Discrete time dynamic estimation model for passenger origin/destination matrices on transit networks

Traonrpn. Ra..&I Vol. 22B. No. 4. pp. 2X-260. 1988 Rimed in Great Britain.

DISCRETE TIME DYNAMIC ESTIMATION MODEL FOR PASSENGER ORIGIN/DESTINATION MATRICES

ON TRANSIT NETWORKS

0191.2615188 13.00+ .a, 9 1988 Pergamon Press pit

S. NGUYEN Centre de Recherche sur les Transports, Universitt de Montreal, Montreal, Canada

E. MORELLO Dipartimento di Pianificazione, Consorzio Trasporti Torinesi, Torino, Italy

S. PALLOT-IINO Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Roma, Italy

(Received 8 December 1986; in revised form 17 June 1987)

Abstract-This paper addresses the problem of estimating or updating a passenger trip matrix for transit networks from passenger counts. An extension of a model to take into account time information contained in the passenger counts is developed. Several promising optimization formulations of the resulting model are presented and implementation issues are examined.

I. INTRODUCTION

A model for estimating or updating passenger trip matrices was recently proposed by Nguyen and Pallottino (1986). An application of this model to the network of the city of Torino, Italy (Inaudi and Morello, 1986), yielded reasonably good results, but also pointed out the difficulty in handling the model’s passenger counts equality construints- the constraint set is almost always infeasible. The incompatibility of the constraints is mainly caused by measurement errors and the temporal variation inherent in traffic counts, but also by the steady state assumption implicit in the assignment model adopted. To overcome this shortcoming, we investigate a dynamic variant of the estimation model based on a partition of the initial time period into several time intervals. The dynamic model is fully supported by the data sets used in the above application.

First, we describe the general problem of estimating or updating an origin/desti- nation (O/D) passenger trip matrix on transit networks, using a transit assignment framework developed by Nguyen and Pallottino (1985). In Sec. 3, we introduce a par- tition of the initial time period for the passenger trip matrix into several time intervals, and develop the fundamental relationship between a link count for a given time interval and trips associated with different time intervals. In Sec. 4, we describe an optimization formulation based on the traditional entropy-maximizing approach and constraint relax- ation. Other formulations based on statistical inference techniques, which do not impose deterministic passenger count constraints are briefly reviewed.

2. ESTIMATING O/D FLOWS ON TRANSIT NETWORKS

A traffic assignment framework. Consider an urban public transportation system consisting of a set of distinct transit

lines and stops where passengers board and alight transit carriers (buses). This transit system is abstracted into a graph (_J:_z?), in which ,Vis the set of nodes (centroids, stops, line nodes) and1 is the set of arcs (line segments, boarding and alighting arcs, centroid connectors). In this graph, each transit line is uniquely identified by a sequence of nodes and arcs-each node corresponds to a line stop and each arc to a line segment between two successive stops. Distinct transit lines are represented by node disjoint paths in the modeling graph. A stop node is also associated with each real stop, whether this latter is served by a single transit line or by multiple lines. A pair of boarding and alighting arcs connects every line node to its corresponding stop node. Finally, walking

251

252 S. NGUYEN er al

arcs connect stop nodes to centroids and to other stop nodes. Pedestrian paths are integrated into the transit system as transit lines with appropriate attributes. The set of boarding arcs will be denoted 2.

It is assumed that each transit line has a fixed line frequency, and this frequency will be associated with the boarding arcs to this transit line. Also, an average in-vehicle travel time or a generalized cost cij is associated with every link (i, j) E 6. We will restrict the analysis to the case with constant average time or cost, since the congestion effect in transit networks does not seem to be well understood at the present time.

