a ternary-arithmetic topological based algebraic method for networks traffic observability

Applied Mathematical Modelling 35 (2011) 5338–5354

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

A ternary-arithmetic topological based algebraic methodfor networks traffic observability

Enrique Castillo a,⇑, Pilar Jiménez b, José María Menéndez b, Ana Rivas b, Inmaculada Gallego b

a Department of Applied Mathematics and Computational Sciences, University of Cantabria, 39005 Santander, Spainb Department of Civil Engineering, University of Castilla La Mancha, 13071 Ciudad Real, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 September 2010Received in revised form 22 April 2011Accepted 26 April 2011Available online 4 May 2011

Keywords:ObservabilityAlgebraic and topological methodsPlate scanning

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.04.044

⇑ Corresponding author.E-mail address: [email protected] (E. Castillo).

In this paper an algebraic method, which shares all the advantages of the topological meth-ods and allows us to obtain the same results as the standard algebraic method with a sub-stantial reduction in memory and cpu requirements, is presented. The main idea consists ofwriting the link, OD and scanned flows in terms of route instead of OD flows. This alterna-tive permits starting the algebraic and topological processes with identical matrices ofzeros and ones. In addition, in most iterations the pivots can be selected in such a way thatthe resulting matrices after each iteration contain only zeroes, ones and minus ones. Thisallows us to design a ternary arithmetic which reproduces the algebraic results exactly,requires only two bits to store each matrix entry and have no precision or non-zero pivotidentification problems. Only when this process cannot be continued, the pure algebraicmethod is used, but only with a very reduced size matrix when compared with the sizeof the initial matrix. The method is illustrated by its application to a very simple networkand to a real network example (the city of Cuenca, Spain).

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

There are some situations in practice in which one needs to know the state of a traffic network by measuring a subset offlows and, based on this information, predicting other flows. These flows are not free to take arbitrary values and to be inagreement with the measured flows they must be subject to the constraints imposed by the networks topology. The observ-ability problem consists of identifying if a set of available (measured) flows is sufficient to calculate other given subset ofunobserved flows. Some examples of observability problems in traffic networks that can be solved by algebraic and topolog-ical techniques are:

1. Determine if a subset of available traffic flows is sufficient to obtain the values of another subset of traffic flows.2. Obtain a minimum subset of data that allow the knowledge of other given subset of flows or the observability of all flows

in the network.3. Identify observable flows, given a subset of observed flows (partial observability).

Though the problem of observability can be stated in a general context, as done in [1], who discuss the problem of observ-ability of a linear system of equations and inequalities in general, most of the existing publications relate to particular fields(see the survey provided by Abur and Gómez Expósito [2], which includes applications to power systems, or [3] for trafficproblems).

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E. Castillo et al. / Applied Mathematical Modelling 35 (2011) 5338–5354 5339

Observability techniques can be classified as:Algebraic. These techniques consider the algebraic relations between the flows and operate them algebraically to draw

observability conclusions (see [2,4–10]). More precisely, they start from the numerical relations among all flows and manip-ulate them to try to express the flows to be observed in terms of those really observed to conclude which of them are ob-servable (they can be calculated in terms of observed flows) and which are not observable, and to obtain the analyticalrelations that allow such calculations when they are possible. For instance, [11] have proposed a ‘‘basis link approach’’ toinfer unobserved link flows (non-basis flows) using the information contained in the basis link flows via the Gaussian elim-ination algorithm.

Compared with topological methods, these methods have the following advantages:

1. They supply more information about the observability problem, that is, the conclusions about observability and unob-servability are on all flows.

2. In addition to answering the observability questions, they provide the algebraic relations that permit obtaining theunknown flows in terms of the known flows.

Topological. These techniques consider only topological and/or qualitative relations between flows to derive observabilityresults (see [12–14,1]). More precisely, they do not use numerical relations but qualitative relations, i.e., the numerical val-ues of the coefficients relating flows are replaced by zeroes and ones. This means that in these methods we take into accountonly if one flow is necessary to calculate other flows, but not the exact numerical relation.

When compared with algebraic methods, they have the following advantages:

1. They require less memory to solve the problems.2. The cpu requirements are much smaller.3. They are faster, that is, they provide results in less time.

Since neither one is best in all senses, the question we ask in this paper is the following: Is it possible to combine the ideasin both methods to obtain an alternative method sharing the advantages of algebraic and topological methods? The answerto this question is positive, and the alternative method having these interesting properties is given in this paper.

Algebraic methods can be used with two aims in mind: (a) obtaining the exact algebraic relation among different flows,such that some can be calculated when others are known, and (b) obtain observability information, that is, determine whichflows can and which flows cannot be calculated when a subset of flows is known, but without seeking the correspondingformulas which allow us the calculations. The second aim is simpler and requires less effort than the first one.

On the contrary, topological methods always look for observability matters and not for algebraic relations, so they takeadvantage of this fact and lead to some important savings, but the price to be paid is that not all questions answered by alge-braic methods can be answered by topological methods.

Previous papers on observability in traffic networks (see [15]) have dealt only with the link and OD flow observabilityproblem in the unusual case of overspecification, that is, when the number of independent link flows is larger than the num-ber of OD flows. However, if the number of OD pairs is larger than the number of independent links, this OD flow observ-ability problem based on link flows has no solution because the problem is underspecified, and there is an infinite set ofsolutions for the OD-pair flows satisfying the conservation laws. Thus, the OD flow observability problem has no solutionin many cases of real practice, and when it has solutions, the situation is not realistic. In order to have a realistic observabilityproblem we must deal with scanned links or other techniques, which supply much more information than the link flows (see[16–18]). Thus, though previous papers on observability have dealt with the OD flow observability problem based oncounted link flows, here we dealt with the problem of observability of link, OD and route flows based on subsets of countedlink flows and/or scanned link flows. In this paper we deal with the observability problem when the number of OD and routeflows is larger than the number of independent link flows, and then other measurements, apart from link counters arenecessary.

The plate scanning approach1 to traffic flow estimation has become frequently used, due to the considerably larger amountof information it contains when compared with other standard methods (see [18] or [19]), and the advantage of being able touse the information gathered by surveillance and traffic control systems based on automatic number plate recognition (ANPR).The idea is to register the plate numbers of the circulating vehicles together with the corresponding times on some subsetsof links and use this information to reconstruct vehicle routes or partial routes from which OD and link flows are evaluated.Since we have plate numbers, we can determine the number of cars traveling through any possible combination of scanned linksand only through them. This implies that having n scanned links permits in theory calculating how many cars travel on each ofthe 2n � 1 possible subsets resulting from scanned links and eliminating the empty set. However, only some of these subsetslead to feasible routes, but the important thing is that there are more of them than n, that is, the information content of scannedlinks is much higher than the information content of the same counted links. Each possible subset of scanned links when

1 Though we refer to plate scanning in the paper, the reader should also understand other electronic devices that provide the same information and which aremuch more reliable. These devices are expected to be implemented in the near future to replace plates.

5340 E. Castillo et al. / Applied Mathematical Modelling 35 (2011) 5338–5354

feasible reports the number of cars wi using the set S of routes that share those scanned links and only those. Thus, the scannedlink information2 is always a set of equations that involves a sum of route flows, i.e., the flows in the routes which share thecorresponding subset of scanned links and only those. If the number of routes is one, say route s, the corresponding scannedinformation provides the number of cars rs traveling that route, because wi = rs. Otherwise, the scanned flow information wi con-tains the cars traveling more than one route, say wi ¼

Pj2Srj, and then the route flows rj; j 2 S cannot be identified with this

information alone, if they are more than one.The plate scanning technique permits not only OD-pair flow observability, but route flow observability, even though the

number of routes is larger than the number of OD flows. So, route observability is dealt with in this paper with some pos-sibilities of success.

Though we concentrate in this paper in the case of scanned and observed links, there are other techniques that also in-crease the amount of information, as, for example, the use of path-ID sensors or AVI counts (see [20–22]), the use of nodesensors (see [23]), etc.

