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## Microsoft word - djukic - final - cst.doc

11th TRAIL Congress

November 2010

** **

**BENCHMARK FRAMEWORK FOR SENSITIVITY ANALYSIS **
**OF DYNAMIC OD DEMAND ESTIMATION METHODS**
**Tamara Djukic MSc, Dr. ir. Hans van Lint, Prof. dr. ir. Serge Hoogendoorn, **
**Prof. dr. Henk van Zuylen **
Faculty of Civil Engineering and Geosciences, Department of Transport and Planning,
Delft University of Technology, the Netherlands

**ABSTRACT **

The indicator for the quality of an OD estimation method is the sensitivity of the method to

(and robustness against) random and structural perturbations of the input (data from sensors,

prior OD matrices) on a few typical test networks. In this paper we propose such an

assessment methodology based on the Latin Hypercube (LHC) method, which is an efficient

alternative for Monte Carlo sampling and particularly suited for high-dimensional estimation

problems. We demonstrate the methodology on a real urban corridor network for one well-

known OD estimation method (the entropy maximization estimation method) to illustrate

which results can be obtained and how these can be used to benchmark different OD

estimation methods.

**KEYWORDS **

Benchmark methodology, Dynamic OD demand estimation, Latin hypercube sampling

**INTRODUCTION **

In recent years a large number of methods have been developed to estimate dynamic origin

destination matrices from traffic counts and other available data. Although many methods

have been proposed to solve the (dynamic) OD estimation problem, much fewer efforts have

been reported to actually evaluate and cross-compare these methods under different

circumstances (e.g. different network structures, different sets of data available in different

qualities). Few studies have focused on evaluation of the reliability and accuracy of the

estimated OD matrices in absence of ground truth OD matrix

*(1)*, and on available ground

truth OD matrix such as

*(2, 3)*. The reason why it is difficult to cross compare and generalize

results from sensitivity analysis undertaken to demonstrate different properties of the methods

is that these methods differ in many aspects.

A structured and generic methodology to benchmark different OD estimation methods under different circumstances (e.g. network lay-out, data availability and quality of prior OD) would enable researchers to pinpoint the strengths and weaknesses of different OD estimation methods and their applicability and validity under different circumstances. This paper presents and demonstrates such a generic OD estimation benchmarking framework. It is generic in the sense that a wide range of OD estimation approaches can be tested under a wide range of different circumstances related to for example data availability and quality, and network lay out. One of the central components is an efficient Monte Carlo sampling method, the so called latin hypercube (LHC) method. In the first part of the paper we will outline the overall framework and discuss its application to the OD estimation problem with various scenarios that are relevant for assessing different OD estimation methods. In the second part of this paper we will demonstrate this evaluation framework for entropy maximization estimation method on real urban corridor network under different scenarios of data quality and availability. The paper closes with a discussion of the results and conclusions.

**BENCHMARKING FRAMEWORK **

As reflected by equation (1), a particular OD estimation method

*h* with a set of parameters

and assumptions

*H*, aims to infer the maximum probable OD matrix given the set of available

inputs (data

*D*, prior and previous OD estimates

*Tprior* and

*Tt*-1,

*Tt*-2,…) and all the assumptions

*T *=

*h*(

*T *,

*T *,.,

*T *,

*D*,

*H * (1)
Typically

*h *is a highly nonlinear and dynamic function of its inputs. Since OD estimation methods differ in many aspects (objective function, optimization method, single or bi-level, etc) and since they may be assessed under many different circumstances (network type with or without route choice, different traffic conditions, varying data availability, etc) the number of simulations required for an exhaustive and statistically sound comparison between different OD estimation methods will quickly become very large. Since the framework has be applicable to a whole range of OD estimation approaches, we consider a simulation-based approach where the OD estimator is considered as black box, providing a certain outcome (OD estimate) given certain input. The obvious choice for a simulation-based approach is to use some sort of constrained or stratified form of Monte Carlo sampling

*(4)*, which does not scale with the input dimensionality. We applied the stratified sampling method (Latin hypercube (LHC) method

*(5)*) that provides a computationally much more efficient alternative to random (Monte Carlo) sampling. We refer to reader to the original papers on LHC method

