1

A Lattice-Computing Ensemble for Reasoning

Based on Formal Fusion of Disparate Data Types,

and an Industrial Dispensing Application

Vassilis G. Kaburlasos and Theodore Pachidis

Department of Industrial Informatics

Technological Educational Institution of Kavala

65404 Kavala, Greece

Emails: {vgkabs,pated}@teikav.edu.gr

Abstract1

By “fusion” this work means integration of disparate types of data including (intervals of) real numbers as2

well as possibility/probability distributions defined over the totally-ordered lattice(R,≤) of real numbers. Such3

data may stem from different sources including (multiple/multimodal) electronic sensors and/or human judgement.4

The aforementioned types of data are presented here as different interpretations of a single data representation,5

namely Intervals’ Number (IN). It is shown that the setF of INs is a partially-ordered lattice(F,¹) originating,6

hierarchically, from(R,≤). Two sound, parametricinclusion measurefunctionsσ : FN × FN → [0, 1] result in the7

Cartesian product lattice(FN ,¹) towards decision-making based on reasoning. In conclusion, the space(FN ,¹)8

emerges as a formal framework for the development of hybrid intelligent fusion systems/schemes. A fuzzy lattice9

reasoning (FLR) ensemble scheme, namelyFLR pairwise ensemble, or FLRpefor short, is introduced here for sound10

decision-making based on descriptive knowledge (rules). Advantages include the sensible employment of a sparse11

rule base, employment of granular input data (to cope with imprecision/uncertainty/vagueness), and employment of12

all-order data statistics. The advantages as well as the performance of our proposed techniques are demonstrated,13

comparatively, by computer simulation experiments regarding an industrial dispensing application.14

Index Terms15

Disparate Data Fusion, Ensemble of Experts, Fuzzy lattice reasoning (FLR), Granular data, Inclusion measure,16

Intervals’ number (IN), Lattice-computing, Lattice theory, Sparse rules17

2

I. I NTRODUCTION18

In the domain ofSoft Computingor, equivalently,Computational Intelligence, the term “hybrid (system/algo-19

rithm)” frequently denotes an integration of different techniques/technologies including artificial neural networks,20

fuzzy systems, evolutionary/swarm computing, etc. towards improving an index of performance in real-world21

applications [1], [15]; the term “intelligence” is pertinent to decision-making, e.g. in pattern classification/recognition22

[81]; moreover, the term “(intelligent) fusion” may signify an aggregate intelligence towards improving decision-23

making [47]. In the aforementioned sense, a “hybrid intelligent fusion system” may be a Multiple Classifier System24

(MCS) [45], [48] also known in the literature asClassifier Ensemble[16], [58], [64], Committee[21], [79], or25

Voting Consensus[5], [50]. Note that a number of MCS architectures/strategies including applications have been26

reported [22], [28], [29], [46], [49], [51], [54], [55], [69], [70], [73], [80], [84], [85]. The MCS techniques are,27

typically, of statistical nature [33] in the Euclidean spaceRN . Nevertheless, a “hybrid intelligent fusion system”28

may be defined otherwise, as explained next.29

The term “fusion” may, alternatively, denote an integration of data stemming from multiple, even heterogeneous,30

sources including (multimodal) electronic devices as well as human judgement [6], [9], [13], [17], [20], [26], [52],31

[56], [63], [65], [67]. In the latter context, there is a keen interest in formal frameworks for unified decision-making32

based on disparate types of data that may accommodate uncertainty [9], [18], [78]. One such a framework has33

been proposed lately [35], in an information engineering context, based on mathematicallattice theoryas follows.34

Different authors have recognized that several types of data of practical interest, includinggranules[61], [83],35

are partially(lattice)-ordered [37], [71]. Hence, lattice theory emerged as a formal framework for the fusion of36

disparate data types [35]. In such context,fuzzy lattice reasoning(FLR) was originally proposed [36], [41], [43]37

as a specific rule-based scheme for classification in a complete lattice(L,¹) data domain including, as a special38

case, the lattice of hyperboxes in the Euclidean spaceRN . In this work, FLR (reasoning) is defined, more widely,39

as any employment of aninclusion measurefunction σ : L × L → [0, 1] for decision-making. Therefore, in the40

context of this work, the term “intelligent” is pertinent to “(FLR) reasoning”.41

Instead of a general mathematical lattice this work considers a specific one originating hierarchically from42

the totally-ordered lattice(R,≤) of real numbers. Note that the latter (lattice) has stemmed, historically, from43

the conventional measurement process of successive comparisons [35], [41]. Our interest in lattice(R,≤) was44

motivated by the existence of vast quantities of real number measurements stored worldwide. Therefore, we sought45

convenient data/information representations based onR. Hence, the complete lattice(F,¹) of Intervals’ Numbers46

(IN) emerged, as detailed below, where a IN is a unified data representation including real numbers, intervals, and47

probability/possibility distributions [59]. In conclusion, the Cartesian product lattice(FN ,¹) is introduced here as48

a formal framework for developing hybrid intelligent fusion systems/schemes, where an element of lattice(FN ,¹)49

is interpreted here as either a rule (of a FLR scheme) or as an input to a FLR scheme.50

In previous work, a FLR scheme for classification has been implemented on theσ-FLNMAP neural network51

architecture [35], [42], [44]. Note that the latter (neural network architecture) was introduced as an enhancement of52

3

the fuzzy-ARTMAP, or FAM for short, neural classifier [11]. More specifically, theσ-FLNMAP has extended the53

applicability domain of FAM from the lattice of hyperboxes inRN to any complete lattice data domain. Moreover,54

even in the Euclidean spaceRN , that is FAM’s sole “applicability domain”, classifierσ-FLNMAP has demonstrated55

significant improvements including tunable nonlinearities as well as the capacity to deal with both nonoverlapping56

hyperboxes and granular (hyperbox) input data [35], [42].57

Due to the fact that both classifiersσ-FLNMAP and FAM areunstable, in the sense that their testing accuracy58

depends on the order of presenting the training data [19], [42], it turns out that both of them make good candidates59

for Voting classification schemes [10], [35], [68]. Indeed, empirical studies have clearly demonstrated an improved60

testing accuracy as well as a more stable testing accuracy for both FAM [3], [12], [60] andσ-FLNMAP [35],61

[44] in RN . Later work has extended the applicability ofσ-FLNMAP from the lattice of hyperboxes to the lattice62

(F,¹) of INs based on FLR [41]. In all, FLR is aLattice-Computingscheme as explained next.63

Lattice-Computing (LC) is a term introduced by Grana [23] to denote any computation in a mathematical lattice.64

Grana and colleagues have demonstrated a number of LC techniques in signal/image processing applications [24],65

[25]. In particular, they have employed mathematical morphology techniques in the totally-ordered lattice of real66

numbers. It turns out that FLR is also a LC scheme, in particular for reasoning, as shown below.67

This paper is based on previously published work on FLR. The novelties of this work include the following.68

First, it presents a space of INs as a formal information fusion framework including a large number of references as69

well as pertinent discussions; a novel mathematical proof is also presented here. Second, it includes mathematical70

notation improvements. Third, it introduces an enhanced definition of FLR. Fourth, it demonstrates the “in principle”71

accommodation of granular inputs. Fifth, it introduces a novel decision-making scheme, that is a descriptive (rule-72

based) FLR ensemble of experts. Sixth, it shows a number of illustrative, new examples including figures. Seventh,73

it demonstrates preliminary (computer simulation) results regarding an industrial application.74

The layout of this work is as follows. Section II presents a formal framework for fusion/integration of disparate75

data types. Section III describes our proposed FLR ensemble scheme. Section IV outlines an industrial application.76

Section V demonstrates, comparatively, preliminary experimental results. Section VI concludes by summarizing77

our contribution. The Appendix presents novel mathematical notation as well as a novel mathematical proof.78

II. A F ORMAL INFORMATION FUSION FRAMEWORK79

This section introduces constructively, in four steps, a formal information fusion framework, namely the80

Cartesian product lattice(FN ,¹) of Intervals’ Numbers (INs). Different interpretations of INs are also presented.81

Note that the four-level hierarchy of lattices presented here is a novelty of this work. For the interested reader,82

useful notions and tools regarding lattice theory are summarized in the Appendix.83