The paper assumes that readers have some familiarity with algebraic and topological methods for observability analysis,aims to produce an improved method having the advantages of both methods, and is organized as follows. Section 3 de-scribes the proposed method and presents an algorithm for its practical computer implementation. In Section 4 one exampleof application is used to illustrate the proposed method and associated methodology. Some conclusions are given in Section5 together with proposals for future work.

2. Determining an optimal set of scanned links

Though the aim of this paper is to provide a method that permits reducing the required cpu time and memory used, to-gether with solving the numerical problems derived from the numerical precision and pivot finding problems, for the sake ofcompleteness we provide in this section some methods, similar to those already reported in the existing literature, to solvethe problem of identifying the minimum number of cameras to be used, and the corresponding subset SC of links to bescanned for obtaining the flows of a given subsetRobs of routes. To his end, we can solve the following optimization problem(see [24]):

2 Byroute (fcorresp

Minimizey;z;nc M ¼ nc; ð1Þsubject to

Xa2Ajdr

aþdr1a ¼1

za P yr; 8ðr; r1Þ 2 R2jr – r1; ð2Þ

Xa2A

zadra P yr ; 8r 2 R; ð3Þ

Xa2A

za ¼ nc; ð4Þ

yr ¼ 1; 8r 2 Robs; ð5Þ

where nc is the number of required cameras, za is a binary variable which takes value 1 if link a is scanned and 0 otherwise, yr

is a binary variable which takes value 1 if route r can be distinguished from other routes using the selected subset of scannedlinks and 0 otherwise, R refers to the set of routes, A is the set of links, dr

a are the elements of the route incidence matrix,which take value 1 if link a is in route r, and 0, otherwise, and Robs is the subset of routes that we want to observe.

In problems (1)–(5), the objective function (1) is used to minimize the number of cameras. Constraint (2), if the binaryvariable yr takes value 1, guarantees that route r can be distinguished by the subset of scanned links from other routes byat least one link. Note that if dr

a þ dr1a ¼ 1, link a belongs only to one of the routes but not to the other, and that ifP

za P yr and yr = 1, at least one scanned link has this property. On the other hand, if yr = 0, the constraint always holds. Con-straint (3), if yr = 1 ensures that route r, which is able to be distinguished by the subset of scanned links from other routes,contains at least one scanned link. Constraint (4) guarantees that nc cameras are used. Constraint (5) guarantees that eachroute in Robs is identified by the subset of scanned links.

If we are interested only in full observability, alternatively we can solve the simpler problem:

Minimizez M ¼Xa2A

za; ð6Þ

subject toX

a2Ajdra–d

r1a

za P 1; 8ðr; r1Þ 2 R2jr – r1; ð7Þ

Xa2A

zadra P 1; 8r 2 R: ð8Þ

and then, the number of cameras is nc ¼P

a2Aza.

‘‘scanning link information’’ we refer to the number of vehicles wi that has been registered for each combination ‘‘i’’ of scanned links compatible with aor an example, see Table 1 and the associated set of scanned links and routes). Each of these numbers is associated with a linear equation that force theonding combination of plate observations to be compatible with the routes.

634

1

2

3

4

5

6

7

521

8

Fig. 1. The elementary example network used for illustrative purposes, showing the nodes, links and the set {1,5,8} of scanned links (with thick arrows).

Table 1Route links and scanned information for two cases.

Route OD nodes OD Route links Scanned info for {1,2,3,4,5,6,8} Number of users Scanned info for {1,5,8} Number of users

1 1–3 1 1, 2 {1,2} w1 {1} w1

2 1–3 1 1, 5, 8 {1,5,8} w2 {1,5,8} w2

3 3–4 6 3 {3} w3 – –

4 2–3 2 2 {2} w4 – –5 2–3 2 5, 8 {5,8} w5 {5,8} w3

6 2–6 4 5, 6 {5,6} w6 {5} w4

7 5–3 7 6, 7 {6} w7 – –8 5–3 7 8 {8} w8 {8} w5

9 2–5 3 5 {5} w9 {5} w4

10 3–1 5 3, 4 {3,4} w10 – –


3. The proposed mixed method

Most previous algebraic methods in the existing literature are stated in terms of OD flows. This implies not only that thealgebraic linear relations between link, OD, route and scanned flows3 have associated flow matrices with coefficients whichare real numbers but that route choice probabilities must be assumed for the different routes of the same OD pair.

To illustrate, consider the simple network4 in Fig. 1, with 6 nodes, 8 links and the routes in Table 1. If we scan linksSC ¼ f1;2;3;4;5;6;8g, that is, we scan the car plates in the links in SC and only there, the corresponding scanned informationis the one shown in Table 1 fifth column. Since we scan 7 links, a user could have one of 27 � 1 possible combinations of links. How-ever, most of them are unfeasible (not compatible with the set of possible routes). For example, the combination {5,8} means thatwe have registered users of links 5 and 8, but not of links 1, 2, 3, 4 and 6. Since there is only one route (route 5) that contains links 5and 8, and does not contain links 1, 2, 3, 4 and 6, this information allows us to associate those users with route r5. The number ofusers leading to this combination is denoted by w5, and then we have the linear equation w5 = r5, where w5 is the number of usersdetected by the plate scanning procedure only in links 5 and 8, and r5 is the number of users of route 5. On the contrary, thecombination {2,4} means that we have registered users of links 2 and 4, but not of links 1, 3, 5, 6 and 8. Since there is no route thatcontains links 2 and 4, and does not contain links 1, 3, 5, 6 and 8, this information is not possible. Similarly, since the combinations{1,2} and {1,5,8} are associated only with routes r1 and r2, respectively, this explains why r1 = w1; r2 = w2 in Eq. (14).

Finally, if we scan the set of links SC ¼ f1;5;8g (see Table 1) and register a user at link 5, that is, the user travels link 5 butnot links 1 and 8, we find that this information is compatible only with routes 6 and 9, but unfortunately we cannot identifywhich of these two routes is the user route. Thus, if we register w4 users of this type, we have the associated linear relation-ship w4 = r6 + r9, where r6 and r9 are the number of users of routes 6 and 9, respectively.

In Table 1 we can see that the set of scanned links5 SC ¼ f1;2;3;4;5;6;8g permits us to identify all routes because theintersection of the sets of route links with the set of scanned links SC leads to disjoint sets. On the contrary, if the set ofscanned links is SC ¼ f1;5;8g, which is the one to be considered in our example, we cannot identify routes 3, 4, 7 and 10,and routes 6 and 9 are confounded, because the intersection sets of SC with the sets of route links coincide in these cases(see the sixth column of Table 1). This means that supplementary information is required to identify these routes. Ouraim is to determine a minimal subset of link counted flows to obtain full observability of the network, that is, a subset oflinks such that if their flows are known, together with the scanned information we can calculate the flows in a given subsetof link, OD or route flows in the network.

3 Scanned flows are the flows that can be obtained from the data resulting from the cameras in the scanned links. This means the number of cars that havepassed through all different combinations of scanned links and not have passed through other combinations of scanned links. We note that knowledge of a carnot passing through a link has the same amount of information that knowing that a car has passed through the same link, and both information types are usedto obtain scanned link flows. Two examples are given in Table 1 columns 4 and 5.

4 This network, though not realistic and very small, has been selected only in order to illustrate the proposed methods, because it leads to a small number offlow equations. Otherwise, it would be impossible to write these equations for illustration in the present paper.

5 By ‘‘scanning link set’’ we mean to the set of scanned links, that is, the set of links in which a camera has been installed to register the vehicle plates and thecorresponding passing times.


