from every other state, be it in one or more than onestep. Every finite, irreducible Markov chain withtransition matrix P has a stationary distribution πsuch that π = πP [1]. However, since the transitionmatrix is precisely what we are uncertain about, notheorems can be applied here, at least in their orig-inal formulation. Therefore, we assume the chaindoes pose a stationary distribution.It is possible to compute the fuzzy stationary dis-

tribution of a fuzzy Markov chain, following the ap-proach described previously. The key idea is thatevery fuzzy stationary probability πi can be con-structed from its α-cuts, and the lower and upperbounds of each α-cut can be computed as the min-imum and maximum possible value for πi when thecrisp transition matrix lies in the feasible solutionspace defined by Dom(α) for each α. Formally,

πi[α] = [πLiα, πUiα] whereπLiα = min{wi|w = wM : M ∈ Dom(α)} (3)πUiα = max{wi|w = wM : M ∈ Dom(α)}(4)

It can be proved [11] that such intervals are closed,connected sets, thus πi can be constructed fromthem by the representation theorem [19].Hence, the process to calculate πLiα and πUiα for a

given α consists in finding, within the matrix spaceDom(α), a matrix that minimizes the i-th compo-nent of its stationary distribution, and another ma-trix maximizing it, respectively. As suggested in[11], this search process should be accomplished us-ing a heuristic constrained optimization techniquesince the objective function, which is the expressionto obtain component wi of the stationary vector wfrom transition matrixM , is a black-box in the gen-eral case of an n-dimensional matrix: no generalanalytical expression exists to express wi as a func-tion of the entries of M , as such expression wouldbe impractical when n grows a bit, e.g. n ≥ 5. De-tails on the optimization algorithm employed hereare provided in the experiment section.Once the α-cuts have been calculated, regression

can be applied to the lower and to the upper bounds(separately) to obtain an analytical expression oftheir membership functions, so that a completelydefined fuzzy number is finally constructed.

3. Linguistic probabilities

What has been explained in the previous section isvalid for any kind of fuzzy number, no matter howthey have been obtained. As mentioned in the in-troduction, most often we have empirical data fromthe phenomenon, and estimates of transition prob-abilities can be derived from them, either crisp orfuzzy. However, in case no data are available, wemay have to elicit probabilities from a human ex-pert. In those cases, it can be quite difficult for himto express his expertise using numerical crisp proba-bilities that, in addition, must meet the requirementof adding to 1. In order to solve this issue, here we

0,0

0,5

1,0

0 0,2 0,4 0,6 0,8 1

Very low chance(VLC)

Small chance(SC)

It may(IM)

Meaningful chance(MC)

Most likely(ML)

Extremely unlikely (EU)

Extremely likely (EL)

Figure 1: Membership functions of linguistic prob-ability labels according to [21].

propose using natural language to express probabil-ities, thus modeling the transition probability as alinguistic variable[20] whose possible values are lin-guistic terms.

The concept of linguistic probability is not new.It was employed, for instance, by Bonissone [21].As he points out, psychological studies cited in thatwork confirm that people are unwilling to give pre-cise numerical estimates of probabilities, so it isreasonable to allow them to provide linguistic esti-mates of the likelihood of given statements that, inthis case, are of the form the system will move fromstate A to state B. More recently, a proposal for alinguistic probability theory [22] explains how to ex-tend the classical probability theory in a linguisticmanner, with an example of a linguistic bayesiannetwork. In the present contribution, we use lin-guistic probabilities as an additional layer that isplaced on top of Buckley’s fuzzy Markov chains ap-proach explained in section 2.

We have defined 7 possible linguistic labels toevaluate the probability that the chain moves fromone state to another. The underlying TrFNs arethe mathematical structure that enables calcula-tions with the labels. Their concrete values arethose proposed in [21] after a psychological study(Fig. 1). Note some of the terms carry more uncer-tainty that others. We also consider crisp Impossibleand Sure, whose TrFNs are respectively the single-tons (0, 0, 0, 0) and (1, 1, 1, 1).Because we are modelling a probability, the con-

straint of being a well-formed probability distri-bution must still hold. In the linguistic case,this is equivalent to the following condition [22].Given a discrete linguistic probability distribution,π1, ..., πn, their sum must contain1 the singletonfuzzy number 1χ, which is defined by µ1χ

