Home > Sensitivity analysis of a gravity-based land use model: the importance of scenarios∗

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E. Borgonovo, M Percoco, R. Polizzi,K. Kockelman December 2014

Abstract Traffic and land use simulation models are complex mathematical machines that reflect social, geographic, economic and demographic factors. They rely on large datasets, and the associated computational complexity makes it difficult for analysts to readily appreciate how variations in one of the exogenous variables or parameters impact the simulation results. In such settings, computationally frugal methods need to be used. Analysts typically run the models through a very limited set of selected scenarios, which help the analyst understand what results the model produces under alternative exogenous contexts. In this work, we address the issue of tackling the scenario analysis of complex computational codes using local sensitivity analysis with interactions that allow analysts to extract new insights when running a model through scenarios. Keywords: Sensitivity analysis; Traffic and Land use modelling; G-LUM.

∗The authors wish to thank Laura Cavalli and Guangmin Wang for useful comments on earlier drafts. Finan-

cial support from the CerTeT Research Centre, within the research grant of Autostrade per L��Italia, is gratefully acknowledged by E. Borgonovo, M. Percoco, L. Cavalli, and R. Polizzi. Affiliations: Emanuele Borgonovo and Ric- cardo Polizzi, Department of Decision Sciences, Bocconi University. Marco Percoco, Department of Policy Analysis and Public Management, Bocconi University, Milan, Italy. Kara Kockelman Department of Civil, Architectural and Environmental Engineering, University of Texas Austin, USA.

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Decision-makers in the transport sector often benefit from the use of quantitative models pre- dicting outcomes of given policy actions (the tolling of a highway, the construction of a tunnel or other interventions that impact transport costs). Land use and transportation forecasting models are usually complex and computationally inten- sive, since they need to account for multiple factors, ranging from the social and economic features of the region of interest to physical layout and congestibility of the transportation network, and the travel times and costs between locations to travellers�� preferences for modes, destination, routes, and times of the day. Realistic traffic simulations are encoded in computer software, where a series of interactive calculations are performed. The specific relationship that binds the endogenous to exogenous vari- ables is not obvious to analysts. A systematic approach is then needed to gauge the response of the endogenous variables or outputs of interest to changes in the exogenous model inputs. Sensitivity analysis plays a crucial role in evaluating the reaction of model outputs to changes in the baseline scenario and/or other assumptions. Changes in model outputs are then key for evaluating the reliability of simulations and the importance of input variables and parameters. In this paper, we propose a systematic approach for the conduct of sensitivity analysis of complex transportation and land use models. Sensitivity analysis of such models is currently gaining attention of transportation scholars, and several recent studies have appeared in the literature. UrbanSim is an urban land use simulation model with nine sub-models describing the spatial configuration of a given city as a function of transport costs and land use regulations (Waddell (2002), Waddell et al. (2003)). Ševc��kov�� et al. (2007) applied Bayesian melding to UrbanSim to assess uncertainty in model forecasts and found that this approach generates better predictions than frequentist model averaging. Ševc��kov�� et al. (2011) used the same approach to assess the effects of Alaskan Way Viaduct on downtown Seattle��s land use and traffic patterns. Here, we consider an analyst who is evaluating a complex simulation model across different scenarios. We rely on a new and different method that produces finite change sensitivity indices for the variation of exogenous variations across scenarios. Calculation of the indices is computationally 1

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2 Sensitivity and Scenario Analyses: An Overview

Scenario analysis and sensitivity analysis imply different but interrelated concepts. Scenarios can be defined as ��different possible future states of a system" (Tietje (2005)), ��stories about how the future might turn out" (O��Brien (2004)). A scenario analysis1 allows an analyst or decision-maker to forecast future outcomes in accordance with his/her state of knowl- edge (Jungermann and Thuring (1988)) as the final goal of a scenario analysis is the consistent elicitation of predictions, the cognitive and methodological aspects of scenario generation have been

1 If it is implemented according to the canons defined in the literature.

2

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2 We refer to O��Brien (2004) and Tietje (2005) for a complete summary of literature findings.