If we express the flow equations corresponding to link counts (vj; j = 1, . . . ,8) and route flows (r‘; ‘ = 1, . . . ,0) in terms ofOD flows (tk; k = 1, . . . ,7), we obtain the system of equations in matrix form

6 SinOD flow

v1

v2

v3

v4

v5

v6

v7

v8

��r1

r2

r3

r4

r5

r6

r7

r8

r9

r10

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

¼

1 0 0 0 0 0 0p11 p12 0 0 0 0 00 0 0 0 1 1 00 0 0 0 1 0 0

1� p11 1� p12 1 1 0 0 00 0 0 1 0 0 p17

0 0 0 0 0 0 p17

1� p11 1� p12 0 0 0 0 1� p17

�� p11 0 0 0 0 0 0

1� p11 0 0 0 0 0 00 0 0 0 0 1 00 p12 0 0 0 0 00 1� p12 0 0 0 0 00 0 0 1 0 0 00 0 0 0 0 0 p17

0 0 0 0 0 0 1� p17

0 0 1 0 0 0 00 0 0 0 1 0 0

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

t1

t2

t3

t4

t5

t6

t7

0BBBBBBBBBB@

1CCCCCCCCCCA

; ð9Þ

where pij is the choice probability of the j OD users to choose the ith route of this OD. We call the matrix in this system theinitial matrix because it corresponds to the flows written in terms of the basic flows (in this case the OD flows). It is worthmentioning that: (a) we cannot write this relation unless, we know these choice probabilities, (b) the rank of the uppersubmatrix in (9) can depend on the choice probabilities, that is, the link flows dependence structure is dependent on theseprobabilities, which values can be considered arbitrary, and (c) the matrix has real numbers as entries.

If alternatively, we express the flow equations corresponding to the scanned link information (wi; i = 1, . . . ,5), link counts(vj; j = 1, . . . ,8) and OD flows (tk; k = 1, . . . ,7), in terms of route flows (r‘; ‘ = 1, . . . ,10), we obtain the matrix equation6

w1

w2

w3

w4

w5

��v1

v2

v3

v4

v5

v6

v7

v8

��t1

t2

t3

t4

t5

t6

t7

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

¼

1 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 1 00 0 0 0 0 0 0 1 0 0� � � � � � � � � �1 1 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 00 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 1 0 0 1 1 0 0 1 00 0 0 0 0 1 1 0 0 00 0 0 0 0 0 1 0 0 00 1 0 0 1 0 0 1 0 0� � � � � � � � � �1 1 0 0 0 0 0 0 0 00 0 0 1 1 0 0 0 0 00 0 0 0 0 0 0 0 1 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 10 0 1 0 0 0 0 0 0 00 0 0 0 0 0 1 1 0 0

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

r1

r2

r3

r4

r5

r6

r7

r8

r9

r10

0BBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCA

; ð10Þ

ce we aim at partial and full observability, we have considered the following types of flows: scanned flows (denoted wi), link counted flows (denoted vj),s (denoted tk) and route flows (denoted r‘).


where the initial matrix now corresponds to the flows written in terms of the basic flows (in this case the route flows). Thus,in this paper we do not write the flows in terms of OD-pair flows, but in terms of route flows.

Note that the scanned link information {w1,w2,w3,w4,w5} has been obtained from column sixth of Table 1, where only w4

is associated with two routes r3 and r4, and thus, w4 = r3 + r4, as it can be seen in (10) (fourth row of the matrix). In otherwords, the ones associated with the scanned link flows wi correspond to the corresponding routes, the ones associated withthe link flow vj correspond to those routes passing through links j, and the ones associated with OD flow tk correspond tothose OD containing the associated routes. All this information is contained in the first, third, fourth and sixth columns ofTable 1.

Because we have selected the set of route flows, denoted ri; i = 1,2, . . . ,10, as a basis, we have: (a) we do not need choiceprobabilities, that is, no arbitrary assumptions are required for writing this matrix, (b) the rank of the three submatrices in(9) are independent of the choice probabilities, i.e. no arbitrary dependencies can occur, and (c) the coefficients of the linearrelations are zeroes and ones (binary numbers).

One important property is that the set of scanned flow equations is always of full rank, and the same occurs with theset of OD flows, but not with the set of link flows, which can be dependent (in fact they are usually dependent). This im-plies that from the three partitioned matrices in (10), the intermediate is normally not full rank, but the other two are fullrank. This implies that matrix A of the system (10) is not assumed to be full rank and has important implications, becauseif two linearly dependent flows are independently observed, they can lead to unfeasibility problems, because the observedflows can have errors and then, the resulting system of equations can be incompatible (they will be incompatible unlesserrors in both compensate). The reader must be aware that inconsistencies are due to linear dependence of observedflows, but do not exist for linear independency. One of the advantages of scanned link flows and OD flows is that theyalways lead to a set of independent equations (see the upper partitioned matrix in (10), for one example), and then whenobserved they do not lead to inconsistencies. This is not true for counted link flows, that lead to well known inconsistencyproblems.

Next, we manipulate this initial system of Eq. (10) and convert it into an equivalent one, in order to see if a givensubset of flows can be calculated in terms of another subset of observed flows. To this end, the observable variables (r, t)are expressed as a function of the available measurements (w,v), that is to transfer columns to rows. In this form, thefirst subset of flows (those to be calculated) is moved to the left hand side of the system (10) and the second (thoseobserved) to the right hand side. For example, assume that we want to know the route flow r1, then, we obtain r1 fromthe first equation in (10) as a function of w1 and replace it into the remaining equations, to obtain the new system ofequations

r1

w2

w3

w4

w5

��v1

v2

v3

v4

v5

v6

v7

v8

��t1

t2

t3

t4

t5

t6

t7

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

¼

1 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 1 00 0 0 0 0 0 0 1 0 0� � � � � � � � � �1 1 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 00 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 1 0 0 1 1 0 0 1 00 0 0 0 0 1 1 0 0 00 0 0 0 0 0 1 0 0 00 1 0 0 1 0 0 1 0 0� � � � � � � � � �1 1 0 0 0 0 0 0 0 00 0 0 1 1 0 0 0 0 00 0 0 0 0 0 0 0 1 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 10 0 1 0 0 0 0 0 0 00 0 0 0 0 0 1 1 0 0



w1

r2

r3

r4

r5

r6

r7

r8

r9

r10

0BBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCA

: ð11Þ

It is important to note that the system (11) with matrix A⁄ is equivalent to system (10), in the sense that both express thesame relations among all the involved flows but in a different form. However, the new system (11) shows that the route flowr1 can be calculated in terms of the scanned link flow w1, that has been observed. This means that r1 can be known once w1 is


known, and then to denote that r1 is observable, we write it boldfaced. Similarly, to denote that w1 has been observed, weboldface it. No other flow in the left hand side matrix in (11) is observable, because no other flow can be written in terms ofw1 alone (see that other flows in this matrix have some nonzero coefficient associated with other route flows which are un-known (r2 to r10)).

The process to get the matrix A⁄ in (11) from the matrix A in (10) can be done using the following pivoting formulas

7 For

a�ij ¼ aij �aajaib

aab; i – a; j – b; a�aj ¼ �

aaj

aab; j – b; a�ib ¼

aib

aab; i – a; a�ab ¼

1aab

; ð12Þ

where aab is the pivot element, where a is the row associated with the observed flow and b is the column of the flow to becalculated (in our example a = b = 1), aij is a general element before transformation, and a�ij is the same element aftertransformation.7

The rationale of this algorithm, expressed in (12) in mathematical terms, consists of trying to write the flows to observe aslinear combinations of observed flows. To this end, we choose as pivot elements aab those elements which a is the row of theobserved flow, and b is the column of the flow to be observed (calculated from observed flows).

