THE OBSERVABILITY PROBLEM IN TRAFFIC MODELS

An Algebraic Step by Step Method

Pilar Jim´enez, Santos S´anchez-Cambronero, Inmaculada Gallego and Ana Rivas

Department of Civil Engineering, University of Castilla La Mancha, 13071, Ciudad Real, Spain

Keywords:

Trafﬁc model, Link ﬂow estimation, OD ﬂow estimation, Algebraic methods, Artiﬁcial intelligence.

Abstract:

This article deals with the problem of observability of trafﬁc networks, understanding as such the problem of

identifying if a set of available ﬂow measurements is sufﬁcient to estimate the remaining ﬂows in the network,

OD-pair or link ﬂows. An algebraic method for solving the observability problems is given. Speciﬁcally, a

step by step procedure allowing updating the information once each item of information (OD-pair or link ﬂow)

becomes available. The method is illustrated by its application to a simple network. The results show that the

proposed method provide useful information on which OD pair or link ﬂows are informative on other OD pair

and link ﬂows, and that the method is applicable to large networks due to its numerical robustness and stability.

1 INTRODUCTION

There are some situations in practice in which one

needs to know the state of a trafﬁcnetwork by measur-

ing a subset of ﬂows and, based on this information,

predicting other ﬂows, which are not free to take ar-

bitrary values but must be subject to constraints, im-

posed by the networks topology, to be in agreement

with the measured ﬂows. In this area, some Artiﬁcial

Intelligence techniques, such as Bayesian networks or

Neural networks, have been used (see (Castillo et al.,

2008c; Yin et al., 2002; Ledoux, (1997; Smith and

Demetsky, 1994), for example). However, before us-

ing these techniques it is important to discuss the ob-

servability problem, which consists of identifying if a

set of available (measured) ﬂows is sufﬁcient to cal-

culate other given subset of ﬂows.

Observability analysis is a previous step to state

estimation. It addresses the question: do we have

enough measurements to estimate the state of a sys-

tem? Observability techniques are essential in many

ﬁelds of knowledge, and in particular in trafﬁc predic-

tion.

Some examples of observability problems in traf-

ﬁc networks 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 observations that al-

low the knowledge of other given subset of ﬂows

or the observability of all ﬂows in the network.

3. Identify observable ﬂows, given a set of observed

ﬂows (partial observability).

Though the problem of observability can be stated

in a general context, as done in (Castillo et al., 2007),

who discuss the problem of observability of linear

system of equations and inequalities, most of exist-

ing publications relate to particular ﬁelds (see, for ex-

ample, the survey provided by (Abur and Exp´osito,

2004), which includes applications to power sys-

tems).

Observability techniques can be classiﬁed as:

1. Algebraic. These techniques consider the al-

gebraic relations between the ﬂows and operate

them algebraically to draw observability conclu-

sions (see (Monticelli and Wu, 1985a; Monticelli

and Wu, 1985b), (Monticelli, 2000), (Abur and

Exp´osito, 2004), (Gou and Abur, 2000; Gou and

Abur, 2001), (Castillo et al., 2005), (Castillo et

al., 2006), (Castillo et al., 2008a), (Castillo et al.,

2008b)).

2. Topological. These techniques consider only

topological and/or qualitative relations be-

tween ﬂows to derive observability results (see

(Clements and Wollenberg, 1975); (Krumpholz

et al., 1980), (Nucera and Gilles, 1991) and

(Castillo et al., 2007; Castillo et al., 2008b)).

Since all these techniques are based on the math-

ematical properties of the systems of equations and

560

Jiménez P., Sánchez-Cambronero S., Gallego I. and Rivas A. (2010).

THE OBSERVABILITY PROBLEM IN TRAFFIC MODELS - An Algebraic Step by Step Method.

In Proceedings of the 2nd International Conference on Agents and Artiﬁcial Intelligence - Artiﬁcial Intelligence, pages 560-567

DOI: 10.5220/0002717805600567

 SciTePress

have the same structure for trafﬁc problems, these ap-

proaches, which have already been applied to “phys-

ical” networks, are equally applicable to trafﬁc net-

works.

