On a Class of General Type-n Normal Fuzzy Sets Synthesized From Subject Matter Expert Inputs

The performance of fuzzy systems has been shown both theoretically and empirically to improve with increasing fuzziness, e.g., when fuzzy values are modeled using interval type 2 (IT2) or interval type 3 (IT3) fuzzy sets. This performance gain comes at a computational cost. We describe a new class of higher order fuzzy sets constructed from IT2 fuzzy sets that supports efficient type reduction. Our focus is on representations that use input from experts (SMEs) in the subject matter of interest. Differing interpretations of a concept by different subject matter experts (SMEs) leads to imprecision, which is captured in an IT2 fuzzy set defined on the relevant domain by approaches such as the Hao–Mendel approach. A second source of imprecision that has largely been ignored is SME confidence in their own judgments. This can also be captured as an IT2 fuzzy set, defined on the unit interval. We present a novel and intuitive construction to combine these representations into an IT3 fuzzy set. We show more generally how imprecision in hierarchical estimates of knowledge can be incorporated into a type-n fuzzy set representation.

On a Class of General Type-n Normal Fuzzy Sets Synthesized From Subject Matter Expert Inputs John T. Rickard , Senior Member, IEEE, Janet Aisbett , Member, IEEE, and J. Tyler Rickard Abstract-The performance of fuzzy systems has been shown both theoretically and empirically to improve with increasing fuzziness, e.g., when fuzzy values are modeled using interval type 2 (IT2) or interval type 3 (IT3) fuzzy sets.This performance gain comes at a computational cost.We describe a new class of higher order fuzzy sets constructed from IT2 fuzzy sets that supports efficient type reduction.Our focus is on representations that use input from experts (SMEs) in the subject matter of interest.Differing interpretations of a concept by different subject matter experts (SMEs) leads to imprecision, which is captured in an IT2 fuzzy set defined on the relevant domain by approaches such as the Hao-Mendel approach.A second source of imprecision that has largely been ignored is SME confidence in their own judgments.This can also be captured as an IT2 fuzzy set, defined on the unit interval.We present a novel and intuitive construction to combine these representations into an IT3 fuzzy set.We show more generally how imprecision in hierarchical estimates of knowledge can be incorporated into a type-n fuzzy set representation.
Index Terms-Computing with words, fuzzy membership functions, fuzzy sets, Hao-Mendel approach (HMA), type reduction, type-n membership functions.
The increased modeling complexity, however, increases the number of design decisions as well as the computational load [4].To simplify computations, secondary and higher order MFs are typically restricted to singleton, triangular, trapezoidal, and truncated Gaussian classes.The literature on type-3 fuzzy sets almost exclusively assumes Gaussian MFs-see, for instance, [1,review Table 3].The focus has been on innovative FLS architectures and optimization strategies, including strategies to determine the Gaussian parameters.
Here, we turn attention to the form of the higher order fuzzy sets themselves, with an emphasis on applications involving group or consensus decisions (e.g., [26], [27]).Rather than start with an analytically convenient representation, our start point is the variability in human representations of ordinal vocabularies, e.g., a set of terms ordered from Very bad to Very good.
Several methods have been proposed to encode interpretations of such words or terms into type-1 and into interval type-2 (IT2) fuzzy representations (e.g., [28], [29], [30]).An objective measure to select appropriate IT2 fuzzy set representations of words is presented in [31].These techniques fall within the category of "computing with words" (CWW).Applications include decision support through "perceptual computing" [32] or through fuzzy cognitive maps whose states and link activations are presented linguistically rather than numerically [33].
The Hao-Mendel approach (HMA) to encoding synthesizes normal trapezoidal IT2 fuzzy sets from intervals within the range [0,10] that respondents have provided to describe terms in a vocabulary.For example, interval estimates for the term Very bad would typically be relatively narrow, with a left endpoint at 0 and a right endpoint much less than 10, while interval estimates for Very good would be relatively narrow with left and right endpoints close to 10, inclusive.Following data collection, the HMA applies a two-part analysis.The first part involves statistical tests to remove bad data such as nonsensical intervals that violate the [0,10] bound and outlier intervals that have too little overlap with other data intervals.The second part operates on the intervals that have survived this data cleaning, to generate normal trapezoidal IT2 fuzzy sets for each word or term.Hao and Mendel [30] have published codebooks for 32-, 15-, 11-, and 6-word vocabularies describing attributes such as amount, degree, goodness, and importance.Additional codebooks for other vocabulary sizes are available [32].
In this article, we define a new class of higher order fuzzy sets which will support efficient type-reduction and defuzzification.Fuzzy values derived from the input of subject matter experts (SMEs) are represented by members of this class, using a novel construction.We assume IT2 representations have been generated either from the intervals that SMEs choose to describe the values, or from their choice of vocabulary terms to describe the values.The idea is to employ respondents' confidence in these choices to elaborate on the IT2 representations.Thus, to build a type-3 fuzzy set representation, we require that respondents provide estimates of domain value memberships in the concepts together with their confidence in these membership estimates.A type-4 fuzzy set representation requires respondents to also nominate their confidence in their confidence estimates, and so on.We assume that respondents are experts in the domain of interest and are committed to the task, so that the initial data cleaning steps of the HMA can be foregone.Our extension intuitively captures the imprecision associated with forming consensus representations from uncertain but expert estimates.
Section II formally defines the new class of fuzzy sets.It then extends the HMA process of encoding interval estimates as IT2 fuzzy sets to the encoding of n-dimensional data as type-(n + 1) fuzzy sets of this class.Section III considers the case when respondents describe fuzzy values using vocabulary terms that have IT2 fuzzy representations.It proposes simple procedures to aggregate words selected by different experts and to interpolate representations provided at discrete domain values.Section IV gives examples of interval type-3 (IT3) representations constructed from word inputs and of interpolation.Section V shows how the class of general type-n fuzzy sets we have identified allows very efficient type-reduction.Section VI reviews our contribution and looks at other work that has used similar constructions, before suggesting future research.