3

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3 Sensitivity measures for scenario decomposition of LUMs

3.1 Methodology

In a scenario analysis, the change from scenario 0 to scenario 1 of the exogenous variables induces the change Ay = y1 −y0 in the endogenous variable. A first way to decompose this change is to suppose that f(x) is r times differentiable at x0 and to use a multivariate Taylor expansion of Ay = y1 − y0: (3) Ay = y1 − y0 = P

=1 f0(x0)Ax + P =1 f

00

(x0)AxAx + ...+

+... + P

1=1 P 2=1 ...P r=1 f

r

12r (x0)Ax1 ...Axr + o(khk

) The first-order derivatives in eq. (3) are the sensitivity measures commonly used for comparative statics (CS) (Samuelson (1947); Quirk (1997); Takayama (1993)), i.e., (4) CS = f0(x0) (k = 1,2,...,K) To be noted is the fact that in our empirical analysis derivatives are computed numerically, so that we do not need to explicitly consider the generating function f(x). 4

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=1 f0(x0)Ax, then the fraction of the change in y across two scenarios is

(5) D = df df = f0(x0)dx P

=1 f0(x0)dx

= CSdx P

=1 CSdx

(k = 1,2,...,K) where D is the differential importance measure, and coincides with the fraction of the differential change in y provoked by a slight perturbation of x [Borgonovo and Apostolakis (2001), Borgonovo (2007)]. However, application of CS and D may lead to partially informative results, if the changes in y are not infinitesimal. Also, if f(x) is not differentiable CS and D are not applicable. To model input changes, we observe that they are generally not infinitesimal in a scenario analysis. A decomposition that accounts for these features is the functional ANOVA decomposition, widely used in the field of mathematical statistics (Efron and Stein (1981), Oakley and O��Hagan (2004), Rabitz and Alis (1999), Saltelli et al. (2000), Sobol�� (2003), Sobol�� (2001), ). The decomposition is based on a sequence of nested integrations. Therefore, the parameter space, ��X, is complemented by a Borel algebra and a measure (��,A,��). The theory has been widely discussed in the literature, and a thorough overview is outside the purpose of the present work. Relevant results of this paper appear through an application of Theorem 1[Borgonovo (2010), Borgonovo and Peccati (2011)]. Theorem 1. Let (��X,A,��) be a measure space and f �� L1(��) be a measurable function. Then, if for any x0 and x1 belonging to �� and for any measure �� satisfying d�� =

Y

=1

d��, the following holds: f(x) = f0 +

X

=1

f(x) +

X

f(x,x) + ... + f12(x1,x2,...,x) = (6) = f0 +

X

=1

X

12k

f12k (x1 ,x2 ,...,xk ) 5

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Y

=1

��(x1

− x0 )dx, one

obtains (8) Ay = f(x1) − f(x0) =

X

=1

Af +

X

Af + ... + A12f where3 (9) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Af = f(x1

,x0 ∼) − f(x0)

Af = f(x1

,x1

,x0 ∼()) − Af − Af − f(x0)

... Eq.(8) suggests that Ay is the sum of 2 −1 terms of increasing dimensionality. All these terms are obtained via a finite difference operator. For instance, Af is the difference between a) f(x1

,x0 ∼

), the value attained by the simulation model when all variables are at scenario 0, but x1

, which

is moved at scenario 1, and b) f(x0). The second order terms, Af are the difference between a) f(x1

,x1 ,x0 ∼()

), the value attained by the simulation model when all variables are at scenario 0, except for x1

and x ; b) f(x0), and c) Af and Af are subtracted. Thus, through the

3 (X1 k, x0 ∼k

) denotes that exogenous variable X is set at the value it assumes in scenario 1, while all other variables are the the values of scenario 0. A similar interpretation holds for (X1

k, x0 ∼k

).

6

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12r

:= A12r f where 12r denotes a group of r indices (r �� K), and ��

12r

is the portion of Ay due to the interaction of exogenous variables corresponding to the selected indices. The indices ��

12r

play the same role in the decomposition of a finite change as Sobol�� indices in the decomposition of variance. Of particular relevance are the first-order finite change sensitivity indices (11) ��1

= Af

and the total order indices (12) ��

= Af + X

Af + ... + A12f ��

[eq.(12)] is the total contribution of x to Ay, and is the sum of the individual contribution of x, plus all the contributions due to the interaction of x with the remaining exogenous variables. The index (13) ��

= �� − ��1

represents the effect of interactions associated with x. ��1

and ��

are generalization of the differential sensitivity measures CS and D. In fact, it can be shown that, if f(x) is smooth, then (14) lim