The present assignment framework assumes that a passenger waiting at a stop, served by several transit lines which can transport this passenger to his destination, always boards the first arriving carrier of a preselected subset of transit lines. The specification of this subset of transit lines, often referred to as the passenger’s attractive set, will depend on the assignment model adopted. However, independent of the as- signment model implemented, the above adaptive component of choice behavior (to board the first arriving carrier of a subset of perceived equivalent lines) cannot be captured by the standard single path assignment framework. Clearly, a new muftiparh concept must be introduced. In the following, a routing alternative is defined as an acyclic subgraph with given arc traversal probabilities and called a hyperpath (Nguyen and Pallottino, 1985). More precisely, consider a single pair of centroids, say pair (r, s).

Definition. A subgraph H = (X, E, P), where X C L i ; E C 1, and P = (plj) is a real value vector of dimension (El, is a hyperpath connecting r and s if:

-H is acyclic with at least one arc, -node r has no predecessor and s no successor, -every node i E X belongs to a path from r to s and, if i is not a stop node, then i

has at most one immediate successor, -the characteristic vector P satisfies:

CPij = 1, Vi E X,

Pij z 09 V(i, j) E E.

Let E: denote the set of arcs incident out of node i E X. For every stop node i of hyperpath H, the set of boarding arcs E: C 33 identifies the attractive set of transit lines for the subgroup of passengers who travel on hyperpath H. Each component pii is equal to the probability that arc (i, j) is traversed by a passenger who arrived at node i. In particular, under the following simplifying assumptions: 1. passengers arrive run- domly at every stop node; 2. all transit lines are statistically independent with a given exponential distribution of heudwuys with mean equal to the inverse of line frequency; the conditional probabilities pij take on specific values for boarding arcs:

pij = cPijl@;, V(i, j) E E,+,

where pii is the frequency of the transit line associated with boarding arc (i, j), and

@= C ‘Pij

(i.i)EE:

is the combined frequency of the attractive set at stop node i. For all other arcs (i, j) E E - E f7B,pij = 1.

Note that in the present case, the mean waiting time pi, at stop node i, is equal to the inverse of the combined frequency: pi = a;‘; and the flow of passengers on hy- perpath H at the stop node i is subdivided among the lines of this attractive set in proportion to their frequency. Similar assumptions were reported in the literature [see for example, Chriqui and Robillard (1975), Marguier (1981), Spiess (1983b)].

Dynamic trip estimation model 253

Let d, denote the probability of travelling on any particular elementary path I of hyperpath I-Z, then:

dl = n p$lf, VI (i.j)EE

(1)

in which 6, equals 1 if path 1 traverses arc (i, j), and 0 otherwise. The path probabilities induce, in turn, the probabilities bi cf traversing node i:

bi = 2 S,ldl, Vi, (2)

where 6i, equals 1 if path I traverses node i and 0 otherwise, and the probabilities Uij of traversing arc (i, j):

aij = C 6,,d,, V(i, j).

The node and arc probabilities satisfy the following equations (Nguyen and Pallot- tino, 1985):

b, = b, = 1,

bj = 2 bipij, Vj E X - {r}, (4)

aij = bipij, V(i, j) ~5 E,

which clearly show that the arc traversal probabilities a;j can be calculated efficiently in a topological order. This result is a crucial feature of the present assignment framework.

Let C denote the average cost of hyperpath H. This satisfies

C = C QijWij,

(i./) (5)

where wij is the travel cost of arc (i, j) of hyperpath H, defined as

Wij = cxp. + Cij, for every boarding arc (i, j) cij, otherwise, (6)

where cr is a user defined conversion parameter, and pi the waiting time associated with stop node i.