II. SYNTHESIS OF MULTIDIMENSIONAL SME INPUTS INTO GENERAL TYPE-N MFS
In this section, we generalize and extend the HMA to create type-n fuzzy set representations of fuzzy values defined on a real-valued domain, call it X.This variable might correspond to a physical quantity, such as predicted wave heights in a basin, or to a normalized variable such as a probability or a term conveying a non-numeric concept such as a "large horse."These representations belong to a particular class of type-n fuzzy sets, which we first describe.We shall refer to this class as RA fuzzy sets to distinguish them from more general type-n fuzzy sets.We then modify the HMA to develop representations of concepts that incorporate experts' confidence in their judgements.

A. RA Class of Type-n Fuzzy Sets
Recall that a type-n fuzzy set can be viewed as a fuzzy set on domain X × [0, 1] n−1 [34].Thus, a Type-3 MF on X can be expressed in the form f (x, y, z) : and is the "footprint of uncertainty" (FOU) of the primary membership.The secondary membership at any domain point x is the IT2 fuzzy set with MF that takes (y For n ≥ 3, a type-n fuzzy set on X has MF f (x, y 1 , . . ., y n−1 ) : The definitions of FOU and of secondary membership generalize defined by: Definition 1: FOU The FOU of a type-n fuzzy set is (x, y 1 ) : max (y 2 ,...,y n−1 )∈[0,1] n−1 f (x, y 1 , . . ., y n−1 ) > 0 .
We use these definitions to define a new class of fuzzy sets.Definition 3: RA class of fuzzy sets For n ≥ 2, a collection C of fuzzy sets of arbitrary order is an RA class if the following conditions are satisfied: 1) the FOU of each member of C is an IT2 fuzzy set in C; 2) each of the k-ary MFs of a member of C is in C.
Clearly Definition 3 requires that any type-2 fuzzy set in an RA class is an IT2 fuzzy set.For example, C might consist of all normal trapezoidal IT2 fuzzy sets together with type-3 fuzzy sets whose FOUs are normal trapezoidal IT2 fuzzy sets and whose secondary MFs are also normal trapezoidal IT2 fuzzy sets.