∆x7��0

��

Ax = lim

∆x7��0

��1

Ax = CS 7

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∆x7��0

��

/Ay Ax = lim

∆x7��0

��1

/Ay Ax = df df = D As to the computation of ��

, it can be shown that (16) ��

= f(x1) − f(x0 ;x1 (∼)

) where f(x1) is the value of the endogenous variable in scenario 1 and (x0

;x1

(∼)

) is the point obtained with all exogenous variables at scenario 1 but x that remains at the base case scenario. Thus, the triplet (��1

,�� ,�� ) can be computed at the cost of 2K simulations, instead of 2. This

computational burden reduction result makes the sensitivity measures applicable also to complex simulation codes. As to the settings, as discussed in the literature, the sensitivity measures for setting 1 are the indices �� in (10), (11) and (12). The sign of the first-order indices (��1

) is the sign change

in y due to the individual change in x. The sign of ��

12r

is the sign of the interaction between the exogenous variables x

1

, x

2

and xr . As to Setting 2, the total-order indices (��

) are the appropriate sensitivity measures, since they deliver not only the individual importance of the factors, but also account for interactions. As to Setting 3, the magnitudes of ��

12r

provide the natural sensitivity measures. If the model is complex, or several exogenous variables are present and it is not possible to compute all the terms. In that case, one obtains a synthetic measure of the relevance of interactions by considering the magnitude of ��

relative to the magnitude of �� . Here,

we recall that if the endogenous variable responds additively to the exogenous variables variation then interactions are not present and ��

= 0. Conversely, if the effect of interaction is strong, we

expect ��

to be close to ��

.

3.2 A numerical example

Before applying the scenario-based sensitivity analysis, it would prove to be beneficial to apply the method to a simpler case. To this end, let us consider the model 8

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3

(18) y2 = x4 ∗ ln(y1)/10 Then, we have 4 exogenous variables and two endogenous variables. Let us evaluate the model under two scenarios: the base case and an alternative scenario. The base case is obtained with all model inputs at their mean value: (19) x0 = [10,10,10,10] and the alternative scenario is (20) x+ = [19.5,19.5,19.5,19.5] Since we have four exogenous variables, we have four individual effects, ¡

4 2¢ = 6 interaction effects

of order two, ¡

4 3¢ = 4 interaction effects of order three and one residual effect of order four for a

total of 16 terms in the decomposition of the finite change. Formally: (21) y1(x+) − y1(x0) =

4

X

=1

��1

+ X

��2

+ X

��3

+ ��4 1234

To compute the first-order effects, we move one factor at a time to x+. The base case value is (22) y1(x0)=1.0020e + 04 Thus, we have (23) ��1

1 = y1(x+ 1 ,x0 −1) − y1(x0)=2x+ 1 + 3/x0 2 + (x0 3)4 − 2x0 1 − 3/x0 2 − (x0 3)4 = 2(19.5 − 10) = 19

9

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2 = −0.1462

and (25) ��1

3 = 1.3459 ∗ 105

Since the model is additive, we have no interaction effects. Note that (26) y1(x+)=1.4463 ∗ 105 so that (27) y1(x+) − y1(x0)=1.4463 ∗ 105 − 1.0020 ∗ 104 = 1.346 1 �� 105 which is the sum of 1.3459 ∗ 105 + 19 − 0.1462 = 1.346 1 �� 105. For y2, instead, we have interactions, because now we have a multiplicative form. Therefore, we obtain that second order terms are also significant now. For instance, we have a strong interaction effect between x3 and x4. In detail, the non-null indices are ��1

1 = 0.0019, ��1 2 = 0, ��1 3 = 2.6694, ��1 4 = 8.7517, ��2 34 = 2.536,

��2

23 = 0

��2

13 = 0

��2

14 = 0.0018, ��2 24 = −0.0018, ��3 134 = −0.0017.

To get, for instance, ��2

34 = 2.536, we need to compute the following:

(28) ��2

34 = y2(x+ 3 ,x+ 4 ,x0 ∼34) − y2(x0) − ��1 3 − ��1 4 = 23.1695 − 9.2124 − 2.6694 − 8.7517 = 2.536

4 The G-LUM Simulation Model

This section provides a concise review of the G-LUM models as introduced in Zhou et al. (2009) and Zhou and Kockelman (2009). G-LUM consists of three sub-models: EMPLOC, RESLOC, and LUDENSITY, calibrated using 10

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HH location forecast Emp location forecast Land Use forecast Next time period

Figure 1: Flow-chart of the G-LUM model an entropy maximization and non-linear least-square principles (similar to maximum likelihood), thereby providing estimates of parameter values that generate the ��best fit�� of forecasts for the base year based on prior year data. Figure 1 shows the model flowchart. The EMPLOC sub-model, to forecast employment��s spatial distribution, relies on: i) accessi- bility to (all) households and jobs of that type, across zones, in prior time periods, ii) zone-to-zone travel costs and, iii) zone sizes. The endogenous variable of interest is the number of jobs of jobs of type p (basic or commercial, retail or service types, for example) in zone i at time t, denoted E

.