Let i E X be any node of hyperpath H, and U, is the average travel cost of the subhyperpath of H connecting i and S, then a computationally oriented alternate defi- nition of travel cost C = LJ, is

uss = 0,

Ui~ = 2 pij(wij + U,s), Vi E X - {S}. (i.j)EE:

(7)

The OID flows estimation problem Let f m denote the average number of trips going from centroid r to centroid s which

reach destination s before the end of a given time period, say [a,,, a,], independent of the departure time. Let F be an array with components fn. F is usually referred to as

2.54 S. NGUYEN et al.

an original-destination (O/D) trip matrix or O/D flows corresponding to time period

[R% RI. Now, let g,, denote the proportion of passenger demand fm traveling on hyperpath

H4. The resulting link flows are

yielding the following aggregated link flows:

(8)

Assume that passenger flows {ir,} on a subset M CL have been observed. Let p denote the vector of these counts. Assume further that additional information on trip demand F-such as ad hoc sample survey estimates or outdated estimates-is also avail- able, the estimation problem addressed here can be stated broadly as: determine an estimate of the OID trip demand matrix F by efficiently combining passenger count based data and all other available information.

Clearly, any mathematical formulation for this problem must use the fundamental relationship (8). The trip proportions g,, in these equations depend on the assignment model adopted. For instance, in the all or nothing assignment model, assuming that the minimum cost hyperpath between any given pair of centroids is uniquely defined, the trip proportions g,,S satisfy:

g,rS = I 1, if q is the minimum cost hyperpath connecting r and s 0, otherwise.

On the other hand, if the perceived cost (?‘s incurred on hyperpath H4 is assumed to be a random variable-reflecting the variation and error in the trip maker’s perception of travel time or cost:

c, = c, •t Eqr

where C, is the average cost defined in eqn (5) and eq is the unobservable residual, then the average trip proportion g,, is given by the probability of choosing hyperpath H, among all hyperpaths connecting r and S:

g,, = Prob (cq < c’h, Vh Z 4). (9)

The value of g,, depends solely on the probability distribution assumed for the error terms zq, and may be computed independent of the trip demand F. For instance, if the error terms l q are assumed to be independent Weibull variates: l q - W(0, a), eqn (9) reduces to the well known multinominal logit model (see, for example, Dial, 1971; Domencich and McFadden, 1975):

emec, g,n = ypi7)

where the parameter 8 is defined as

Dynamic trip estimation model 255

Alternatively, if the errors eq are assumed to follow a joint multivariate normal distri- bution with moments

JQq) = 0,

Var(E,) = PC,,

COV(Eq, 4 = PCqnh

where Cqnh is the travel cost of the common part of hyperpaths H4 and H,,, the mean trip proportions are then given by the probit model (see Daganzo and Sheffi, 1977; Daganzo, 1979).

For the sake of clarity, we will assume in the sequel that only a single (aggregated) hyperpath is utilized between each O/D pair with link traversal probabilities

leaving the choice of an assignment model to the user.

3. PARTITIONING THE TIME COORDINATE INTO INTERVALS

In this section, we develop the fundamental relationship between a link flow in a given time interval and trips associated with different time intervals. This requires careful choice of the notation used.

We first divide the time coordinate into intervals of length t.~ indexed by T, for 7 = 1 * . , ~nla.x* A typical value for the interval length p, would be 20 min. Let t-(T) and t’(i) denote the time origin and end of interval T, respectively. The origin of the first interval, and the end of the last one will be identified later. Naturally, the choice of P must be supported by the data available. Let M be a subset of links, M C 2, and assume that for every link I = (i, j) E M there is a set of passenger counts ir;, for T = 1, . . . , T,, (each component 0; is equal to the number of passengers observed on link I during interval T).

Let U, denote the average travel time incurred on the utilized (aggregated) hyper- path connecting node j to destination s. For every link 1 = (i, j), let & = Ujx.

Let I(&) denote the number of intervals covered by &.

I(&) = : . 11 (11)

It will be assumed that a passenger, observed on link 1 in interval T, will reach destination s in interval (T + I(&)). In practice, depending on the values of CL, S, and different errors, the actual arrival interval may be the one immediately preceding or following the assumed interval (7 + I(&)).

