Fuzzy data treated as functional data: A one-way ANOVA test approach

Computational Statistics and Data Analysis 56 (2012) 943–955

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Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda

Fuzzy data treated as functional data: A one-way ANOVA test approach Gil González-Rodríguez a,∗ , Ana Colubi b , María Ángeles Gil b a

European Centre for Soft Computing, Edificio Científico-Tecnológico, 33600 Mieres, Asturias, Spain

b

Departamento de Estadística, I.O. y D.M., Universidad de Oviedo, 33071 Oviedo, Spain

article

info

Article history: Received 14 January 2010 Received in revised form 13 June 2010 Accepted 13 June 2010 Available online 22 June 2010 Keywords: Functional data Fuzzy data k-samples test ANOVA statistic Hilbert space Convex cone Bootstrap Local alternatives

abstract The use of the fuzzy scale of measurement to describe an important number of observations from real-life attributes or variables is first explored. In contrast to other well-known scales (like nominal or ordinal), a wide class of statistical measures and techniques can be properly applied to analyze fuzzy data. This fact is connected with the possibility of identifying the scale with a special subset of a functional Hilbert space. The identification can be used to develop methods for the statistical analysis of fuzzy data by considering techniques in functional data analysis and vice versa. In this respect, an approach to the FANOVA test is presented and analyzed, and it is later particularized to deal with fuzzy data. The proposed approaches are illustrated by means of a real-life case study. © 2010 Elsevier B.V. All rights reserved.

1. Introduction and motivation Data which cannot be exactly described by means of numerical values, such as evaluations, medical diagnosis or quality ratings, to name but a few, are frequently classified as either nominal or ordinal. A well-known example is the so-called Likert scales (cf. Likert, 1932; Allen and Seaman, 2007) in which categories are labeled with numerical values. Using these scales, the statistical analysis is limited. Many parameters and techniques cannot be directly used or, when they can, the interpretation and reliability of the conclusions are considerably reduced (see, for instance, Stevens, 1946, or Chimka and Wolfe, 2009). Additionally, the transition from one category to another is rather abrupt (see, for instance Ammar and Wright, 2000). A third concern is that categories are not perceived in the same manner by different observers, so that the variability and accuracy cannot always be well captured. A new easy-to-use representation of such data through fuzzy values is to be considered. The measurement scale of fuzzy values includes, in particular, real vectors and set values as special elements. It is more expressive than ordinal scales and more accurate than rounding or using real or vectorial-valued codes. The arithmetic and the metric to be used make it possible to extend naturally many of the usual statistical measures and techniques. The transition between closely different values can be made gradually, and the variability, accuracy and possible subjectiveness can be well reflected in describing data. Although fuzzy data could be directly viewed as special functional data, the arithmetic and the metric being coherent with their meaning do not coincide with the usual ones for functional data. Nevertheless, the so-called support function establishes a useful embedding of the space of fuzzy values into a cone of a functional Hilbert space (see Puri and Ralescu, 1983, 1985; Körner and Näther, 2002).

∗

Corresponding author. Tel.: +34 985456545; fax: +34 985456699. E-mail address: [email protected] (G. González-Rodríguez).

0167-9473/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2010.06.013

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Fig. 1. Application to express the perception on the relative length of segments.

A guideline suggesting how to collect/describe fuzzy data associated with random experiments will be presented in Section 2. In Section 3, the natural arithmetic, the support function and a family of metrics on the space of fuzzy values will be recalled. Random fuzzy sets and their connection with Hilbert space-valued random elements will be described in Section 4. In Section 5, a one-way ANOVA test for functional data will be introduced, and it will be particularized to deal with fuzzy data. Specifically, an asymptotic test and its behaviour under local alternatives, as well as a bootstrap procedure, will be analyzed. A case study introduced in Section 2 will be considered in Section 6 to illustrate the approach. 2. Collecting fuzzy data from random experiments The space Fc (Rp ) of fuzzy values to be considered contains the mappings  U : Rp → [0, 1] so that for each α ∈ (0, 1] the