(x) = 1if x = 1, and 0 otherwise. This statement is validregardless of the type of fuzzy numbers involved orhow the sum has been defined, although in our ex-periments we employ Eq. (1). In a Markov chain,the use of linguistic probabilities leads to a linguistic

1A ⊇ B ↔ µA(x) ≥ µB(x)∀x ∈ R

709

transition matrix. When expressing his judgmentsthrough linguistic terms, the expert should checkthe constraint explained above for every row:

pi1 ⊕ pi2 ⊕ ...⊕ pin ⊇ 1χ,∀i = 1, ..., n (5)

The reason is the following. The set Domi(α) isnot empty when there exists at least one probabilitydistribution whose elements fulfill all the interval(α-cuts) constraints of row i, and this happens ∀α ∈[0, 1] if, and only if, the fuzzy numbers of row i,whose α-cuts constitute the box constraints, satisfyEq. (5), as expressed in the following lemma.

Lemma 3. Let S = pi1 ⊕ ... ⊕ pin. Then,Domi(α) 6= ∅ ⇐⇒ S ⊇ 1χ, i.e. iff Eq. (5) holds.

Proof. a) If ∀α,Domi(α) is not empty, then∃(pi1, ..., pin) ∈ Rn : pij ∈ [pLijα

, pUijα] ∧∑j pij =

1. Because pLijα≤ pij ≤ pUijα

∀j = 1, ..., n, then∑j p

Lijα≤∑j pij ≤

∑j p

Uijα

, i.e.,∑j p

Lijα≤ 1 ≤∑

j pUijα

, which means that 1 ∈ [∑j p

Lijα,∑j p

Uijα

],i.e. 1 ∈ [SLα , SUα ], and this happens ∀α, henceS ⊇ 1χ and Eq. (5) is satisfied.

b) If Eq. (5) is satisfied, then ∀α, 1 ∈ [SLα , SUα ]so∑j p

Lijα≤ 1 ≤

∑j p

Uijα

. It follows that, for thelast element n, pLinα

≤ 1 −∑j<n p

Lijα

. Moreover,1 ≤ pUinα

so 1−∑j<n p

Lijα≤ pUinα

. Putting both to-gether,

(1−

∑j<n p

Lijα

)∈ [pLinα

, pUinα]. Thus we

can easily obtain an element of Domi(α) whosecomponents belong to the desired intervals, for in-stance (pLi1α

, pLi2α, ..., pLi,(n−1)α

, 1−∑j<n p

Lijα

) ∈ Rn.Since we have found an element of Domi(α), itmeans that Domi(α) 6= ∅.

The above lemma shows the connection betweentwo apparently different proposals on fuzzy proba-bilities, namely [11] and [22].

3.1. Output retranslation process

Since our system allows for a linguistic input, it isalso expected that the system is able to provide alinguistic output. Therefore, after first computingthe α-cuts of the stationary fuzzy probabilities andsubsequently applying regression, it is necessary toassign a linguistic probability label to every fuzzynumber obtained. This is known as retranslationand has been studied previously, for instance in [23].The idea is to assign to every output TrFN the labelof the closest TrFN of the reference set (Fig. 1).Although other variants are possible, at this stage

of research we use a simple measure of distance be-tween TrFNs defined in Eq. (2). Note this metriccan be replaced by any other that better fits theneeds of the problem.

4. Example of application and results

We depart from the Markov chain of Fig. 2 whosetransition matrix is known to us. It represents the

The general case is an easy extension. Our solving algorithm

develops into two stages.

In the first stage we check if there exists at least one

patroller’s strategy such that stay-out is a best response for

the intruder. If such a strategy exists, then the patroller

will follow it, being its payoff maximum when the intruder

abstains from the intrusion (recall that X0 ≥ Xi for all i).This stage is formulated as the following bilinear feasibility

problem in which αi,js are the unknown variables (C \ i is

the set obtained by removing the element i from C):

αi,j ≥ 0 ∀i, j ∈ C (1)X

j∈C

αi,j = 1 ∀i ∈ C (2)

αi,j ≤ ti,j ∀i, j ∈ C (3)

γ1,wi,j = αi,j ∀w, i, j ∈ C, j 6= w (4)

γh,wi,j =

X

x∈C\w

“

αx,jγh−1,wi,x

” ∀h ∈ {2, . . . , d},∀w, i, j ∈ C, j 6= w

(5)