*(5, 6)* for more details. Given such an efficient sampling method exists, the key properties of a benchmarking framework for OD estimators are the following:
1. It needs to sample / vary over all the relevant inputs, i.e.:
a. Network topology, considering a number of typical networks (e.g. a freeway
corridor without route choice, a freeway network with a few dominant routes and a grid network), for which ground truth OD data are available
b. Traffic conditions, considering various degrees of congestion c. Data availability and quality, considering many possible data scenarios
i. availability/quality of local sensors (vehicle counts)
ii. availability/quality of probe vehicle data / travel reports
iii. availability / quality of a priori OD matrices
2. It must be able to assess and compare either a single or multiple OD estimators, and
possibly sample different parameter settings for each method
3. It needs to be able to generate all relevant (conditional) output distributions from the
results on the basis of which a wide range of performance indicators can be derived. We identify three levels of aggregation:
a. The total OD demand (summed over all OD flows). Clearly, OD methods
should at least be robust in deriving this quantity (as a function of departure time) even in cases where little data (or poor quality) is available
b. The OD demand summed over particular destinations (or origins). Here not just
the total amount, but also the distribution over the destinations may be assessed
c. OD demand per origin destination pair, i.e. the entire OD matrix. In this case
particularly the preservation of the matrix structure (i.e. the dominant OD relations versus the near zero ones) is of interest.
Requirement 1 essentially determines which input scenarios, varying in terms of network topology, traffic conditions, and data availability, need to be considered during the assessment. Requirement 2 describes which methods and variations of methods are to be assessed and compared. Requirement 3 describes the output required for assessment, cross-comparison and analysis.

**EXPERIMENTAL SET UP **

In this section we will describe the input data and assessment scenarios used in this paper.

This case study has a purpose to demonstrate the main features of the proposed assessment

framework for one particular OD estimation method. A benchmarking exercise of several OD

estimators will be the subject of further studies. The entropy maximization model proposed by

*(7)* is applied in this study. We use the solution algorithm based on an advanced iterative

algebraic reconstruction method, referred to as the Revised Multiplicative Algebraic

Reconstruction Technique (RMART), as proposed by

*(8)*. For more detail explanation of this

algorithm we refer to the papers

*(8)*.

* *

By implementing the LHC method described in previous section, we can simulate the fact that

*d* may contain errors. We consider the following three scenarios:
1.

*Scaled (total) demand variation scenario*: This scenario illustrates how the variation of
demand flows across the sequence of departure intervals can influence the estimation results without changes in OD matrix pattern (the all demand values within the same departure interval are scaled with a factor uτ );
2.

*Random demand variation scenario*: This scenario is based on the assumption that the
prior OD matrix is the best estimate of the mean of the dynamic OD matrices. In this case

*d* is varied by adding uniformly random components to the ground truth OD
matrix, representing the difference between the smoothed historical estimate and the particular daily realization;
3.

*Systematic demand variation scenario:* This scenario addresses situations where the
prior OD demand might contain other, structural errors besides the random daily fluctuations. The demand per each origin over destinations is generated from positively and negatively skewed mean values of distribution from random demand scenario.
For a comprehensive benchmark study it is important to take into account the real state and conditions (in terms of output, errors, location, etc.) of the detectors on the network. Therefore we assume the random perturbations of traffic counts measurements using the following two detector layouts:

*1. Detector layout based on maximum coverage of all links in network: *In this “best”
case we want to obtain the benchmark results for comparison of OD estimates with other detector layouts.

* *
2.

*Detector layout based on the maximum link flows*: The choice of link traffic detectors
is according to the maximum link flow measurements such that links with highest volume have priority. First, we identified the links with highest volume. Then detector layout is defined randomly removing the detectors from links to obtain tree scenarios with 80%, 60% and 40% coverage of total number of links on network.
The true link measurements are randomly perturbed following the same approach proposed in random demand variation scenario. We evaluate the performance of the maximum entropy method on a real corridor network. This network was chosen because of the availability and quality of the empirical detector data on network, and because a calibrated OD matrix was available. The network consists of 214 OD pairs and 118 corresponding links. The evening peak hour reflecting the congested state at the network which was divided in 6 departure time intervals of 10 minutes is chosen for experiments.