4

A. The Complete Lattice (R,≤)84

The setR of real numbers is a totally-ordered, non-complete lattice denoted by(R,≤). It turns out that(R,≤)85

can be extended to a complete lattice by including both symbols “−∞” and “+∞”. In conclusion, the complete86

lattice (R,≤) emerges, whereR = R∪{−∞, +∞}. Note that previous work has, erroneously, assumed that lattice87

(R,≤) is complete [37], [59]. Even though the aforementioned error is not critical, this work considers, instead,88

the complete lattice(R,≤) 1. We remark that complete lattices are important not only in defining aninclusion89

measurefunction, as shown in the Appendix, but they are also important inmathematical morphology[57], [66].90

On the one hand, any strictly increasing functionv : R → R is a positive valuation in the complete lattice91

(R,≤). Motivated by the two constraints presented in the Appendix (subsection B), here we consider positive92

valuation functionsv : R → R≥0 such that bothv(−∞) = 0 andv(+∞) < +∞. On the other hand, anybijective93

(i.e. one-to-one), strictly decreasing functionθ : R → R is a dual isomorphic function in lattice(R,≤). We will94

refer to functionsθ(.) and v(.) simply asdual isomorphicand positive valuation, respectively. Useful extensions95

to the corresponding lattice of intervals are presented next.96

B. The Complete Lattice (∆,¹) Induced from (R,≤)97

A generalized intervalis defined in lattice(R,≤) as follows.98

Definition 1: Generalized intervalis an element of the product lattice(R,≤∂)× (R,≤).99

Recall that≤∂ in Definition 1 denotes thedual (i.e. converse) of order relation≤ in lattice (R,≤), i.e.≤∂≡≥.100

Product lattice(R,≤∂)× (R,≤) ≡ (R× R,≥ × ≤) will be denoted, simply, by(∆,¹).101

A generalized interval will be denoted by[x, y], wherex, y ∈ R. It follows that themeet(f) and join (g) in102

lattice (∆,¹) are given, respectively, by[a, b] f [c, d] = [a ∨ c, b ∧ d] and [a, b] g [c, d] = [a ∧ c, b ∨ d].103

The set ofpositive (negative) generalized intervals[a, b], characterized bya ≤ b (a > b), is denoted by∆+104

(∆−). It turns out that(∆+,¹) is a poset, namelyposet of positive generalized intervals. Note that poset(∆+,¹)105

is isomorphicto the poset(τ(R),¹) of conventional intervals (sets) inR, i.e. (τ(R),¹) ∼= (∆+,¹). We augmented106

poset(τ(R),¹) by a least (empty) interval, denoted byO = [+∞,−∞] – We remark that agreatestinterval107

I = [−∞,+∞] already exists inτ(R). Hence, the complete lattice (τO(R) = τ(R) ∪ {O},¹)∼= (∆+ ∪ {O},¹)108

emerged. In the sequel, we will employ isomorphic lattices(∆+ ∪ {O},¹) and (τO(R),¹), interchangeably. We109

point out that a trivial interval[x, x] is anatomin the complete lattice(τO(R),¹), where an atom[x, x] by definition110

satisfies both[+∞,−∞] = O ≺ [x, x] and there is no interval[a, b] ∈ (τO(R),¹) such thatO ≺ [a, b] ≺ [x, x].111

Consider both a positive valuation functionv : R → R≥0 and a dual isomorphic functionθ : R → R. Then,112

proposition 6.2 (in the Appendix) implies that functionv∆ : ∆ → R given by v∆([a, b]) = v(θ(a)) + v(b) is a113

1Personal communication with Peter Sussner in the context of the Hybrid Artificial Intelligence Systems (HAIS ’2010) InternationalConference, 23-25 June 2010, San Sebastian, Spain. It is understood that the authors here assume full responsibility for possible errors.

5

positive valuation in lattice(∆,¹). There follow bothv∆(O = [+∞,−∞]) = 0 andv∆(O = [−∞, +∞]) < +∞.114

Therefore, based on Theorem 6.1 (in the Appendix), the following two inclusion measures emerge in lattice(∆,¹).115

(1) σf([a, b] ¹ [c, d]) = v(θ(a∨c))+v(b∧d)v(θ(a))+v(b) , and116

(2) σg([a, b] ¹ [c, d]) = v(θ(c))+v(d)v(θ(a∧c))+v(b∨d) .117

The above inclusion measures are extended to the lattice(τO(R),¹) of intervals (sets) as follows.118

(1) σf([a, b] ¹ [c, d]) = v(θ(a∨c))+v(b∧d)v(θ(a))+v(b) , if a ∨ c ≤ b ∧ d; otherwise,σf([a, b] ¹ [c, d]) = 0, and119

(2) σg([a, b] ¹ [c, d]) = v(θ(c))+v(d)v(θ(a∧c))+v(b∨d) .120

Functionsθ(.) andv(.) can be selected in different ways; for instance, choosingθ(x) = −x andv(.) such that121

v(x) = −v(−x) it follows v∆([a, b]) = v(b)− v(a). Here, we select a pair of parametric functionsv(x) andθ(x)122

so as to satisfy equalityv∆([x, x]) = v(θ(x)) + v(x) = Constant required for atoms by a popular FLR algorithm123

[42], [43]. Eligible pairs of functionsv(x) and θ(x) include, first, v(x) = A1+e−λ(x−µ) and θ(x) = 2µ − x,124

where A, λ ∈ R≥0, µ, x ∈ R and, second,v(x) = px and θ(x) = Q − qx, where p, q,Q > 0, x ∈ [0, A].125

Since it was assumedv(θ(x)) + v(x) = Constant, for the latter pair of functionsv(x) and θ(x) it follows126

v(θ(x)) + v(x) = p[Q + (1− q)x] = Constant; therefore,q = 1.127

C. The Complete Lattice(F,¹) Induced from (∆,¹)128

Based on generalized interval analysis above, this subsection presentsintervals’ numbers(INs). A more129

general number type is defined in the first place, next.130

Definition 2: Generalized interval number, or GIN for short, is a functionG : (0, 1] → ∆.131

Let G denote the set of GINs. It follows complete lattice(G,¹), as the Cartesian product of complete lattices132

(∆,¹). Our interest here focuses on thesublattice2 of intervals’ numbersdefined next.133

Definition 3: An Intervals’ Number, or IN for short, is a GINF such that bothF (h) ∈ (∆+ ∪ {O}) and134

h1 ≤ h2 ⇒ F (h1) º F (h2).135

Let F denote the set of INs. It follows that(F,¹) is a complete lattice with least elementO = O(h) =136

[+∞,−∞], h ∈ (0, 1] and greatest elementI = I(h) = [−∞,+∞], h ∈ (0, 1]. Conventionally, a IN will be137

denoted by a capital letter in italics, e.g.F ∈ F.138

Definition 3 implies that a INF is a function from interval(0, 1] to the setτ(R)∪ {[+∞,−∞]} of intervals,139

i.e. F (h) = [ah, bh], h ∈ (0, 1], where both interval-endsah andbh are functions ofh ∈ (0, 1].140

The following two inclusion measures emerge, respectively, in the complete lattice (F,¹) of INs [34], [35]:141

(1) σf(F1 ¹ F2) =1∫0

σf(F1(h) ¹ F2(h))dh.142

(2) σg(F1 ¹ F2) =1∫0

σg(F1(h) ¹ F2(h))dh.143

2A sublatticeof a lattice(L,¹) is another lattice(S,¹) such thatS ⊆ L.