This endogenous variable is estimated from: (29) E

=

X

=1

H−1A

−1

U

−1

cp

exp(pc) + (1 − )E −1

11

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−1

= [

X

=1

(E

−1

)p Lp

cp exp(ßc)]−1

and (31) U

−1

= (E

−1

)p (L)p where H is total households in the region , c is the travel cost, L is the land area, t represents the year/time period, and U

−1

represents the attractiveness of the zone per employment type at t − 1. a,, b,p, ß, are parameters, which are the result of a calibration, which we are to discuss. The second submodel, RESLOC, concerns the household spatial distribution. It is based on: i) access to all households in all categories and jobs of all types in prior time periods, ii) zone-to-zone travel costs, and iii) land use condition (in prior period). (For further details see Putman (1995)). The corresponding equations are (32) H

= ��

X

=1

Q

B W cd exp(ßc) + (1 − ��)N −1

where H

is the number of households of type d residing in zone i at time t, and

(33) Q = P aE

, B = [P =1 W cd exp(ßc)]−1

and W

= (L )d

(��

)d

(L

)d Q0 Ã1 + H0

P H

!d

d0

where Q converts employment to households, and W

represents the attractiveness of zone i for

household type d at t − 1, and ; c is impedance (travel time and/or cost) between zones, a is the number of type d households per type p employee in the region under investigation; L

is vacant

developable land in zone i; ��

is the proportion of developable land already developed; L

is residential land and a,��,s and b are parameters estimated in model calibration. Finally, LUDENSITY (land per household or job) relies on: i) land use conditions in the current time period, and ii) the share of households and jobs by type in the current period. It estimates 12

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5 Sensitivity Analysis of GLUM

5.1 A baseline example

This section describes the scenario decomposition method as applied to G-LUM. The Austin, Texas data used in Zhou and Kockelman (2009) are used here with Z = 1074 zones. G-LUM 13

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: number of jobs across all zones, for each job type at t = 1,2,3,4,5. The job types are

Basic, Retail and Service; H

: contains the forecasted number of households across all zones at t = 1,2,3,4,5, for each

household of the four types (low, medium, medium-high and high income); LB: amount of land allocated for basic jobs in all zones at t = 1,2,3,4,5 after time period; LC: amount of land allocated for commercial jobs in all zones at t = 1,2,3,4,5; LR: amount of land allocated for residential use in all zones, at t = 1,2,3,4,5. Overall, we have M = 10 endogenous variables. As exogenous variables, we consider the following three datasets that are used as model inputs. The employment dataset (EMP), that contains jobs basic, retail, service for each zone in the base year; it is a matrix of size 1074 �� 3. The households dataset (HH), that contains the number of households per type in each zone, in the base year; it is a matrix of size 1074 �� 4. The link impedance dataset (TT), containing the travel times between each pair of zones in the base year. It is a 1074 �� 1074 matrix. The supplied values form our base case scenario, so that x0 = (EMP0,HH0,TT0).4 In Paul and Kockelman (2005), the model is run over three alternative scenarios, with EMP, HH and TT increased by 50% of the their distribution respectively, each one at a time. In our framework, we consider the following scenario that encompasses the three variations, namely x1 = (EMP1,HH1,TT1). Then, in accordance with eq. (8), the change of each model out- put from x0 to x1 can be decomposed into eight terms that account for the individual change in EMP, HH and TT, to their interactions in pairs and in the residual term that contains their overall and residual interaction Thus, we have the following sensitivity indices: ��1

, ��1 , ��1 ,

��2

,��2 ,��2 , and ��3 and the total indices

(34) ��

= ��1 + ��2 + ��2 + ��3

��

= ��1 + ��2 + ��2 + ��3

��

= ��1 + ��2 + ��2 + ��3

4 It should be mentioned that our working scenarios are proposed only for expositional purposes and do not intend

to describe necessarily specific situations.