This paper is focused on algebraic methods, which

can be used with two aims in mind: (a) obtaining the

exact algebraic relation among different ﬂows, such

that some can be calculated when other are known,

and (b) obtain observability information, that is, de-

termine which ﬂows can and which ﬂows cannot be

calculated when a subset of ﬂows is known, but with-

out seeking the corresponding formulas which allow

us the calculations. The second aim is simpler and

requires less effort than the ﬁrst one.

This paper deals with observability problems, as-

suming that the matrix relating link and OD ﬂows is

given. However, though the solution of the system of

equations implied by OD-link relations is clearly de-

pendent on the particular selection of the choice prob-

abilities or proportions of the users selecting different

paths, the observability problem is not in general. Of

course, there are many especial cases in which some

linear equations can become linear combinations of

other, but what we are really interested in is in struc-

tural linear dependencies and not accidental depen-

dencies. Thus, ﬁxing the probabilities (unless they

are ﬁxed to zero values) is not a problem. In fact, they

should be ﬁxed at random with the aim of avoiding ac-

cidental dependencies (with zero probability in prac-

tice). Another different matter is to ﬁx some proba-

bilities to zero or not zero. This means incorporating

or deleting paths, and then, the observability problem

is clearly dependent on these selections.

Note that the technique developed is applicable

not only for observingtotal link ﬂows, but also for dis-

aggregated link ﬂows, by origin, destinations, or any

type of disaggregation as that associated with plate

scanning. However, for the sake of simplicity we il-

lustrate the method with total link ﬂows.

This paper is organized as follows. In section 2

the observability problem is stated and the algebraic

method is given. Section 3 illustrates the proposed

method by their application to a simple network. Fi-

nally, in Section 4 some conclusions are given.

2 THE OBSERVABILITY

PROBLEM

Consider a trafﬁc network (N , A ) where N is the set

of nodes and A is the set of links. Let v

be the ﬂow of

link a, p

the probability of a user to choose path k of

the OD-pair i, t

the OD-pair ﬂow i, δ

the incidence

matrix, that is, δ

= 1 if link a belongs to path k of

OD-pair i, and 0, otherwise.

Then, for compatibility of OD-pair and link ﬂows,

i.e., for the conservation law to be satisﬁed, one must

have:

∑

, (1)

where F is a matrix with elements f

deﬁned as

∑

;

∑

= 1; ∀i. (2)

The system of equations (1) in matrix form becomes

V = FT, (3)

where V and T are the column matrices of link and

OD-pair ﬂows with dimensions m × 1 and n × 1, re-

spectively.

It is important to note that given the OD ﬂows t

Equation (3) allows us to calculate the link ﬂows v

Thus, given the topology of the network and the OD

path ﬂows, the OD ﬂows are the minimum number of

data items needed to determine the remaining ﬂows

of the network. In fact, enumerating paths can be

avoided if the matrix F, which gives the proportions

of OD ﬂows travelling through each link, is known.

However, since OD ﬂows are not normally known,

each of them needs to be replaced by link ﬂows that

are practically observable. As we will see, the alge-

braic approach proceeds by replacing the OD ﬂows

with the observed links, until all of them have been

replaced.

Therefore, it is assumed that a subset T

of T and

a subset V

of V are observed (known) and the com-

plementary subsets T

of T and V

of V are not.

Then, the system (3) can be partitioned as

− − −

| F

− − − + −− −

| F

− − −

, (4)

In order to join (write together) the unknown ﬂow

variables T

and V

, the system (4) can be written

in the alternative equivalent form

D =



−F

− − − − −

−F

+ V



= Bz =



| −I

− − − + − − −

| 0



− − − − −



(5)

a system where the unknowns appear on the right

hand side and the observations on its left hand side.

The coefﬁcient matrix is B, the independent term col-

umn matrix is D and z contains the unknown ﬂows T

and V

We ﬁnally note that for illustrative purposes and

for the sake of facilitating the understanding of this

example, the OD-pair and link ﬂows have been dis-

tinguished as two different items, but from a mathe-

matical point of view they are undistinguishable. In

other words, the data and the unknown sets can con-

tain any subset of variables.