If using this transformation sequentially we obtain r2, r5, r6 and r8 from Eqs. (2)–(5) in (11), and replace these values intothe remaining equations, as before, we obtain the system of equations

r1

r2

r5

r6

r8

��v1

v2

v3

v4

v5

v6

v7

v8

��

t1

t2

t3

t4

t5

t6

t7

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

¼

1 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 1 0 0 �1 0

0 0 0 0 0 0 0 1 0 0

� � � � � � � � � �1 1 0 0 0 0 0 0 0 0

1 0 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 1

0 1 0 0 1 1 0 0 0 0

0 0 0 0 0 1 1 0 �1 0

0 0 0 0 0 0 1 0 0 0

0 1 0 0 1 0 0 1 0 0

� � � � � � � � � �

1 1 0 0 0 0 0 0 0 0

0 0 0 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 1 0 0 �1 0

0 0 0 0 0 0 0 0 0 1

0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 0 0

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

w1

w2

r3

r4

w3

w4

r7

w5

r9

r10

0BBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCA

; ð13Þ

which is equivalent to systems (10) and (11), where now we assume that we have observed all the scanned links, that is, weknow the flows w1,w2, . . . ,w5 (indicated by boldfacing these flows) and we observe that because of this, the flows r1, r2, r5, r8,v1, v5, v8 and t1 can be calculated (indicated as boldfaced). The remaining flows r6, v2, v3, v4, v6, v7 and t2 to t7 cannot be cal-culated from w1,w2, . . . ,w5 alone, as can be seen by inspecting the matrix in (13) and observing that their corresponding ele-ments associated with flows r3, r4, r7, r9 and r10 are not all null.

Finally, if sequentially we obtain r4, r3, r10, r7 and r9 from Eqs. (7)–(9), (11) and (12) in (13), and replace these values intothe remaining equations, as before, we obtain the system of equations

a detailed description of pivoting transformations see [25,26].


r1

r2

r5

r6

r8

��v1

r4

r3

r10

v5

r7

r9

v8

��t1

t2

t3

t4

t5

t6

t7



¼

1 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 0 1 0 �1 00 0 0 0 0 0 0 1 0 0� � � � � � � � � �1 1 0 0 0 0 0 0 0 0�1 0 0 1 0 0 0 0 0 00 0 1 0 0 0 0 0 0 �10 0 0 0 0 0 0 0 0 10 1 0 0 1 1 0 0 0 00 0 0 0 0 0 0 0 1 00 0 0 0 0 1 �1 0 1 00 1 0 0 1 0 0 1 0 0� � � � � � � � � �1 1 0 0 0 0 0 0 0 0�1 0 0 1 1 0 0 0 0 00 0 0 0 0 1 �1 0 1 00 0 0 0 0 0 1 0 �1 00 0 0 0 0 0 0 0 0 10 0 1 0 0 0 0 0 0 �10 0 0 0 0 0 0 1 1 0



w1

w2

v3

v2

w3

w4

v6

w5

v7

v4

0BBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCA

; ð14Þ

which shows that all route, OD and link flows v1, v5 and v8 can be calculated from the scanned flows w1, w2, w3, w4, w5 andthe counted link flows v2, v3, v4, v6 and v7. In addition, the matrix Eq. (14) provides the formulas for these calculations. Oneimportant observation is that all coefficients in this matrix are zeroes, ones or minus ones.

This observation motivates the present paper. In fact, we ask ourselves if this is a coincidence or a general property of thisprocess. As we will see, we can keep this property for a long time during the substitution process, and only in a few or noiterations we will violate the property. The important practical consequence is that we can save memory space, computertime to perform the operations, and that we do not have precision problems.

In addition, we must emphasize that in spite of the fact that the final system of Eq. (14) provides full observability, thepaper also deals with partial observability. To this end we need only to stop in previous iterations and see what flows areobservable and what flows are not (see Eq. (13)). It is important to note that all the partial observability information is alsocontained in the last table of the process, because observable flows do not change its flow equations (rows of the A matrix).

We remind the reader that topological methods were based on replacing non-null coefficients by ones, which lose thequantitative information contained in the coefficients, but not the corresponding qualitative information; this is why thesame conclusions are not possible for the topological and the algebraic approaches. In this paper we extend the topologicalmethod to include zeroes, ones and minus ones in order not to lose information and obtain exact results with the new topo-logical method.

This paper deals with the observability problem and finds out that choosing a more convenient basis for writing the flowrelations, a new algebraic-topological method can be designed so that it shares the advantages of both methods, that is, with-out renouncing to complete information and obtaining the final structural relations among the flows. In other words, theobservability problem can be solved using much less memory and cpu requirements, which indirectly implies faster results,and without loss of information and precision in the results.

In summary, the key feature of the method to be proposed is the use of route flows as the basis for writing the algebraicrelations for link, OD-pair and scanned flows, instead of using the standard basis of OD-pair flows, and the main advantage isthat all coefficients in the initial matrix of coefficients are ones or zeroes and that the probabilities of choosing different pathsare not needed. An important implication is that the initial matrix needs no transformation for the topological method, be-cause all the matrix elements are already zeroes or ones. However, the price we pay is that the dimension of the flow (inci-dence) matrices increases because there are more paths than ODs. Nevertheless, the proportion of non-null elements is verysmall, so that scarce and banded matrix methods can be used to minimize this inconvenience.

Another important and especially relevant feature is that when pivoting, it is observed that most entries of the matrix arezeroes, ones and minus ones and very seldom other positive or negative integer numbers. When only zeroes, ones and minusones occur in the incidence matrix, we can replace the real number operations, which are time consuming, by much fasterternary arithmetic operations, because only three numbers are involved in the calculations. This is what we call ternaryarithmetic stage. Otherwise, we say that we are in the real arithmetic stage. In the first case, each entry or element of thematrices can be stored in two-bits, and the pivoting operation (12) can be done faster. In the second case, we need more


memory to store the matrix elements, operations require more computing effort, and we can be subjected to precision prob-lems and/or pivot non-zero identification problems (how small an element must be to be considered null).

Table 2 illustrates the ternary arithmetic operations with the initial aij and the transformed values a�ij after the pivotingprocess for a pivot element aab, together with the relative frequencies of each case for the Cuenca example to be developedlater. The boldfaced values correspond to elements a�ij which change during the pivoting process, that is, when they are dif-ferent from aij. Note that these combinations that need to be calculated seldom appear (less than six times in every ten mil-lion cases of the total number of combinations in this example), which allows us a substantial reduction in the number ofoperations if this fact is controlled in the computer implementation. Note also that some elements are 2 and �2, indicating atransition from ternary to integer arithmetic, but that these cases appear rarely (less than twice in every ten million cases inthe Cuenca example).

The basic idea of the method consists of keeping control of up to when the resulting matrices contain entries with onlyzeroes, ones and minus ones. If this is so, not only the matrix can be stored in a reduced size (two bits per entry) instead ofthe memory required for integer or real numbers, but the pivoting operations can be simplified. To this end, when one of thepivot candidates leads to leaving the ternary stage, it is discarded and stored in a list for later use. In this way, we can proceedwith the ternary arithmetic and corresponding storage stage until a very advanced phase of the process, and only then ini-tiate the more complex and more resource (memory and cpu) demanding real arithmetic.

It is important to note that the proposed method obtains exactly the same results as the algebraic method, but much fas-ter (ternary arithmetic) and with less computational and memory effort and, in addition, the ternary arithmetic phase is lessdemanding than the standard algebraic method, which uses real number arithmetic. Note also that no precision or non-zeropivot identification problems need to be dealt with during the ternary stage, because exact values are obtained for all matrixelements, which is not true if OD flows are used as the basis.

It is important to realize that in the Cuenca network less than one in one million values have aij – a�ij, which means thatonly a very reduced fraction of all possible calculations need to be done, as one can see from the two frequency columns inTable 2. To illustrate, the case aij ¼ aaj ¼ ai;b ¼ a�ij ¼ 0; aab ¼ 1 occurs in 98% of the cases, and since a�ij ¼ aij no calculationsneed to be done after identifying this case.

3.1. Proposed algorithm

The following algorithm summarizes the ideas explained above and explains in detail the steps to be followed to imple-ment the proposed method. With this algorithm the flows to be calculated are written in terms of scanned or counted linkflows, which is the main aim of the algorithm.

INPUT: The topology of the network, a set SF of scanned flows (which can be obtained from the set of scanned links andwritten in terms of the set of routes), and a subset OF of counted links.

OUTPUT: The set of link, OD and route flows which are observable in terms of the flows in the sets SF and OF , and thealgebraic formulas which allow to calculate these flows in terms of the flows in SF and OF .