α -level set  Uα = {x ∈ Rp :  U (x) ≥ α} is a nonempty compact convex set of Rp . When p = 1 the fuzzy values are referred to as fuzzy numbers. Formally, fuzzy values are [0, 1]-valued upper semicontinuous functions with nonempty convex bounded α -levels. Real, vectorial, interval and set-valued data can be viewed as particular fuzzy data, by identifying them with the associated indicator functions. A fuzzy value  U ∈ Fc (Rp ) models an ill-defined subset of Rp , so that for each x ∈ Rp the value  U (x) can be interpreted as ‘degree of membership’ of x to  U. Alternatively,  U may be interpreted as the ‘degree of compatibility’ of x with an illdefined property  U. In practice, fuzzy data usually come from either a pre-established classification, such as the danger of forest fires (see Colubi and González-Rodríguez, 2007), or from a designed experiment. This is the case of the expert evaluation of the trees in a reforestation analyzed in Colubi (2009), where the ill-defined characteristic ‘quality’ is individually described through a fuzzy set. Obviously, accuracy and variability of data are much better captured by using individual fuzzy assessments than by considering a pre-fixed list of fuzzy values. The main concepts and methods are to be illustrated by means of a case study which is now introduced along with some guidelines to describe fuzzy data. The case study regards an experiment in which people have been asked for their perception of the relative length of different line segments with respect to a fixed longer segment that is used as a standard for comparison. Fig. 1 displays the screen of the application. On the center top of the screen the pattern (longest line segment) is drawn in black. At each trial a gray shorter line segment is generated and placed below the pattern one, parallelly and without considering a concrete location (i.e., indenting or centering). After an explanation of the fuzzy values, participants are asked by their judgment of relative length for each of several line segments in two ways. First, to choose a label from a Likert-like list, {very small, small, medium, large, very large}. Second, to describe the perception through a trapezoidal fuzzy number with support included in [0, 100] (0% indicating the minimum relative length and 100% maximum the one). The support is to be chosen as the set of all values that the participant subjectively considers to be compatible with the relative length of the generated segment to a greater or lesser extent. The 1-level has to be the set of all values that the participant considers to be completely compatible with his/her perception about the relative length of the generated segment. The trapezoidal fuzzy set is formed by the linear interpolation of both intervals, although it is possible to change the shape. In other words, out of the support are the values that the participant is not willing to accept as possible values for the relative length at all. The membership degree is linearly increasing from the minimum of the support to the first value for which the participant would say that it is the relative length of the line (see Fig. 1). However, since the participant may have doubts, often there is not a unique value in these conditions, but an

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interval. This interval with full membership degree is the 1-level set. Analogously, from the last value in this set and the maximum of the support, the membership degree is linearly decreasing. In Section 6 data corresponding to 17 participants will be analyzed. The relative length can be considered as an underlying real-valued random variable which could be physically measured. However, in Section 6 the interest will only refer to the fuzzy-valued variable corresponding to the judgment of the relative length. Since each participant can express his/her perception and degree uncertainty, this fuzzy valued-based description provides with a richer information than traditional ones, and the variability of collected data is better captured, which entails an informational gain. Other real-life experiments where experts have employed this fuzzy scale to represent their opinions can be found in a forestry study in Colubi (2009) and in a flood analysis in Fernández et al. (in press). 3. Fuzzy data viewed as functional data The operations to be used are the sum and the product by a scalar (Zadeh, 1975; Nguyen, 1978), which extend the corresponding ones for sets and pay attention to the fuzzy meaning. Given  U,  V ∈ Fc (Rp ), and γ ∈ R,  U + γ V ∈ Fc (Rp ) is defined so that for each α ∈ [0, 1]:

  ( U + γ V )α =  Uα + γ  Vα = y + γ z : y ∈  Uα , z ∈  Vα . This arithmetic does not coincide with the usual one for functions. The application of the functional arithmetic in Fc (Rp ) may lead to elements out of this space, and the fuzzy meaning would be lost. The space (Fc (Rp ), +, ·) has not a linear (but a semilinear-conical) structure, because the sum extends level-wise the Minkowski sum of sets. 3.1. The support function: a functional representation of fuzzy values Consider the space H = L2 (Sp−1 × (0, 1], λp × λ) of the L2 -type real-valued functions defined on the unit sphere Sp−1 of Rp times the interval (0, 1] with respect to the corresponding normalized Lebesgue measures denoted by λp and λ. Taking inspiration in Trutschnig et al. (2009), the mid/spr decomposition of a function f ∈ H can be defined as f = mid f + spr f where, for all u ∈ Sp−1 and α ∈ (0, 1], mid f (u, α) =

f (u, α) − f (−u, α) 2

,

spr f (u, α) =

f (u, α) + f (−u, α) 2

,

where if u = (u1 , . . . , up ) ∈ Sp−1 , −u denotes the element (−u1 , . . . , −up ) ∈ Sp−1 . It can be proven that mid f , spr f ∈ H , mid f is an odd function and spr f is an even one, w.r.t. the first component. On this basis, a valuable inner product in H can be defined. More precisely, let θ ∈ (0, +∞) and let ϕ be a weighting measure formalized as an absolutely continuous probability measure on ([0, 1], B[0,1] ) with positive mass function in (0, 1). For f , g ∈ H consider the value



ϕ

f , g θ = mid f , mid g



ϕ

 ϕ + θ spr f , spr g ,

where f,g



ϕ





= (0,1]

Sp−1

f (u, α)g (u, α) dλp (u) dϕ(α).

Then, the following properties are satisfied:

ϕ

ϕ

(i) f , g θ is an inner product in H , for which the associated norm is denoted by ∥ · ∥θ . (ii) The mid/spr decomposition of a function f ∈ H is orthogonal. ϕ (iii) (H , ⟨·, ·⟩θ ) is a separable Hilbert space.



