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Fuzzy numbers, visually

Each figure on this page shows one idea about fuzzy numbers. They are drawn with bulanik.plot, so you can make the same pictures from your own numbers. bulanik computes every value in them: α-cuts, sums, products, scores and aggregates. The figures follow your colour scheme; switch between light and dark mode to see the other variant.

How these figures are made

The scripts are in docs/scripts/. Each one calls bulanik.plot and then adds titles and annotations. Regenerate all the figures with uv run --group docs python docs/scripts/build_figures.py. Each figure is written as a light and a dark SVG, and the output is deterministic.

Triangular and trapezoidal numbers

Membership functions of TFN(2, 4, 7) and TrFN(1, 3, 5, 8), with support, core and the α-cut at 0.5 marked Membership functions of TFN(2, 4, 7) and TrFN(1, 3, 5, 8), with support, core and the α-cut at 0.5 marked
Anatomy. A triangle (l, m, u) and a trapezoid (a, b, c, d) are fixed by their corner points. The support is where the membership is above 0, and the core is where it equals 1: one point for a triangle, a plateau for a trapezoid. Every α-cut between them is an interval; here the cut at α = 0.5 is alpha_cut(0.5). See Triangular numbers.
α-cuts of A = TFN(1, 2, 4), B = TFN(2, 3, 5), A + B and A × B, stacked on one x axis α-cuts of A = TFN(1, 2, 4), B = TFN(2, 3, 5), A + B and A × B, stacked on one x axis
Arithmetic, one α at a time. Fuzzy arithmetic works level by level on α-cuts, as interval arithmetic. At α = 0.5 the sum's cut is the sum of the two operand cuts, [1.5, 3] + [2.5, 4] = [4, 7]. Addition keeps a triangle exact. The exact product has curved sides (the outline marked "exact cuts"), so A * B returns the triangle through its corner products: at α = 0.5 that gives [4, 13] where the exact cut is [3.75, 12]. See TFN.mul.
TFN(1, 2, 7) with stems at its graded mean 2.667 and its centroid and mean 3.333 TFN(1, 2, 7) with stems at its graded mean 2.667 and its centroid and mean 3.333
Defuzzification. score turns a number into one crisp value, and the method you choose matters for a skewed number. The graded mean weights the peak four times, so it stays near m. For a triangle the centroid equals the mean (l + m + u)/3, which the long right tail pulls further out.
The seven TFN terms of EXAMPLE_TFN_7 on 0 to 10, with a rating of 6.5 belonging to MG at 0.75 and to F at 0.25 The seven TFN terms of EXAMPLE_TFN_7 on 0 to 10, with a rating of 6.5 belonging to MG at 0.75 and to F at 0.25
A linguistic scale. EXAMPLE_TFN_7 maps seven words to triangles. Neighbouring terms overlap, so a crisp rating belongs partly to two words: 6.5 is "MG" to degree 0.75 and "F" to degree 0.25. Drawn with plot.variable; see Sets and scales.

Orthopair numbers

Nested feasible regions for q = 1, 2, 3 and 5 in the μ–ν plane, with one IFN, PFN, FFN and q = 5 number and their hesitancy segments Nested feasible regions for q = 1, 2, 3 and 5 in the μ–ν plane, with one IFN, PFN, FFN and q = 5 number and their hesitancy segments
One family, nested regions. An orthopair number (μ, ν) of rung q must satisfy μq + νq ≤ 1. Raising q enlarges the region: intuitionistic (q = 1) ⊂ Pythagorean (2) ⊂ Fermatean (3) ⊂ q = 5. Each sample number sits outside the region of the rung below its own. The short segment from a point to its own boundary grows with the hesitancy π. See Orthopair numbers.
Arrows from IFN x to the algebraic and Einstein sums and products of x and y Arrows from IFN x to the algebraic and Einstein sums and products of x and y
Operator families. Adding y to x moves toward full membership (bottom right); multiplying moves toward full non-membership (top left). The Einstein operators (ops="einstein") move further than the algebraic default in both directions: the Einstein t-conorm is larger than the algebraic one, and its t-norm is smaller. See QROFN.add and QROFN.mul.
Three weighted IFN ratings with their weighted average WA and weighted geometric mean WG Three weighted IFN ratings with their weighted average WA and weighted geometric mean WG
WA vs WG. Both wa and wg combine the three ratings with the same weights. They disagree in a systematic way. The weighted average (IFWA) leans to the high membership of the heaviest rating. The geometric mean (IFWG) is pulled toward the weak ratings, so it lands at lower μ and higher ν. See Aggregation.
The seven IFN terms of EXAMPLE_IFN_7 in the μ–ν plane, joined in scale order The seven IFN terms of EXAMPLE_IFN_7 in the μ–ν plane, joined in scale order
An intuitionistic scale. EXAMPLE_IFN_7 runs from "EL" (extremely low) to "EH" (extremely high), and the hairline joins the terms in scale order. The distance of a term from the boundary μ + ν = 1 is its hesitancy. "L" and "M" have the most (π = 0.1), and "EH" lies on the boundary (π = 0).