Step 0: Initialization. Build the incidence matrix A of scanned, link and OD flows. This matrix can be interpreted as apartitioned matrix, which columns are associated with the basic flows, which initially represent route flows (denoted by r),and the rows are the non-basic flows, which are partitioned into three blocks associated with three different sets of flows,which initially are sets of scanned flows (denoted by w), link flows (denoted by v) and OD flows (denoted by t). The result is amatrix of zeroes and ones, in which each route represents a column, and the rows of each matrix contain a one if the cor-responding scanned, link or OD flow is associated with the corresponding route (column). Next, we build the initial list ofbasic B and non-basic NB variable names. As indicated, initially the first one contains all route flows, and the second, thesets SF ;OF and the OD (see the first row (list B) and the first column (list NB) of iteration 0 in Table 3). When a flowhas been observed, its name is boldfaced in list B, and when the flow is observable, that is, it can be calculated in termsof observed flows, its name is also boldfaced in list NB (see, for example, the lists of basic B (first row) and non-basic NB(first column) flows in iterations 6 to 10 in Table 4).

Step 1: Choose a pivot. Choose an unobservable flow inNB \ SF \ OF , that is, a row i of matrix A and a route in B, that is,a column j of the same matrix, such that the corresponding value aij – 0 and when transformed to A⁄ by the pivoting processdoes not produce a matrix with some 2 or �2 entries after pivoting,8 and go to Step 2. If such a element aij does not exist,choose the pivot even though a�ij < �1 or a�ij > 1 and do not use this condition in posterior iterations. If not possible, stop theprocess, and return matrix A, as the matrix coefficients of the linear algebraic formulas which allow us to calculate theobservable flows in NB in terms of the observed scanned and link flows in B, and the lists of link and OD observable flows.

Step 2: Pivoting. Perform the pivoting process (see Eq. (12)) only with non-observable flows (associated rows),9 that is,calculate the transformed matrix A⁄, and exchange the list of basic and non-basic pivot flows at this stage. In other words,incorporate the flow in position i of NB into position j of list B, and the route flow in position j of B into position i of list NB.

8 In fact this situation is identified when implementing the ternary arithmetic, so that it is not necessary to test it initially. The idea consists of assuming thatthis does not occur and only if it takes place we do go back and undo the operations. Note that the frequency of appearance of this case is very small and we willhave to undo it in very few instances.

9 Notice that the rows associated with observable flows do not change during the pivoting process.

Table 2Ternary arithmetic for the Cuenca network. Boldfaced figures are those which change ða�ij – aijÞ during the pivoting process. In this table aab is the pivotelement, and aij is a generic element of the matrix A being transformed. The frequencies columns refer to the frequency of appearance of each case in theCuenca example.

aa,b aij aaj ai,b a�ij Frequency aa,b aij aaj ai,b a�ij Frequency

�1 �1 �1 �1 0 0.0000000000 1 �1 �1 �1 �2 0.0000000000�1 �1 �1 0 �1 0.0000000061 1 �1 �1 0 �1 0.0000000623�1 �1 �1 1 �2 0.0000000112 1 �1 �1 1 0 0.0000002568�1 �1 0 �1 �1 0.0000000069 1 �1 0 �1 �1 0.0000000094�1 �1 0 0 �1 0.0000010767 1 �1 0 0 �1 0.0000119448�1 �1 0 1 �1 0.0000000055 1 �1 0 1 �1 0.0000002814�1 �1 1 �1 �2 0.0000001799 1 �1 1 �1 0 0.0000000315�1 �1 1 0 �1 0.0000000416 1 �1 1 0 �1 0.0000000288�1 �1 1 1 0 0.0000000000 1 �1 1 1 �2 0.0000000000�1 0 �1 �1 1 0.0000000062 1 0 �1 �1 �1 0.0000000020�1 0 �1 0 0 0.0000004405 1 0 �1 0 0 0.0000132414�1 0 �1 1 �1 0.0000000125 1 0 �1 1 1 0.0000001404�1 0 0 �1 0 0.0000100586 1 0 0 �1 0 0.0000086791�1 0 0 0 0 0.0003023892 1 0 0 0 0 0.9816888954�1 0 0 1 0 0.0000064855 1 0 0 1 0 0.0095048791�1 0 1 �1 �1 0.0000000651 1 0 1 �1 1 0.0000000045�1 0 1 0 0 0.0000089500 1 0 1 0 0 0.0000010049�1 0 1 1 1 0.0000000443 1 0 1 1 �1 0.0000000191�1 1 �1 �1 2 0.0000000177 1 1 �1 �1 0 0.0000002099�1 1 �1 0 1 0.0000000031 1 1 �1 0 1 0.0000001551�1 1 �1 1 0 0.0000000000 1 1 �1 1 2 0.0000000000�1 1 0 �1 1 0.0000001408 1 1 0 �1 1 0.0000000839�1 1 0 0 1 0.0000055372 1 1 0 0 1 0.0076857928�1 1 0 1 1 0.0000000557 1 1 0 1 1 0.0007484351�1 1 1 �1 0 0.0000000000 1 1 1 �1 2 0.0000000000�1 1 1 0 1 0.0000001760 1 1 1 0 1 0.0000000164�1 1 1 1 2 0.0000000779 1 1 1 1 0 0.0000000376


Step 3: Update the list of basic and non-basic flows. If a non-boldfaced flow in NB (row of A) has null coefficients in allits columns associated with non-observed basic flows (columns of A), the flow is observable, and then its name is boldfaced,that is, added to the set of observable flows. Next, go to Step 1.

3.2. Illustrative example

Table 3 in iteration 0, which is equivalent to Eq. (9), shows scanned, link and OD flows written in terms of route flows, thatis, the flow information but choosing routes as the basic information. Note that the initial matrix is full rank (the rank coin-cides with the number of routes), but we can start with a non-full rank matrix if partial observation is looked for.

Tables 3 and 4 show the resulting matrices corresponding to all steps of the pivoting algorithm, which is explained below.We note that the tables in iterations 5 and 8 correspond to Eqs. (13) and (14), respectively, and that the column matrices onthe right hand sides appear in the first row of the tables of the corresponding iterations (this is more convenient than thematrix approach because it permits identifying better the corresponding columns).

Initially, we assume that we scan links 1, 5 and 8, that is SC ¼ f1;5;8g, and we look for a subset of links from the setOF ¼ f2;3;4;6;7g to be counted in order to get full observability. According to Table 1 eighth column, we have 5 differentscanned flows {w1,w2,w3,w4,w5}, which correspond to the subsets of links {1}, {1,5,8}, {5,8}, {5} and {8}, respectively. Inother words, w2 is the number of cars scanned at links 1, 5 and 8 and only in those links, w4 is the number of cars scannedat link 5 and only at this link, etc.

Step 0: Initialization. From this information we build the upper submatrix in matrix A (see iteration 0 of Table 3). Next,we build the central and lower submatrices of matrix A, based on the link flows and the OD flows, respectively (see iteration0 of Table 3), and construct the initial B list, which contains the routes (see the first row of Table 3 iteration 0), and the NBlist, which contains the scanned, link and OD flows, in this order (see the first column of Table 3 iteration 0). Note that noneof them is boldfaced, because at this stage we assume that there are no observations.

Step 1: Choose a pivot. We choose w1 as an unobservable flow in NB, that is, row i = 1 of matrix A and route r1 in B, thatis, column j = 1 of the same matrix, because the corresponding value a11 = 1 – 0 and perform the pivoting process (see (12))to check that it does not produce a matrix A⁄ with some 2 or �2 entries, and, since this is true, we go to Step 2.

Step 2: Pivoting. We consolidate the pivoting process, that is, replace matrix A with the transformed matrix A⁄, and ex-change the list of basic and non-basic flows at this stage. In other words, incorporate the flow w1 of NB into position 1 of listB and boldface it because it is observed, and the route flow r1 of B into position 1 of NB.

Step 3: Update the list of basic and non-basic flows. Since flow r1 in NB (rows of A) has null coefficients in all its col-umns associated with non-observed basic flows (columns of A), flow is observable, and then its name is boldfaced, that is,added to the set of observable flows. Next, we go to Step 1 again.