B. Using Experts' Sentiment to Moderate the Representation of Uncertain Concepts
Suppose one or more SMEs have an interest in modeling an area of their expertise.We want these SMEs to nominate intervals that represent their interpretation of fuzzy values in some real-valued domain.Additionally, we want them to nominate intervals that represent the imprecision in their interpretation, which we take to be a fuzzy variable reflecting respondents' sentiment toward their estimates of the domain variable.
Instead of soliciting a set of one-dimensional intervals to describe the fuzzy variable, we therefore request the SMEs to prescribe a set of rectangles, or even hyper-rectangles.The first dimension of the rectangle (or hyper-rectangle) represents values in the primary domain X, and so can be on an arbitrary scale.The second dimension of the rectangle must lie within the interval [0,1], since it describes imprecision in the domain estimate.Any higher order dimensions also lie within [0,1], since they describe imprecision in the estimate in the dimension one order below.
For example, an SME might select a rectangle whose width extends over [20,50] and height extends over [0.3, 0.5] in the 2-D space, where the x-dimension represents forecast storm wave height in feet and the y-dimension captures the sentiment in their prediction.Multiple SMEs would provide such rectangles describing their wave height forecasts and their sentiment in their predictions.Fig. 1 illustrates.
We then employ a variation on HMA to create two trapezoidal IT2 MFs, respectively, synthesized from the extents of the rectangles in the two dimensions.We relax the stipulated HMA scale [0,10] to allow arbitrary scales [ , r] for the domain variable, where −∞ < < r < ∞, and to use the interval [0, 1] for the sentiment variable.Fig. 2 shows FOUs derived from Fig. 1 inputs.
Suppose the trapezoidal IT2 fuzzy set formed from the rectangle widths has MF f 0 : X × [0, 1] → [0, 1].For each x ࢠ X, the support of f 0 (x, y) in [0,1] is an interval, call it I 0 (x).The trapezoidal IT2 fuzzy set formed from the rectangle heights representing sentiment in the predictions is defined on [0,1].Suppose this latter set has MF f 1 (y 1 , y 2 ) ∈ [0, 1], where y 1 ∈ [0, 1] is its primary variable and y 2 ∈ [0, 1] is its secondary variable.We want shifted and scaled versions of this MF to be the secondary membership functions of an IT3 fuzzy set on X.To achieve this, use the transformation defined by the following: where I 0 (x) and Ī0 (x) are the lower and upper endpoints of the primary membership interval I 0 (x).This transformation has support in X × I 0 (x) and range [0,1].We now model the consensus of the SME inputs for the primary variable and sentiment as the IT3 fuzzy set with MF function g 3 (x, y 1 , y 2 ), where Clearly, for any pair x, y 1 the support of g 3 (x, y 1 , y 2 ) lies within the FOU of the trapezoidal IT2 function on X.Note that the secondary MFs of this type-3 fuzzy set preserve the shape of f 1 (y 1 , y 2 ), but their domain and support are now inherently coupled to the domain intervals I 0 (x) determined by the primary interval inputs of the SMEs.Thus, if the consensus of the SMEs is that the sentiment is low, so that f 1 (y 1 , y 2 ) has support on lower values of y 1 , the support of g 3 (x, y 1 , y 2 ) will lie in relatively low values of y 1 within I 0 (x) for each x ࢠ X.Conversely, high sentiment would make the support lie in the upper range of these intervals.An aggregation of differing SME sentiments, such as described in Section III-B, would broaden the resulting support of g 3 (x, y 1 , y 2 ), making it a larger portion of the I 0 (x) interval.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Formally, suppose [J, J] is the restriction of the FOU of consensus SME sentiment to its (primary) y 1 -axis; for instance, [0.27, 0.38] in the example Fig. 2(b).From (2), g 3 (x, y 1 , y 2 ) = 0 unless s 0 (x, y 1 ) ∈ [J, J] and y 1 ∈ I 0 (x).Thus, from (1), the FOU of the type-3 fuzzy set is as follows: That is, the upper and lower boundaries of the FOU in the type-3 fuzzy set are weighted sums of the upper and lower MFs of the IT2 fuzzy set on the domain of interest built using the HMA.Moreover, the weights are derived from the IT2 fuzzy set formed from sentiment.The FOU is trapezoidal because the weighted sums of two lines that do not intersect is a line lying between them.Fig. 3 illustrates the type-3 fuzzy set generated by this process from the IT2 fuzzy sets depicted in Fig. 2.
The green shaded area is the FOU of the primary variable as shown in Fig. 2(a).At the value x = 42.5, the primary membership interval is I 0 (x) = [0.27,0.65].The vertical blue-shaded FOU depicts the SME sentiment depicted in Fig. 2(b) as mapped onto the primary membership interval I 0 (x), with its support indicated by the yellow line as a subinterval of I 0 (x).The red line on the secondary FOU, corresponding to the value y 1 = 0.545, depicts the interval I 1 (y 1 ), over which the level-3 membership is unity.
According to Definition 3, the type-3 fuzzy sets generated by this process belong to an RA class of trapezoidal fuzzy sets.