**RESULTS AND DISCUSSION **

We use two criteria to check whether estimated OD matrices from the entropy method are on

average equal to the true value, or show at least only a small variation around this value. The

first criterion relates to the sensitivity of the total demand estimates under different scenarios

of counting errors and errors in the prior OD matrix. Secondly we look at the sensitivity of

certain OD pairs to the same error scenarios.

The results provided new insights into the sensitivity of EM estimation method with respect to

the scale in the prior OD matrix. We found that errors in link measurements lead to the larger

range of errors resulting in underestimation and overestimation of total demand and per total

origin demand as well. The results finally also showed that the use of less accurate link

measurements can dominate the accuracy of entropy method especially when the link flow

measurements availability is limited.

As we were expected, in all tested scenarios we observed that the minimum absolute error for

some OD pairs is close to zero while the maximum error is usually over 35%. It was thus

interesting to identify the properties of the paths and magnitude of the demand associated with

such ‘problematic’ OD pairs. These data are presented in Figure 1a (influence of path length)

and 1b (influence of magnitude of the demand).

**Figure 1: The average errors a) path length per OD pair and b) value of OD demand **
From Figure 1 we observe that the entropy method furthermore turned out to be sensitive to path lengths and the demand per OD pair. As a result, it can be concluded that it is harder to estimate OD pairs that have longer paths or higher demand accurately, since these showed larger error in the link measurements.

**CONCLUSIONS **

This paper proposes a new framework to assess and cross-compare the sensitivity of OD

estimation methods to different types of uncertainty in the input data, and the parameters of

the OD estimation methods. Central to this benchmarking framework is a well-known

simulation-based stratified sampling technique that allows assessing the sensitivity without

the need to perform an unfeasible number of simulations. The contribution of the proposed

benchmark methodology is to enhance the understanding in how the abovementioned aspects

impact dynamic OD estimation methods in terms of robustness and accuracy. Using this

benchmark framework, both researchers and practitioners will gain insight in which OD

estimation methods are suitable for both off-line applications, such as ex-ante simulation

studies, and on-line applications, such as short-term traffic prediction, decision support

systems, etc.

**REFERENCES **
Yang, H., Y. Iida, T. Sasaki (1991) An analysis of the reliability of an origin-destination trip matrix estimated from traffic counts, in:

*Transportation Research Part B: Methodological* *25(5),* pp. 351-363. Marzano, V., A. Papola, F. Simonelli (2009) Limits and perspectives of effective O-D matrix correction using traffic counts, in:

*Transportation Research Part C: Emerging Technologies, 2009. 17(2)*, pp. 120-132. Stathopoulos, A., T. Tsekeris (2003) Framework for analysing reliability and information degradation of demand matrices in extended transport networks, in:

*Transport Reviews: A Transnational Transdisciplinary Journal, 23(1),* pp. 89 - 103. Reuven, Y.R., (1981)

*Simulation and the Monte Carlo Method*, John Wiley \& Sons, Inc. 304.
McKay, M.D., R.J. Beckman, W.J. Conover (1979) A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code, in:.

*Technometrics, 21(2)*, pp. 239-245. Budiman, M., B.M. Alex, (2006) A conditioned Latin hypercube method for sampling in the presence of ancillary information, in:

*Comput. Geosci., 32(9)*, pp. 1378-1388. Van Zuylen, H.J., L.G. Willumsen (1980) The most likely trip matrix estimated from traffic counts, in:

*Transportation Research Part B: Methodological*, 14(3), pp. 281-293. Wu, J. (1997) A real-time origin-destination matrix updating algorithm for on-line applications, in:

*Transportation Research Part B: Methodological, 31(5)*, pp. 381-396.

Source: http://rstrail.nl/sites/default/files/Djukic.pdf

UNIVERSIDAD LAICA ELOY ALFARO DE MANABI TESIS DE GRADOS 2010 – 2011 FACULTAD DE MEDICINA TEMA : CERVICITIS BACTERIANA DIAGNOSTICADA CON PAPANICOLAU Y SU RELACION CON ENFERMEDAD INFLAMATORIA PELVICA EN MUJERES MULTIGESTAS COMPRENDIDAS DE 25 A35 AÑOS DE EDAD ATENDIDAS EN LA CONSULTA EXTERNA ESPECIALIZADA DE L HOSPITALVERDI CEVALLOS BALDA DE PORTOVIEJO EN EL PERIODO 2009 -2010. AU

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