The support function of  U ∈ Fc (Rp ) (see Puri and Ralescu, 1985) extends level-wise the notion of the support function of p−1 a set (see, for instance Castaing and Valadier, 1977) and it is given by the mapping s × (0, 1] → R defined so that U : S s U (u, α) = sup ⟨u, v⟩ v∈ Uα

for all u ∈ Sp−1 , α ∈ (0, 1], where ⟨·, ·⟩ denotes the inner product on Rp . In general, one can state that s U (u, α) represents the ‘‘oriented’’ distance from 0 ∈ Rp to the supporting hyperplane of  Uα which is orthogonal to u. 2 p p   The mapping s : Fc2 (Rp ) → H , such that s( U ) = s U for all U ∈ Fc (R ) = {U ∈ Fc (R ) : s U ∈ H }, is semilinear, that is, it transforms the fuzzy arithmetic to the functional arithmetic in the corresponding cone. Let  U ∈ Fc2 (Rp ), then, from Trutschnig et al. (2009) we have that (i) for all α ∈ (0, 1] the projection of  Uα over a direction u ∈ Sp−1 is given by the interval

  Πu  Uα = −s U (−u, α), s U (u, α) ;

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(ii) mid s U (u, α) =

s (u,α)−s (−u,α) U U 2 s (u,α)+s (−u,α) U U

= mid-point/center of Πu  Uα ;  (iii) spr s = spread/radius of Πu Uα ; U (u, α) = 2 (iv) if mid  U (•, ⋆) = mid-point of Π•  U⋆ , spr  U (•, ⋆) = spread of Π•  U⋆ , then   s U = mid U + spr U . Consequently, the mid and the spr of a fuzzy value can be interpreted as a kind of functional measurements of its ‘location’ and ‘shape’, respectively. 3.2. Distance between fuzzy values and isometrical embedding ϕ

The above-established connection induces a family of L2 metrics on Fc2 (Rp ) from that associated with the norms ∥ · ∥θ on H . Specifically, Trutschnig et al. (2009) have introduced the following family of metrics. Let θ ∈ (0, +∞) and let ϕ be an absolutely continuous probability measure on ([0, 1], B[0,1] ) with positive mass function ϕ in (0, 1). Then, the mapping Dθ : Fc2 (Rp ) × Fc2 (Rp ) → [0, +∞) such that for any  U,  V ∈ Fc2 (Rp ) ϕ

U,  V) Dθ (



2

 ϕ ϕ 2 = ⟨sU − sV , sU − sV ⟩θ = ∥sU − sV ∥θ

satisfies that ϕ

(i) Fc2 (Rp ), Dθ is a separable L2 -type metric space.



(ii) The support function s : Fc2 (Rp ) → H states an isometrical embedding of Fc2 (Rp ) onto a closed convex cone of H . ϕ

As a result, data in the fuzzy setting with the fuzzy arithmetic and the metric Dθ can be systematically translated into ϕ data in the setting of functional values with the functional arithmetic and the metric based on the norm ∥ · ∥θ . In this way, although fuzzy data should not be treated directly as functional data, they can be treated as functional data by considering the identification with their support functions. Many developments in functional data analysis could be applied to fuzzy data by using the appropriate   identifications and correspondences, whenever it can be guaranteed that the elements which should belong to s Fc2 (Rp ) are well-defined within it. ϕ

The Dθ metric on Fc2 (Rp ) can be equivalently expressed as follows: ϕ

Dθ ( U,  V) =



  ϕ 2 ϕ 2 ∥mid  U − mid  V ∥1 + θ ∥spr  U − spr  V ∥1 .

For each level, the choice of θ allows us to weight the effect of the deviation between spreads (which can be intuitively translated into the difference in ‘shape’ or ‘imprecision’), in contrast to the effect of the deviation between mid’s (intuitively translated into the difference in ‘location’). On the other hand, the choice of ϕ enables to weight the relevance of different levels. 4. Random fuzzy sets and relevant parameters Random fuzzy sets (for short RFS) were introduced by Puri and Ralescu (1986), as a mathematical model associating a fuzzy value with each outcome of a random experiment and extending level-wise the concept of random set. They are often referred to in the literature as fuzzy random variables in Puri and Ralescu’s sense. From Colubi et al. (2001, 2002) and Trutschnig et al. (2009), several measurability conditions are deduced to be equivalent. Namely, Theorem 4.1. Given a probability space (Ω , A, P ), consider the mapping X : Ω → Fc2 (Rp ). Then, the following statements are equivalent: (i) X is a RFS, that is, for all α ∈ (0, 1] the α -level set-valued mapping

Xα : Ω → Kc (Rp ) = {nonempty compact convex sets of Rp }, ω → (X(ω))α ,

(ii) (iii) (iv) (v)

is a compact convex random set (that is, A|β− measurable, where β is the Borel σ -field associated with the Hausdorff metric on Kc (Rp )). ϕ X is a Borel measurable mapping w.r.t. A and the Borel σ -field generated by the topology induced by the metric Dθ on 2 p Fc (R ). sX : Ω → H is an H -valued random element, that is, a Borel measurable mapping w.r.t. A and the Borel σ -field generated ϕ by the topology induced by ∥ · ∥θ on H . For all α ∈ (0, 1] and u ∈ Sp−1 , the function sX (u, α) : Ω → R is a real-valued random variable. For all α ∈ (0, 1] and u ∈ Sp−1 , the functions mid sX (u, α) : Ω → R and spr sX (u, α) : Ω → [0, +∞) are real-valued random variables.