C Generalizations
The above process can be extended inductively to higher dimensions when SME input is in the form of hyper-rectangles.For example, suppose SMEs provide 3-D input, and the HMA approach applied to the third dimension yields the trapezoidal IT2 fuzzy set with MF ) is the support of f 1 (y 1 , y 2 ) at y 1 and if the transformation s 1 is defined on [0,1] × [0,1] analogously to (1), then the above procedure yields a trapezoidal interval type-3 fuzzy set with MF Again, this will have a trapezoidal FOU.Attach this type-3 fuzzy set representing consensus SME sentiment to the IT2 fuzzy set representing the concept as before, by applying the function (1).This gives the type-4 fuzzy set defined by the following: 0, else . ( The support of the secondary MF is an interval [J, J] = {y 1 : g 4 (x, y 1 , y, y ) = 1 for some y, y ∈ [0, 1]} that is independent of x.The FOU of the MF in ( 5) is again given by (3), and again is trapezoidal.
Generally, if SMEs provide n-dimensional hyper-rectangles, we synthesize n trapezoidal interval type-2 fuzzy sets with MFs . . .
Use the procedure presented in Section II-B to construct a trapezoidal interval type-3 fuzzy set from the IT2 fuzzy sets with respective MFs f n−3 , f n−2 .By induction, and using transformations analogous to (1), we form the interval type-n fuzzy set whose MF has support x ∈ X, y 1 ∈ I 0 (x), y j+1 ∈ I j (y j ), j = 1, . . ., n − 3, where it is defined by At each stage, the FOU will be trapezoidal, and so the constructed fuzzy sets form an RA class.
Requiring SMEs to use a uniform interval width to describe their sentiment or confidence over the entire range of their forecast wave heights, say, is unnecessarily restrictive.For example, in the 2-D case, an SME may consider the more extreme values of their primary interval to have lower (or higher) membership values than those in the middle range of the primary interval.We therefore generalize the above approach to allow the SMEs to specify any bounded region in n-dimensional space that intersects every axis of the (n-1)-dimensional hyperplane perpendicular to the primary variable axis in a convex set.This condition ensures that these intersections generate intervals within [0,1] for each value of the fuzzy variable in each of the remaining dimensions.
Fig. 4 illustrates a region satisfying this condition in the 2-D case, where the SME's membership intervals have variable widths and are slightly lower for the extreme values of wave height, relative to the membership intervals for central values.
We discretize the range of the primary variable and, where required, discretize the secondary, tertiary, and higher k-ary membership intervals.We then iteratively calculate trapezoidal IT2 fuzzy sets at each of the discrete values that capture consensus SME views of the concept or of their sentiments.Of course, these fuzzy set MFs may now have different shaped trapezoids at each of the discrete points, i.e., the various consensus IT2 MFs the discretized lower order variables x, y 1 , . . ., y k−1 , as are the primary membership intervals I k (y k ).Denote these respectively by f k (x, y 1 , . . ., y k+1 ) and I k (x, y 1 , . . ., y k ).Then, we again perform translation/scaling operations to map these IT2 MFs onto their appropriate domains, noting that the transformations are also dependent on lower order variables.
The FOUs of these MFs are now also functions of the lower order variables.For instance, the primary FOU is as follows: Let [J(x, y 1 ), J(x, y 1 )] be the restriction of the FOU of f 2 (x, y 1 , y 2 , y 3 ) to the primary axis, i.e., [J(x, y 1 ), J(x, y 1 )] = {y 2 : f 2 (x, y 1 , y 2 , y) > 0 for some y}.Then, the FOU of the secondary membership at x is as follows: .
Although each of these FOUs is an IT2 fuzzy set lying in the FOU of the consensus IT2s, they may no longer be trapezoidal.They do, however, belong to the general RA class in which C is generated by arbitrary IT2 fuzzy sets.Thus, the generalization from hyper-rectangles is more realistic but also more labor-intensive.It requires each SME to specify secondary (and possibly higher order membership intervals) for each primary value x, or for subranges of that domain.Section III addresses this problem by providing an intuitive approach to characterizing the imprecise MF specifications in each dimension.

III. COMPUTING WITH WORDS APPROACH
Section II presented a general approach to constructing type-n fuzzy MFs from interval data.In this section, we present an approach that makes use of words represented by IT2 trapezoidal MFs, as described in [30], to construct interval type-n MFs.This approach thus lies within the context of CWW and provides an intuitive and user-friendly method for synthesizing input into type-n MFs.
Our approach employs translated/scaled MFs representing terms that describe SME's sentiment regarding an interval they nominated.These terms can be applied to any order of fuzzy memberships from type-2 to type-n, where the sentiment described for the nth order dictates the general membership values for the (n-1)th order.Thus, for example, Very low, High, and Neutral represented by a type-2 MF might describe sentiment in the membership ranges of the primary fuzzy MF at three different primary values x, whereas words specified for a general type-3 MF additionally describe sentiment regarding the secondary membership ranges (which are subsets of [0,1]) at these primary values of x.