14

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1

, m = 1,2,...,10, t = 1,2,..5. Each of these sensitivity measures will then assume a different value for each of the ten outputs of interest. We present here as a first illustration the results obtained for the endogenous variables of the type hyi, m = 1,2,...,10 and t = 1,2,...,5. We start with first-order sensitivity indices. Figure 2 displays the first-order sensitivity indices of EMP. Figure 2 reads as follows. On the vertical axis (z), one finds the magnitude and sign of the sensitivity measures. On the first horizontal axis (y) one finds the model outputs (from basic to LR), and on the second horizontal axis (x) one finds time, from year one to five. Figure 2 allows us to appreciate that the increase in the values in the dataset EMP, alone, impacts upon predictions on future jobs, especially in year one and two, more heavily than the households counts, which are almost insensitive to this variation. The amount of land for residential use (LR) is also much more impacted than the amount of land for commercial and basic jobs. We observe that the effect here is negative; that is, to an increase in EMP there corresponds a decrease in the average of land allocated to residential use. Figure 3 reports the first-order effects of HH on each of the exogenous variables. One notes the almost null effects on the predicted numbers of jobs and on household counts. However, we register a strong (negative) influence of HH on LR, with a decrease in the value of land allocated 15

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1

, m = 1,2,...,10, t = 1,2,..5. Figure 4: ��

1

, m = 1,2,...,10, t = 1,2,..5. to residential use. Figure 4 reports the first-order effects associated with the change in the 1074 �� 1074 dataset TT. One observes that the exogenous variables are almost insensitive to this variation, with the exception of a slight change in LR. We now consider the effect of interactions. Figure 5 reports the results for the interactions between EMP and HH. We observe here that interaction effects are negligible for jobs and household counts. However, positive interactions are registered on endogenous variables LC and LR. All signs are positive, 16

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2

, m = 1,2,...,10, t = 1,2,..5. Figure 6: ��

2

, m = 1,2,...,10, t = 1,2,..5. signalling that, when varied together, EMP and HH amplify their individual effects. Figure 6 shows the interaction effects between EMP and TT. They are negligible for all exogenous variables. A similar result is encountered for the interactions between HH and TT (Figure 7). Figure 7 shows non-negligible, albeit slight interaction effects only on LR. Overall, the message of Figures 5, 6, and 7 is that the endogenous variables respond additively to the finite changes, with the sole exception of LR. Then, we come to total effects. They are displayed in Figure 8. Figure 8 shows that EMP is the most influential group on job predictions exogenous variables, 17

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2

, m = 1,2,...,10, t = 1,2,..5. Figure 8: Total order indices, ��

(upper left), ��

(upper right), ��

(lower centered). 18

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5.2 A further example: a change in total counts

Let us now consider an homogeneous change input variables by +50% of total counts. The sensitivity analysis is carried out through the use of finite change sensitivity indices deriving from the spatial average of zonal outputs generated, for each endogenous variable, prediction period and scenario, by changes in the exogenous variables EMP, HH and TT. The first two variables (hereinafter respectively EMP and HH) represent the control totals for jobs and households in the region under investigation, while the third variable refers to the travel times between each pair of zones. As a result of individual variations of the three input files (i.e., the exogenous variables), the model produced output files containing the endogenous variables used to calculate first-order sen- sitivity indices. Results relative to the individual variation of the exogenous variable EMP are reported in Figure 9, where the horizontal axis displays the set of endogenous variables and the vertical axis shows the magnitude of the sensitivity indices. According to the figure, the individual variation of EMP has a significant effect on the number of future jobs, with an upward trend throughout prediction periods On the contrary, both the number of future households and land use variables, for every category, appear not be affected and are almost insensitive to this variation. Figure 10 reports the first-order sensitivity indices relative to the individual variation of HH. The figure clearly shows that the increase in the exogenous variable HH has a strong effect on the forecasted number of households and the amount of land allocated to residential use (LR), while there is almost no impact on the number of future jobs. Differently to the two previous variables, first-order effects relative to the variation of the ex- ogenous variable TT are null, except for very slight changes in the amount of land allocated to residential use. Similar considerations apply to second- and third-order effects; that is, changes in endogenous variables due to interactions of two or all three exogenous variables. Interactions deriving from simultaneous variations of EMP-HH and EMP-TT have only a modest impact on 19