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On the basis of Theorem 4.1 it is concluded that notions like the distribution induced by an RFS or the stochastic independence of RFSs are the usual ones for Borel measurable mappings in metric spaces. The mean value of an RFS can be presented in two equivalent ways, either as an extension of the set-valued Aumann expectation or induced from the expectation of an H -valued random element (see Puri and Ralescu, 1983, 1985, 1986). Thus, Definition 4.1. Given a probability space (Ω , A, P ) and an associated RFS X such that sX ∈ L1 (Ω , A, P ), the (Aumann type) mean value or expected value of X is the fuzzy value  E (X) ∈ Fc (Rp ) such that for all α ∈ (0, 1]

   E (X) α = Aumann integral of Xα   p 1 = X (ω) dP (ω) for all X : Ω → R , X ∈ L (Ω , A, P ), X ∈ Xα a.s. [P ] Rp

or, equivalently, such that s E (X) = E (sX ). In case p = 1, if X is an RFS such that max | inf X0 |, | sup X0 | ∈ L1 (Ω , A, P ), we have that for each α ∈ [0, 1]:





   E (X) α = [E (inf Xα ), E (sup Xα )] . This definition for the mean value is coherent with the considered arithmetic and satisfies the usual properties of linearity. ϕ Moreover, it is Fréchet’s expectation w.r.t. Dθ . Following Lubiano et al. (2000) and Körner and Näther (2002) the variance of an RFS will be based on Fréchet’s approach. The (θ, ϕ)-Fréchet variance is conceived as a measure of the error in approximating or estimating the values of the RFS through the corresponding mean value. The real-valued quantification of the dispersion will enable to compare random elements, populations, samples, estimators, etc. by simply ranking real numbers. Due to the properties of the support function and the Hilbertian random elements, the considered variance satisfies the usual properties for this concept. Definition 4.2. Given a probability space (Ω , A, P ) and an associated RFS X such that sX ∈ L2 (Ω , A, P ), the (θ , ϕ)-Fréchet variance of X is the real number

σX2 = E



ϕ

2 

E (X) Dθ X, 



or, equivalently,



ϕ  2 



 2  ∥sX − E (sX )∥ϕθ = Var(sX )     = E [∥mid X − E (mid X)∥ϕ ]2 + θ E [∥spr X − E (spr X)∥ϕ ]2 = Var(mid X) + θ Var(spr X).

σX = E 2

 sX − s

E (X) θ

=E

5. One-way FANOVA test and its particularization to a one-way ANOVA test for fuzzy data There are some key distinctive features of the statistical inference from fuzzy data in contrast to the case of random variables/vectors. Firstly, the lack of realistic and operational ‘parametric’ families of probability distribution models for RFSs (a model for normal RFSs was suggested in Puri and Ralescu (1985), but it is very restrictive and unrealistic). Secondly, the lack of Central Limit Theorems for RFSs being directly applicable for inferential purposes. There exist some results assuring that the limit in law of the normalized distance between the sample and population fuzzy means is the norm of a Gaussian random element with values belonging to a wide functional space (usually not belonging to the cone). This section aims to develop first an ANOVA test for functional data, and to particularize it later to an ANOVA test for fuzzy data by using the identification explained in Section 3. Special care has to be taken in guaranteeing that the result remains in the considered cone; the bootstrap approach will guarantee it. 5.1. An ANOVA test approach for functional data The statistical study of functional data has received much attention in the last years (see, for instance Ramsay and Silverman, 1997, 2002; Ferraty and Vieu, 2006; Yao et al., 2005; Müller, 2005; Cuevas et al., 2006; for an overview of some of the recent trends, see the special issue edited by González-Manteiga and Vieu, 2007, and for some very recent publications see Cao and Ramsay, 2009; Ferraty and Vieu, 2009; Müller and Yang, 2010). The problem of testing the equality of means of k independent Hilbert space-valued random elements is now to be considered. The framework is quite close to, although more general than, the one analyzed in Cuevas et al. (2004). Let H

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be a separable Hilbert space with inner product ⟨·, ·⟩ and associated with a norm ∥ · ∥, and let H1 , . . . , Hk be k independent H -valued random elements for which there exist the expected values m1 , . . . , mk , and the covariance functions K1 , . . . , Kk , respectively. The aim is to test H0 : m1 = · · · = mk

versus Ha : ∃ i1 ̸= i2 with mi1 ̸= mi2 .

In Cuevas et al. (2004) an asymptotic ANOVA test for functional data based on a statistic quantifying the sum of the pairwise differences between the sample means has been determined. In contrast, the test statistic to be used in this subsection is a natural extension of the ANOVA statistic for real-valued data (its potential use has been suggested by Cuevas et al., although they have declined to enter its study). Additionally, an analysis of the consistency of the test under local alternatives and the bootstrap approximations for both the asymptotic procedure and the consistency result will be developed. The class of Hilbert space-valued random elements the approach applies to is slightly wider than that in Cuevas et al. (2004). n Let {Hij }j=i 1 be a simple random sample obtained from the Hilbert space-valued random element Hi for all i ∈ {1 . . . , k}, and let n ∈ N be the overall sample size, that is, n = n1 + · · · + nk . The natural extension of the ANOVA statistic is given by An =

k 

2

ni Hi· − H··  ,



i=1

where Hi· = An =

ni

j =1

k  i =1

Hij /ni and H·· =

2 

 

ni Hic· − H··c  +

k ni

k 

i=1

j =1

Hij /n. This statistic can be decomposed as follows:

ni ∥ m i − µ n ∥ 2 + 2

k 

i=1





ni Hic· − H··c , mi − µn ,

i=1

where = Hij − mi , µn = i=1 ni mi and ⟨·, ·⟩ stands for the corresponding inner product. As well, the first term can be expressed in functional form as follows: 1 n