Y0

0

@1 −X

i∈C\wγd,wz,i

1

A + Yw

X

i∈C\wγd,wz,i ≤ 0 ∀z, w ∈ C (6)

Constraints (1)-(2) express that probabilities αi,js are well

defined; constraints (3) express that the patroller can only

move between two adjacent free cells; constraints (4)-

(5) express the Markovian hypothesis over the patroller’s

decision policy; constraints (6) express that no action

enter-when(w, z) gives to the intruder an expected utility

larger than that of stay-out. Notice that the non-linearity is

due to the constraints (5). If the above problem admits a

solution, the resulting αi,js are the optimal patrolling strategy

for the robot. When the above problem is unfeasible, we pass

to the second stage of the algorithm.

In the second stage we find the best response i of the

intruder such that the patroller’s expected utility is maximum.

This is formulated as a multi bilinear programming problem,

where the single bilinear problem in which enter-when(q, s)is the best response for the intruder is defined as follows:

max Xq

X

i∈C\qγd,qs,i + X0

0

@1 −X

i∈C\rγd,qs,i

1

A

s.t.

constraints (1)-(5)

Y0

0

@1 −X

i∈C\qγd,qs,i

1

A + Yq

X

i∈C\qγd,qs,i ≥

≥ Y0

0

@1 −X

i∈C\wγd,wz,i

1

A + Yw

X

i∈C\wγd,wz,i

∀z, w ∈ C (7)

The objective function is the maximization of the pa-

troller’s expected utility. Constraints (7) express that no

action enter-when(w, z) gives a larger value to the intruder

than action enter-when(q, s). We can formulate n2 above

problems, for all the possible enter-when(q, s) actions (q, s ∈{1, 2, . . . , n}). If a problem is feasible, its solution is a set

of αi,js, namely a possible patrolling strategy for the robot.

From all the solutions of feasible problems, we pick out the

one that gives the patroller the maximum expected utility.

We report in Fig. 2 the optimal patroller’s strategy for the

setting of Fig. 1, as calculated with the algorithm described

here. The expected utility for the patroller is 0.805 and

the corresponding induced best response for the intruder is

enter-when(04,01), namely to enter cell 04 when the patroller

is in 01. Cells 05, 08, and 13 are excluded from the route

of the patroller. Indeed, visiting these cells would allow the

intruder to perform an always successful action, for example

enter-when(06,08) (see Fig. 1).

01 02 03 04 05

06 07 08

09 10 11 12 13

0.05

0.95

0.41

0.59

0.23

0.44

0.33

1

0.65

0.35

0.55

0.450.53

0.47 0.52

0.48 0.35

0.35

0.3

1

0

0

0

0

0

0

0

0

Fig. 2. Patroller’s optimal strategies in the setting of Fig. 1

D. Augmenting Patroller’s Sensing Capabilities

In this section we extend the sensing model of the pa-

trolling robot by considering that it can sense the presence

of the intruder beyond its current cell. We introduce a matrix

V (n × n) where vi,j = 1 if cell j can be sensed by the

patroller from cell i and vi,j = 0 otherwise. Matrix Vembeds a model of the detecting sensor of the robot. A

general sensing model of the patroller can be considered in

our approach by substituting constraints (4)-(5) above with:

γ1,wi,j = αi,j (1 − vj,w) ∀w, i, j ∈ C, j 6= w (8)

γh,wi,j = (1 − vj,w)

X

x∈C\w

“

αxj γ

h−1,wi,x

” ∀h ∈ {2, . . . , d},∀w, i, j ∈ C, j 6= w

(9)

In this case γh,wi,j is the probability that the patroller reaches

cell j in h steps, starting from cell i and not having sensed

cell w. For example, let us consider that the patroller is able

to sense its current cell and the free cells that are one cell

away from it (r = 2). For instance, in Fig. 1, from cell 05, it

can sense cells 04 and 08; from cell 06, it can sense cells 01

and 09; from cell 11, it can sense cells 07, 10, and 12; and

so on. With this sensor model, the optimal patrolling strategy

for the robot is reported in Fig. 3. Comparing with Fig. 2,

cells 04 and 12 have been excluded from the patrol route.

This makes sense, since the patroller, due to the augmented

sensing capabilities, is able to patrol them from adjacent cells

that are more “central” (03 and 11, respectively).

