Editing Fuzzy set (section)

==Definition==

A fuzzy set is a pair <math>(U, m)</math> where <math>U</math> is a set (often required to be [[empty set|non-empty]]) and <math>m\colon U \rightarrow [0,1]</math> a membership function. 
The reference set <math>U</math> (sometimes denoted by <math>\Omega</math> or <math>X</math>) is called '''universe of discourse''', and for each <math>x\in U,</math> the value <math>m(x)</math> is called the '''grade''' of membership of <math>x</math> in <math>(U,m)</math>. 
The function <math>m = \mu_A</math> is called the '''membership function''' of the fuzzy set <math>A = (U, m)</math>.

For a finite set <math>U=\{x_1,\dots,x_n\},</math> the fuzzy set <math>(U, m)</math> is often denoted by <math>\{m(x_1)/x_1,\dots,m(x_n)/x_n\}.</math>

Let <math>x \in U</math>. Then <math>x</math> is called 
* '''not included''' in the fuzzy set <math>(U,m)</math> if {{nowrap|<math>m(x) = 0</math>}} (no member), 
* '''fully included''' if {{nowrap|<math>m(x) = 1</math>}} (full member), 
* '''partially included''' if {{nowrap|<math>0 < m(x) < 1</math> (fuzzy member).<ref>{{Cite web|url=http://www.aaai.org/aitopics/pmwiki/pmwiki.php/AITopics/FuzzyLogic|archive-url=https://web.archive.org/web/20080805071058/http://www.aaai.org/aitopics/pmwiki/pmwiki.php/AITopics/FuzzyLogic|url-status=dead|title=AAAI|archive-date=August 5, 2008}}</ref>}}
The (crisp) set of all fuzzy sets on a universe <math>U</math> is denoted with <math>SF(U)</math> (or sometimes just <math>F(U)</math>).{{cn|date=December 2024}}

===Crisp sets related to a fuzzy set===
For any fuzzy set <math>A = (U,m)</math> and <math>\alpha \in [0,1]</math> the following crisp sets are defined:
* <math>A^{\ge\alpha} = A_\alpha = \{x \in U \mid m(x)\ge\alpha\}</math> is called its '''α-cut''' (aka '''α-level set''') 
* <math>A^{>\alpha} = A'_\alpha = \{x \in U \mid m(x)>\alpha\}</math> is called its '''strong α-cut''' (aka '''strong α-level set''') 
* <math>S(A) = \operatorname{Supp}(A) = A^{>0} = \{x \in U \mid m(x)>0\}</math> is called its '''support''' 
* <math>C(A) = \operatorname{Core}(A) = A^{=1} = \{x \in U \mid m(x)=1\}</math> is called its '''core''' (or sometimes '''kernel''' <math>\operatorname{Kern}(A)</math>).

Note that some authors understand "kernel" in a different way; see below.

===Other definitions===
* A fuzzy set <math>A = (U,m)</math> is '''empty''' (<math>A = \varnothing</math>) [[iff]] (if and only if)
::[[Universal quantification#notation|<math>\forall</math>]]<math> x \in U: \mu_A(x) = m(x) = 0</math>

* Two fuzzy sets <math>A</math> and <math>B</math> are '''equal''' (<math>A = B</math>) iff
::<math>\forall x \in U: \mu_A(x) = \mu_B(x)</math>

* A fuzzy set <math>A</math> is '''included''' in a fuzzy set <math>B</math> (<math>A \subseteq B</math>) iff
::<math>\forall x \in U: \mu_A(x) \le \mu_B(x)</math>

* For any fuzzy set <math>A</math>, any element <math>x \in U</math> that satisfies
::<math>\mu_A(x) = 0.5</math> 
:is called a '''crossover point'''.