Hijc

k  i=1

2 

 

k √

ni Hic· − H··c  = ηn ( n1 · H1c· , . . . ,

√

nk · Hkc· ),

with ηn : H × · · · × H → R mapping (h1 , . . . , hk ) into

 2 k k      n  αli hl  ,  hi −   l =1 i=1  √ k where αlin = nl /ni / r =1 (nr /ni ). Proposition 5.1. If ni → ∞, ni /n → pi > 0 as n → ∞ for all i ∈ {1, . . . , k}, and the null hypothesis H0 is fulfilled, then, An converges in law to the distribution of

2  k k       αli Zl  , A= Zi −   l =1 i=1 √

where αli = pl /pi / r =1 (pr /pi ) for all i, l ∈ {= 1, . . . , k}, and Z1 , . . . , Zk are independent centered Gaussian processes in H with covariance functions K1 , . . . , Kk , respectively.

k

Proof. If H0 is true, then the above-mentioned decomposition of An is reduced to

√

An = ηn ( n1 · H1c· , . . . ,

√

nk · Hkc· ).

√

The Central Limit Theorem in Hilbert spaces (see, for instance Laha and Rohatgi, 1979) assures that ( n1 · H1c· , . . . , nk · Hkc· ) converges in law to (Z1 , . . . , Zk ), where Z1 , . . . , Zk are centered independent Gaussian processes with the same covariance function as H1 , . . . , Hk respectively. Consider

√

 2 k  k      η(h1 , . . . , hk ) = αli hl  .  hi −   i =1 l =1 Since η only involves linear combinations and the L2 norm, then it is continuous, whence if (hn1 , . . . , hnk ) converges to (h01 , . . . , h0k ) as n → ∞ in H × · · · × H , then ηn (hn1 , . . . , hnk ) converges to η(h01 , . . . , h0k ) as n → ∞. √ Consequently, Slutsky theorem and the continuous mapping theorem guarantee the convergence in law of An = ηn ( n1 · H1c· , . . . ,

√

nk · Hkc· ) to A = η(Z1 , . . . , Zk ).



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To complete the asymptotic study of the statistic An the behaviour under local alternatives can be analyzed. To this purpose, consider

δn

mi = m∗ + √ m∗i n

with m∗ , m∗i ∈ H ,

√

for δn ∈ R+ and n ∈ N, and so that there exist i1 ̸= i2 with m∗i1 ̸= m∗i2 . If δn / n → 0 then ∥mi − mj ∥2 → 0 as n → ∞ for

√

all i, j ∈ {1, . . . , k}, that is, although H0 is not fulfilled for all n ∈ N it is approached with ‘‘speed’’ δn / n. Thus,

√

Proposition 5.2. If ni → ∞ and ni /n → pi > 0 as n → ∞ for all i ∈ {1, . . . , k}, δn → ∞ and δn / n → 0 as n → ∞, then P (An ≤ t ) → 0 as n → ∞ for all t ∈ R. Proof. To apply arguments analogous to those in Proposition 5.1, by taking into account the expression of the considered population means mi , the decomposition of An can be written as follows:

√

An = ηn ( n1 · H1c· , . . . ,

√

nk · Hkc· ) + δn2

k  ni

n

i=1

∥m∗i − µ∗n ∥2

√ √ + δn ζn ( n1 · H1c· , . . . , nk · Hkc· , m∗i − µ∗n , . . . , m∗k − µ∗n ) k ∗ −1

where µ∗n = n

i=1

ni mi and

k 

ζn (h1 , . . . , hk , g1 , . . . , gk ) =



ni

 hi −

n

i =1

k 

 α

n li hl

, gi

l =1

for all (h1 , . . . , hk , g1 , . . . , gk ) ∈ [H ]2k . The first term has been proved to converge in law to η(Z1 , . . . , Zk ) in Proposition 5.1. Regarding the second term, we k k k ∗ have that i=1 ni ∥m∗i − µ∗n ∥2 /n converges to i=1 pi ∥m∗i − µ∗ ∥2 , with µ∗ = i=1 pi mi . Thus, the second term divided by ∗ ∗ 2 2 δn converges to a positive quantity, since there exist ii ̸= i2 with ∥mi1 − mi2 ∥ > 0. Concerning the third term, if

ζ (h1 , . . . , hk , g1 , . . . , gk ) =

k  √



p i hi −

k 

i=1

 αli hl , gi

l =1

√

√

for all (h1 , . . . , hk , g1 , . . . , gk ) ∈ [H ]2k , analogously to Proposition 5.1, one has that ζn ( n1 · H1c· , . . . , nk · Hkc· , m∗1 − µ∗n , . . . , m∗k − µ∗n ) converges in law to ζ (Z1 , . . . , Zn , m∗1 − µ∗ , . . . , m∗k − µ∗ ). Consequently, it can be easily concluded that 3/2

An /δn t ∈ R.

