Fuzzy PLS Path Modeling: A New Tool For Handling Sensory Data ...

Fuzzy PLS Path Modeling: A New Tool For Handling Sensory Data Francesco Palumbo1 , Rosaria Romano2 and Vincenzo Esposito Vinzi3 1 2 3

University of Macerata, Italy [email protected] University of Copenhagen, Denmark [email protected] ESSEC Business School of Paris, France [email protected]

Abstract. In sensory analysis a panel of assessors gives scores to blocks of sensory attributes for profiling products, thus yielding a three-way table crossing assessors, attributes and products. In this context, it is important to evaluate the panel performance as well as to synthesize the scores into a global assessment to investigate differences between products. Recently, a combined approach of fuzzy regression and PLS path modeling has been proposed. Fuzzy regression considers crisp/fuzzy variables and identifies a set of fuzzy parameters using optimization techniques. In this framework, the present work aims to show the advantages of fuzzy PLS path modeling in the context of sensory analysis.

1 Introduction In sensory analysis a panel of assessors gives scores to blocks of sensory attributes for profiling products, thus yielding a three-way table crossing assessors, attributes and products. This type of data are characterized by three different sources of complexity: complex structure of relations among the variables (different blocks), three directions of information (samples, assessors, attributes) and influential human beings’ involvement (assessors’ evaluations). Structural Equation Models (SEM) (Bollen, 1989) consist of a network of causal relationships among Latent Variables (LV) defined by blocks of Manifest Variables (MV). The main idea behind SEM is that the features on which the analysis would focus cannot be properly measured and are determined through the measured variables. In a recent contribution (Tenenhaus and Esposito-Vinzi, 2005), SEM have been successfully used to analyze sensory data. When SEM are based on the scores of a set of assessors, they are generally based on the mean scores. However, it is important to analyze if there exist individual differences between assessors. Even if assessors are carefully trained to adopt the same yardstick, this cannot completely protect us against their single sensibility.

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When human estimation is influential and the observations cannot be described accurately but we can give only an approximate description of them, fuzzy approach is more useful and convenient than the classical one (Zadeh, 1965). Fuzzy sets allow us coding and treating many different kinds of imprecise data. Recently, a fuzzy approach to SEM has been proposed (Romano, 2006) and successively used for comparing different SEM (Romano and Palumbo, 2006b). The present paper proposes to use the new fuzzy structural equation models for handling the different sources of information and uncertainty arising from sensory data. First a brief introduction to the methodology of reference (Romano, 2006) will be given. Then an application to data from sensory profiling will be presented.

2 Fuzzy PLS path modeling Fuzzy PLS Path Modeling is a new methodology to dealing with system complexity. It allows us taking into account both complexity in information codification and in structures of relations among the variables. Fuzzy codification and structural equations are combined to handling these different sources of complexity, respectively. The strategy allowing imprecision in codification for reducing complexity is appropriately expressed by Zadeh’s principle of incompatibility (Zadeh, 1973). The main idea is that the traditional techniques for analyzing systems are not well suited to dealing with human systems. In human thinking, the key elements are not numbers but classes of objects or concepts in which the membership of each element to the class is gradual (fuzzy) rather than sharp. For instance, the concept of sweet coffee does not correspond to an exact amount of sugar in the coffee. But it is possible to define the classes sweet coffee, normal coffee, bitter coffee. On the other hand, the descriptive complexity of a system can also be reduced by breaking the system into its appropriate subsystems. This is the general principle behind Structural Equation Models (SEM) (Bollen, 1989). The basic idea is that different subsets of variables are the expression of different concepts, belonging to the same phenomenon. These concepts are named latent variables (LV) as they are not directly observable but measurable by means of a set of manifest variables (MV). The aim of SEM is to study the system of relations between each LV and its MV, and among the different LV inside the system. Considering one by one each part forming the whole system, and analyzing the relations among the different parts, the system complexity is reduced allowing a better description of the main system characteristics. F-PLSPM consists in introducing fuzzy models inside SEM, by means of a twostage procedure. This allows dealing with system complexity using both an approach which is tolerant to imprecision and a well suited methodology to link the different parts into which the system may be decomposed. 2.1 Interval data, fuzzy data and fuzzy models It is very common to measure statistical variables in terms of single-values. However, for many reasons, and in many situations exact measures are very hard (or even

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impossible) to achieve. A rigorous study of interval data is given by Interval Analysis (Alefeld and Herzenberger, 1987). In this framework, an interval value is a bounded subset of real numbers [x] = [x, x], formally: [x] = {x ∈ R| x ≤ x ≤ x}