THE OBSERVABILITY PROBLEM IN TRAFFIC MODELS - An Algebraic Step by Step Method

561

2.1 The Algebraic Approach: A

Step-by-step Procedure

The step-by-step procedure allows us to discover what

happens, in terms of observability, as some new items

of information (OD-pair or link ﬂows) become avail-

able.

The rationale for the algorithm below is basically

to express the observable ﬂows in terms of the actu-

ally observed ﬂows. That is, to transfer “columns to

rows” and vice versa in matrix F. If all variables can

be expressed as linear combinations of measurements,

the network is observable; otherwise, it is not. The

actual operations are based on the orthogonal trans-

formation algorithm reported in (Castillo et al., 2000;

Castillo et al., 2002). The proposed algorithm pro-

vides two sets and one matrix of interest:

1. Set C of cardinality n, whose elements are c

, con-

tains the list of a minimum set of required mea-

surements to attain observability of all variables.

These measurements are denominated essential

measurements or basic measurements.

2. Set B of cardinality m, whose elements are b

contains the list of redundant measurements for

observability purposes, that is, even if the mea-

surements are lost, the network remains observ-

able.

3. Matrix F

of dimension m× n contains the coefﬁ-

cients of the linear combinations of the redundant

measurements in terms of the required (essential)

ones.

Algorithm 1 (Basic observability procedure).

INPUT. The set of links A , the set of OD-pairs,

two disjoint subsets B and C of the set H of all ﬂows

(OD-pair and link ﬂows) such that H = B ∪ C , B ∩

C =

0 and |C | = n, an initial matrix F giving the ﬂows

in B in terms of the ﬂows in C .

OUTPUT. A transformed matrix F

∗

associated

with the transformed

B and

C sets.

Step 1: Choose a pivot. Choose an unobservable el-

ement in B , that is, a row i of matrix F that we called

α, and an element in C , that is a column j of the same

matrix, called β, such that the corresponding value

αβ

6= 0, and go to Step 2. If such an element f

αβ

null, stop the process informing on the impossibility

of exchanging the supplied elements b

and c

Matrix F can be easily obtained from the network

topology. For example, it can be the matrix F in (3) or any

other resulting after manipulation of this one by exchanging

the variables in B and C .

Step 2: Pivoting. Perform the pivoting process, that

is, calculate the transformed matrix F

∗

using the fol-

lowing transformation, being α and β the i row and j

column, respectively, chosen for pivoting in Step 1,

∗











−

αj

αβ

iβ

if i 6= α; j 6= β

−

αj

αβ

if i = α; j 6= β

iβ

αβ

if i 6= α; j = β

αβ

if i = α; j = β

(6)

Next, exchange the ﬂow c

in terms of the ﬂow b

and all other ﬂows in C at this stage. In other words,

incorporate the ﬂow in position α of B into position β

of list C, and the ﬂow in position β of C into position

α of list B.

Once replaced c

in all equations associated with

the system

B = FC,

we obtain the new matrix F

∗

such that

∗

= F

∗

where the asterisk refers to the new situation, i.e. after

the interchange of b

and c

has been done.

Note that the systems of equations B = FC and

∗

= F

∗

are equivalent in the sense that they have

the same solutions.

Step 3: Update the list of essential and redundant

ﬂows. Return matrix F

∗

and the updated ﬂow subsets

∗

and C

∗

. If a non-boldfaced ﬂow in B (row of F

∗

)

has null coefﬁcients in all its columns associated with

non-observable basic ﬂows (columns of F

∗

), the ﬂow

is observable, and then its name is boldfaced, that is,

added to the set of observable ﬂows.

Since it appears to be more convenient and infor-

mative to update the ﬂow knowledge as soon as each

unit of information is obtained, an algorithm for this

step-by-step process is given below. It uses Algorithm

1 as its main tool.

Algorithm 2 (Observability updating procedure).