→ ∞ in probability, and, hence, An → ∞ in probability, which implies that P (An ≤ t ) → 0 as n → ∞ for all 

The proposed statistic can be studentized by considering as usual a denominator related to the overall variability. The strong law of large numbers and Slutsky theorem, along with the preceding propositions, enable to get immediately the next result. Theorem 5.3. If ni → ∞ and ni /n → pi > 0 as n → ∞ for all i ∈ {1, . . . , k} and Hi is non-degenerated for some i ∈ {1, . . . k}, then Bn =

ni k 1 

n i=1 j=1

∥Hij − Hi· ∥2 →

k 

pi E ∥Hi − mi ∥2 > 0 a.s. [P ].

i=1

Consequently, if the null hypothesis H0 is true, then the following convergence in law holds k  i=1 1 n

k 

ni ∥Hi· − H·· ∥2

ni k   i =1 j =1

→ ∥Hij − Hi· ∥2

∥Zi −

i=1 k 

k 

αli Zl ∥2

l =1

.

pi E ∥Hi − mi ∥2

i =1

Moreover, under the local alternatives above-described, we have that if δn → ∞ as n → ∞, then P (An /Bn ≤ t ) → 0 as n → ∞ for all t ∈ R. On the basis of Theorem 5.3, an asymptotic ANOVA test generalizing the usual one to the functional data analysis case is derived. A bootstrap approximation is now considered to improve asymptotic results by means of re-sampling techniques. In this respect, the Bootstrap Central Limit Theorem by Giné and Zinn (1990) ensures that if for all i ∈ {1, . . . k} the ni functional

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random variables Hij∗ randomly chosen from {Hi1 − Hi· , . . . , Hini − Hi· } are considered, one has that a.s.−[P ] as ni → ∞. Consequently, if the bootstrap statistic is defined as ∗

An =

k  i=1

√

ni · Hi∗· → Zi in law

 2 k    ∗  n ∗ αli Hl·  , ni Hi· −   l =1

by the same arguments than those in Proposition 5.1, the following result is obtained. Theorem 5.4. If ni → ∞ and ni /n → pi > 0 as n → ∞ for all i ∈ {1, . . . , k}, then A∗n converges in law a.s. −[P ] to the distribution of A in Proposition 5.1 (thus, the distribution of An under H0 can be approximated by the one of A∗n ). 5.2. An ANOVA test approach for fuzzy data The results in Section 5.1 can be particularized to fuzzy data by using the identification in Section 3. Thus, an extension of the ANOVA introduced in Gil et al. (2006) (valid only for simple random fuzzy sets in the 1-dimensional setting) is obtained. Consider the problem of testing the equality of means of RFSs. Let X1 , . . . , Xk be k independent RFSs with values in Fc2 (Rp ). i ∈ Fc2 (Rp ), the (θ , ϕ)-Fréchet variances Assume that for each i ∈ {1, . . . , k} there exist the fuzzy mean values  E (Xi ) = m 2 σXi , i ∈ {1, . . . , k}, and the covariance functions K1 , . . . , Kk of sX1 , . . . , sXk , respectively. The aim is to test

1 = · · · = m k H0 : m

i1 ̸= m  i2 , versus Ha : ∃ i1 ̸= i2 with m n

on the basis of a simple random sample {Xij }j=i 1 obtained from the RFS Xi for all i ∈ {1 . . . , k}, Theorem 5.5. If ni → ∞, ni /n → pi > 0 as n → ∞ for all i ∈ {1, . . . , k} and Xi is non-degenerated for some i ∈ {1, . . . k}, then if the null hypothesis H0 is true, the following convergence in law holds k  i =1

Tn = 1 n

ϕ

ni Dθ (Xi· , X·· )



2 →

ni  k   i=1 j=1

ϕ Dθ (Xij , Xi· )

 ϕ 2  k k    Zi −  αli Zl   

i=1

θ

l =1

k 

2

i=1

,

pi σXi 2

where Z1 , . . . , Zk are independent centered Gaussian processes in H with covariance functions K1 , . . . , Kk , Xi· =

ni

j =1

Xij /ni

k ni

and X·· = j=1 Xij /n. Moreover, under the local alternatives, if δn → ∞ as n → ∞, then P (Tn ≤ t ) → 0 as n → ∞ i=1 for all t ∈ R. n Let {X∗ij }j=i 1 be randomly chosen from {Xi1 , . . . , Xini }. If the bootstrap statistic is defined as A∗n =

k 



2

ϕ

ni Dθ (X∗i· + X·· , Xi· + X∗·· )

,

i=1

then the distribution of An under H0 can be approximated by the one of A∗n . As a consequence, H0 will be rejected at a given significance level α whenever distribution of A∗n .