Table 3Resulting matrices corresponding to all steps of the pivoting algorithm.

Iteration 0 Iteration 1 Iteration 2

r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 w1 r2 r3 r4 r5 r6 r7 r8 r9 r10 w1 w2 r3 r4 r5 r6 r7 r8 r9 r10

w1 1 0 0 0 0 0 0 0 0 0 r1 1 0 0 0 0 0 0 0 0 0 r1 1 0 0 0 0 0 0 0 0 0w2 0 1 0 0 0 0 0 0 0 0 w2 0 1 0 0 0 0 0 0 0 0 r2 0 1 0 0 0 0 0 0 0 0w3 0 0 0 0 1 0 0 0 0 0 w3 0 0 0 0 1 0 0 0 0 0 w3 0 0 0 0 1 0 0 0 0 0w4 0 0 0 0 0 1 0 0 1 0 w4 0 0 0 0 0 1 0 0 1 0 w4 0 0 0 0 0 1 0 0 1 0w5 0 0 0 0 0 0 0 1 0 0 w5 0 0 0 0 0 0 0 1 0 0 w5 0 0 0 0 0 0 0 1 0 0v1 1 1 0 0 0 0 0 0 0 0 v1 1 1 0 0 0 0 0 0 0 0 v1 1 1 0 0 0 0 0 0 0 0v2 1 0 0 1 0 0 0 0 0 0 v2 1 0 0 1 0 0 0 0 0 0 v2 1 0 0 1 0 0 0 0 0 0v3 0 0 1 0 0 0 0 0 0 1 v3 0 0 1 0 0 0 0 0 0 1 v3 0 0 1 0 0 0 0 0 0 1v4 0 0 0 0 0 0 0 0 0 1 v4 0 0 0 0 0 0 0 0 0 1 v4 0 0 0 0 0 0 0 0 0 1v5 0 1 0 0 1 1 0 0 1 0 v5 0 1 0 0 1 1 0 0 1 0 v5 0 1 0 0 1 1 0 0 1 0v6 0 0 0 0 0 1 1 0 0 0 v6 0 0 0 0 0 1 1 0 0 0 v6 0 0 0 0 0 1 1 0 0 0v7 0 0 0 0 0 0 1 0 0 0 v7 0 0 0 0 0 0 1 0 0 0 v7 0 0 0 0 0 0 1 0 0 0v8 0 1 0 0 1 0 0 1 0 0 v8 0 1 0 0 1 0 0 1 0 0 v8 0 1 0 0 1 0 0 1 0 0t1 1 1 0 0 0 0 0 0 0 0 t1 1 1 0 0 0 0 0 0 0 0 t1 1 1 0 0 0 0 0 0 0 0t2 0 0 0 1 1 0 0 0 0 0 t2 0 0 0 1 1 0 0 0 0 0 t2 0 0 0 1 1 0 0 0 0 0t3 0 0 0 0 0 0 0 0 1 0 t3 0 0 0 0 0 0 0 0 1 0 t3 0 0 0 0 0 0 0 0 1 0t4 0 0 0 0 0 1 0 0 0 0 t4 0 0 0 0 0 1 0 0 0 0 t4 0 0 0 0 0 1 0 0 0 0t5 0 0 0 0 0 0 0 0 0 1 t5 0 0 0 0 0 0 0 0 0 1 t5 0 0 0 0 0 0 0 0 0 1t6 0 0 1 0 0 0 0 0 0 0 t6 0 0 1 0 0 0 0 0 0 0 t6 0 0 1 0 0 0 0 0 0 0t7 0 0 0 0 0 0 1 1 0 0 t7 0 0 0 0 0 0 1 1 0 0 t7 0 0 0 0 0 0 1 1 0 0


w1 w2 r3 r4 w3 r6 r7 r8 r9 r10 w1 w2 r3 r4 w3 w4 r7 r8 r9 r10 w1 w2 r3 r4 w3 w4 r7 w5 r9 r10

r1 1 0 0 0 0 0 0 0 0 0 r1 1 0 0 0 0 0 0 0 0 0 r1 1 0 0 0 0 0 0 0 0 0r2 0 1 0 0 0 0 0 0 0 0 r2 0 1 0 0 0 0 0 0 0 0 r2 0 1 0 0 0 0 0 0 0 0r5 0 0 0 0 1 0 0 0 0 0 r5 0 0 0 0 1 0 0 0 0 0 r5 0 0 0 0 1 0 0 0 0 0w4 0 0 0 0 0 1 0 0 1 0 r6 0 0 0 0 0 1 0 0 �1 0 r6 0 0 0 0 0 1 0 0 �1 0w5 0 0 0 0 0 0 0 1 0 0 w5 0 0 0 0 0 0 0 1 0 0 r8 0 0 0 0 0 0 0 1 0 0v1 1 1 0 0 0 0 0 0 0 0 v1 1 1 0 0 0 0 0 0 0 0 v1 1 1 0 0 0 0 0 0 0 0v2 1 0 0 1 0 0 0 0 0 0 v2 1 0 0 1 0 0 0 0 0 0 v2 1 0 0 1 0 0 0 0 0 0v3 0 0 1 0 0 0 0 0 0 1 v3 0 0 1 0 0 0 0 0 0 1 v3 0 0 1 0 0 0 0 0 0 1v4 0 0 0 0 0 0 0 0 0 1 v4 0 0 0 0 0 0 0 0 0 1 v4 0 0 0 0 0 0 0 0 0 1v5 0 1 0 0 1 1 0 0 1 0 v5 0 1 0 0 1 1 0 0 0 0 v5 0 1 0 0 1 1 0 0 0 0v6 0 0 0 0 0 1 1 0 0 0 v6 0 0 0 0 0 1 1 0 �1 0 v6 0 0 0 0 0 1 1 0 �1 0v7 0 0 0 0 0 0 1 0 0 0 v7 0 0 0 0 0 0 1 0 0 0 v7 0 0 0 0 0 0 1 0 0 0v8 0 1 0 0 1 0 0 1 0 0 v8 0 1 0 0 1 0 0 1 0 0 v8 0 1 0 0 1 0 0 1 0 0t1 1 1 0 0 0 0 0 0 0 0 t1 1 1 0 0 0 0 0 0 0 0 t1 1 1 0 0 0 0 0 0 0 0t2 0 0 0 1 1 0 0 0 0 0 t2 0 0 0 1 1 0 0 0 0 0 t2 0 0 0 1 1 0 0 0 0 0t3 0 0 0 0 0 0 0 0 1 0 t3 0 0 0 0 0 0 0 0 1 0 t3 0 0 0 0 0 0 0 0 1 0t4 0 0 0 0 0 1 0 0 0 0 t4 0 0 0 0 0 1 0 0 �1 0 t4 0 0 0 0 0 1 0 0 �1 0t5 0 0 0 0 0 0 0 0 0 1 t5 0 0 0 0 0 0 0 0 0 1 t5 0 0 0 0 0 0 0 0 0 1t6 0 0 1 0 0 0 0 0 0 0 t6 0 0 1 0 0 0 0 0 0 0 t6 0 0 1 0 0 0 0 0 0 0t7 0 0 0 0 0 0 1 1 0 0 t7 0 0 0 0 0 0 1 1 0 0 t7 0 0 0 0 0 0 1 1 0 0

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Table 4Resulting matrices corresponding to all steps of the pivoting algorithm.