6

The following Proposition derives from [37].144

Proposition 2.1: Consider a continuousdual isomorphicfunction θ : R → R and a continuouspositive145

valuation function v : R → R≥0. Let X0(h) = [x0, x0], h ∈ (0, 1] be a trivial (point) IN, moreover letE(h),146

h ∈ (0, 1] be a IN withupper-semicontinuousmembership functionmE : R → R. Thenσf(X0 ¹ E) = mE(x0).147

We remark that Proposition 2.1 couples a IN’s two different representations, namely theinterval-representation148

and themembership-function-representation. The principal advantage of the former (interval) representation is149

that it enables useful algebraic operations, whereas the principal advantage of the latter (membership function)150

representation is that it enables convenient interpretions, e.g. fuzzy logic interpretions, etc.151

D. Extensions to More Dimensions152

A N -tuple IN will be denoted by a capital letter in bold, e.g.F = (F1, ..., FN ) ∈ FN . Lattice (FN ,¹) is153

the “fourth level” in a hierarchy of complete lattices whose “first level”, “second level” and “third level” include154

lattices (R,≤), (∆,¹) and (F,¹), respectively.155

The following Proposition derives from [37].156

Proposition 2.2: ConsiderN complete lattices(Li,¹), i ∈ {1, ..., N} each one equipped with an inclusion157

measure functionσi : Li× Li → [0, 1], respectively. ConsiderN -tuplesx = (x1, . . . , xN ) andy = (y1, . . . , yN ) in158

L = L1×· · ·×LN . Furthermore, consider the conventional lattice orderingx ¹ y ⇔ xi ¹ yi, ∀i ∈ {1, ..., N}. Then,159

both functions (1)σ∧ : L× L → [0, 1] given byσ∧(x ¹ y) = mini∈{1,...,N}

{σi(xi ¹ yi)}, and (2)σΠ : L× L → [0, 1]160

given byσΠ(x ¹ y) = Πi∈{1,...,N}

σi(xi ¹ yi), are inclusion measures in lattice(L,¹).161

We remark that Propositions 2.1 and 2.2 establish that, for trivial inputs, an inclusion measure reduces to162

standard fuzzy inference system (FIS) practices [37].163

E. IN Interpretations, Representation Issues & More, Useful Results164

The complete lattice(F,¹) of INs has been studied in a series of publications [34], [38], [40], [41], [59],165

[62]. In particular, it has been shown that a IN is a mathematical object, which may admit different interpretations166

as follows. First, based on the “resolution identity theorem” [82], a INF (h), h ∈ (0, 1] may be interpreted as167

a fuzzy number, whereF (h) is the correspondingα-cut for α = h. Hence, a INF : (0, 1] → τO(R) may,168

equivalently, be represented by anupper-semicontinuousmembership functionmF : R → (0, 1] – Note that a169

number of authors have employedα-cuts and/or intervals in fuzzy logic applications [2], [74], [75], [76], [77].170

There follows equivalencemF1(x) ≤ mF2(x) ⇔ F1(h) ¹ F2(h), wherex ∈ R, h ∈ (0, 1] [59]. Second, a IN171

F (h), h ∈ (0, 1] may also be interpreted as a probability distribution such that intervalF (h) includes100(1−h)%172

of the distribution, whereas the remaining100h% is split even both below and above intervalF (h).173

Fig.1 explains how a IN can be constructed from a population of (real number) data samples using algorithm174

CALCIN [34], [35], [39], [59], [62]. More specifically, Fig.1(a) displays the data itself. Fig.1(b) displays a histogram175

of the data in Fig.1(a) in10 steps of length∆x = 0.04. Hence, the histogram of Fig.1(b) may be thought of as176

7

a probability density function(pdf) approximation, which (histogram) asymptotically tends to the corresponding177

pdf when both∆x → 0 and the number of data samples tends to infinity. Fig.1(c) displays the corresponding178

cumulative distribution function (PDF). Finally, Fig.1(d) displays a IN computed from the PDF of Fig.1(c) using179

the algebraic formulas shown within Fig.1(d); that is, algorithm CALCIN.180

Fig.2 shows the two different representations of the IN (F ) computed in Fig.1(d). More specifically, Fig.2(a)181

displays the membership-function-representation of INF , whereas Fig.2(b) displays the corresponding interval-182

representation forL = 32 different levels spaced evenly over the interval(0, 1]. Triangular INs are of particular183

significance in practice, therefore they are studied next.184

Consider both the triangular INF , with membership functionmF (x), and the trivial INV0 in Fig.3. IN F is185

specified by the three parametersm, wL andwR. A horizontal line at heighth ∈ (0, 1] intersects INF at points186

ah and bh; moreover, it intersects trivial INV0 at pointsch and dh, wherech = dh = V0. Since the left line of187

the triangular membership functionmF (x) equalsy = [x − (m − wL)]/wL and the right line ofmF (x) equals188

y = [(m + wR) − x]/wR, it follows ah = wLh + (m − wL), moreoverbh = −wRh + (m + wR). Next, we189

analytically calculate inclusion measuresigma-joinσg(F ¹ V0) =1∫0

v(θ(ch))+v(dh)v(θ(ah∧ch))+v(bh∨dh)dh usingv(x) = px and190

θ(x) = Q− x. Integral∫

1ax+bdx = 1

a ln|ax + b|+ C0 will be useful in the following calculations.191

(1) For m + wR ≤ V0, it follows192

σg(F ¹ V0) =1∫0

Q−ch+dh

Q−ah+dhdh = −Q

1∫0

1wLh+[(m−wL)−(Q+V0)]

dh = QwL

ln (Q+V0)−m+wL

(Q+V0)−m .193

(2) For m ≤ V0 ≤ m + wR, it follows194

σg(F ¹ V0) =h0∫0

Q−ch+dh

Q−ah+bhdh+

1∫h0

Q−ch+dh

Q−ah+dhdh = −Q

h0∫0

1(wL+wR)h−(Q+wL+wR)dh−Q

1∫h0

1wLh−[Q−(m−wL)+V0]

dh =195

QwL+wR

ln Q+wL+wR

(Q+wL+wR)−(wL+wR)h0+ Q

wLln [Q−(m−wL)+V0]−wLh0

[Q−(m−wL)+V0]−wL, whereh0 = mF (V0).196

(3) For m− wL ≤ V0 ≤ m, it follows197

σg(F ¹ V0) =h0∫0

Q−ch+dh

Q−ah+bhdh+

1∫h0

Q−ch+dh

Q−ch+bhdh = −Q

h0∫0

1(wL+wR)h−(Q+wL+wR)dh−Q

1∫h0

1wRh−[Q−V0+(m+wR)]dh =198

QwL+wR

ln Q+wL+wR

(Q+wL+wR)−(wL+wR)h0+ Q

wRln [Q−V0+(m+wR)]−wRh0

[Q−V0+(m+wR)]−wR, whereh0 = mF (V0).199

(4) For V0 ≤ m− wL, it follows200

σg(F ¹ V0) =1∫0

Q−ch+dh

Q−ch+bhdh = −Q

1∫0

1wRh−[Q−V0+(m+wR)]dh = Q

wRln (m+Q−V0)+wR

m+Q−V0.201

A triangular IN’s edge corresponds to a uniform pdf as shown in Fig.4(a) as well as in Fig.4(b). Letp1(x)202

andp2(x) be the latter pdfs corresponding to INsF1 andF2, respectively. More specifically, it is203

pi(x) =

12wL

, mi − wL ≤ x ≤ mi

12wR

, mi ≤ x ≤ mi + wR

, for i ∈ {1, 2},204

wherewL and wR represent the ranges of the uniform pdf located to the left and to the right, respectively,205

of the medianmi, i ∈ {1, 2}; hence, in Fig.4(a) it iswL = r, wR = R, whereas in Fig.4(b) it iswL = R,206

wR = r. Note that themedian“m” of a pdf p(x) is defined here such thatm∫−∞

p(x)dx = 0.5 =+∞∫m

p(x)dx. Next,207

we compute the means as well as the variances of pdfsp1(x) and p2(x) corresponding to the INsF1 and F2,208

respectively.209

µ1 =+∞∫−∞

xp1(x)dx =m1∫

m1−r

x 12r dx +

m1+R∫m1

x 12Rdx = m1 + R−r

4 .210

8

µ2 =+∞∫−∞

xp2(x)dx =m2∫

m2−R

x 12Rdx +

m2+r∫m2

x 12r dx = m2 − R−r

4 .211

σ21 =

+∞∫−∞

(x− µ1)2p1(x)dx =m1∫

m1−r

(x− µ1)2 12r dx +

m1+R∫m1

(x− µ1)2 12Rdx = 5r2+5R2+6Rr

48 .212

σ22 =

+∞∫−∞

(x− µ2)2p2(x)dx =m2∫

m2−R

(x− µ2)2 12Rdx +

m2+r∫m2

(x− µ2)2 12r dx = 5r2+5R2+6Rr

48 .213

We remark thatwL = wR implies bothµ =+∞∫−∞

xp(x)dx =m+wR∫m−wL

x 1wL+wR

dx = m andσ2 = (wL+wR)2

12 as214

expected for a uniform pdf – Recall also that a uniform pdf corresponds to an isosceles triangular IN [34], [35].215