INPUT. A list D of ﬂows to be observed, the

initial matrix F, and a partition B and C of all the

ﬂow variables H (OD-pair and link ﬂows) such that

H = B ∪ C , B ∩ C =

0 and |C | = n.

OUTPUT. The updated matrix F

∗

and sets B

∗

and C

∗

, together with the sets B and C of all ﬂows

that become known in B and C , respectively, due to

the observed ﬂows in each step.

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562

Step 0: Initialization step. Initialize the sets B and

C of known ﬂows in B and C , respectively, to empty

sets.

Repeat the following steps for each ﬂow d

in the

list D .

Step 1: Update observability matrix. If the ﬂow d

is already in set C , i.e. d

coincides with some c

∈ C ,

simply add d

to list C . Otherwise, it must coincide

with some b

∈ B , and then use Algorithm 1 to

incorporate the ﬂow variable d

≡ b

to set C . To this

end, select a ﬂow c

of C not in C to be exchanged

with d

. If this is not possible because there is no

non-null element f

αj

, inform on this impossibility

of observing d

(it must already have a ﬁxed value,

which can be calculated in terms of already observed

ﬂows) increase r in one unit and continue in Step 1.

Otherwise, update matrix F to F

∗

, exchange ﬂows c

and d

in sets B and C , respectively, using Algorithm

1 and add d

to list C .

Step 2: Identify all known ﬂows in set B . Find the

known ﬂows b

∈ B , i.e. the rows of F such that the

are null for all j associated with the unknown

ﬂows in C , and add them to set B .

Step 3: Return observability information. Return

matrix F

∗

, the sets B

∗

and C

∗

, and the subsets of

known ﬂows B and C , increase r in one unit and con-

tinue in Step 1.

Both algorithms 1 and 2 return matrices F

∗

, which

give the symbolic relation between the ﬂows in B and

those in C , that is, the linear formulas that permit the

ﬂows in B to be written in terms of those in C . Note

that once calculated, they can be used many times (for

observations taken at different days, hours, etc.).

Note that the technique in Algorithms 1 and 2 does

not require the actual values of the observed ﬂows.

This is an important advantage from a practical point

of view because one is not subject to observation er-

rors that in quite a few cases can and certainly do lead

to the incompatibility of the system of equations (3).

Since this method works using algebraic opera-

tions with real numbers, it is subject to rounding er-

rors. In particular, testing for zero ﬂows must be re-

placed by testing for small numbers, which can give

numerical problems for large networks.

With respect to use the second algorithm with

a given list D of observable ﬂows, it is convenient

to study the rank associated with the rows of D , to

avoid observing redundant ﬂows, that is, a set of ﬂows

which contains the same information as a proper sub-

set of it. This check permits the unnecessary observa-

tions, i.e., those observations that are linear combina-

tions of other observations, to be eliminated.

2.2 Computational Issues

In this section we deal with some computational is-

sues.

Since the proposed method is a pivoting process

analogous to the Gauss elimination, all the gained ex-

perience for Gauss methods is applicable to the pro-

posed method. This implies that it is applicable to

very large networks.

For some particular cases of the choice probabil-

ities and network topologies, the system can become

ill-conditioned, and then, the standard methods used

to solve this problem can be applied. In particular one

can use the different well known partial and complete

pivot strategies, which are very useful.

The complexity of algorithm 2 is similar to the

complexity of inverting a matrix. If one has n essen-

tial variables, the number of operations required for

obtaining a minimum set of observed links for observ-

ability of the whole system is (3n−1)n

, which gives

a clear idea of how the size of the problem affects the

computational time.

3 EXAMPLE OF APPLICATION

In this section we illustrate the proposedmethod by its

application to a simple network in order to be able to

show the results in form of tables of reasonable size.

Though this method was tested using other networks,

as the Nguyen-Dupuis network (13 nodes, 38 links, 8

OD pairs, 50 routes), and the real Ciudad Real net-

work (102 nodes, 218 links, 72 OD pairs, 179 routes),

for example, and similar results were obtained. Since

the algebraic techniques used are very well known

from the point of view of their numerical robustness

and stability, it can be said that they behave very well

for very large networks.