k

i =1

ϕ

2

ni Dθ (Xi· , X·· )



takes on a value greater than the estimated 100(1 − α) fractile of the

In particularizing the asymptotic results in Section 5.1 to the case of fuzzy data, the Gaussian processes involved in the asymptotic result is not Fc2 (Rp )-valued. However, the suggested bootstrap approximation has been expressed in terms of sums and distances, and it does not involve difference of fuzzy values, whence the bootstrap statistic is defined on the sampling cone. 6. Illustrative example: application to a case study The survey described in Section 2 was electronically distributed. A sample of 17 people from different countries, professions, ages and sexes (8 women and 9 men) have participated in. Each of the 17 people have performed a sequence of 27 trials. The generation of the gray line segment was made at random and so that one from 9 different relative sizes can be chosen in each trial. The 9 relative sizes are equally spaced in such a way that possibilities from very small line segment to very large are covered. Each of the 9 relative sizes appears 3 times in the sequence, although positions and locations vary within each sequence and between sequences (very slight changes can appear in the list of the 9 relative sizes depending on the screen resolution). Variations in the data could be imputed to either variations in the perceptions or variations due to the line segment size; showing the same 9 relative sizes 3 times to each of the participants is useful to control the variations associated with the line segment size. In order to compare the overall perception of relative lengths, an not only the perceptions associated with a particular length, tests about the average perception, as a summary measure,

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Fig. 2. Dotted and dashed lines: two fuzzy data for line segments of 26.76% and 96.27% relatives chose by two participants (left—female, right—male). Thick line: average of the 27 trials of both participants.

Fig. 3. Distances between the two couple of trials for the female (dotted lines), for the male (dashed lines) and between the averages of both individuals (thick line) in Fig. 2.

will be carried out. Since all the participants have evaluated the same set of lengths, the mean values should coincide except for differences in perception. Additionally, to show the results for each length, avoiding in this way possible compensations, the 9 relative sizes are considered separately to test differences in perception between men and women. The dataset to be analyzed and the application providing it can be found in http://bellman.ciencias.uniovi.es/SMIRE/perceptions.html (web of research group SMIRE). In Fig. 2 two of the fuzzy data obtained for line segments of 26.76% and 96.27% relative lengths for 2 individuals (female and male), as well as the average of the 27 trials of each individual (thick line) are shown. The mid-points of the level sets are linked to the location, as the class mark of grouped and interval data. The spreads are linked to the imprecision, in the sense that when the participants have more doubts, they choose wider intervals. For both individuals, the imprecision is greater for the 26.76% length, especially for the female. Concerning the average, since the mid (respectively the spread) of the mean perception is the mean of the mids (respectively the spreads), we can conclude that the imprecision is greater for the female in mean, although the average locations of the perceptions seem quite similar. To corroborate numerically this assertion, distances are determined. The distance between fuzzy sets is now computed as a function of the weighted distances between mids and spreads, by averaging the results of each α -level. Thus, differences between fuzzy sets can be associated with differences between mids (location) and differences between spreads (imprecision). To make this fact clearer, the metric (with ϕ = λ = Lebesgue measure on (0, 1]) will be expressed as a convex linear combination of the squared distances between mids and between spreads in the following way D2τ ( U,  V ) = (1 − τ ) ∥mid  U − mid  V ∥λ1



2

 2 + τ ∥spr  U − spr  V ∥λ1

for all  U,  V ∈ Fc2 (Rp ), where τ ∈ (0, 1) represents the weight of the spreads against the mids. In Fig. 3 the distances between some fuzzy sets in Fig. 2 are displayed as functions of the weight τ . The dotted lines correspond to the perceptions of the female, the dashed lines to those of the male, and the thick line is associated with the distance of the averages of both individuals. For both individuals, the larger distance corresponds to the case of relative lengths of 26.76%, which seems to be more difficult to evaluate. In this case, the difference in location (the lower value of τ , the more weighted the location) is greater for the male, who chose for the two trials fuzzy sets located in rather different places although with very similar imprecision (the larger value of τ , the more weighted the imprecision). Whereas distances between the trials decreases as τ increases, the distance between the averages increases with the spread weight, due to the above-mentioned difference in imprecision.

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Fig. 4. Sample means for the women group (dotted line) and for the men group (dashed line) for the relative lengths of 26.76%, 96.27% and the average of the 9 lengths.

Fig. 5. p-values of the ANOVA tests for individuals (k = 17), and sex groups (two-sample) as a function of the proportion ϱ.

Several ANOVA tests have been studied, namely, a first one for individual analysis (i.e, k = 17 samples of sizes ni = 27), a second one for sex analysis (i.e., k = 2 with n1 = 27 · 8 = 216 and n2 = 27 · 9 = 243), the third and the fourth individually for women (i.e., k = 8 samples of sizes ni = 27) and men (i.e., k = 9 samples of sizes ni = 27), respectively, and 9 for each one of the relative sizes grouped by sex (i.e., k = 2 with n1 = 3 · 8 = 24 and n2 = 3 · 9 = 27). The bootstrap approach in Theorem 5.5 has been applied to approximate the corresponding p-values with 10,000 bootstrap replications. For simplicity, the sample means are represented only for the overall sex analysis and the relative lengths of 26.76% and 96.27% (see Fig. 4). The test results depend on Dτ , and therefore, on the parameter τ . To analyze the differences in the fuzzy means by taking advantage of the decomposition of the variance of X in the part related to the location, mid X, and the part related to the imprecision, spr X, the p-values have been computed as a function of the proportion ϱ of the total variation that is due to the (weighted) ‘variation in imprecision’, that is,

ϱ=

τ Var(spr X) . (1 − τ )Var(mid X) + τ Var(spr X)