In Fig.4(c), pdfsp1(x) and p2(x) were placed such thatµ1 = µ = µ2; the corresponding INs, respectively,216

F1 and F2 are also shown in Fig.4(c). On the one hand, note that both the first- and the second- order statistics217

of pdfs p1(x) and p2(x) are identical, i.e.µ1 = µ2 and σ1 = σ2. Nevertheless, pdfsp1(x) and p2(x) differ in218

their third-order statistic, namely theirskewness. More specifically,p1(x) is skewed to the left, whereasp2(x) is219

skewed to the right. On the other hand, recall that an inclusion measure function can detect all-order statistics [39],220

[40], [41]. Hence, in Fig.4(c), an inclusion measure can discriminate between INsF1 and INF2 induced from pdfs221

p1(x) andp2(x), respectively, as demonstrated below.222

Furthermore, let us define the following two alternative conditions/specifications (S1)|mi− V0| ≤ T and (S2)223

|µi − V0| ≤ T , for a user-defined threshold valueT , whereV0 and mi, µi for i ∈ {1, 2} as well asR, r are224

shown in Fig.4. From both Fig.4(a) and Fig.4(b) it follows that exactly0.5 of the distribution does not satisfy (S1).225

Moreover, first, from Fig.4(a) it follows that0.5+(R−r)/8R of the distribution does not satisfy (S2) and, second,226

from Fig.4(b) it follows that0.5− (R− r)/8R of the distribution does not satisfy (S2). Note also that the truth of227

inequalitymi < µi (mi > µi) indicates that the corresponding pdf is skewed to the left (right).228

III. A F UZZY LATTICE REASONING (FLR) ENSEMBLE SCHEME229

Fuzzy lattice reasoning (FLR) is a term proposed originally for a concrete classification scheme [43], where230

an inclusion measure functionσ(A ¹ B) was employed, in the lattice of hyperboxes inRN , to compute a (fuzzy)231

degree of inclusion of a hyperboxA to another oneB. It was also shown that an inclusion measureσ(., .)232

supports two different modes of reasoning, namelyGeneralized Modus Ponensand Reasoning by Analogy. More233

specifically, on the one hand,Generalized Modus Ponensis supported as follows: Given both a rule “IFvariable234

V0 is E THEN propositionp” and a proposition “variable V0 is Ep” such thatEp ¹ E, where bothEp and E235

are elements in a lattice(L,¹), it reasonably follows “propositionp”. On the other hand,Reasoning by Analogy236

is supported as follows: Given both a set of rules “IFvariable V0 is Ek THEN propositionpk”, k ∈ {1, . . . , K}237

and a proposition “variable V0 is Ep” such thatEp � Ek, for k ∈ {1, . . . , K}, it follows “propositionpJ ”, where238

J.= arg max

k∈{1,...,K}{σ(Ep ¹ Ek) < 1}.239

A FLR extension to the lattice of INs has been possible according to the following rationale. We know (see in240

section II-C) that a IN can, equivalently, be represented either by a membership function or by a set of intervals.241

Therefore, since an interval is a hyperbox in spaceR1, it follows that an inclusion measure function can be extended242

from spaceR1 to the spaceF of INs by a single integral operation. Further enhancements are proposed next.243

9

A. FLR Enhancements244

Here we propose using the term FLR to denote any decision-making based on an inclusion measure function245

σ(., .). Note that advantages of using an inclusion measureσ(., .) include, first, accommodation of nontrivial246

(granular) input data, second, activation of a rule by an input outside the rule’s support (hence, asparserule-247

base becomes “sensibly” usable) and, third, a capacity to employ alternative positive valuation functions than248

v(x) = x (the latter positive valuation is exclusively employed in the literature, implicitly). We point out that249

a parametric positive valuation function may introduce tunable nonlinearities by optimal parameter estimation250

techniques; likewise, for aparametricdual isomorphic function.251

Recent work [37] has demonstrated that conventional fuzzy inference systems (FISs) [27], [30], [53], [72]252

apply “in principle” FLR, in lattice(FN ,¹), as follows.253

A FIS, typically, includesK rulesRk, k = 1, . . . K, of the following form254

Rule Rk : IF (variable V1 is Fk,1).AND. . . . .AND.(variable VN is Fk,N ) THEN propositionpk,255

where the antecedent of ruleRk is the conjunction ofN simple propositions “variableVi is Fk,i”, i = 1, . . . N ,256

moreover the consequent “propositionpk” of rule Rk is typically either a likewise proposition (e.g. in a Mamdani257

type FIS [53]) or a polynomial (e.g. in a Sugeno type FIS [72]). Our interest here focuses on rule antecedents.258

In particular, we assume that the degree of activation of a simple proposition “variable Vi is Fk,i”, i = 1, . . . N259

by another one “variable Vi is F0,i” equalsσg(F0,i ¹ Fk,i). The following examples demonstrate some technical260

application details.261

B. FLR Examples in lattice(F,¹)262

In this work we employ solely inclusion measureσg(., .) rather thanσf(., .) because only inclusion measure263

σg(., .) is non-zero for non-overlapping INs; hence, onlyσg(., .) can reason based on asparserule base.264

Example - 1265

INs F andV0 referred to, in this example, are shown in Fig.3.266

Fig.5 plots inclusion measureσg(F ¹ V0) versus the medianm of IN F from m = 0.5 to m = 9.5 using267

parameter valueswL = wR = 0.5 andV0 = 4.6; moreover, both the linear positive valuationv(x) = px and dual268

isomorphic functionθ(x) = Q − x were used with parameter valuesp = 1, Q = 10. Equality wL = wR = 0.5269

implies that triangular INF has, in particular, an isosceles triangular shape – Recall that an isosceles triangular270

IN corresponds to a uniform pdf. Since the median (m) equals the mean (µ) of a uniform pdf it follows that, for271

an isosceles triangular IN, thex-axis in both Fig.5 and Fig.6, denotesm as well asµ.272

Fig.6 plots inclusion measureσg(F ¹ V0) versus its medianm from m = 0.5 to m = 9.5 using parameter273

valueswL = wR = 0.5 andV0 = 4.6. Moreover, both positive valuationv(x) = 11+e−0.5(x−4.6) and dual isomorphic274

function θ(x) = 2(4.6)− x were employed.275

10

Notice the similarity of Fig.5 and Fig.6, where each figure was generated using a different positive valuation276

function v(x). In particular, Fig.5 was generated using alinear positive valuation, whereas Fig.6 was generated277

using asigmoidpositive valuation. In all our experiments, in the context of this work, we empirically confirmed that278

for any linear positive valuationv`(x) there is a sigmoid positive valuationvs(x), which produces an “identical”,279

for all practical purposes, inclusion measureσg(., .) function. A sigmoid positive valuation is preferable because it280

is defined over the whole setR of real numbers, therefore no truncation/normalization is necessary. In conclusion,281

unless otherwise specified, in the remaining of this work we employ sigmoid positive valuation functions.282

Example - 2283

The previous example has dealt with isosceles (triangular) INs. This example considers non-isosceles triangular284

IN shapes towards demonstrating that an inclusion measure can effectively detect higher-order statistics.285

Fig.7(a) displays inclusion measureσg(F1 ¹ V0) versus its medianm1 from m1 = 3 to m1 = 90 using IN286

F1 parameter valueswL = r = 3, wR = R = 10 andV0 = 65; Fig.7(b) shows the latter figure in the vicinity of287

its global maximum atm1 = 65. Likewise, Fig.7(c) displays inclusion measureσg(F2 ¹ V0) versus its median288

m2 from m2 = 10 to m2 = 97 using IN F2 parameter valueswL = R = 10, wR = r = 3 andV0 = 65; Fig.7(d)289

shows the latter figure in the vicinity of its global maximum atm2 = 65. Where, INsF1, F2 andV0 are shown in290