The simple network consists of 9 nodes and 18

links, as shown in Figure 1. We use an example of

bidirectional ﬂow, i.e., we assume the existence of

symmetric links, i.e. any pair of nodes i and j is con-

nected in both directions by links ℓ

and ℓ

, respec-

tively.

We have assumed the following OD-pairs ﬂows

(elements of the matrix T):

} ≡ {t

where the subindices refer to the OD-pair or the node

numbers, respectively, and the paths (given in terms

THE OBSERVABILITY PROBLEM IN TRAFFIC MODELS - An Algebraic Step by Step Method

563

Figure 1: The simple network, showing the nodes and links.

of nodes):

OD-pair 1− 4 : {1, 2,5, 4}, {1,2, 5,6, 4}, {1,5, 4},

{1, 5,6, 4}, {1,6, 4}, {1,6, 5,4},

{1, 3,6, 4}, {1,3, 6,5, 4},

OD-pair 2− 4 : {2, 5,4}, {2, 5,6, 4},

OD-pair 3− 4 : {3, 6,5, 4}, { 3,6, 4},

OD-pair 4− 1 : {4, 5,2, 1}, { 4,6, 5,2, 1}, {4,5, 1},

{4, 6,5, 1}, {4,6, 1}, {4,5, 6,1},

{4, 6,3, 1}, {4,5, 6,3, 1},

OD-pair 4− 2 : {4, 5,2}, {4, 6,5, 2},

OD-pair 4− 3 : {4, 5,6, 3}, { 4,6, 3}.

To illustrate the step-by-step method we explain

our input data and give a detailed explanation of the

different steps in Algorithm 2 below.

Example 1 (The step-by-step approach) .

INPUT. It consists of the following list D of ﬂows

to be observed (only link ﬂows are observed in this

example):

D ≡ {v

the initial matrix F, which has been obtained using

(1), and is given in Table 1, iteration 0, and as

partition of the ﬂow variables H we take B as the

set of link ﬂows, and C as the set of OD-pair ﬂows,

respectively.

Step 0: Initialization step. We initialize the sets of

known ﬂows to empty sets, that is, B =

0 and C =

We show the initial sets B and C and their updated

versions B

∗

and C

∗

in the ﬁrst rows and columns, re-

spectively, of the tables, and we distinguish the sets B

and C with boldfaced letters.

With all this, we have the initial observability

information in Table 1, iteration 0. Note that in this

and the following tables, the matrix F and the sets B ,

C , B and C are shown.

Next, we repeat the following steps for each ﬂow

in the list D , starting with ﬂow v

Step 1: Update observability matrix. Since the

ﬂow v

is not in set C but is in set B , Algorithm 1

is used to incorporate the ﬂow variable v

to set C.

To this end, we select the ﬂow c

≡ t

of C to be

exchanged with v

, and update matrix F to F

∗

and

add v

to list C (boldfaced) (see the right part of

Table 1).

Step 2: Identify all known ﬂows in set B . At this

stage, there are known ﬂows b

∈ B , i.e. there are

rows of matrix F with null values in the columns of

unknown ﬂows in C , {t

}. So, the ﬂows

} are added to B .

Step 3: Return observability information. We

produce this information in the right part of Table 1.

Since repeating all the steps for all the iterations

would be too reiterative and space consuming, we

assume that we are at the beginning of iteration 4 (see

Table 3), i.e., when we will observe v

, and continue

with Steps 1 to 3, as follows.

Step 1: Update observability matrix. Since the

ﬂow v

is not in set C but is in set B , Algorithm

1 is used to incorporate the ﬂow variable v

to set

C. To this end, we select the ﬂow c

≡ t

of C to be

exchanged with v

. Note that this is not the only

option, because there are other non-null f

αj

values

associated with the unobserved ﬂows in C , such as

or v

. Then, we update matrix F to F

∗

and add

to list C (we boldface it, as shown in Table 3),

iteration 5.