IV. A SYNCHRONIZED MULTIROBOT SETTING

In this section we show (a) that our model can easily

capture settings with multiple synchronized robots and (b)

that the patroller’s strategy produced by our approach, when

the patroller and the intruder have no preferences over the

cells, and by the approach presented in [3] are the same.

Hence, since our approach is able to capture more general

Figure 2: Markovian patrolling scheme of an au-tonomous robot in a map with 13 cells. Reproducedfrom [24]. Non accessible states 5, 8 and 13 were notconsidered for the transition matrix.

1 2 3 4 5 6 7 8 9 101 - VLC - - ML - - - - -2 IM - IM - - - - - - -3 - SC - SC - IM - - - -4 - - EL - - - - - - -5 MC - - - - - SC - - -6 - - IM - - - - - IM -7 - - - - IM - - IM - -8 - - - - - - IM - IM -9 - - - - - SC - IM - SC10 - - - - - - - - EL -

Table 1: Linguistic transition matrix estimatedfrom a sequence of 200 observations.

randomized movement of an autonomous robot thatis patrolling a bi-dimensional area divided in cellsagainst an intruder who wants to attack some (un-known) location. The movement from one cell toany of the adjacent locations is randomized in orderto be able to patrol a larger environment withoutletting any location always unvisited. In this way,an intruder who learns the robot’s movement, evenif he notices that the patrolling scheme is random-ized and learns the robot’s movement probabilities,does not have a guarantee that he will be able tosuccessfully attack any given location, since thereis always a non-zero probability of being caught.The movement probability to an adjacent locationonly depends on the current location, and not on thepath followed by the robot to reach current location.Therefore, it is Markovian by definition, with eachstate of the chain representing one location withinthe grid. More details on this problem and the so-lution method that led to this Markov chain can befound in [24].

In a real instance of this problem, the transitionprobabilities shown in the figure would be unknown,so an expert that either has been watching the robotor knows (for some reason) its patrolling schemein an approximate way would provide a linguistictransition matrix directly.

However, instead of asking an expert, in this pre-liminary study we have proceeded as follows. Wegenerated a sequence of 200 observations of the evo-lution of the chain. Each observation is an integernumber indicating the state at that time instant.

710

0.05 0.15 0.25 0.350.

00.

40.

8

State 1 : VLC

x

0.00 0.04 0.08 0.12

0.0

0.4

0.8

State 2 : EU

x

0.00 0.10 0.20

0.0

0.4

0.8

State 3 : VLC

x

0.00 0.04 0.08

0.0

0.4

0.8

State 4 : EU

x

0.1 0.2 0.3 0.4

0.0

0.4

0.8

State 5 : SC

x

0.00 0.05 0.10 0.15 0.20

0.0

0.4

0.8

State 6 : VLC

x

0.05 0.10 0.15 0.20 0.25

0.0

0.4

0.8

State 7 : VLC

x

0.05 0.15 0.25

0.0

0.4

0.8

State 8 : VLC

x

0.05 0.15 0.25

0.0

0.4

0.8

State 9 : VLC

x

0.00 0.04 0.08 0.12

0.0

0.4

0.8

State 10 : EU

x

Extremely unlikely (EU)Very low chance (VLC)Small chance (SC)Output TrFN obtained after regressionSmall circles: -cuts

Figure 3: Fuzzy stationary probabilities (TrFNs) obtained and linguistic labels assigned (above each plot).For each state, we show the TrFN obtained, together with the closest labels of the reference set (Fig. 1).

Then, with this sequence, we estimated the tran-sition probabilities as the proportion of times thechain moves from each state to another. Finally, weconverted such numeric estimates into linguistic la-bels, by replacing each crisp probability estimate bythe linguistic label to which it belongs most, accord-ing to Fig. 1. For instance, a crisp probability of 0.4belongs simultaneously to fuzzy sets Small chanceand It may, but the membership degree of the latteris higher, so it would be replaced by It may (IM).The linguistic transition matrix obtained is shownin Table 1, where “-” stands for an impossible transi-tion in the crisp sense. We made some adjustmentsto make sure Eq. (5) holds for every row, such asreplacing in some cases the label for which a crispestimate has the highest membership, by the imme-diately greater or immediate smaller label. Table 1was obtained as a result of this conversion process.The TrFNs underlying the labels of the table arethe input of our method.