Assume that the extremities of the time period of interest [ao, ai] are multiples of p and

where II is a positive integer. Define

s max = max max &, IEM ,

(12)

l??(B) 22:L-B

256 S. NGUYEN et al.

then any passenger who is observed on link 1 before time (a0 - S,) will reach his destination prior to uo, thus, we may set the origin of the time axis at

r-(l) = a0 - l.LZ(S,,).

Similarly, there is no need to consider passenger counts corresponding to any interval T such that t+(7) L a,, therefore

7 max = II + I(&&

We now decompose the initial trip matrix F into a sum of n matrices Fk:

F= f: F’, k-l

where fi is equal to the number of passengers from O/D pair (r, s), who reach desti- nation s in interval T,, - n + k.

Note that a passenger who is observed on link 1 in interval T, for any 1 5 T 5 T,,, may still reach his destination prior to or after interval [uo, ai]. These passengers will be associated with a residual trip matrix P, which will be defined shortly.

For the sake of clarity, consider, for the moment, trip matrices for all time intervals, and let j$ denote the number of passengers of O/D pair (r, s) who reach destination s in interval k; k = 1, . . . , K, where K = n + 21(&J. Note that

The passenger counts ir;, 1 E A4 decompose into trips ending in different intervals as follows:

(13)

in which

AL = A ,,r, if k = T + I(&) 0, otherwise.

Since we are only interested in the trips which end within the initial time period

[co, 4:

[jk], for all 1 + I(&,,) 5 k 5 n + I(&,),

all other trips will be aggregated into a single residual matrix P for efficiency. More precisely

However, this aggregation may require a corresponding relaxation of eqns (13) to pre- serve the feasibility of the constraint set. In fact, for any given link 1, all equations:

n k

such that there exists

Dynamic trip estimation model 257

where

kl 5 I( S,,), kt 2 I(&,) + n + 1

must be replaced by inequalities (1). Assuming that a proper processing of eqns (13) has been carried out, we may now

rewrite these equations directly in terms of [f:J for k = 0, 1, . . . , n as follows:

2 Xlrsf~fl”) = (r)D;, for all 1 5 -r 5 T,,,, and 1 E M, rs

where

- ms(T, 1) = I T + I(&) I(&lA if 1 5 7 + I(S,) - I(&,) 5 n 0 9 otherwise.

4. MODEL FORMULATIONS

In this section, we will examine both the standard maximum entropy formulation, and optimization formulations based on the maximum likelihood approach.

Formulations with deterministic constraints The standard optimization formulation with deterministic constraints can be stated

as follows (Nguyen, 1983):

Min C(F, Y) (14)

subject to 2 X&$‘,O = (r)fi;, for all 1 5 T 5 T,,,,~, and 1 E M (15) IS

Fr 0,

FE I-, (16)

where Y = (yi) represents the a priori information on trip demand F, and constraint (16) will take into account any additional information on F which can be expressed by linear constraints. The analytical objective function C( F, Y) will depend on the approach followed. For instance, the entropy-maximizing or minimum information method (Wil- son, 1970; Van Zuylen and Willumsen, 1980) yields

C(F, Y) = i c fk In 5. t-o I, Is

(17)

On the other hand, in the constrained maximum likelihood approach, if Y represents a set of sample counts with given sampling fraction 8, then under the assumption that the likelihood _L’( Y(F) of observing Y given F can be expressed as a product of Poisson probability functions with parameters ofi, we obtain the following expresssion for the log-likelihood:

ln-L’(YIF) = i 2 [y$ In (0fi) - 13fQ + const, (18) k-0 rs

258

yielding the objective function

S. NGumN et al.

C(F, Y) = - i: 2 [y!i In (Of:) - efi]. k=O IS

(19)

The above constrained maximum likelihood approach was previously proposed by Lan- dau, Hauer and Geva (1982) and by Spiess (1983a). Note that with the above objective function (19), a zero cell yk does not force the corresponding fi to zero. This is an advantage over the previous maximum entropy approach. On the other hand, Ben- Akiva, Macke and Hsu (1985) found that a maximum entropy trip estimation model for a single transit line, with a Bayesian seeding of nonstructural zero cells of Y (see, for example, Kirby and Leese, 1978), is more effective than constrained maximum likelihood and least-squares models.