Fig.4. Finally, Fig.7(e) displays both inclusion measuresσg(F1 ¹ V0) and σg(F2 ¹ V0) versus their (identical)291

meanµ. More specifically, Fig.7(e) demonstrates thatσg(F2 ¹ V0) reaches its global maximumbeforeV0 = 65, as292

expected, because INF2 is skewed to the right; whereas,σg(F1 ¹ V0) reaches its global maximumafter V0 = 65,293

also as expected, because INF1 is skewed to the left.294

C. FLRpe: A Pairwise FLR Ensemble Scheme for Reasoning295

Based on an expert-supplied propositionp : “Variable V equalsx” the question here is to decide whether296

another propositionp0 : “Variable V equalsx0” is true or not, where bothx andx0 are INs. We responded to the297

aforementioned question by computing a (fuzzy) degree of fulfillment of implication “p → p0” by σg(x ¹ x0).298

More specifically, ifσg(x ¹ x0) ≥ T , whereT ∈ [0, 1] is user-defined, only then propositionp0 is accepted.299

Since a single expert propositionp may be prone to errors, hence it may be unreliable, we assumed an ensemble300

of N experts each one of whom supplied one propositionpk : “Variable V equalsxk ’, k ∈ {1, . . . , N}. Our basic301

assumption is that at least 2 out of theN experts are reliable. In conclusion, FLR is carried out by considering all302

different pairs of experts as shown in Algorithm 1, that is the FLRpe scheme.303

We remark that the FLRpe scheme accepts propositionp0 if and only if the corresponding implicationspk → p0,304

k ∈ {1, . . . N} of any two expertsk ∈ {i, j} are jointly accepted, in the sense that it isσg(xk ¹ x0) ≥ T for two305

different expertsk ∈ {i, j} as indicated in the mathematical expression in the last step of Algorithm 1; the latter306

(expression) derives from Proposition 2.2. In other words, propositionp0 is accepted if and only if the maximum307

(∨

) inclusion measureσg(.) of all different pairs of experts is above a user-defined thresholdT ∈ [0, 1]. Apparently,308

the FLRpe is a “collective reasoning” scheme based on an ensemble of experts.309

11

Algorithm 1 FLRpe: A Pairwise FLR Ensemble Scheme1: Consider a propositionp0 : “Variable V equalsx0” and a thresholdT ∈ [0, 1]. Furthermore, consider

N expert-supplied propositionspk : “Variable V equalsxk”, k ∈ {1, . . . , N}, wherex0, xk are INs,k ∈ {1, . . . , N}.

2: Consider one implicationrk, k ∈ {1, . . . , N} per expert as follows:Implication rk : IF pk THEN p0, symbolicallypk → p0.

3: Compute the degreeσg(xk ¹ x0) of fulfillment of each implicationrk : pk → p0, k ∈ {1, . . . , N}.4: Accept propositionp0 if and only if∨

i,j∈{1,...,N},i6=j

σ∧([xi, xj] ¹ [x0, x0]) =∨ { ∧

i,j∈{1,...,N},i6=j

{σg(xi ¹ x0), σg(xj ¹ x0)} } ≥ T

IV. A N INDUSTRIAL DISPENSINGAPPLICATION310

This section outlines an industrial application.311

A. The Industrial Problem312

Ouzois a popular Greek liquor, whose final stage production involves dispensing three different liquids, namely313

water, spirit, andyeast, to a “mixing” tank. More specifically,water is typically supplied by a local utility company,314

spirit is a commercial product whoseGs = 96% volume is pure ethanol, moreover theyeast, whoseGy volume315

(in the range40%− 80%) is pure ethanol, is prepared according to a local recipe.316

The Greek law calls for a specific percentage (Gb1) of ethanol in the final (ouzo) product, e.g.Gb

1 = 38% or317

Gb1 = 40%, etc. Furthermore, the law calls for a specific ratiopy

1 : ps1, wherepy

1 denotes the final product’s ethanol318

percentage stemming-from-yeast andps1 denotes the corresponding percentage stemming-from-commercial-spirit;319

it is py1 + ps

1 = 1. In the context of this work, we call pair(Gb1, p

y1 : ps

1) alcoholic identityof the (ouzo) product.320

Currently, the production of ouzo is largely empirical, therefore it is prone to errors as explained next.321

Typically, a skilled worker (manually) calculates the volumes of water (V w1 ), spirit (V s

1 ), and yeast (V y1 ) required322

to produce a specific volumeV b1 of ouzo ofalcoholic identity(Gb

1, py1 : ps

1). Nevertheless, when a different volume323

V b2 6= V b

1 is requested, at the absence of a skilled worker to compute the corresponding volumesV w2 , V s

2 , andV y2 ,324

then errors may occur. Another source of errors regards the manual dispensing of volumesV w1 , V s

1 , and V y1 to325

the mixing tank. Hence, thealcoholic identityof the final (ouzo) product might be outside specifications. It is of326

practical interest to keep, an automated ouzo production, within specifications.327

Work is, currently, under way towards automating the production of ouzo for a local beverage company in328

the Greek Macedonia region. Note that the problem of industrial dispensing has been treated also by other authors329

[14] using conventional modeling techniques; moreover, fuzzy regression techniques have been employed [32]. We330

applied the FLRpe scheme via a novel software platform, developed for the needs of this work as described next.331

B. A Novel Software Platform332

A novel software platform, namely XtraSP.v1 (Fig.8), was developed for the needs of this work using the333

Labview environment of the National Semiconductors Company. XtraSP.v1 operates as a user-friendly interface for334

12

controlling all the required electromechanical equipment, including four valves and one pump, via a NI USB-6501335

device. The latter (USB) is a Universal Serial Bus to digital I/O device which also measures the flow, in the range336

6 − 120 `t/min, to the mixing tank by counting pulses generated by aflowmeterusing a 32 bit long counter.337

Mounted (inside) on the upper side of the mixing tank there is anultrasonic level meter(U.L.M.) device, which338

measures the liquid level in the mixing tank with accuracy in the range3− 6 mm by transmitting short ultrasonic339

pulses to the liquid surface. In addition, there is a transparentcommunicating tube(C.T.) connected to the side of340

the mixing tank, which (tube) functions as an indicator of the liquid level (in the mixing tank) by operating on the341

principle of communicating tubes. The overall physical system architecture is shown in the upper half of Fig.8.342

In worksheet cells of XtraSP.v1 a user can specify (a) A label, e.g. for a tank, (b) An initial quantity of343

a liquid in a tank, (c) The percentage of ethanol in both the (commercial) spirit and the yeast, (d) The total344

percentage of ethanol in the undisposed ouzo, (e) The percentages of ethanol in the undisposed ouzo stemming,345

respectively, from (commercial) spirit and yeast, (f) The desired percentage of (pure) ethanol in the mixing tank,346

(g) The desired percentages of ethanol in the mixing tank stemming, respectively, from (commercial) spirit and347

yeast. Box “DECISION-SUPPORT & PARAMETERS” allows the user to specify useful rules & parameters.348

Software platform XtraSP.v1 can automatically carry out any required calculation/action on user demand.349

Furthermore, a number of safety instructions as well as warning messages can be issued. Note also that software350

platform XtraSP.v1 can operate either in a SIMULATION mode or in a real-world OPERATION mode, where the351

latter (mode) can be either MANUAL or AUTOMATIC.352

C. Implementation of the FLRpe Scheme353

An expert-based reasoning scheme, which may also accommodate uncertainty/ambiguity, is of particular interest354

in an industrial application. Furthermore, the capacity to effectively cope with an unreliable expert is a specification355

of critical importance because an unreliable expert may result in a final product outside specifications. The proposed356

FLRpe scheme appears to satisfy the aforementioned specifications, therefore it was applied as described next.357

The volume of a liquid being dispensed to the mixing tank was estimated simultaneously by three different358

“experts” including, first, a flowmeter measurement device, second, an ultrasonic level meter measurement device359

and, third, a human expert who visually consults the transparent tube connected to the side of the mixing tank. We360

employed the following (binary) decision rule.361

Rule R : IF volumev (of the liquid being dispensed) equalsV0 THEN stop dispensing,362