(1)

where x and x are called lower and upper bound, respectively. Alternatively, an interval value may by expressed in terms of width (or radius), xw , and center (or midpoint), xc : xw = 12 |x − x| and xc = 12 |x + x|. A fuzzy set is a codification of the information allowing us to represent vague concepts expressed in natural language. Formally, given the universe of objects :, Z as the generic element, a fuzzy set A˜ in : is defined as a set of ordered pairs: A˜ = {(Z, zA˜ (Z))|Z ∈ :}

(2)

where the value zA˜ (Z0 ) expresses the membership degree for a generic element Z0 ∈ ˜ If :. The larger the value of zA˜ (Z), the higher the degree of membership of Z in A. the membership function is permitted to have only the values 0 and 1 then the fuzzy set is reduced to a classical crisp set. The universal set : may consist of discrete (ordered and non ordered) objects or it can be a continuous space. A fuzzy set in the real line that satisfies both the conditions of normality and convexity is a fuzzy number. It must be normal so that the statement “real number close to r" is fully satisfied by r itself, i.e. zA˜ (r) = 1. In addition, all its D−cuts for D ≡ 0 must be closed intervals so that the arithmetic operations on fuzzy sets can be defined in terms of operations on closed intervals. On the other hand, if all its D−cuts are closed intervals, it follows that the fuzzy number is a convex fuzzy set. In possibility theory (Zadeh, 1978), a branch of fuzzy set theory, fuzzy numbers are described by possibility distributions. A possibility distribution SA˜ (Z) is a function which satisfies the following conditions (Tanaka and Guo, 1999): i) there exists an Z such that SA˜ (Z) = 1 (normality); ii) D−cuts of fuzzy numbers are convex; iii) SA˜ (Z) is piecewise continuous. Particular fuzzy numbers are the symmetrical fuzzy numbers whose possibility distribution may be denoted as:      Z − ci q  (3) SA˜ i (Z) = max 0, 1 −  ri  Specifically, (3) corresponds to triangular fuzzy numbers when q = 1, to square root fuzzy numbers when q = 1/2 and parabolic fuzzy numbers when q = 2. It is easy to show that (3) corresponds to intervals when q = +f. It is worth noticing that fuzzy variables are associated with possibility distributions in the similar way that random variables are associated with probability distributions. Furthermore, possibility distributions are numerically equal to membership functions (Zadeh, 1978).

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In the early 80’s, Tanaka proposed the first fuzzy linear regression model, moving on from fuzzy sets theory and possibility theory (Tanaka et al., 1980). The functional relation between dependent and independent variables is represented as a fuzzy linear function whose parameters are given by fuzzy numbers. Tanaka proposed the first Fuzzy Possibilistic Regression (FPR) using the following fuzzy linear model with crisp input and fuzzy parameters: y˜n = E˜ 0 + E˜ 1 xn1 + . . . + E˜ p xnp , + . . . + E˜ P xnP

(4)

where the parameters are symmetric triangular fuzzy numbers denoted by E˜ p = (c p ; w p )L with c p and w p as center and the spread, respectively. Differently from statistical regression, the deviations between data and linear models are assumed to depend on the vagueness of the parameters and not on measurement errors. The basic idea of Tanaka’s approach was to minimize the uncertainty of the estimates, by minimizing the total spread of the fuzzy coefficients. Spread minimization must be pursued under the constraint of the inclusion of the whole given data set, which satisfies a degree of belief D (0 < D < 1) defined by the decision maker. The estimation problem is solved via a mathematical programming approach, where the objective function aims at minimizing the spread parameters, and the constraints guarantee that observed data fall inside the fuzzy interval: minimize

P N  

w p |xnp |

(5)

n=1 p=0

subject to the following constraints: " # " # c0 + Pp=1 c p xnp + (1 − D) w0 + Pp=1 w p |xnp | ≥ yn " # " # c0 + Pp=1 c p xnp − (1 − D) w0 + Pp=1 w p |xnp | ≤ yn w p ≥ 0, c p ∈ R, xn0 = 1, n = (1, . . . , N), p = (1, . . . , P) where xn0 = 1 (n = 1, . . . , N), w p ≥ 0 and c p ∈ R (p = 1, . . . , P). 2.2 The F-PLSPM algorithm The F-PLSPM follows the component based approach SEM-PLS, alternatively defined PLS Path Modeling (PLS-PM) (Tenenhaus et al., 2005). The reason is that fuzzy regression and PLS path modeling share several characteristics. They are both soft modeling and data oriented approaches. Specifically, fuzzy regression joins PLS-PM in its final step, allowing for a fuzzy structural model (see, Figure 1) but a still crisp measurement model. This connection implies a two stage estimation procedure: •

stage 1: latent variables are estimated according to the PLS-PM estimation procedure (Wold, 1982);

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Fig. 1. Fuzzy path model representation

•

stage 2: FPR on the estimated latent variables is performed so that the following fuzzy structural model is obtained: [h = E˜ h0 +