Step 2: Identify all known ﬂows in set B . At this

stage, there are ﬂows b

∈ B which become known,

i.e. there are rows of the matrix F with null values in

the column of t

, the only unknown ﬂow in C . In this

iteration, v

is the only new known ﬂow added to B .

Step 3: Return observability information. We

return the information in Table 3, iteration 5.

Finally, in the next iteration all the ﬂows become

known (see that in Table 4 all the ﬂows are boldfaced).

Note also that the coefﬁcients in this table permit

ICAART 2010 - 2nd International Conference on Agents and Artificial Intelligence

564

Table 1: Initial F matrix and F after observing {v

Iteration 0

0.3 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 0.0

0.3 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 0.0

0.3 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 0.0

0.3 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 0.0

0.3 0.0 0.7 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 1.0

0.3 0.0 0.3 0.3 0.5 0.0

0.3 0.3 0.0 0.3 0.0 0.5

0.0 0.0 0.0 0.3 1.0 0.0

0.3 0.7 0.0 0.0 0.0 0.0

0.5 0.3 0.3 0.0 0.0 0.0

0.0 0.0 0.0 0.5 0.5 0.5

0.5 0.3 0.3 0.0 0.0 0.0

0.0 0.0 0.0 0.5 0.5 0.5

Iteration 1

4.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 0.0

1.0 0.0 0.7 0.0 0.0 0.0

0.0 0.0 0.0 0.3 0.0 1.0

1.0 0.0 0.3 0.3 0.5 0.0

1.0 0.3 0.0 0.3 0.0 0.5

0.0 0.0 0.0 0.3 1.0 0.0

1.0 0.7 0.0 0.0 0.0 0.0

2.0 0.3 0.3 0.0 0.0 0.0

0.0 0.0 0.0 0.5 0.5 0.5

2.0 0.3 0.3 0.0 0.0 0.0

0.0 0.0 0.0 0.5 0.5 0.5

Table 2: F after observing {v

} and {v

Iteration 2

4.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 4.0 0.0 0.0

1.0 0.0 0.7 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 1.0

1.0 0.0 0.3 1.0 0.5 0.0

1.0 0.3 0.0 1.0 0.0 0.5

0.0 0.0 0.0 1.0 1.0 0.0

1.0 0.7 0.0 0.0 0.0 0.0

2.0 0.3 0.3 0.0 0.0 0.0

0.0 0.0 0.0 2.0 0.5 0.5

2.0 0.3 0.3 0.0 0.0 0.0

0.0 0.0 0.0 2.0 0.5 0.5

Iteration 3

4.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 4.0 0.0 0.0

1.0 0.0 0.7 0.0 0.0 0.0

0.0 0.0 0.0 -1.0 0.0 1.0

1.0 0.0 0.3 1.0 0.5 0.0

1.0 0.3 0.0 0.5 0.0 0.5

0.0 0.0 0.0 1.0 1.0 0.0

1.0 0.7 0.0 0.0 0.0 0.0

2.0 0.3 0.3 0.0 0.0 0.0

0.0 0.0 0.0 1.5 0.5 0.5

2.0 0.3 0.3 0.0 0.0 0.0

0.0 0.0 0.0 1.5 0.5 0.5

obtaining all the ﬂows of B in terms of the observed

ﬂows, that now are all the ﬂows in C :

= 4v

= v

.. . . .. . . . .. . .. . . . . ... .. .

= −v

−2v

= −v

.. . . .. . . . .. . .. . . . . ... .. .

= v

−v

Next, we illustrate the general problems stated in the

introduction with some particular cases.