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6 Conclusions

Transport-land use models are widely utilized to inform decision-making. Those analytical tools consider estimation and/or calibration of a large number of equations and are computationally intensive also in the optimization stage. Once model outputs are obtained, policy makers are often interested into changes in model inputs or parameters (e.g., a change in transport costs or in the structure of demand). In this paper, we have proposed the use of a local sensitivity analysis technique that has a direct interpretation in terms of comparative statics as it relies on the computation of derivatives and relative changes describing given scenarios. The proposed approach has the advantage of being extremely inexpensive in terms of model runs and provides useful information on the relative importance of single and groups of variables. Furthermore, by relying on a simple Taylor expansion of the equations of interest, it also quantifies the synergies between variables. The methodology has been applied to the well-known G-LUM model of Zhou and Kockelman (2009). Using the scenario decomposition technique, we have found that, over the given scenarios, the endogenous variables respond almost additively to variations in the model inputs. Changes in the base year data concerning employments also influence future predictions on the number of jobs and land use. Our approach to sensitivity analysis allows the decision-maker to assess the relevance of several scenarios at a low cost in terms of model runs. In particular, in cases of extended cost-benefit analysis, when general equilibrium effects are crucial our approach to local sensitivity analysis may provide important information for understanding the sources of changes in social welfare. 22

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Paul B. and Kockelman K., 2005: Gravity-based land use model, A Presentation for Texas MPOs, University of Texas Austin. Borgonovo E. and Apostolakis G.E., 2001: ��A New Importance Measure for Risk-Informed Decision-Making��, Reliability Engineering and System Safety, 72 (2), pp. 193-212. Borgonovo E., 2007: ��Differential Importance and Comparative Statics: an Application to Inventory Management��, International Journal of Production Economics, forthcoming in 2007. Borgonovo E., 2010: ��Sensitivity Analysis with Finite Change: Application to Modified EOQ Models��, European Journal of Operational Research, 200, pp. 127-138. Borgonovo E. and L. Peccati, 2011: ��Managerial Insights from Service Industry Models: A New Scenario Decomposition Method,�� Annals of Operations Research, 185 (1), pp. 161-179. Ciriello V., Di Federico V., Riva M., Cadini F., De Sanctis J., Zio E. and Guadagnini A., 2013: Polynomial Chaos Expansion for Global Sensitivity Analysis Applied to A Model of Radionuclide Migration in a Randomly Heterogeneous Aquifer, Stochastic Environmental Re- search and Risk Assessment, 27(4), pp. 945-954. Efron B. and Stein C., 1981: ��The Jackknife Estimate of Variance,�� The Annals of Statistics, 9 (3), pp. 586-596. Eschenbach T.G., 1992: ��Spiderplots versus Tornado Diagrams for Sensitivity Analysis,�� Inter- faces, 22, pp. 40-46. Helton J.C., 1993: ��Uncertainty and Sensitivity Analysis Techniques for Use in Performance Assessment for Radioactive Waste Disposal,�� Reliability Engineering and System Safety, 42, pp. 327-367. Howard R.A., 1988: ��Decision Analysis: Practice and Promise,�� Management Science, 346, pp. 679-695. Jungermann H. and Thuring M., 1988: ��The Labyrinth of Experts�� Minds: Some Reasoning Strategies and Their Pitfalls��, Annals of Operations Research, 16, pp. 117-130. 23

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Symbol Meaning y Endogenous variables (model outputs) y

()

m endogenous variable at time t in zone i,j hyi V [y] Max{y} Min{y} Average, variance, maximum and minimum of yvalues x =(x1,x2,...,x), exogenous variables K Number of exogenous variables (model inputs, including parameters) ��X Exogenous variable space f(x) Simulation exogenous-endogenous variable mapping x0, x1 Any two scenarios Ax,Ay Change in x and y across scenarios (x1

;x0

(−)

) [(x0

;x1

(−)

)] Point of ��X with x at scenario 1[0] and theremaining variab. at scen.0[1] S{·} Local sensitivity operator CS Comparative statics sensitivity measure of x D Differential importance of x f12k Generic term in the functional ANOVA expansion of f(x) A12k f Orthogonalized change in f due to the changes in x1 ,x2 ,...,xk ��

12r

Finite change sensitivity index of order r ��1

First-order finite change sensitivity index of x ��

Total order finite change sensitivity index of x I

: = S{f (x)}

Vector containing the K sensitivity measures of endogenous variable y E

Number of jobs of type p in zone i at time t H

Households count of type d in zone i at time t 27

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We offer here an analysis of the local sensitivity analysis framework as it will help us in framing the sensitivity analysis quests that we are to conduct. A local sensitivity measure is a mathematical operator that acts on f(x) to produce the sensi- tivity indicator of exogenous variable x, k = 1,2,...,K. Formally, we write (35) S : ��S �� R In eq. (35), ��S is the set of functions with domain ��X for which the application of the operator S is well posed. That is, ��Sis the domain of S. We suppose that ��S is contained in a linear space of functions. For instance, if S is a differential operator, then ��S must be enclosed in the set of differentiable functions on ��X. We then write: I : = S{f(x)} where I defines the importance of the k exogenous variable, k = 1,2,...,K. That is, after application of the operator S, we obtain a vector of k sensitivity measures. Thus, acting on f(x), S maps each of the indices 1,2,...,K into a corresponding value in R. Such a value is called the importance (sensitivity) measure of x [Helton (1993)]. This framework applies to generic local sensitivity measures, when the simulation output is a scalar quantity. However, in realistic applications, the simulation model produces several quantities of interest (exogenous variables) and often they are spatially and temporally distributed. The fact that these models are becoming increasingly popular calls for an extension of the previous framework, if we wish to set up the sensitivity analysis exercise in a rigorous context. The first step is the extension to the case in which the model computes M quantities of interest to the analyst/decision-maker. The model output is, then, a vector of quantities of interest. The mapping in eq. (1) becomes: (36) y = f(x), f : ��X �� R 28

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= f (x), m = 1,2,..,M, i = 1,2,...,z and j = 1,2,...,z

where y

is the m output calculated in region (i,j).

Application of the operator S to eq. (38) produces a K �� M �� z �� z array of sensitivity measures. We write I

: = S{f (x)}, m = 1,2,..,M, i = 1,2,...,z and j = 1,2,...,z

where I

is the K-component vector of sensitivity measures of the exogenous variable with respect

29

Page 31 |

that contains the

sensitivity measures of the M exogenous variables simulated by the model over the spatial region of interest. Finally, the framework is further extended if the simulation produces the time-evolution of the M endogenous variables, in each region. If a discrete time simulation is chosen (we will restrict to this case for simplicity of an already complicated notation), we have that eq. (1) becomes also parameterized by the time t: (39) y

= f (x), t = 1,2,..,T, i = 1,2,...,z, j = 1,2,...,z and m = 1,2,..,M

Correspondingly, we have I

: = S{f (x)}, t = 1,2,..,T, m = 1,2,..,M, i = 1,2,...,z and j = 1,2,...,z

which represents the vector of sensitivities of the m − i − j endogenous variables at time t, with respect to the K exogenous variables. The above-mentioned discussion should help in evidencing a potential problem. The analyst/decision- maker runs the risk of becoming flooded with sensitivity measures, especially if the number of re- gions or model outputs of interest is high. One then needs to find a way to understand the insights that can be obtained from the sensitivity measures and how results can be represented (visually or graphically) in the most convenient way. This need is already highlighted in earlier works, such Eshenbach (1992), that suggest that one should avoid to ��overwhelm managers with data.�� Therefore, one needs a systematic approach to the extraction of insights and the selection of the quantities of interest. As to insights, we examine them in the context of the following three settings. - sign of change: the sensitivity measures in I

convey the sign of change in the m − i −

j endogenous variable, given the variations in the exogenous variables over the predetermined scenarios; - key-drivers: the magnitudes of the sensitivity measures in I

can be used to understand

which exogenous variable impacts upon the m − i − j endogenous variable the most; 30

Page 32 |

ij

X

=1=1

y

ZZ The spatial variance is given by: (41) V [y] =

ij

X

=1=1

(y

)2

ZZ − ¡hyi¢2 The spatial maximum and minimum of y are: (42) Max{y} = max=12i=12j {y

} and Min{y} =

min

=12i=12j{y }

Here, we need to recognize that each of these quantities depends on the scenario on which we run the model. Then, each of them is an endogenous variable depending on x. The choice of which quantity to use is on a case-by-case basis and is driven by the overall purposes of the analy- sis. For instance, in hydraulic applications concerning the modelling of floodings, the maximum 31

Page 33 |

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