Three-part numbers

Four picture fuzzy numbers on the μ–η–ν ternary triangle, with halos sized by their refusal degree Four picture fuzzy numbers on the μ–η–ν ternary triangle, with halos sized by their refusal degree
Picture fuzzy numbers. A PictureFN splits a judgement into positive μ, neutral η and negative ν. The rest, ρ = 1 − μ − η − ν, is refusal. Each point sits at its parts rescaled to sum to 1, (μ, η, ν)/(1 − ρ), so the grid reads the rescaled shares and the labels give the parts themselves: D (0.3, 0.25, 0.15) sits on the η ≈ 0.36 line. The halo grows with ρ, and D leaves the most unsaid (ρ = 0.3). See Three-part numbers.
Three spherical fuzzy numbers A, B, C inside the positive octant of the unit sphere Three spherical fuzzy numbers A, B, C inside the positive octant of the unit sphere
Spherical fuzzy numbers. A SphericalFN (μ, ν, π) only needs μ² + ν² + π² ≤ 1, so it can be any point of the unit ball's positive octant. A = (0.8, 0.3, 0.4), B = (0.4, 0.7, 0.5) and C = (0.3, 0.3, 0.85) are drawn with drop lines to the μ–ν floor so you can read their depth. C sits high on the π axis: mostly hesitant.
Two neutrosophic numbers as triangles on truth, indeterminacy and falsity spokes Two neutrosophic numbers as triangles on truth, indeterminacy and falsity spokes
Neutrosophic numbers. The truth, indeterminacy and falsity parts of a NeutrosophicFN are independent; each can be anywhere in [0, 1]. A radar triangle shows all three at once. A is a confident "true", and B is mostly indeterminate.

Interval-valued orthopair numbers

An interval-valued intuitionistic, Pythagorean and Fermatean number as rectangles over the q = 1, 2 and 3 feasible regions An interval-valued intuitionistic, Pythagorean and Fermatean number as rectangles over the q = 1, 2 and 3 feasible regions
Rectangles instead of points. An interval-valued orthopair number gives membership and non-membership as intervals, so it is the rectangle [μL, μU] × [νL, νU] of all the pairs it admits. The IVIFN, IVPFN and IVFFN shown here (IVIFN, IVPFN, IVFFN) each have their upper-right corner on the boundary of their own rung: μU = 0.3 on q = 1, 0.5 on q = 2 and 0.88 on q = 3. Only that corner has to satisfy μUq + νUq ≤ 1, and then the whole rectangle lies inside the region.
Two interval-valued intuitionistic numbers A and B with their sum and product as rectangles Two interval-valued intuitionistic numbers A and B with their sum and product as rectangles
Arithmetic on rectangles. With the algebraic operators, A ⊕ B moves toward high membership and low non-membership, and A ⊗ B toward low membership and high non-membership. Both results are again rectangles inside the region. The corner formulas act on the interval ends, so no interval has to be sampled. See IVQROFN.add and IVQROFN.mul.

Interval type-2 numbers

A seven-term interval type-2 trapezoidal scale from VL to VH with the DTraT score of each term A seven-term interval type-2 trapezoidal scale from VL to VH with the DTraT score of each term
An interval type-2 scale. Each term is an IT2TrFN: an upper trapezoid (bold line), a lower trapezoid that is nested inside it (thin line) and, between them, the footprint of uncertainty (the wash). The numbers under the axis are the DTraT scores that bulanik computes for each term, which turn the type-2 term into one crisp value. The seven terms form an illustrative scale in the style of the interval type-2 TOPSIS literature (Chen and Lee 2010, doi:10.1016/j.eswa.2009.09.012). The values are not a transcription of any one published table.

DEMATEL

DEMATEL cause-effect diagram of five factors, with the strongest influences as arrows DEMATEL cause-effect diagram of five factors, with the strongest influences as arrows
Cause and effect. The 5 × 5 direct-relation matrix of the vignette of the R package dematel goes through dematel. Each factor is placed by its prominence D + R (how involved it is) and its relation D − R (above the line, a net giver of influence). F1, F4 and F5 are causes, and F2 and F3 are effects. Arrows mark the total influences above 0.45; the default threshold, the mean of T (0.33), would draw 13 of them. See plot.dematel.