A. Review of Specification and HMA Construction of IT2 MFs Describing Words
An IT2 trapezoidal MF is described entirely by the two row vectors These parameters are calculated from the endpoints of the intervals provided by respondents as descriptions of a word, following the HMA data cleaning referred to in the Introduction.The overlap parameters o , o r are simply the largest of the left endpoints and the smallest of the right endpoints of the intervals that remain after cleaning.The parameters a , a r are calculated from the mean and standard deviation of the midpoints of the intervals after truncating at o ; similarly, for b , b r after truncating the intervals to only have values larger than o r (a modification to this calculation using the area of the FOU rather than the standard deviations of the endpoints is presented in [1]).
Equations ( 5) and ( 6) as given in [30] admit cases in which a l > a r and/or b l > b r and thus should be amended as follows: These corrections ensure proper x-intercept orderings for the UMF and LMF of the trapezoidal tails of the IT2 MFs [37].
We were graciously provided with datasets on a scale of [0,10] corresponding to a seven-word vocabulary [37] which Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE II SEVEN-WORD CODEBOOK FOR SENTIMENT
we adopted as corresponding to degrees of sentiment.We followed the HMA procedure on these interval datasets to compute trapezoidal MFs.Table II lists the words in the vocabulary and their representations, using the notation at the start of this section and scaled to [0,1].
We shall use this codebook to illustrate our approach in the examples of Section IV.However, note that custom codebooks can be constructed using the HMA from sets of intervals solicited for sentiment words in any given attribute and for any desired vocabulary size.

B. Dealing With Variable SME Word Evaluations
Suppose multiple SMEs provide different words to describe their sentiment.How should these be aggregated into a single MF describing the consensus of the SME sentiments?This aggregation problem can be addressed in numerous ways.But we suggest a very simple method whereby the respective trapezoidal MF parameters of each SME's word chosen from Table II are averaged (perhaps with importance weights to discriminate among the SME's relative expertise) to produce an average trapezoidal MF representing the consensus sentiment.
Formally, suppose m SMEs provide assessments of word sentiments using words that have IT2 fuzzy set representations (UMF k , LMF k ), k = 1 . . .m, where each pair represents a rowvector entry pair from Table II.Suppose further that the SME's relative expertise ranked on an arbitrary scale is given by an m-vector W having positive entries W k .
Then, we can form an aggregate sentiment trapezoidal MF with parameters UMF and LMF given by the matrix-vector multiplications (Note that the constructions of UMF and LM F in ( 11) and ( 12) produce normal trapezoidal IT2 MFs, thus preserving the structure of the MFs of codebook words in Table II, albeit with different parameters).
Obviously, this approach can be applied to vocabularies of arbitrary size.Furthermore, individual SMEs can apply this approach to their own sentiment evaluations if they have some ambiguity about which word best describes their sentiment.In the latter case, the kth SME would select a group of (typically adjacent) words spanning their ambiguity and prescribe a weight vector W for aggregating their sentiment over these words.They would then use (11) and (12) with this vector to calculate their aggregate trapezoidal parameter sets UMF k and LMF k that would then be employed as their corresponding column entries of the matrices in (11) and (12) for aggregation over the SMEs.