We used the R programming language [25] forimplementing the optimization problems of Eq. (3)and (4), and also for plotting the results. The opti-

mization algorithm employed was an R version [26]of Differential Evolution [27]. The computation ofthe α-cuts took approximately 10 minutes in a par-allelized implementation that exploits all the coresover an Intel Core-i7 processor at 2.67 GHz with6 GB RAM. We made use of the plotting facilitiesprovided by the FuzzyNumbers package [28]. Theresults are displayed in Fig. 3.

A number of issues should be pointed out. Thefirst one is that the membership function of the ob-tained TrFNs (thick line) is an almost perfect line atboth sides of the core. This is just as expected if wetake into account that the input were TrFNs withstraight lines as well. However, in addition, thisphenomenon confirms that the optimization processto compute the α-cuts is working fine, since the re-sulting lower bounds and upper bounds are visuallyperfectly aligned. If the expressions of the mem-bership functions of input probabilities were more“exotic”, then generalized TrFNs with other kind ofmembership functions at both sides (not lines butcurves) would have been obtained because the α-cuts calculated would not be aligned. This does

711

not represent a problem since it is enough to definea proper distance measure that can cope with anykind of TrFN in order to assign a label to them.It also proves that the general method outlined in

[11] is effective in practice, provided the optimizercan deal properly with constrained optimization tomake sure the solutions (i.e., matrices) evaluatedbelong to Dom(α) in each case.

Secondly, in this case only three different labelswere required for all states. This depends heav-ily on the nature of the Markov chain being ob-served. Some Markov chains, as the one of thisexample, have very spread stationary probabilities,while other chains tend to be more concentratedaround one state which is visited far more often thanthe rest. Therefore, this is not an issue caused byour method.

Note there are some difficult cases when assign-ing a linguistic to the TrFN, such as states 2 or 5.In those cases, a different distance measure mighthave yielded different results but, as mentioned inthe previous section, our method proposal is inde-pendent of the distance measure employed and thusadmits any function that fits the user’s needs.

Finally, it should be recalled that, regardless theshape of the input probabilities, the output is lin-guistic and hence, easier to interpret than numeri-cal values. In addition, since the transition matrixis uncertain, crisp stationary probabilities obtainedwith poor numerical punctual estimates of the tran-sition probabilities may be misleading and lead toerroneous conclusions when comparing values of sta-tionary probabilities that are very similar, or whenwe have too little information to do a reliable com-parison. In those cases, our method will most likelyassign the same linguistic label to those fuzzy sta-tionary probabilities, which is probably more desir-able and more robust.

5. Conclusions and further work

For the first time, we have developed a methodto compute linguistic stationary probabilities of aMarkov chain when the information about the tran-sition probabilities is given in linguistic terms. Wehave addressed the uncertainty that is present innatural language judgments by using fuzzy num-bers in the transition matrix. Furthermore, thefuzzy probabilities taken as reference set were ob-tained after a psychological study to better capturethe uncertainty behind the linguistic evaluation ofthe likelihood of real-life events. We have employedan existing proposal to calculate the α-cuts of fuzzystationary probabilities. Trapezoidal fuzzy numbersare built from the obtained α-cuts through a linearregression procedure to compute the membershipfunction at both sides of the core. Finally, the out-put fuzzy numbers are assigned a linguistic termfrom the reference set.Our proposal has been implemented in the R pro-

grammin language and has been successfully appliedto a sample Markov chain. The obtained TrFNs pre-serve the shape of the input fuzzy numbers, and theoutput linguistic stationary probabilities are easierto interpret. In future studies, formal sensitivityanalyses should be conducted to demonstrate therobustness of the method in problems where smallvariations in the transition probability matrix causea big change in the stationary probabilities. Similarrobustness studies have been conducted in [6] fol-lowing the alternative fuzzy Markov chain approachthat is based on fuzzy relations. Furthermore, otherkind of measures from the Markov chain, such asfirst-passage time of a given state, could be investi-gated with the fuzzy linguistic methodology, as wellas the application to real problems such as roboticpatrolling.

Acknowledgments

This work has been partially funded by projectsTIN2011-27696-C02-01 from the Spanish Ministryof Economy and Competitiveness, and P11-TIC-8001 from the Andalusian Government. The firstauthor acknowledges support from a FPU scholar-ship from the Spanish Ministry of Education.

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