* Given a fuzzy set <math>A</math>, any <math>\alpha \in [0,1]</math>, for which <math>A^{=\alpha} = \{x \in U \mid \mu_A(x) = \alpha\}</math> is not empty, is called a '''level''' of A. 
* The '''level set''' of A is the set of all levels <math>\alpha\in[0,1]</math> representing distinct cuts. It is the [[image (mathematics)|image]] of <math>\mu_A</math>:
::<math>\Lambda_A = \{\alpha \in [0,1] : A^{=\alpha} \ne \varnothing\} = \{\alpha \in [0, 1] : {}</math>[[Existential quantification|<math>\exist</math>]]<math>x \in U(\mu_A(x) = \alpha)\} = \mu_A(U)</math>

* For a fuzzy set <math>A</math>, its '''height''' is given by
::<math>\operatorname{Hgt}(A) = \sup \{\mu_A(x) \mid x \in U\} = \sup(\mu_A(U))</math>
:where <math>\sup</math> denotes the [[Infimum and supremum|supremum]], which exists because <math>\mu_A(U)</math> is non-empty and bounded above by 1. If ''U'' is finite, we can simply replace the supremum by the maximum.

* A fuzzy set <math>A</math> is said to be '''normalized''' iff
::<math>\operatorname{Hgt}(A) = 1</math>
:In the finite case, where the supremum is a maximum, this means that at least one element of the fuzzy set has full membership. A non-empty fuzzy set <math>A</math> may be normalized with result <math>\tilde{A}</math> by dividing the membership function of the fuzzy set by its height:
::<math>\forall x \in U: \mu_{\tilde{A}}(x) = \mu_A(x)/\operatorname{Hgt}(A)</math>
:Besides similarities this differs from the usual [[normalizing constant|normalization]] in that the normalizing constant is not a sum.

* For fuzzy sets <math>A</math> of real numbers <math>(U \subseteq \mathbb{R})</math> with [[bounded set|bounded]] support, the '''width''' is defined as
::<math>\operatorname{Width}(A) = \sup(\operatorname{Supp}(A)) - \inf(\operatorname{Supp}(A))</math>
:In the case when <math>\operatorname{Supp}(A)</math> is a finite set, or more generally a [[closed set]], the width is just
::<math>\operatorname{Width}(A) = \max(\operatorname{Supp}(A)) - \min(\operatorname{Supp}(A))</math>
:In the ''n''-dimensional case <math>(U \subseteq \mathbb{R}^n)</math> the above can be replaced by the ''n''-dimensional volume of <math>\operatorname{Supp}(A)</math>. 
:In general, this can be defined given any [[Measure (mathematics)|measure]] on ''U'', for instance by integration (e.g. [[Lebesgue integration]]) of <math>\operatorname{Supp}(A)</math>.

* A real fuzzy set <math>A (U \subseteq \mathbb{R})</math> is said to be '''convex''' (in the fuzzy sense, not to be confused with a crisp [[convex set]]), iff
::<math>\forall x,y \in U, \forall\lambda\in[0,1]: \mu_A(\lambda{x} + (1-\lambda)y) \ge \min(\mu_A(x),\mu_A(y))</math>.
: Without loss of generality, we may take ''x'' ≤ ''y'', which gives the equivalent formulation
::<math>\forall z \in [x,y]: \mu_A(z) \ge \min(\mu_A(x),\mu_A(y))</math>.
: This definition can be extended to one for a general [[topological space]] ''U'': we say the fuzzy set <math>A</math> is '''convex''' when, for any subset ''Z'' of ''U'', the condition 
::<math>\forall z \in Z: \mu_A(z) \ge \inf(\mu_A(\partial Z))</math>
: holds, where <math>\partial Z</math> denotes the [[Boundary (topology)|boundary]] of ''Z'' and <math>f(X) = \{f(x) \mid x \in X\}</math> denotes the [[image (mathematics)|image]] of a set ''X'' (here <math>\partial Z</math>) under a function ''f'' (here <math>\mu_A</math>).

===Fuzzy set operations===
{{main|Fuzzy set operations}}
Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity.
* For a given fuzzy set <math>A</math>, its '''complement''' <math>\neg{A}</math> (sometimes denoted as <math>A^c</math> or <math>cA</math>) is defined by the following membership function:
::<math>\forall x \in U: \mu_{\neg{A}}(x) = 1 - \mu_A(x)</math>.
* Let t be a [[t-norm]], and s the corresponding s-norm (aka t-conorm). Given a pair of fuzzy sets <math>A, B</math>, their '''intersection''' <math>A\cap{B}</math> is defined by:
::<math>\forall x \in U: \mu_{A\cap{B}}(x) = t(\mu_A(x),\mu_B(x))</math>,
:and their '''union''' <math>A\cup{B}</math> is defined by:
::<math>\forall x \in U: \mu_{A\cup{B}}(x) = s(\mu_A(x),\mu_B(x))</math>.