C. Interpolation of SME Sentiment Across the Primary Variable Range
An analogous approach to that of the previous section can be applied to the primary variable by using interpolation of the trapezoidal coordinates of sentiment words between adjacent "pegged" word MFs to produce a smoothly continuous IT2 secondary MF.This interpolation is performed after the UMF and LMF aggregations of ( 11) and ( 12) have determined the consensus sentiment MFs at a set of discrete primary variable values.Here, we present the mathematical details for this operation.Section IV provides an example.
Let UMF i , LMF i , i = 1, . . ., p denote the trapezoidal parameters of the aggregate MFs of the IT2 fuzzy sets synthesized using ( 11) and ( 12) from individual word MFs selected from Table II by the SMEs at discrete domain values x i , i = 1, . . ., p.As normal trapezoidal MFs, each of these functions is described, as in (8), by four parameters contained in [0,1] These values detail the respective intercept points (a (i) , 0), (a r , 0), (b (i) , 0), and (b r , 0) of the left and right legs of the UMF and LMF with the y = 0 axis and the left and right endpoints (o (i) , 1) and (o r ] of unity membership for both UMF and LMF (Note that for left-shoulder MFs, a (i) = b (i) = o (i) = 0, whereas for right-shoulder MFs, ).To ensure that the UMF support interval [s , s r ] of the primary variable is enforced, we assign the first and last of the domain x i values to be x 1 = s and x p = s r , with corresponding trapezoidal parameters: These values also correspond to all x < x 1 and x > x p .We also require that the x-coordinates of the endpoints [o (i) , o r ] of the interval of unity membership be included in the set of x i values.(In the case of a triangular MF, only the single value at the peak is included.)The corresponding trapezoidal parameters are as follows: where j, j + 1 are the adjacent indices corresponding to the [o (i) , o r ] interval endpoints.The remaining x i can be arbitrarily specified, but are sorted such that x i < x i+1 for all i.
To obtain UMF and LMF trapezoidal MFs for intermediate values of x = x i , we interpolate between the respective Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
coordinates of the UMFs and LMFs associated with adjacent pegged values x i , x i+1 such that x i < x < x i+1 .The first step is to calculate the primary MF intervals [y i , y i ] and [y i+1 , y i+1 ] corresponding to x i , x i+1 .Next, map the trapezoidal coordinates of UMF i , LMF i and UMF i+1 , LMF i+1 in ( 13) and ( 14) on [0, 1] onto their respective primary MF intervals.This mapping is accomplished by the function where y and y are the lower and upper interval bounds.Let UMF i , LMF i and UMF i+1 , LMF i+1 denote the mapped trapezoidal coordinate row vectors determined by the function in (17), where Then, the interpolated trapezoidal coordinate values UMF(x) and LMF(x) for x i < x < x i+1 are calculated from the functions Here, x i < x < x i+1 .and j ∈ 1, . . ., 4 indexes the corresponding columns of the row vectors in ( 18)- (21).
Equations ( 18)-( 23) define a 3-D region that, at a given value x, specifies an IT2 MF along the y-axis that is a secondary MF with unity tertiary membership values.An example of this construction is provided in Section IV.

IV. EXAMPLES
Consider a fuzzy value with trapezoidal UMF and LMF defined by the following parameters: The MF of the fuzzy value is shown in Fig. 5. Consider the domain value x = 1 250 000 in this figure, which has a primary membership interval range of [0, 0.487].Fig. 6(a)-(d) show the secondary MFs corresponding to the terms very low, somewhat low, somewhat high, and very high sentiment, disposed over the interval [0, 0.487].
Note that, in each case, the sentiment MFs are disposed over the interval [0, 0.487] via the scaling operation described by ( 17)- (21).This scaling becomes more compressed if we reduce the primary MF value to x = 1 150 000, as illustrated in Fig. 7. And, of course, the sentiment MF collapses to unity at all values for which the primary MF value is unity.As further examples, suppose that four SMEs provide differing assessments of sentiment for the value x = 1 250 000 in the previous example.In particular, suppose that the SME's assessments are very low, low, somewhat low, and neutral, respectively, as described in Table II.Initially, assume that all SMEs have equal credibility, such that the weight vector W used in ( 11) and ( 12) can be taken as W = [1, 1, 1, 1] T .From ( 11) and ( 12) we then obtain the following: This aggregate MF is shown in Fig. 8.Note that this MF is visually similar to that of somewhat low in Fig. 6(b), but it has narrower tails and a wider interval of unity values.Now assume that, for the same set of sentiments, the first SME (whose assessment is very low) has ten times the credibility of the others, so that the aggregation weight vector is given by W = [10, 1, 1, 1] T .In this case, we obtain from (11) The resulting aggregate MF described is shown in Fig. 9.As expected, the aggregate MF is now disposed over a lower range of values, with a considerably wider interval of unity values.
As a final example, we illustrate the interpolation of SME sentiment trapezoidal MF coordinates specified at a set of values x i , i = 1, . . ., p within the primary variable support interval [x 1 , x p ], as described in the previous section.For simplicity, but without loss of generality, we assume these MFs correspond to words drawn from Table II together with the words Zero and Unity represented, respectively, by (15) and (16).The interpolation can of course be performed between arbitrary adjacent pairs of specified MF parameters.
To this end, suppose that for the primary MF defined by ( 24) and (25) and shown in Fig. 5, SMEs have provided their confidence in their predictions at the values x i , i = 2, . . ., p − 1 shown in the first column of the following array.Suppose the consensus sentiment at each value is the vocabulary word shown Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Verylow   where the entries in the second column refer to the words from Table II.This array is augmented with a first row at [0.8465 • 10 6 Zero ] and a last row at [1.392 • 10 6 Zero ], corresponding to the endpoints of the UMF support interval [x 1 , x p ], where Zero is the zero word MF with trapezoidal parameters given by (15).Performing the interpolation operations described by ( 17)-( 23), we obtain two surface manifolds in (x, y 1 , y 2 ) space.These correspond to the normal trapezoidal UMF and LMF of the secondary MF, as shown in color contour plots in Fig. 10(a) and (b).At any vertical slice on the x-axis of the volume described by these figures, the corresponding (y 1 , y 2 ) plane defines an IT2 secondary MF with an interval-valued tertiary membership.
From these examples, it should be obvious that we can easily construct secondary and higher order fuzzy MFs of quite variable shapes belonging to the RA class using the intuition provided by our approach.While the above examples represent IT3 MFs, it is straightforward to see that the same approach can be replicated to higher order RA class MFs of type-n, where n is an arbitrary integer of at least 2.