w1 w2 r3 v2 w3 w4 r7 w5 r9 r10 w1 w2 v3 v2 w3 w4 r7 w5 r9 r10 w1 w2 v3 v2 w3 w4 r7 w5 r9 v4

r1 1 0 0 0 0 0 0 0 0 0 r1 1 0 0 0 0 0 0 0 0 0 r1 1 0 0 0 0 0 0 0 0 0r2 0 1 0 0 0 0 0 0 0 0 r2 0 1 0 0 0 0 0 0 0 0 r2 0 1 0 0 0 0 0 0 0 0r5 0 0 0 0 1 0 0 0 0 0 r5 0 0 0 0 1 0 0 0 0 0 r5 0 0 0 0 1 0 0 0 0 0r6 0 0 0 0 0 1 0 0 �1 0 r6 0 0 0 0 0 1 0 0 �1 0 r6 0 0 0 0 0 1 0 0 �1 0r8 0 0 0 0 0 0 0 1 0 0 r8 0 0 0 0 0 0 0 1 0 0 r8 0 0 0 0 0 0 0 1 0 0v1 1 1 0 0 0 0 0 0 0 0 v1 1 1 0 0 0 0 0 0 0 0 v1 1 1 0 0 0 0 0 0 0 0r4 �1 0 0 1 0 0 0 0 0 0 r4 �1 0 0 1 0 0 0 0 0 0 r4 �1 0 0 1 0 0 0 0 0 0v3 0 0 1 0 0 0 0 0 0 1 r3 0 0 1 0 0 0 0 0 0 �1 r3 0 0 1 0 0 0 0 0 0 �1v4 0 0 0 0 0 0 0 0 0 1 v4 0 0 0 0 0 0 0 0 0 1 r10 0 0 0 0 0 0 0 0 0 1v5 0 1 0 0 1 1 0 0 0 0 v5 0 1 0 0 1 1 0 0 0 0 v5 0 1 0 0 1 1 0 0 0 0v6 0 0 0 0 0 1 1 0 �1 0 v6 0 0 0 0 0 1 1 0 �1 0 v6 0 0 0 0 0 1 1 0 �1 0v7 0 0 0 0 0 0 1 0 0 0 v7 0 0 0 0 0 0 1 0 0 0 v7 0 0 0 0 0 0 1 0 0 0v8 0 1 0 0 1 0 0 1 0 0 v8 0 1 0 0 1 0 0 1 0 0 v8 0 1 0 0 1 0 0 1 0 0t1 1 1 0 0 0 0 0 0 0 0 t1 1 1 0 0 0 0 0 0 0 0 t1 1 1 0 0 0 0 0 0 0 0t2 �1 0 0 1 1 0 0 0 0 0 t2 �1 0 0 1 1 0 0 0 0 0 t2 �1 0 0 1 1 0 0 0 0 0t3 0 0 0 0 0 0 0 0 1 0 t3 0 0 0 0 0 0 0 0 1 0 t3 0 0 0 0 0 0 0 0 1 0t4 0 0 0 0 0 1 0 0 �1 0 t4 0 0 0 0 0 1 0 0 �1 0 t4 0 0 0 0 0 1 0 0 �1 0t5 0 0 0 0 0 0 0 0 0 1 t5 0 0 0 0 0 0 0 0 0 1 t5 0 0 0 0 0 0 0 0 0 1t6 0 0 1 0 0 0 0 0 0 0 t6 0 0 1 0 0 0 0 0 0 �1 t6 0 0 1 0 0 0 0 0 0 �1t7 0 0 0 0 0 0 1 1 0 0 t7 0 0 0 0 0 0 1 1 0 0 t7 0 0 0 0 0 0 1 1 0 0

Iteration 9 Iteration 10

w1 w2 v3 v2 w3 w4 v6 w5 r9 v4 w1 w2 v3 v2 w3 w4 v6 w5 v7 v4

r1 1 0 0 0 0 0 0 0 0 0 r1 1 0 0 0 0 0 0 0 0 0r2 0 1 0 0 0 0 0 0 0 0 r2 0 1 0 0 0 0 0 0 0 0r5 0 0 0 0 1 0 0 0 0 0 r5 0 0 0 0 1 0 0 0 0 0r6 0 0 0 0 0 1 0 0 �1 0 r6 0 0 0 0 0 0 1 0 �1 0r8 0 0 0 0 0 0 0 1 0 0 r8 0 0 0 0 0 0 0 1 0 0v1 1 1 0 0 0 0 0 0 0 0 v1 1 1 0 0 0 0 0 0 0 0r4 �1 0 0 1 0 0 0 0 0 0 r4 �1 0 0 1 0 0 0 0 0 0r3 0 0 1 0 0 0 0 0 0 �1 r3 0 0 1 0 0 0 0 0 0 �1r10 0 0 0 0 0 0 0 0 0 1 r10 0 0 0 0 0 0 0 0 0 1v5 0 1 0 0 1 1 0 0 0 0 v5 0 1 0 0 1 1 0 0 0 0r7 0 0 0 0 0 �1 1 0 1 0 r7 0 0 0 0 0 0 0 0 1 0v7 0 0 0 0 0 �1 1 0 1 0 r9 0 0 0 0 0 1 �1 0 1 0v8 0 1 0 0 1 0 0 1 0 0 v8 0 1 0 0 1 0 0 1 0 0t1 1 1 0 0 0 0 0 0 0 0 t1 1 1 0 0 0 0 0 0 0 0t2 �1 0 0 1 1 0 0 0 0 0 t2 �1 0 0 1 1 0 0 0 0 0t3 0 0 0 0 0 0 0 0 1 0 t3 0 0 0 0 0 1 �1 0 1 0t4 0 0 0 0 0 1 0 0 �1 0 t4 0 0 0 0 0 0 1 0 �1 0t5 0 0 0 0 0 0 0 0 0 1 t5 0 0 0 0 0 0 0 0 0 1t6 0 0 1 0 0 0 0 0 0 �1 t6 0 0 1 0 0 0 0 0 0 �1t7 0 0 0 0 0 �1 1 1 1 0 t7 0 0 0 0 0 0 0 1 1 0

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All this process is repeated several times. We arrive at the end of iteration 10, in which, for the first time, all the flows areboldfaced. This implies that for full observability, in addition to the set of scanned links SC ¼ f1;5;8g, we need to observelinks in the set {v2,v3,v4,v6,v7}, which are those which appear boldfaced in set B at this stage.

Note that the matrices in iterations 1 to 10 contain only zeroes, ones and minus ones, that is, no matrix has two or minustwo entries.

Though in Tables 3 and 4 we have included the rows of the observable flows in NB, they can be eliminated, because nochange takes place when pivoting once a flow becomes observable. This implies a very important reduction in the calcula-tions, a fact that takes special relevance in the second stage of the process, when elements in matrix A appear which are notzeroes, ones or minus ones, that is, when more resources are required for storage of matrix A and the pivoting process.

12

3

4

5

6

10

18

19

2732

3444

5052 54 55

6064

68

85

99102

120

127 128136

148152

158

160 172

180182

189190

192196

199

201

217220

221225

229231

237241

242

249 251252

253

257261 262

265268

271272

275

278

280

282283286

290

295299 300

302 303

305

311 313

317318

321322

324325

326

327

329330

331

335

337

343344

347

354

356

372

381

382386

389390

391392

395 396

399401

404

406

409

415416

419425426

429

431

434

437

440

441442

443444

455

462

467

473

478

482

489

502

509

511

517

520521

528530

533 534

539

543 546

547548

552

555

556

557566

568

575576

581 582

597

601602

604608

614

618

619

621622

625

631632

645

647

649

654

659660

669

672

Fig. 2. The Cuenca network showing its nodes and links, together with a minimum subset (outlined in the figure) of links to be scanned for fullobservability. Solid lines refer to this minimum subset of scanned links, dashed lines refer to other links, arrows indicate the flow direction, and numbersrefer to link numbers.


We note that Tables 3 and 4 provide important partial observability information. One flow is observable if all its non-nullcoefficients (in most cases 1 or �1) involve observed flows. In particular, they show the sets of flows that are observable aftersequentially observing the flows in the subset {w1,w2,w3,w4,w5,v2,v3,v4,v6,v7} and in the indicated order. These observableflows appear boldfaced in the first column of the corresponding tables, which also show the matrix of coefficients to be usedfor calculating the observable flows in terms of the corresponding observed column flows, which appear identified in the firstrow. As one example, at iteration 5 in Table 3, that is, when only the scanned information {w1,w2,w3,w4,w5} is available,flows {r1,r2,r5,r8,v1,v5,v8, t1} are observable and from the corresponding incidence matrix we get

Fig. 3.lines re

r1 ¼ w1; r2 ¼ w2; r5 ¼ w3; r8 ¼ w5; v1 ¼ w1 þw2;

v5 ¼ w2 þw3 þw4; v8 ¼ w2 þw3 þw5; t1 ¼ w1 þw2;ð15Þ

that is, the formulas of the observable flows in terms of the scanned flows.