By the definition of the t-norm, we see that the union and intersection are [[commutative]], [[monotonic]], [[associative]], and have both a [[absorbing element|null]] and an [[identity element]]. For the intersection, these are ∅ and ''U'', respectively, while for the union, these are reversed. However, the union of a fuzzy set and its complement may not result in the full universe ''U'', and the intersection of them may not give the empty set ∅. Since the intersection and union are associative, it is natural to define the intersection and union of a finite [[Indexed family|family]] of fuzzy sets recursively. It is noteworthy that the generally accepted standard operators for the union and intersection of fuzzy sets are the max and min operators:
* <math>\forall x \in U: \mu_{A\cup{B}}(x) = \max(\mu_A(x),\mu_B(x))</math> and <math>\mu_{A\cap{B}}(x) = \min(\mu_A(x),\mu_B(x))</math>.<ref name="BGFuzzy">{{cite journal|last1=Bellman|first1=Richard|last2=Giertz|first2=Magnus|date=1973|title=On the analytic formalism of the theory of fuzzy sets|journal=Information Sciences|volume=5|pages=149–156|doi=10.1016/0020-0255(73)90009-1}}</ref>

* If the standard negator <math>n(\alpha) = 1 - \alpha, \alpha \in [0, 1]</math> is replaced by another [[t-norm#Non-standard negators|strong negator]], the fuzzy set difference (defined below) may be generalized by
::<math>\forall x \in U: \mu_{\neg{A}}(x) = n(\mu_A(x)).</math>
* The triple of fuzzy intersection, union and complement form a '''De Morgan Triplet'''. That is, [[De Morgan's laws]] extend to this triple.
:Examples for fuzzy intersection/union pairs with standard negator can be derived from samples provided in the article about [[t-norm]]s.

:The fuzzy intersection is not [[Idempotence|idempotent]] in general, because the standard t-norm {{math|min}} is the only one which has this property. Indeed, if the arithmetic multiplication is used as the t-norm, the resulting fuzzy intersection operation is not idempotent. That is, iteratively taking the intersection of a fuzzy set with itself is not trivial. It instead defines the '''''m''-th power''' of a fuzzy set, which can be canonically generalized for non-[[integer]] exponents in the following way:

* For any fuzzy set <math>A</math> and <math>\nu \in \R^+</math> the &nu;-th power of <math>A</math> is defined by the membership function:
::<math>\forall x \in U: \mu_{A^{\nu}}(x) = \mu_{A}(x)^{\nu}.</math>
The case of exponent two is special enough to be given a name.
* For any fuzzy set <math>A</math> the '''concentration''' <math>CON(A) = A^2</math> is defined
::<math>\forall x \in U: \mu_{CON(A)}(x) = \mu_{A^2}(x) = \mu_{A}(x)^2.</math>
Taking <math>0^0 = 1</math>, we have <math>A^0 = U</math> and <math>A^1 = A.</math>

* Given fuzzy sets <math>A, B</math>, the fuzzy set '''difference''' <math>A \setminus B</math>, also denoted <math> A - B</math>, may be defined straightforwardly via the membership function:
::<math>\forall x \in U: \mu_{A\setminus{B}}(x) = t(\mu_A(x),n(\mu_B(x))),</math>
:which means <math>A \setminus B = A \cap \neg{B}</math>, e. g.:
::<math>\forall x \in U: \mu_{A\setminus{B}}(x) = \min(\mu_A(x),1 - \mu_B(x)).</math><ref name="Vemuri2014">N.R. Vemuri, A.S. Hareesh, M.S. Srinath: [http://www.math.sk/fsta2014/presentations/VemuriHareeshSrinath.pdf Set Difference and Symmetric Difference of Fuzzy Sets], in: Fuzzy Sets Theory and Applications 2014, Liptovský Ján, Slovak Republic</ref>

:Another proposal for a set difference could be: 
::<math>\forall x \in U: \mu_{A-{B}}(x) = \mu_A(x) - t(\mu_A(x),\mu_B(x)).</math><ref name="Vemuri2014" />

