential choices. This is a single estimate for all levels combined, and it may

0.47 0.66 1.23 1.92 0.83 0.91 −5.91

0.07 0.22 0.29 0.29 0.30 0.377 0.32

t-Statistic 4.22 6.57 2.99 4.19 6.63 2.71 2.41 −18.01

Note: Number of cases = 288; number of observations = 6,224; L(0) = −3,309; L(C) = −1,225; L(β) = −722.43; ρ2 = .41.

TABLE 4 Estimation Results for TDNLM Model for External Validation Explanatory Variable Name

Estimate

Standard Error

Lognormal (time-dependent distance) Hurricane category Evacuation order TOD1 TOD2 TOD3 Storm surge Constant Lambda

1,358.52

268.74

5.05

0.72 1.82 2.30 2.77 1.24 0.16 −8.46 0.65

0.12 0.40 0.54 0.44 0.55 0.33 1.19 0.08

5.73 4.48 5.12 2.80 4.18 0.48 −7.09 8.05

t-Statistic

Note: Number of cases = 288; number of observations = 6,224; L(0) = −2,067; L(C) = −1,959; L(β) = −708; ρ2 = .64.

140


120

Number of Evacuations

100 80 60

TDSLM TDNLM

40

Observed 20 0 0

5

10

15

20

25

-20 Time Intervals FIGURE 3 Comparison of predictions from TDSLM and TDNLM on Hurricane Georges.

be that the correlation could be higher in certain time periods than in others. However, the correlation may be one explanation for the relatively good performance of the TDSLM, which assumes independence between sequential choices. External validation of both models, TDSLM and TDNLM, demonstrated that the two models were equally accurate in predicting the time-dependent evacuation demand of Hurricane Georges. Acknowledgment The authors express their gratitude to the Louisiana Transportation Research Center for financially supporting this research. References 1. Fu, H., and C. G. Wilmot. Sequential Logit Dynamic Travel Demand Model for Hurricane Evacuation. In Transportation Research Record: Journal of the Transportation Research Board, No 1882, Transportation Research Board of the National Academies, Washington, D.C., 2004, pp. 19–26. 2. Wilmot, C., and B. Mei. Comparison of Alternative Trip Generation Models for Hurricane Evacuation. Natural Hazards Review, Vol. 5, No. 4, 2004, pp. 170–178. 3. Hasan, S., S. Ukkusuri, H. Gladwin, and P. Tuite. Behavioral Model to Understand Household-Level Hurricane Evacuation Decision Making. ASCE Journal of Transportation Engineering, Vol. 137, 2011, pp. 341–348. 4. Wolshon, B., E. Urbina, C. Wilmot, and M. Levitan. Review of Policies and Practices for Hurricane Evacuation. I: Transportation Planning Preparedness, and Response. Natural Hazards Review, Vol. 6, Aug. 2005, pp. 143–161. 5. Gudishala, R. Development of a Time-Dependent, Audio-Visual, Stated Choice Method of Data Collection for Hurricane Evacuation Behavior. PhD dissertation. Louisiana State University, Baton Rouge, 2011. 6. Liu, Y., X. Lai, and G. Chang. Two-Level Integrated Optimization System for Planning of Emergency Evacuation. ASCE Journal of Transportation Engineering, Vol. 32, No. 10, 2006, pp. 800–807. 7. Yuan, F., L. D. Han, S. Chin, and H. Hwang. Proposed Framework for Simultaneous Optimization of Evacuation Traffic Destination and Route Assignment. In Transportation Research Record: Journal of the

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