We assumed that the degree of truth of a RuleR equals the degree of truth of its antecedent. Hence, we “stop363

dispensing” if the antecedent propositionp0: “volume v (of the liquid being dispensed) equalsV0” is true. The364

latter (antecedent) degree of truth was calculated from the degrees of fulfillmentσg(Vi ¹ V0) of implications365

rk : IF “volume v is Vk” THEN “volume v is V0”,366

where one implicationrk, k ∈ {1, 2, 3} was supplied per expert.367

13

Therefore, the FLRpe scheme was applied as described in Algorithm 1. We point out that dispensing stops368

if and only if at least two volume IN estimates, supplied by two different experts, approximate volumeV0 in an369

inclusion measure “σg(., .) ≥ T ” sense for a user-defined thresholdT .370

V. EXPERIMENTS AND RESULTS371

We carried out comparative simulation experiments as described in this section.372

A. Disparate Data Representation and Fusion373

Recall that the FLRpe scheme here consists of an ensemble of three experts including Expert-1, that is a374

flowmeter measurement device, Expert-2, that is an ultrasonic level meter device and, Expert-3, that is a human375

expert supervisor of the industrial dispensing procedure.376

First, a dispensed (liquid) volume estimate supplied by Expert-1 was represented by a triangular IN (Fig.9)377

as follows. Even though our flowmeter device supplies a precise measurement, there is uncertainty regarding the378

dispensed volume due to both time-delays and the storage capacity of the pipes used to drive a fluid to the mixing379

tank. The latter uncertainty was modeled by two adjacent uniform pdfs, respectively, one above- and the other below-380

an obtained flowmeter measurement. For instance, let a flowmeter measurement be eitherm1 (Fig.4(a)) orm2381

(Fig.4(b)). The aforementioned two adjacent uniform pdfs are shown in Fig.4(a) as well as Fig.4(b). In conclusion,382

an estimate for a dispensed liquid volume by Expert-1 had a triangular shape as in Fig.9. The corresponding383

inclusion measure functionσg(F ¹ V0), for V0 = 65, is plotted in Fig.10 versus the medianm.384

Second, a dispensed (liquid) volume estimate supplied by Expert-2 was represented by an irregularly shaped IN385

(Fig.11) as follows. In a short sequence, we obtained a number ofN = 9 successive measurements of the liquid level386

in the mixing tank resulting in a population ofN = 9 estimates of the dispensed liquid volume. In conclusion, from387

the aforementioned population, we induced a IN (Fig.11) using algorithm CALCIN. The corresponding inclusion388

measure functionσg(F ¹ V0), for V0 = 65, is plotted in Fig.12 versus the medianm.389

Third, a dispensed (liquid) volume estimate supplied by Expert-3 was represented by a trapezoidal IN (Fig.13)390

as follows. A human supervisor of the industrial procedure, based on visual inspection of the transparent tube391

connected to the side of the mixing tank (Fig.8) as well as based on personal judgement, supplied a numeric392

estimatem of the middle of an interval[m − w, m + w] which (interval) is the core of a trapezoidal fuzzy set.393

Furthermore, both trapezoidal tailswL and wR in Fig.13 were suggested by Expert-3. Fig.14 displays a typical394

estimate for a dispensed liquid volume given by Expert-3, wherew = 1, wL = 5 andwR = 2. The corresponding395

inclusion measure functionσg(F ¹ V0), for V0 = 65, is plotted in Fig.15 versus the medianm.396

We remark that both curves in Fig.10 and Fig.15 are smooth because they have been computed analytically using397

equations in section II-E; whereas, the curve in Fig.12 is not smooth due to the irregularly shaped IN of Fig.11.398

Furthermore note that, first, the triangular IN (Fig.4) supplied by Expert-1 represents a probability distribution399

includinga priori information; in particular, the two adjacent iniform pdfs in either Fig.4(a) or Fig.4(b) representa400

14

priori information supplied by the user. Second, the irregularly shaped IN (Fig.11) supplied by Expert-2 represents401

a distribution of measurements and, third, the trapezoidal IN (Fig.14) supplied by Expert-3 represents a fuzzy set.402

Hence, each expert interprets differently the IN it supplies. In the latter sense, disparate data fussion takes place.403

B. Comparative Experimental Results and Discussion404

We carried out, comparatively, preliminary computer simulation experiments, using a standard commercial405

software package (MATLAB), as described in the following.406

First, we compared an employment of the meanµ versus the medianm of a distribution. Note that a standard407

practice in the industry is to employ the average/mean valueµ of a population of measurements instead of the408

corresponding median valuem as it was demonstrated above (see in section III-B, Example-2). However, the409

theoretical discussion above (see in the last paragraph of section II-E) has shown that an employment of inequality410

m < µ, for skewed pdfs, can increase the probability of a dispensed liquid volume “being inside the specifications”.411

In a series of Monte-Carlo computer experiments we confirmed, for both Expert-1 and Expert-2, that a combined412

employment ofm andµ results in fewer violations of the specifications. The latter is significant for our industrial413

application. Nevertheless, a conceptual problem arises regarding the employment of a medianm computed for the414

fuzzy set supplied by Expert-3 because a medianm is meaningless for a fuzzy set. However, due to the one-to-415

one correspondence between INs and pdfs [34], [35], [39], [40], it follows that for any IN amedian equivalent416

(parameter)m can be defined. Moreover, compared with the medianm of a pdf, inclusion measureσg(.) has the417

advantage that onlyσg(.) can capture higher-order data statistics; in fact,σg(.) can capture all-order data statistics418

[39], [40], [41].419

Second, we comparatively evaluated the performance of our proposed FLRpe scheme. The latter (scheme) was420

tested in a number of computer simulation experiments assuming a single unreliable expert. More specifically, we421

assumed that two experts were able to supply accurate (dispensed) liquid volume INs, whereas the third expert422

supplied a IN either at random or lagging/leading the correct volume. In other words, we used “intact” two of the423

three inclusion measuresσg(F ¹ V0) curves shown in Fig.10, Fig.12 and Fig.15, whereas we used either random424

samples of the third curve or a left/right-translated version of the third curve. In conclusion, analternative decision425

schemehas employed the average of the three inclusion measures values supplied by the three experts.426

Each one of the three inclusion measuresσg(F ¹ V0) curves shown in Fig.10, Fig.12 and Fig.15 was sampled427

at specific values of the parameterm – Note that successive parameterm samples correspond to successive time428

instances. Then, both the FLRpe and the aforementionedalternative decision schemewere applied at every (data)429

sampling instance. We confirmed, using thresholdT = 0.93, that the FLRpe scheme always accurately stops430

dispensing, whereas the alternative decision scheme may fail even at all (data) sampling instances. Note also that431

a single expert never performed better than the FLRpe scheme. Such reliable decision-making, as the FLRpe can432

provide, can be of critical importance in our industrial application due to the fact that one of the three experts may,433

occasionally, fail as it will be detailed in a future publication.434

15

VI. CONCLUSION435

Automated as well as accurate dispensing towards retaining a competitive product quality is of interest in a436

wide range of industrial applications including plastics, chemicals, dyeing, pharmaceuticals, and foods. This work437

has demonstrated a novel scheme, namelyFuzzy Lattice Reasoning pairwise ensemble, or FLRpe for short, for438

industrial dispensing based on (FLR) reasoning, which may accommodate imprecision/uncertainty/vagueness in the439

data. The FLRpe operates by considering, pairwise, all combinations of a number of expert implications based on440

the sigma-joinσg(., .) inclusion measure. Preliminary experimental results have been encouraging.441

This work has also presented a formal information fusion framework, namely the Cartesian product lattice442

(FN ,¹) of Intervals’Numbers (INs), towards an integration of disparate types of data including (intervals of)443

real numbers as well as probability/possibility distributions. Furthermore, a number of mathematical improvements444

were presented. Several illustrative examples have demonstrated practical advantages of the proposed techniques445

including the employment of granular input data as well as the sensible employment of a sparse rule base.446