E˜ hh [h

(6)

h

where E˜ hh refers to the generic fuzzy path coefficient, [h and [h are adjacent latent variables and h, h ∈ [1, . . . , H] vary according to the model complexity. It is worth noticing that the structural model from this procedure is different with respect to the traditional structural model. Here the path coefficients are fuzzy numbers and there is no error term, as a natural consequence of a FPR. In the analysis of a statistical model one should always, in one way or another, take into account the goodness of fit, above all in comparing different models. The proposal is then to use the FPR. The estimation of fuzzy parameters, instead of single-valued (crisp) parameters, permits us to gather both the structural and the residual information. The characteristic to embed the residual in the model via fuzzy parameters (Tanaka and Guo, 1999) permits to evaluate the differences between assessors (panel performance) as well as the reproducibility of each assessor (assessor performance) (Romano and Palumbo, 2006b).

3 Application The data set comes from sensory profiling of 14 cheese samples by a panel of 12 assessors on the basis of twelve attributes in two replicates. The final data matrix consists of 336 rows (12 assessors × 14 samples × 2 replicates) and 12 columns (attributes: intensity odour, acidic odour, sun odour, rancid odour, intensity flavour, acidic flavour, sweet flavour, salty flavour, bitter flavour, sun flavour, metallic flavour, rancid flavour). Two blocks of variables describe the latent variables odour and flavour. First the hierarchical PLS model proposed by Tenenhaus and Vinzi (2005) will be used to estimate a global model after averaging over the assessors and the replicates (see, Figure 2). Thus, collapsing the data structure into a two-way table (samples × attributes). Then fuzzy PLS path modeling will

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provide two sets of synthesized assessments: the overall latent scores for each product and the partial latent scores for the different blocks of attributes. The synthesis of scores into a global assessment permits to investigate differences between products. However, in such a way, we lose all the information on the individual differences between assessors. At this aim, as many path models as assessors will be considered and compared in terms of fuzzy path coefficients so as to detect eventual heterogeneity in the panel. Figure 2 shows the global path model. As can be seen, the latent variable global depends on the two latent variables odour and flavour. The F-PLSPM

Fig. 2. Global model

algorithm is used to estimate the fuzzy path coefficients (E˜ 1 and E˜ 2 ). Crisp path coefficients in Table 1 show that the global quality of the products mostly depends on the flavour rather than on the odour. Furthermore, fuzzy path coefficients describe a worse panel performance for the flavour emphasized by a more imprecise estimate (wider fuzzy interval). Therefore, the F-PLSPM algorithm enriches the results of the classical PLSPM crisp approach by providing information on the imprecision of path coefficients. At the same time, the coherence of results is granted as the crisp estimates are comprised within the fuzzy intervals. Table 1. Global Model Path Coefficients Latent Variable crisp path coefficients fuzzy path coefficients Odour Flavour

0.4215 0.6283

[0.3952; 0.4517] [0.6043; 0.7817]

The most interesting result coming from the proposed approach is in Figure 3, which compares the interval valued estimates on the different assessors. Figure 3 reports the fuzzy path coefficients for the 12 local models referred to each assessor. By looking within each plot (flavour and odour) separately, the assessor performance and the coherence between assessors can be evaluated: a) the wider

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Fig. 3. Local fuzzy path coefficients

the interval, the less consistent is the assessor; b) the closer the intervals between them, the more coherent are the assessors. In the example, for the odour, assessor 7 is the least consistent assessor while assessor 12, being positioned far away from the rest of the assessors, is the least coherent as compared to the panel. Finally, by comparing the two plots, differences in the way each assessor perceives flavour and odour may be detected: for instance, assessor 7 is the most imprecise for the odour while it is extremely consistent for the flavour; assessor 12 is similarly consistent for both flavour and odour but, in both cases, it is in clear disagreement with the panel (a much higher influence of the odour as opposed to a much lower influence of the flavour).

4 Conclusion The joint use of PLS component-based approach to structural equation modeling and fuzzy possibilistic regression has yielded promising results in the framework of sensory data analysis. Namely, while taking into account the multi-block feature of sensory data, the proposed Fuzzy-PLSPM leads to a fuzzy estimation of the path coefficients. Such an estimation provides information on the precision of the classical estimates and allows a thorough comparison of the sensory evaluations between assessors and within assessors for different products. Future directions of research aim to extend the fuzzy approach also to the measurement model by introducing an appropriate fuzzy possibilistic regression in the external estimation phase of the PLSPM algorithm. This further development has a twofold interest: allowing for fuzzy input data; yielding fuzzy estimates of the loadings, of the outer weights and, as a consequence, of the latent variable scores, thus embedding the measurement error that naturally affects sensory assessments.

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