* Proposals for symmetric fuzzy set differences have been made by Dubois and Prade (1980), either by taking the [[absolute value]], giving
::<math>\forall x \in U: \mu_{A \triangle B}(x) = |\mu_A(x) - \mu_B(x)|,</math>
:or by using a combination of just {{math|max}}, {{math|min}}, and standard negation, giving
::<math>\forall x \in U: \mu_{A \triangle B}(x) = \max(\min(\mu_A(x), 1 - \mu_B(x)), \min(\mu_B(x), 1 - \mu_A(x))).</math><ref name="Vemuri2014" />

:Axioms for definition of generalized symmetric differences analogous to those for t-norms, t-conorms, and negators have been proposed by Vemur et al. (2014) with predecessors by Alsina et al. (2005) and Bedregal et al. (2009).<ref name="Vemuri2014" />

* In contrast to crisp sets, averaging operations can also be defined for fuzzy sets.

===Disjoint fuzzy sets===
In contrast to the general ambiguity of intersection and union operations, there is clearness for disjoint fuzzy sets:
Two fuzzy sets <math>A, B</math> are '''disjoint''' iff
:<math>\forall x \in U: \mu_A(x) = 0 \lor \mu_B(x) = 0</math>
which is equivalent to
:[[Existential quantification#Negation|<math>\nexists</math>]] <math>x \in U: \mu_A(x) > 0 \land \mu_B(x) > 0</math>
and also equivalent to 
:<math>\forall x \in U: \min(\mu_A(x),\mu_B(x)) = 0</math>
We keep in mind that {{math|min}}/{{math|max}} is a t/s-norm pair, and any other will work here as well.

Fuzzy sets are disjoint if and only if their supports are [[disjoint sets|disjoint]] according to the standard definition for crisp sets.

For disjoint fuzzy sets <math>A, B</math> any intersection will give ∅, and any union will give the same result, which is denoted as
:<math>A \,\dot{\cup}\, B = A \cup B</math> 
with its membership function given by
:<math>\forall x \in U: \mu_{A \dot{\cup} B}(x) = \mu_A(x) + \mu_B(x)</math>
Note that only one of both summands is greater than zero.

For disjoint fuzzy sets <math>A, B</math> the following holds true:
:<math>\operatorname{Supp}(A \,\dot{\cup}\, B) = \operatorname{Supp}(A) \cup \operatorname{Supp}(B)</math>

This can be generalized to finite families of fuzzy sets as follows:
Given a family <math>A = (A_i)_{i \in I}</math> of fuzzy sets with index set ''I'' (e.g. ''I'' = {1,2,3,...,''n''}). This family is '''(pairwise) disjoint''' iff
:<math>\text{for all } x \in U \text{ there exists at most one } i \in I \text{ such that } \mu_{A_i}(x) > 0.</math>

A family of fuzzy sets <math>A = (A_i)_{i \in I}</math> is disjoint, iff the family of underlying supports <math>\operatorname{Supp} \circ A = (\operatorname{Supp}(A_i))_{i \in I}</math> is disjoint in the standard sense for families of crisp sets.

Independent of the t/s-norm pair, intersection of a disjoint family of fuzzy sets will give ∅ again, while the union has no ambiguity:
:<math>\dot{\bigcup\limits_{i \in I}}\, A_i = \bigcup_{i \in I} A_i</math> 
with its membership function given by
:<math>\forall x \in U: \mu_{\dot{\bigcup\limits_{i \in I}} A_i}(x) = \sum_{i \in I} \mu_{A_i}(x)</math>
Again only one of the summands is greater than zero.

For disjoint families of fuzzy sets <math>A = (A_i)_{i \in I}</math> the following holds true:
:<math>\operatorname{Supp}\left(\dot{\bigcup\limits_{i \in I}}\, A_i\right) = \bigcup\limits_{i \in I} \operatorname{Supp}(A_i)</math>