This proportion ϱ is an increasing function of τ and satisfies that ϱ → 0+ iff τ → 0+ , ϱ = .5 iff (1 − τ )Var(mid X) = τ Var(spr X) (i.e., half of the overall variation is due to the variation of the spreads, and the other half by the one of the mids), ϱ → 1− iff τ → 1− , and so on. In practice Var(mid X) is usually much greater than Var(spr X), due to the magnitude difference. Consequently, in most of the real-life situations, assessments of weights τ ≤ 0.5 are associated with very small values of ϱ. The population variances in the definition of ρ have been approximated by using analogue estimates based on the total sample in each case. Fig. 5 (respectively Fig. 6) displays the p-values of the two first (respectively two second) bootstrap ANOVA tests as a function of the proportion ϱ. Two vertical reference lines cross the figures; the one on the right corresponds to ϱ = 0.5, whereas the left-side one corresponds to a very small value of ϱ (the one associated with τ = 0.5). Figs. 5 and 6 indicate that the greater the weight associated with the variation in imprecision, the lower the p-value of both tests. Significant differences are more easily detectable in case of comparing the mean perception after grouping by sex (respectively women) than in case of comparing individually (respectively men). This can be justified because the variation in imprecision between sex groups (respectively women) is substantially greater than the variation in imprecision between individuals (respectively men). To interpret the results two representative situations have been considered. Table 1 contains the p-values for the cases ϱ = 0.05 and ϱ = 0.25. Roughly speaking, in the first situation, the test is more focused on differences in locations of

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Fig. 6. p-values of the ANOVA tests for women’s individuals (k = 8), and men’s individuals (k = 7) as a function of the proportion ϱ. Table 1 p-values of the four ANOVA tests for two representative values of ρ = 0.25.

ϱ = 0.05 ϱ = 0.25

Individuals

Males/females

Males

Females

0.734 0.000

0.111 0.000

0.979 0.165

0.384 0.000

Table 2 Contingency table for the labels classified by gender.

Men Women

very small (%)

small (%)

medium (%)

large (%)

very large (%)

18.93 12.96

20.17 25.92

24.28 21.30

22.22 20.30

14.40 18.52

the average perception, because the imprecision is low weighted. In this case, no significant differences are detected. In the second situation, the imprecision is more weighted by considering ϱ = 0.25. Thus, the test is more affected by differences in imprecision. In this case, significant differences have been detected among individuals, between men and women (see Fig. 4 for the sample results) and among females, but they are not among males, which are concluded to be more uniform in expressing their average perception. The conclusions concerning the differences related to the imprecision in the average perception due to the gender are obtained because of the use of the fuzzy scale, which allows to indicate the degree of uncertainty that the participants have when he or she evaluates each instance. To illustrate this fact, the data of the Likert-like labels chosen by the participants classified according to the gender was analyzed. In Table 2 the sample results of 253 data for men and 216 for women are shown. The p-value of the χ 2 -test is 0.204. If data are understood to come from a percentage-type evaluation, the p-value of the corresponding bootstrap ANOVA test would be 0.271. Both tests fail to reject the null hypothesis, and no significant differences are found. It should be underlined that since the perception for small and large relative lengths may be different, the average over the 9 relative lengths may compensate and obscure the results. For this reason, the analysis of each one of the 9 relative sizes by sex are shown in Figs. 7–9. No significant differences are found for the cases of 26.76% and 96.27% relative sizes (see Fig. 4). For the rest of the cases, the conclusions are the same than those for the average perceptions of all lengths. 7. Concluding remarks An approach has been presented to develop statistics with fuzzy data by identifying them as special functional data. Based on this identification, the random mechanisms producing fuzzy data and relevant associated parameters have been formalized by using concepts for Hilbert space-valued random elements (see also Körner (2000), Körner and Näther (2002) and González-Rodríguez et al. (2006b)). Nevertheless, special attention should be paid in particularizing the methods from functional to fuzzy data, since fuzzy data have a conical structure instead of the linear one shown by functional data. For this reason, some previous statistical techniques to deal with fuzzy data have been designed ad hoc (cf. Montenegro et al., 2001, 2004; Gil et al., 2006; González-Rodríguez et al., 2006b). An ANOVA test for functional data has been introduced. The proposed test statistic is an extension of the classical one for real-valued data. The ANOVA approach has been later particularized to an ANOVA test for fuzzy data; it has been remarked that whereas the asymptotic approach would not involve a fuzzy-valued random element, the bootstrap one is well-developed. Several open related problems can be considered, namely, the one-way ANOVA test for dependent samples,

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Fig. 7. p-values of the ANOVA tests for the small relative sizes, and sex groups as a function of the proportion ϱ.

Fig. 8. p-values of the ANOVA tests for the medium relative sizes, and sex groups as a function of the proportion ϱ.

Fig. 9. p-values of the ANOVA tests for the large relative sizes, and sex groups as a function of the proportion ϱ.

the factorial ANOVA test, the posterior tests, the experimental design analyses, and so on. On the other hand, the results in Theorem 5.3 could be applied along with the fuzzy representation of real-valued random variables (see González-Rodríguez et al., 2006a) to develop an ANOVA test for distributions. Acknowledgements This research has been partially supported by/benefited from the Spanish Ministry of Science and Innovation Grants MTM2009-09440-C02-01 and MTM2009-09440- C02-02, the Principality of Asturias Grants IB09-042C1 and IB09-042C2,

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