Frequently in practice, due to measurement errors and temporal variations, the constraint set (15)-(16) does not have any feasible solution. This requires the use of a cumbersome data smoothing procedure. We prefer a relaxation approach which replaces the equality constraints (15) by two sets of inequalities:

where [I is a user defined tolerance which can be tailored appropriately to reflect variation in the accuracy of passenger counts. The main disadvantage of this relaxation scheme is an increase in the number of constraints, although this should not produce significant additional computational effort.

Formulations with extended objective function To take into account measurement errors, random variation in trip demand and the

trip maker route selections over time, and inherent errors in the network and assignment models, we may assume that the passenger counts O;, 1 E M, are observations of random variables with mean values equal to values u; predicted by the assignment. This can be expressed as

i’; = c h&p’) + E/, E(Q) = 0. (21) IS

Assume further that the two sets of data, Y and v, are statistically independent so that the likelihood of observing both sets can be expressed as

sf(Y, V’IF) = &(YIF) -@IF). (22)

The maximum likelihood estimator FML of trip demand F may then be obtained by maximizing expression (22) or, more conveniently, its natural logarithm:

FM = arg rnz; {ln L(YIF) + In l(~lF)}, (23)

where I is the set of feasible F. Unless explicitly specified, I will simply be the set of all non-negative F.

The computation of FML will depend on the probability distribution assumed for the sample count vector Y and that of the passenger count vector v [or equivalently the residual E in eqn (21)]. In practice, the Poisson is, perhaps, the prevailing distribution

Dynamic trip estimation model

choice for the error E (Van Zuylen and Branston, 1982), yielding:

259

In -@IF) = 2 C (ir; In IJ; - u;), r=l /EM

(24)

where u; is given by the left-hand side of eqn (15). In addition to Poisson distribution, multivariate normal distribution has also been considered, giving rise to a quadratic expression for the log-likelihood of P (see, for example, Cascetta, 1984). If the first term in the objective function of problem (23) is given by (18), then, in either case we have a convex program subject to simple constraints (non-negativity of F). For a detailed derivation of alternative objective functions, see the review by Nguyen and Cascetta (1986).

Algorithms Problem (14) with objective function (17) or (19) are convex programs with linear

constraints, and therefore can be solved using adaptations of a large variety of existing mathematical programming techniques (see, for example, Luenberger, 1973). However, with the entropy function (17), we can use Bregman’s efficient algorithm (Bregman, 1967; see also Lamond and Stewart, 1981), which is particularly advantageous for prob- lems with large numbers of equality and inequality constraints.

With stochastic passenger count data [eqns (21)], and under Poisson assumptions, we have a convex program with simple constraint (23). This problem may be solved efficiently with an adaptation of Bertsekas’s projected Newton algorithm (Bertsekas, 1982). Numerical experimentation with these methods is presently being carried out.

5. CONCLUSION

This paper presents a dynamic model for estimating or updating an O/D trip matrix for transit networks. Several optimization formulations based on the entropy-maximizing and the maximum likelihood approaches are described. Potential efficient solution al- gorithms for the maximum entropy model with inequality constraints, and the simple constrained maximum likelihood model with Poisson distributions are suggested.

The dynamic model was formulated with n + 1 trip matrices. For real life networks and typical value of n, say 3 or 4, we may be confronted with a very large number of variablesfk. If this leads to computational difficulties then a partial aggregation of the matrices Fk, k = 1, . . . , n, into fewer matrices should be considered. From a practical point of view, the variant with only two trip matrices (F’ and the residual Fo), may prove to be the most effective model, and produce significantly more accurate estimates than those obtained from models proposed previously.

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