Future plans include, first, a study of implicationp → q based on both inclusion measuresσf(., .) andσg(., .)447

and, second, an industrial application of the FLRpe scheme for automated ouzo production. The mathematical448

instruments presented here may also be especially useful for the design of dynamically evolving fuzzy systems [4],449

as well as for fuzzy regression analysis [8].450

APPENDIX451

This Appendix summarizes useful notions and tools regarding lattice theory [7], [35], [43], [59] using an452

improved mathematical notation [31], [37].453

A. Mathematical Background454

Given a setP , a binary relation (¹) in P is called partial order if and only if it satisfies the following455

conditions:x ¹ x (reflexivity), x ¹ y and y ¹ x ⇒ x = y (antisymmetry), and x ¹ y and y ¹ z ⇒ x ¹ z456

(transitivity) – We remark that theantisymmetrycondition may be replaced by the following equivalent condition:457

x ¹ y andx 6= y ⇒ y � x. If both x ¹ y andx 6= y then we writex ≺ y. A partially ordered set, or posetfor458

short, is a pair(P,¹), whereP is a set and¹ is a partial order relation inP . Note that, in this work, we employ459

an improved mathematical notation using, first, “curly” symbolsg, f, ¹, ≺, etc. for general poset/lattice elements460

and, second, “straight” symbols such as∨, ∧, ≤, <, etc. for real numbers, i.e. elements of the totally-ordered461

lattice (R,≤).462

A lattice is a poset(L,¹) any two of whose elementsx, y ∈ L have both agreatest lower bound, or meetfor463

short, and aleast upper bound, or join for short, denoted byx f y andx g y, respectively. Two elementsx, y ∈ L464

in a lattice(L,¹) are calledcomparable, symbolicallyx ∼ y, if and only if it is eitherx ¹ y or x  y. A lattice465

(L,¹) is calledtotally-orderedif and only if x ∼ y for any x, y ∈ L. If x � y holds for two elementsx, y ∈ L of466

a lattice(L,¹) thenx andy are calledincomparableor, equivalently,parallel, symbolicallyx||y.467

16

Given a lattice(L,¹) it is known that(L,¹∂) ≡ (L,º) is also a lattice, namelydual (lattice), where¹∂ denotes468

the dual (i.e. converse) of order relation¹. Furthermore, it is known that the Cartesian product(L1,¹)× (L2,¹),469

of two lattices(L1,¹) and (L2,¹), is a lattice with order(x1, x2) ¹ (y1, y2) ⇔ x1 ¹ y1 and x2 ¹ y2. In the470

latter Cartesian product lattice it holds both(x1, x2) f (y1, y2) = (x1 f y1, x2 f y2) and (x1, x2) g (y1, y2) =471

(x1 g y1, x2 g y2). It follows that the Cartesian product(L,º) × (L,¹) ≡ (L × L,º × ¹) is a lattice with472

order (x1, x2) ¹ (y1, y2) ⇔ x1 º y1 and x2 ¹ y2; moreover,(x1, x2) f (y1, y2) = (x1 g y1, x2 f y2) and473

(x1, x2) g (y1, y2) = (x1 f y1, x2 g y2). An element of lattice(L × L,º × ¹) will be denoted by a pair ofL474

elements within square brackets, e.g.[a, b].475

Our interest, here, is incompletelattices. Recall that a lattice(L,¹) is calledcompletewhen each of its subsets476

X has both a greatest lower bound and a least upper bound inL; hence, forX = L it follows that a complete477

lattice has both aleastand agreatestelement. In the interest of simplicity, here we use the same symbolsO and478

I to denote the least and the greatest element, respectively, in any complete lattice. Likewise, we use the same479

symbol¹ to denote the partial order relation in any (complete) lattice. Consider the following definition.480

Definition 4: Let (L,¹) be a complete lattice with least and greatest elementsO and I, respectively. An481

inclusion measurein (L,¹) is a functionσ : L× L → [0, 1], which satisfies the following conditions482

I0. σ(x,O) = 0, ∀x 6= O.483

I1. σ(x, x) = 1,∀x ∈ L.484

I2. x f y ≺ x ⇒ σ(x, y) < 1.485

I3. u ¹ w ⇒ σ(x, u) ≤ σ(x,w).486

We remark that an inclusion measureσ(x, y) can be interpreted as the fuzzy degree to whichx is less thany;487

therefore notationσ(x ¹ y) may be used instead ofσ(x, y).488

B. Useful Mathematical Instruments489

Two different inclusion measures are presented next, based on apositive valuation3 function.490

Theorem 6.1:Let functionv : L → R be a positive valuation in a complete lattice(L,¹) such thatv(O) = 0;491

then both functionssigma-meetσf(x, y) = v(xfy)v(x) andsigma-joinσg(x, y) = v(y)

v(xgy) are inclusion measures.492

Due to practical restrictions, we introduce two constraints on positive valuation functions, next. First, in order493

to satisfy condition I0 of Definition 4, our interest is in positive valuation functions such that “v(O) = 0”. Second,494

since a positive valuation functionv : L → R implies a metric (distance) functiond : L × L → R≥0 given by495

d(a, b) = v(a g b)− v(a f b), furthermore infinite distances between lattice elements are not desired, our second496

constraint is “v(I) < +∞”. Our interest, in the context of this work, focuses solely on inclusion measure functions.497

3Positive valuationin a general lattice(L,¹) is a real functionv : L× L → R that satisfies bothv(x) + v(y) = v(x f y) + v(x g y) andx ≺ y ⇒ v(x) < v(y).

17

A bijective (i.e. one-to-one)dual isomorphic4 function θ : L → L such thatx ≺ y ⇔ θ(x) Â θ(y), in a498

lattice (L,¹), can be used for extending an inclusion measure from a lattice(L,¹) to the corresponding lattice of499

intervals. Given a dual isomorphic functionθ : L → L there follow, by definition, bothθ(xf y) = θ(x)g θ(y) and500

θ(x g y) = θ(x) f θ(y). The latter equalities are handy in the proof of the following Proposition.501

Proposition 6.2: Let real functionv : L → R be apositive valuationin a lattice(L,¹); moreover, let bijective502

function θ : L → L be dual isomorphicin (L,¹), i.e. x ≺ y ⇔ θ(x) Â θ(y). Then, functionv∆ : L× L → R given503

by v∆(a, b) = v(θ(a)) + v(b) is a positive valuation in lattice(L× L,º × ¹).504

Proof505

1. First, we show thatv∆(a, b) + v∆(c, d) = v∆((a, b) f (c, d)) + v∆((a, b) g (c, d)) as follows.506

v∆(a, b) + v∆(c, d) = [v(θ(a)) + v(b)] + [v(θ(c)) + v(d)] = [v(θ(a)) + v(θ(c))] + [v(b) + v(d)] = [v(θ(a) f507

θ(c))+v(θ(a)gθ(c))]+[v(bfd)+v(bgd)] = [v(θ(agc))+v(θ(afc))]+[v(bfd)+v(bgd)] = [v(θ(agc))+508

v(bfd)]+[v(θ(afc))+v(bgd)] = v∆(agc, bfd))+v∆(afc, bgd) = v∆((a, b)f(c, d))+v∆((a, b)g(c, d)).509

2. Second, we show that(a, b) ≺ (c, d) ⇒ v∆(a, b) < v∆(c, d) as follows.510

(a, b) ≺ (c, d) ⇒ either (a  c and b ¹ d) or (a º c and b ≺ d) ⇒ either (θ(a) ≺ θ(c) and b ¹ d)511

or (θ(a) ¹ θ(c) and b ≺ d) ⇒ either (v(θ(a)) < v(θ(c)) and v(b) ≤ v(d)) or (v(θ(a)) ≤ v(θ(c)) and512

v(b) < v(d)) ⇒ v(θ(a)) + v(b) < v(θ(c)) + v(d) ⇒ v∆(a, b) < v∆(c, d).513

The latter completes the proof of Proposition 6.2.514

We remark that Proposition 6.2 has been proven, quite restrictively, for a totally-ordered lattice(L,¹) in [43].515

Acknowledgement516

This work has been supported, in part, by a project Archimedes-III contract.517

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22

1.3 1.4 1.484 1.6 1.70

1

a data samples population

1.3 1.4 1.484 1.6 1.70

0.5

1

PDF: integral of the finest histogram

(a) (c)

1.3 1.4 1.484 1.6 1.70

0.1

0.2

0.3a histogram (of frequencies)

1.3 1.4 1.484 1.6 1.70

1

2*PDF

2*(1-PDF)

the corresponding IN

(b) (d)

Fig. 1. Calculation of a IN from a population of data samples. (a) The data samples with medianm = 1.484. (b) A histogram of thedata. (c) The corresponding cumulative distribution function (PDF). (d) Computation of a IN from the corresponding PDF; that is, algorithmCALCIN.