V. TYPE REDUCTION
Type reduction of fuzzy MFs of order n ≥ 2 is generally a critical operation in any fuzzy systems application, since the eventual output of such a system often must be reduced to a scalar variable such as a decision score, a control input, a time series prediction, a fair market value estimate, etc. Type reduction is an iterative process, whereby a fuzzy MF of order n is successively reduced to fuzzy MFs of order nk, k = 1, . . ., n, so that the final MF is an order 0 MF, i.e., a scalar value.
A compelling advantage of the RA construction approach in this article is that, by employing IT2 MFs to describe fuzziness at each order, we can employ highly computationally efficient algorithms to perform type reduction.For example, the enhanced Karnik-Mendel (EKM) algorithm [35] achieves superexponential convergence.Other approximate type-reduction algorithms are often even more computationally efficient [36].
The procedure for type reducing an interval type-n MF of the class constructed in this article is as follows.
1) Beginning with the n -ary MF that describes an IT2 fuzzy variable, calculate the centroid of this MF using the EKM algorithm.This results in intervals of unity membership at multiple values of fuzzy variables of order n -1, thus corresponding to an IT2 MF at order n -1 (for example, an interval type-3 MF as illustrated in Fig. 10 would be reduced to intervals of unity membership at multiple x values for the primary variable, representing a type-reduced IT2 MF for the primary variable).2) Apply the interpolation algorithm of Section III between the resulting intervals to produce additional intervals of finer granularity at order n -1. 3) Iteratively repeat steps 1 and 2 to calculate the centroids of the IT2 MFs of order n − 2, . . ., 0 using the EKM or other efficient algorithms at each stage.This results in an IT2 MF for the primary variable.For order 1, we obtain an interval type-1 MF, whose midpoint is the desired order 0 (scalar) output.In the example shown by the UMF and LMF manifolds in Fig. 10, we discretized the UMF support interval into 1000 increments using the interpolation approach of Section III-C.We then applied the EKM algorithm to the trapezoidal IT2 MF corresponding to each x increment value.This produced the type-reduced IT2 MF whose UMF and LMF are shown in Fig. 11.The computation required less than 1 s to run on a laptop computer using the Mathcad programming environment.
We note that the narrowness of the intervals between the UMF and LMF of this IT2 MF is due to the assumption that all SMEs were consistent in their assignments of sentiment words at the x values shown in (30).Thus, the support intervals of the consensus sentiments were restricted to the relatively smaller supports of the individual words in (30), rather than the larger support intervals that would have resulted from the aggregation of differing sentiments, as described in Section III-B.By a further application of the EKM algorithm to this IT2 MF, we typereduce it to an interval type-1 MF in the primary variable.This has unity MF over the interval [1.070574 • 10 6 , 1.073364 • 10 6 ].The interval midpoint, at 1.071969 • 10 6 , is the ultimate order 0 type-reduced value.
This ability to perform computationally efficient typereduction of higher order fuzzy MFs will hopefully encourage employment of type-n MFs in fuzzy systems applications.