===Scalar cardinality===
For a fuzzy set <math>A</math> with finite support <math>\operatorname{Supp}(A)</math> (i.e. a "finite fuzzy set"), its '''cardinality''' (aka '''scalar cardinality''' or '''sigma-count''') is given by
:<math>\operatorname{Card}(A) = \operatorname{sc}(A) = |A| = \sum_{x \in U} \mu_A(x)</math>.
In the case that ''U'' itself is a finite set, the '''relative cardinality''' is given by 
:<math>\operatorname{RelCard}(A) = \|A\| = \operatorname{sc}(A)/|U| = |A|/|U|</math>.
This can be generalized for the divisor to be a non-empty fuzzy set: For fuzzy sets <math>A,G</math> with ''G'' ≠ ∅, we can define the '''relative cardinality''' by:
:<math>\operatorname{RelCard}(A,G) = \operatorname{sc}(A|G) = \operatorname{sc}(A\cap{G})/\operatorname{sc}(G)</math>,
which looks very similar to the expression for [[conditional probability]]<!--and because of that, sc(A|G) is used instead of sc(A/G)- this isn't **set** division, isn't it? -->.
Note:
* <math>\operatorname{sc}(G) > 0</math> here.
* The result may depend on the specific intersection (t-norm) chosen.
* For <math>G = U</math> the result is unambiguous and resembles the prior definition.

===Distance and similarity===
For any fuzzy set <math>A</math> the membership function <math>\mu_A: U \to [0,1]</math> can be regarded as a family <math>\mu_A = (\mu_A(x))_{x \in U} \in [0,1]^U</math>. The latter is a [[metric space]] with several metrics <math>d</math> known. A metric can be derived from a [[Norm (mathematics)|norm]] (vector norm) <math>\|\,\|</math> via
:<math>d(\alpha,\beta) = \| \alpha - \beta \|</math>.
For instance, if <math>U</math> is finite, i.e. <math>U = \{x_1, x_2, ... x_n\}</math>, such a metric may be defined by: 
:<math>d(\alpha,\beta) := \max \{ |\alpha(x_i) - \beta(x_i)| : i=1, ..., n \}</math> where <math>\alpha</math> and <math>\beta</math> are sequences of real numbers between 0 and 1. 
For infinite <math>U</math>, the maximum can be replaced by a supremum.
Because fuzzy sets are unambiguously defined by their membership function, this metric can be used to measure distances between fuzzy sets on the same universe:
:<math>d(A,B) := d(\mu_A,\mu_B)</math>,
which becomes in the above sample:
:<math>d(A,B) = \max \{ |\mu_A(x_i) - \mu_B(x_i)| : i=1,...,n \}</math>.
Again for infinite <math>U</math> the maximum must be replaced by a supremum. Other distances (like the canonical 2-norm) may diverge, if infinite fuzzy sets are too different, e.g., <math>\varnothing</math> and <math>U</math>.

Similarity measures (here denoted by <math>S</math>) may then be derived from the distance, e.g. after a proposal by Koczy:
:<math>S = 1 / (1 + d(A,B))</math> if <math>d(A,B)</math> is finite, <math>0</math> else, 
or after Williams and Steele:
:<math>S = \exp(-\alpha{d(A,B)})</math> if <math>d(A,B)</math> is finite, <math>0</math> else
where <math>\alpha > 0</math> is a steepness parameter and <math>\exp(x) = e^x</math>.{{Cn|date=December 2024}}

===''L''-fuzzy sets===
Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) [[algebraic structure|algebra]] or [[structure (mathematical logic)|structure]] <math>L</math> of a given kind; usually it is required that <math>L</math> be at least a [[poset]] or [[lattice (order)|lattice]]. These are usually called '''''L''-fuzzy sets''', to distinguish them from those valued over the unit interval. The usual membership functions with values in [0,&thinsp;1] are then called [0,&thinsp;1]-valued membership functions. These kinds of generalizations were first considered in 1967 by [[Joseph Goguen]], who was a student of Zadeh.<ref>{{cite journal | doi=10.1016/0022-247X(67)90189-8 | title=L-fuzzy sets | date=1967 | last1=Goguen | first1=J.A |author-link=Joseph Goguen | journal=Journal of Mathematical Analysis and Applications | volume=18 | pages=145–174 }}</ref> A classical corollary may be indicating truth and membership values by {f,&thinsp;t} instead of {0,&thinsp;1}.

An extension of fuzzy sets has been provided by [[Krassimir Atanassov|Atanassov]]. An '''intuitionistic fuzzy set''' (IFS) <math>A</math> is characterized by two functions:
:1. <math>\mu_A(x)</math> – degree of membership of ''x''
:2. <math>\nu_A(x)</math> – degree of non-membership of ''x''
with functions <math>\mu_A, \nu_A: U \to [0,1]</math> with <math>\forall x \in U: \mu_A(x) + \nu_A(x) \le 1</math>.