23

1.3 1.4 1.484 1.6 1.70

1F

x

mF(x

)

(a)

1.3 1.4 1.484 1.6 1.70

1F

F(x)

h

(b)

Fig. 2. The two different representations of a IN F from Fig.1(d). (a) The membership-function-representationmF (x). (b) The interval-representation forL = 32 different levels spaced evenly over the interval(0, 1].

mF

R

y

m-wL m+wRm

h

V0

ch = dhx

F1

ah bh

Fig. 3. Two INs including a triangular INF with membership functionmF , specified by the three numbersm− wL, m, m + wR, and atrivial IN V0. A horizontal line at heighth ∈ (0, 1] intersects INF at pointsah andbh, moreover it intersects trivial INV0 at ch = dh = V0.

24

V0

y

m1-r m1+Rm1x

F1

1

1

1

2r

1

2R

(a)

V0

y

m2-R m2+rm2x

F2

1

2

1

2r

1

2R

(b)

V0

y

F2

m2m1x

F1

1

(c)

Fig. 4. Triangular INsF1, F2 and trivial IN V0. (a) IN F1 corresponds to a piecewise-uniformp1(x) pdf such thatp1(x) = 12r

form1 − r ≤ x ≤ m1, whereasp1(x) = 1

2Rfor m1 ≤ x ≤ m1 + R. (b) IN F2 corresponds to a piecewise-uniformp2(x) pdf such that

p2(x) = 12R

for m2 −R ≤ x ≤ m2, whereasp2(x) = 12r

for m2 ≤ x ≤ m2 + r. (c) INs F1 andF2 were placed so as the correspondingpdfs p1(x) andp2(x), respectively, have identical means, i.e.µ1 = µ = µ2. Note that the standard deviations ofp1(x) andp2(x) are alsoidentical, i.e.σ1 = σ2.

25

1 2 3 4 4.6 5 6 7 8 9 100.5

0.6

0.7

0.8

0.9

1

h

m

(a)

3 4 4.6 5 60.8

0.9

1

h

m

(b)

Fig. 5. (a) Inclusion measureσg(F ¹ V0) is plotted versus its medianm, where INsF andV0 are shown in Fig.3, using parameter valueswL = wR = 0.5 and V0 = 4.6; moreover, both the linear positive valuationv(x) = x and the dual isomorphic functionθ(x) = 10 − xwere used. (b) The above figure is shown in the vicinity of its global maximum atm = 4.6.

26

0 1 2 3 4 4.6 5 6 7 8 9 100.5

0.6

0.7

0.8

0.9

1

h

m

(a)

3 4 4.6 5 60.8

0.9

1

h

m

(b)

Fig. 6. (a) Inclusion measureσg(F ¹ V0) is plotted versus its medianm, where INsF andV0 are shown in Fig.3 using parameter valueswL = wR = 0.5 and V0 = 4.6; moreover, both the sigmoid positive valuationv(x) = 1

1+e−0.5(x−4.6) and the dual isomorphic functionθ(x) = 2(4.6)− x were used. (b) The above figure is shown in the vicinity of its global maximum atm = 4.6.

27

0 10 20 30 40 50 60 65 70 80 90 1000.5

0.6

0.7

0.8

0.9

1

h

m1

0 10 20 30 40 50 60 65 70 80 90 1000.5

0.6

0.7

0.8

0.9

1

h

m2

(a) (c)

60 65 700.9

0.91

0.92

0.93

0.94

0.95

h

m1

60 65 700.9

0.91

0.92

0.93

0.94

0.95h

m2

(b) (d)

60 65 700.9

0.91

0.92

0.93

0.94

0.95

h

(e)

Fig. 7. INs F1, F2 and V0 are shown in Fig.4 withr = 3 and R = 10, moreover trivial INV0 is located at65. (a) Inclusion measureσg(F1 ¹ V0) is plotted versus its medianm1. (b) The latter figure is shown in the vicinity of its global maximum atm1 = 65. (c) Inclusionmeasureσg(F2 ¹ V0) is plotted versus its medianm2. (d) The latter figure is shown in the vicinity of its global maximum atm2 = 65. (e)Inclusion measuresσg(F1 ¹ V0) andσg(F2 ¹ V0) are shown, comparatively, in the vicinity of their global maximum versus their identicalmeanµ.

28

Fig. 8. A fully functional software platform, namely XtraSP.v1, has been developed, in the context of this work, towards an industrialproduction ofouzo (alcoholic) beverage by automating the corresponding liquid dispensing application. Cell label “U.L.M.” stands forUltrasonic Level Meter, moreover cell label “C.T.” stands for Communicating Tube.

29

35 40 45 50 55 60 650

0.2

0.4

0.6

0.8

1

m [lt]

(a)

35 40 45 50 55 60 650

0.2

0.4

0.6

0.8

1

m [lt]

(b)

Fig. 9. Expert-1, that is a flowmeter measurement device, supplied a triangular IN estimate of a dispensed volume as detailed in the text.(a) The membership-function-representation of a dispensed volume estimate. (b) The corresponding interval-representation.

0 10 20 30 40 50 60 65 70 80 90 1000.5

0.6

0.7

0.8

0.9

1

h

m [lt]

(a)

60 65 70 75

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

h

m [lt]

(b)

Fig. 10. (a) Inclusion measureσg(F ¹ V0) is plotted versus its medianm, where INF is shown in Fig.9, moreoverV0 = 65. (b) Theabove figure is shown in the vicinity of its global maximum atm = 65.

30

35 40 45 50 55 60 650

0.2

0.4

0.6

0.8

1

m [lt]

(a)

35 40 45 50 55 60 650

0.2

0.4

0.6

0.8

1

m [lt]

(b)

Fig. 11. Expert-2, that is a ultrasonic level meter measurement (U.L.M.) device, supplied a population of measurements resulting in a INof irregular shape as an estimate of a dispensed volume as detailed in the text. (a) The membership-function-representation of a dispensedvolume estimate. (b) The corresponding interval-representation.

0 10 20 30 40 50 60 65 70 80 90 1000.5

0.6

0.7

0.8

0.9

1

h

m

(a)

60 65 70 75

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

h

m

(b)

Fig. 12. (a) Inclusion measureσg(F ¹ V0) is plotted versus its medianm, where INF is shown in Fig.11, moreoverV0 = 65. (b) Theabove figure is shown in the vicinity of its global maximum atm = 65.

31

R

y

m-w-wL m+w+wRm

h

V0

ch=dhx

F1

ah bhm+wm-w

Fig. 13. Two INs including a trapezoidal INF, specified by the four numbersm− w − wL, m− w, m + w, m + w + wR (note thatmis the average of numbersm− w andm + w), and a trivial INV0. A horizontal line at heighth ∈ (0, 1] intersects INF at pointsah andbh, moreover it intersects trivial INV0 at pointch = dh = V0.

35 40 45 50 55 60 650

0.2

0.4

0.6

0.8

1

m [lt]

(a)

35 40 45 50 55 60 650

0.2

0.4

0.6

0.8

1

m [lt]

(b)

Fig. 14. Expert-3, that is a human expert, supplied a trapezoidal IN estimate of a dispensed volume as detailed in the text. (a) Themembership-function-representation of a dispensed volume estimate. (b) The corresponding interval-representation.

32

0 10 20 30 40 50 60 65 70 80 90 1000.5

0.6

0.7

0.8

0.9

1

h

m [lt]

(a)

60 65 70 75

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

h

m [lt]

(b)

Fig. 15. (a) Inclusion measureσg(F ¹ V0) is plotted versus its corresponding median parameterm, where IN F is shown in Fig.14,moreoverV0 = 65. (b) The above figure is shown in the vicinity of its global maximum atm = 65.