VI. CONCLUSION
This work makes two key contributions.First, it introduces a new class of type-n fuzzy sets that allow efficient type-reduction and defuzzification while supporting complex modeling of imprecision.Members of this class must have FOUs that are IT2 fuzzy sets in the class, and all their secondary, tertiary, and any other k-ary MFs must belong to the class.Second, it argues that the sentiment that respondents hold about their interpretations of a fuzzy value should be incorporated into MFs derived from those interpretations, and it shows how to do this.
A fuzzy value is initially represented as an IT2 fuzzy set that captures differences in understanding of what the concept represents, including experts' self-assessments alters this representation.Our construction is based on trapezoidal IT2 fuzzy sets, which have been extensively applied for their simple specification and ability to represent both symmetric and unsymmetric imprecision.
The IT3 fuzzy sets constructed in works such as [5] and [17] resemble the sets we have constructed, although they use truncated Gaussian rather than trapezoidal MFs.Their motivation and construction differ from ours, however, as the FOUs of the truncated Gaussian secondary membership functions are deployed over the entire primary FOU.The additional fuzzy parameters are introduced to handle noisy system inputs from multiple sources such as imperfect sensors, quantization errors, environmental factors, and so on.The effects of any one noise source are not independently modeled.
In contrast, our construction of IT3 fuzzy sets requires modeling a source of imprecision (the experts' self-assessments of their judgements) independently of the imprecision due to differing concept interpretations.The input from this independent source then restricts the primary FOU in the concept representation.
The RA class of trapezoidal type-n fuzzy sets that we have identified appears useful, given the intuition of modeling imprecision consistently across orders of membership.Any family of fuzzy numbers F (e.g., specified with truncated Gaussian or pentagonal MFs) could be used to define an RA class of type-n fuzzy sets on domain X by requiring that all the FOUs have UMFs and LMFs that are in F. However, note that our procedure for aggregating two or more IT2 fuzzy sets would not result in an IT3 fuzzy set in the class if F were, for instance, fuzzy numbers having truncated Gaussian MFs.This is because the weighted sum of normalized Gaussian MFs is not a Gaussian MF, even when they have the same peak value.
The performance of fuzzy predictive systems based on IT3 fuzzy sets, constructed from expert input as we have described, needs to be compared with that of systems based on IT2 fuzzy sets constructed using conventional HMA.Comparisons of the predictive performance of systems based on IT3 fuzzy sets derived either from expert input or from training data would also be worthwhile.
The extension to type-4 and higher order fuzzy sets is of theoretical interest at this stage as we have not yet investigated practical benefits of the richer modeling.However, the computational feasibility of performing type reductions on the RA class of type-n MFs opens many possibilities for such modeling.

Fig. 3 .
Fig. 3. Illustration of the construction of an IT3 MF derived from the IT2 MFs in Fig. 2.

Fig. 4 .
Fig. 4. Illustration of SME input using a region whose intersection with every vertical line is convex.
) for its upper MF (UMF) and lower MF (LMF) trapezoidal vertices, respectively.Here, [a l , b r ] is the support of the UMF and [a r , b l ] is the support of the LMF, and [o l , o r ] is the interval with unity primary and secondary memberships.In the case of a Left-shoulder type FOU, a l = a r = o l = 0, and for a Rightshoulder FOU, under the HMA protocol, o r = b l = b r = 10.

Fig. 5 .
Fig.5.Fuzzy MF of a primary variable, e.g., production from a mine for a given year, in kilograms of copper.

Fig. 6 . 6
Fig. 6.General type-2 MFs of the fuzzy variable shown in Fig. 5 at the value x = 1 250 000, with corresponding sentiments of very low, somewhat low, somewhat high, and very high.(a) Very low.(b) Somewhat low.(c) Somewhat high.(d) Very high.

Fig. 7 .
Fig. 7. General type-2 MFs of the fuzzy variable shown in Fig. 5 at the primary variable value x = 1 150 000, with sentiments of very low, somewhat low, somewhat high, and very high.

Fig. 8 .
Fig. 8. Aggregate general type-2 MFs of the fuzzy variable shown in Fig. 5 at the value x = 1 250 000, with differing SME sentiments of very low, low, somewhat low, and neutral, and with equal credibility of all SMEs.

Fig. 9 .
Fig. 9. Aggregate general type-2 MFs of the fuzzy variable shown in Fig. 5 at the value x = 1 250 000, with differing SME sentiments of very low, low, somewhat low, and neutral, respectively, but with SME 1 having ten times the credibility of the other SMEs.

TABLE I APPLICATIONS
OF TYPE-3 FUZZY SYSTEMS