This resembles a situation like some person denoted by <math>x</math> voting 
* for a proposal <math>A</math>: (<math>\mu_A(x)=1, \nu_A(x)=0</math>), 
* against it: (<math>\mu_A(x)=0, \nu_A(x)=1</math>), 
* or abstain from voting: (<math>\mu_A(x)=\nu_A(x)=0</math>).
After all, we have a percentage of approvals, a percentage of denials, and a percentage of abstentions.

For this situation, special "intuitive fuzzy" negators, t- and s-norms can be defined. With <math>D^* = \{(\alpha,\beta) \in [0, 1]^2 : \alpha + \beta = 1 \}</math> and by combining both functions to <math>(\mu_A,\nu_A): U \to D^*</math> this situation resembles a special kind of ''L''-fuzzy sets.

Once more, this has been expanded by defining '''picture fuzzy sets''' (PFS) as follows: A PFS A is characterized by three functions mapping ''U'' to [0,&thinsp;1]: <math>\mu_A, \eta_A, \nu_A</math>, "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition <math>\forall x \in U: \mu_A(x) + \eta_A(x) + \nu_A(x) \le 1</math>
This expands the voting sample above by an additional possibility of "refusal of voting".

With <math>D^* = \{(\alpha,\beta,\gamma) \in [0, 1]^3 : \alpha + \beta + \gamma = 1 \}</math> and special "picture fuzzy" negators, t- and s-norms this resembles just another type of ''L''-fuzzy sets.<ref>Bui Cong Cuong, Vladik Kreinovich, Roan Thi Ngan: [http://digitalcommons.utep.edu/cgi/viewcontent.cgi?article=2050&context=cs_techrep A classification of representable t-norm operators for picture fuzzy sets], in: Departmental Technical Reports (CS). Paper 1047, 2016</ref>

===Pythagorean fuzzy sets===
One extension of IFS is what is known as Pythagorean fuzzy sets. Such sets satisfy the constraint <math>\mu_A(x)^2 + \nu_A(x)^2 \le 1</math>, which is reminiscent of the Pythagorean theorem.<ref>{{Cite book|last=Yager|first=Ronald R. |title=2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS) |chapter=Pythagorean fuzzy subsets |date=June 2013 |pages=57–61|doi=10.1109/IFSA-NAFIPS.2013.6608375|isbn=978-1-4799-0348-1|s2cid=36286152}}</ref><ref>{{Cite journal|last=Yager|first=Ronald R|date=2013|title=Pythagorean membership grades in multicriteria decision making|journal=IEEE Transactions on Fuzzy Systems|volume=22|issue=4|pages=958–965|doi=10.1109/TFUZZ.2013.2278989|s2cid=37195356}}</ref><ref>{{Cite book|title=Properties and applications of Pythagorean fuzzy sets.|last=Yager|first=Ronald R.|publisher=Springer |location=Cham|date=December 2015|isbn=978-3-319-26302-1|pages=119–136}}</ref> Pythagorean fuzzy sets can be applicable to real life applications in which the previous condition of <math>\mu_A(x) + \nu_A(x) \le 1</math> is not valid. However, the less restrictive condition of <math>\mu_A(x)^2 + \nu_A(x)^2 \le 1</math> may be suitable in more domains.<ref name="CADsurvey">{{cite journal | vauthors = Yanase J, Triantaphyllou E| title = A Systematic Survey of Computer-Aided Diagnosis in Medicine: Past and Present Developments. | journal = Expert Systems with Applications | volume = 138 | pages = 112821 | date = 2019 | doi = 10.1016/j.eswa.2019.112821 | s2cid = 199019309 }}</ref><ref name="SevenChallenges">{{Cite journal|vauthors = Yanase J, Triantaphyllou E|date=2019|title=The Seven Key Challenges for the Future of Computer-Aided Diagnosis in Medicine.|doi=10.1016/j.ijmedinf.2019.06.017|pmid=31445285|journal= International Journal of Medical Informatics|volume=129|pages=413–422|s2cid=198287435 }}</ref>