Table 3: F after observing {v

} and

Iteration 4

4.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 4.0 0.0 0.0

1.0 0.0 0.7 0.0 0.0 0.0

0.0 0.0 0.0 -1.0 0.0 1.0

-2.0 0.0 -0.7 -2.0 2.0 0.0

1.0 0.3 0.0 0.5 0.0 0.5

-2.0 0.0 -0.7 -1.0 2.0 0.0

1.0 0.7 0.0 0.0 0.0 0.0

2.0 0.3 0.3 0.0 0.0 0.0

-1.0 0.0 -0.3 0.5 1.0 0.5

2.0 0.3 0.3 0.0 0.0 0.0

-1.0 0.0 -0.3 0.5 1.0 0.5

Iteration 5

4.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 4.0 0.0 0.0

1.0 0.0 0.7 0.0 0.0 0.0

0.0 0.0 0.0 -1.0 0.0 1.0

-2.0 0.0 -0.7 -2.0 2.0 0.0

-3.0 3.0 0.0 -1.5 0.0 -1.5

-2.0 0.0 -0.7 -1.0 2.0 0.0

-1.0 2.0 0.0 -1.0 0.0 -1.0

1.0 1.0 0.3 -0.5 0.0 -0.5

-1.0 0.0 -0.3 0.5 1.0 0.5

1.0 1.0 0.3 -0.5 0.0 -0.5

-1.0 0.0 -0.3 0.5 1.0 0.5

Problem 1. Determine if the subset of link ﬂows

B ≡ {v

} is sufﬁcient to determine the subset

of trafﬁc ﬂows G ≡ {v

Solution. The answer to this problem is positive, be-

cause supplying the given set B as the input list to

Algorithm 2, it returns what it is in Table 2, iter-

ation 2, where we can see that all the ﬂows in the

given set G are boldfaced, i.e. the ﬂows in G are

observable.

Moreover, if we compare these results with Fig-

ure 1, we are able to deduce that the ﬁrsts eight

links of the network (upper area) are lineal com-

binations. So, any pair of links of this set

} allows us to know the

ﬂow of all these links.

Problem 2. Determine a minimum set of observa-

tions (OD-pair and/or link ﬂows) that allow ob-

servability of the network.

Solution. Since the rank of matrix F in Table 3 is

n = 6, then, the minimum subset must contain

6 ﬂows. Thus, after applying Algorithm 2 with

D ≡ {v

} we obtain Iteration

6, Table 4, where all observable ﬂows are bold-

faced, i.e., those whose coefﬁcients of unobserved

essential variables are null (this means that the lin-

ear combinations can be calculated, because all

nonnull terms are known).

Problem 3. Identify observable ﬂows, given the set

of observed ﬂows {v

Solution. In Iteration 4 of Table 3, after incorpo-

rating all observed ﬂows, it can be seen that

and v

have a zero in the

THE OBSERVABILITY PROBLEM IN TRAFFIC MODELS - An Algebraic Step by Step Method

565

Table 4: F after observing {v

Iteration 6

4.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 4.0 0.0 0.0

-1.0 -2.0 2.0 1.0 0.0 1.0

0.0 0.0 0.0 -1.0 0.0 1.0

0.0 2.0 -2.0 -3.0 2.0 -1.0

-3.0 3.0 0.0 -1.5 0.0 -1.5

0.0 2.0 -2.0 -2.0 2.0 -1.0

-1.0 2.0 0.0 -1.0 0.0 -1.0

-3.0 -3.0 3.0 1.5 0.0 1.5

0.0 1.0 -1.0 0.0 1.0 0.0

0.0 0.0 1.0 0.0 0.0 0.0

0.0 1.0 -1.0 0.0 1.0 0.0

column positions of t

and t

, which are the only

unknown OD ﬂows in set C .

4 CONCLUSIONS

The problem of exact observability of trafﬁc ﬂows can

be dealt with in a simple form even for large net-

works. To solve this problem an algebraic step-by-

step method is given. This method allows determining

the set of observable ﬂows and permits updating the

observability information every time we have a new

item of information, so that a detailed observability

analysis can be done at each step of the process.

The illustration of the proposed methodology us-

ing a simple network shows that the proposed method

is efﬁcient and practically valid.

Finally, some suggestions for future work are the

application of topological techniques to trafﬁc net-

works, and developing an alternative method sharing

the advantages of both observability techniques alge-

braic and topological.

ACKNOWLEDGEMENTS

The authors are indebted to the Spanish Ministry of

Science and Technology (Project BIA2005-07802-

C02-01) and to the Council of Education and Science

of Castilla-La Mancha (Project A06-016) for partial

support of this work.

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