85

372

311

34

313

401

618

303

521

280

18

1955

196

272

331

335

621

354

305329

502

3

172

517

533

44

473

321

10

32

60

102

241

283

434

530

552

415

543

608

120

322

330

220

5

27

237

275

390

509

654

64

455

482

4

199

343

286

622

1

429

555

604

278

262

52

649

625

317

201

443

520

632

645

381

582

399 271

68

318

597

391

602

556

337

386382

290

659

128

669

99

2

229

395

534

672

511

426

257

404

152

548

54

268282

327

251

631

127

425

441

489

356

409

136

325

442

462

217

160

225

299

344

261

253

242

406

182557

249

158

416

601

478

419

231

444

6

180

148

192

546

221

300

619

396

265

189190

392

547

660

566

581575

576

The Cuenca network showing the subset (outlined in the figure) of 158 scanned links. Solid lines refer to this subset of 158 scanned links, dashedfer to other links, arrows indicate the flow direction, and numbers refer to link numbers.


Similarly, at iteration 7 in Table 4, that is, after knowing the scanned information and the link flows v2 and v3, the subsetof observable flows is {r1,r2,r5,r8,v1,r4,v5,v8, t1, t2,}, that is, in addition to (15), we have (see Table 4):

Fig. 4.with thnumbe

r4 ¼ v2 �w1; t2 ¼ v2 þw3 �w1:

3.3. Some additional advantages of the proposed method

Apart from the interesting properties of the proposed method, which is based on writing all flows in terms of route flows,it has other important advantages, as follows:

8

2831

697071

235

236

248 254

256

273

287288

295296

310

323

326

342

440

507

528

541

563

596

598

599

124

185

252

336

384

508

The Cuenca network showing the set (outlined in the figure) of extra links to be counted in order to get full observability of the network togethere 158 scanned links. Solid lines refer to subset of extra counted links, dashed lines refer to other links, arrows indicate the flow direction, and

rs refer to link numbers.

Table 5A comparison of the required number of products/quotients np and np1

� �and sums for the simplified ns and ns1

� �and the real arithmetic

methods.

Standard Simplified

Example np ns np1ns1

Illustrative 1710 1440 0 36Cuenca 7532518140 7524271590 0 15496


1. Partial observability information is contained in all iterations.2. Full observability can be derived directly from the initial incidence matrix A, at Iteration 0. In fact, any subset of rows of

matrix A with full rank (the same rank as A), leads to complete observability of the traffic network. The important numer-ical property of this matrix is that it contains only zeroes and ones, and consequently, ranks, inverses and determinantscan be calculated exactly, without the appearance of numerical problems.

3. The process is reversible, i.e. once one row flow has been exchanged with one column flow, it can be exchanged again toreturn to its initial position, and this can be done, at least during the ternary arithmetic process, with neither error prop-agation nor numerical problems.

4. Since most of the row elements are null, we can store only non-null values, saving a lot of memory space in largenetworks.

5. Matrix A at any step (iteration) contains the same information, but written in a different form. This permits updatingresults starting from any of the iterations, by exchanging row and column flows as desired, that is, without the needof starting from Iteration 0 again.

4. The Cuenca network

In this section we show that the proposed method can be applied to real size networks such as the Cuenca network (seeFig. 2), which consists of 503 links, 232 nodes, 139 OD-pairs and acyclic 2190 routes, which have been obtained by the SUEassignment, with h = 0.5, using the software VISUM10 for a demand of 11,419 trips in the afternoon peak hour of the city. Thisdemand was the result of the household surveys and the travel standards of Cuenca. In this network, the minimum subset ofscanned links sufficient for full observability contains 175 links and is shown in Fig. 2. This means 34% of the existing links. Thisset of 175 links was obtained using the methods described in Section 2.

However, we have selected a subset of only 158 links to be scanned, which is shown in Fig. 3, and we have looked for acomplementary subset of links to be counted, so that together with the 158 scanned links they provide full observability ofthe network. These 158 scanned links allow observing 2127 route flows, 125 OD pair flows and 407 independent link flows.

The algorithm described in Section 3 was applied to this network and set of scanned links, and then, the ternary arith-metic step could be used in 2184 iterations. This means that only ones, zeros and minus ones were involved in the updatedmatrices during the process.

Next, the real arithmetic step was used until full observability was attained at iteration 2190 (only 2190–2184 = 6 iter-ations), that is, only 0.3% of the total number of iterations. The final result is that in addition to the 158 scanned links, a sub-set of 34 extra links need to be counted. This subset is given in Fig. 4.

Table 5 shows a comparison of the required number of products/quotients (np and np1) and sums (ns and ns1 ) for the sim-

plified (ternary arithmetic) and the real arithmetic methods, respectively. We have used p for products or quotients and s forsums, while the subindex 1 refers to the simplified method. As can be seen it is a huge reduction.

In order to have an idea of the cpu time required, the algebraic implementation of the Cuenca network took 992 s of cpu.The reduction produced by the proposed method using the ternary representation depends on the computer implementation(Matlab, C, C++, etc.), but Table 5 indicates that the reduction will be very important. An optimized implementation of theternary arithmetic approach, which is complex and depends on the language selected, is the aim of a future project.

5. Conclusions and future work

The main conclusions that can be drawn from this paper are:

1. Choosing route flows as basic flows to write link, OD and scanned links is more convenient than choosing OD flows, fromthe point of view of simplicity and because only zeroes and ones appear in the incidence matrix. This makes the initialmatrices for the algebraic and topological method coincident, and consequently of the same rank. The price to pay is thatincidence matrices become large in size.

2. The pivoting process can be carried out using matrices containing zeroes, ones and minus ones in a large proportion of allthe required iterations, and this implies less memory size, faster computation and no precision nor pivot selection

10 VISUM is a flexible software system for transportation planning, travel demand modeling and network data management.


problems. Note that no precision or non-zero pivot identification problems must be dealt with, contrary to what happenswhen writing the flows in terms of OD flows.

3. As the number of observable flows increases with the number of observed flows, the number of operations required forthe pivoting process simultaneously decreases, because only non-observable flow rows change, that is, need to beupdated.

4. When the pivoting forces entries different from zeroes, ones or minus ones, real arithmetic must be used, which requiresmore memory and cpu resources. However, when this occurs, the size of the matrix that needs to be updated is muchsmaller than the initial matrix.

5. The proposed methods are valid not only for full observability, but for partial observability. In fact, the intermediate iter-ations contain this partial observability information.

6. The illustrative example and the real case example of the Spanish city of Cuenca show that the proposed method is appli-cable not only to toy examples but to real problems.

7. Since incidence and flow matrices are scarce and banded, and its size is very large, scarce and banded matrix implemen-tations of the proposed method are required for large networks.

8. Since the size of the incidence matrices explodes with the size of the network, the proposed methods have a limit of itsapplicability. In order to apply the proposed method to large networks, it must be implemented taking into account: (a)the memory requirements, that is, storing only the non-null elements in the matrices, (b) the cpu requirements, that is,optimizing the pivoting process by considering only the matrix elements that change during this process (a very reducedproportion of all matrix elements), (c) the frequencies of their appearance (this leads to substantial savings), etc. This isthe aim of a future paper to be developed by the group.

Acknowledgments

The authors are indebted to the Spanish Ministry of Science and Technology (Project BIA2005-07802-C02–01 and TRA2010-17818) and to the Council of Education and Science of Castilla-La Mancha (A06-016) for partial support of this work.

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a ternary-arithmetic topological based algebraic method for networks traffic observability

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