Gaussian Centroid Defuzzification (Gaussian Centroid Method)

In some fuzzy systems, membership degree is defined not as a triangle or a trapezoid but as a Gaussian (bell) curve: a peak point (the highest membership) and a spread that decreases smoothly the further one moves from that peak. This definition is particularly favoured for sensor measurements and continuous physical quantities, because natural measurement error tends to follow a smooth distribution rather than a sharp-cornered one. This card describes the defuzzification rule that reduces such a Gaussian fuzzy number to a single crisp number. Its output is a single crisp score; it produces no weights, it only converts a Gaussian fuzzy number into a number. This rule is the Gaussian-curve adaptation of the centroid used for classical triangular numbers, and it is frequently used in engineering and fuzzy-control applications.

The idea is the same as the centroid for triangular fuzzy numbers: finding the centre of mass of the area beneath the membership curve. The Gaussian curve has a special property: if the curve is symmetric (opening with the same spread to the left and the right), the centre of mass falls exactly on the peak, with no extra calculation needed. This is what makes the Gaussian centroid, in most cases, as simple as "take the peak". But in real applications a measurement's margin of error rarely spreads equally in both directions; a sensor, for instance, may rarely undershoot the true value but often overshoot it. In such cases the curve opens to different widths on the two sides and is no longer symmetric; the centre of mass then shifts away from the peak.

This philosophy carries a consequence: the Gaussian centroid is a rule that is very simple when the assumption of "symmetric spread" holds true, and easy to overlook when it does not. If, in an application, measurement error is known to be wider in one particular direction (say, consistently downward), ignoring this asymmetry and reporting only the peak understates the true uncertainty.

The method proceeds through two steps.

First, finding the centre of mass of a symmetric Gaussian curve. A Gaussian fuzzy number is defined by a peak point (where the highest membership value sits) and a spread width. Because the curve itself is symmetric, the area's centre of mass is algebraically exactly equal to the peak; no additional calculation is needed.

Second, finding the centre of mass of a Gaussian curve that spreads differently to the two sides (asymmetric). In some applications the curve's left and right spread differ; this is defined by joining two half-Gaussian curves at a single peak point. In this case the curve is no longer symmetric, and the centre of mass departs from the peak by an amount proportional to the difference between the two spreads. The shift runs towards the more widely spread side.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The number this rule gives is the Gaussian curve's area-weighted average. If the curve is symmetric, this number is identical to the peak and can also be read as the "most likely value". If the curve is asymmetric, this number departs from the peak, and it no longer means only "the most likely value" but "a balanced summary that also accounts for the spread". Even if two alternatives' peaks sit very close to one another, their defuzzified scores can diverge markedly if their spreads differ.

Thus instead of writing:

"The defuzzified score is the measurement's most likely (peak) value"

the report should read:

"The defuzzified score is a joint summary of the measurement's most likely value and its spread; if the spread is symmetric the two coincide, if not the defuzzified score shifts towards the more widely spread side"

This rule works with fuzzy data, specifically fuzzy numbers defined in Gaussian (bell-curve) form. In DecisionMind, this building block has no extension of its own; it is separate from the centroid building block that works with triangular fuzzy numbers and is used only with Gaussian-type fuzzy inputs. You need a peak point (mean) and one or two spread values (a single one if symmetric, separate left and right values if asymmetric); the spread values must be greater than zero. This data typically comes from a sensor's calibration uncertainty, the distribution of repeated measurements, or an expert's judgement of the kind "roughly this value, but it may deviate by this much".

If your fuzzy data is naturally defined by a bell curve rather than a triangle or trapezoid (sensor measurement error or repeated experimental results, for instance), this rule is the right choice. If the spread is known not to be equal in both directions (a device with a known tendency to drift in one direction, for instance), the asymmetric form should be used, and the peak alone should not be reported. If your data is given as a triangular or trapezoidal fuzzy number (sharp-cornered, of the kind an expert describes as "at least, most likely, at most"), the classical centroid, bisector or MOM rules are more suitable; applying the Gaussian formula to triangular data is a wrong type match.

Bell-curve (Gaussian) measurement uncertainty, symmetric → Gaussian Centroid (peak directly)

Bell-curve measurement uncertainty, asymmetric → Gaussian Centroid (asymmetric formula)

Triangular/trapezoidal fuzzy number (expert judgement, "at least, most likely, at most") → Centroid, Bisector or MOM

Type-2 fuzzy number (upper and lower bound together) → Type Reduction

This rule's clearest strength is how simple the calculation is in the symmetric case: taking the peak is enough, no extra formula is needed. The Gaussian curve represents natural measurement error more realistically than a triangle or trapezoid, because physical processes tend to follow a smooth rather than a sharp-cornered distribution. The asymmetric form can carry directly into the result the information that a measurement is differently reliable in the two directions, for instance that a sensor is more prone to drifting downward.

This rule's limitations stem from its data requirements. Within Zadeh's (1965) general fuzzy-set framework, the Gaussian form is a specific choice; whether the data genuinely fits this form must be checked, otherwise the analysis proceeds on a wrong assumption. Runkler (1997), comparing which application properties different defuzzification rules suit, showed that a mismatch between the curve's shape and the defuzzification rule can distort the result. The asymmetric Gaussian formula is less well known than the symmetric case and is easy to forget or misapply by hand. Finally, whether the spread values (sigma) come from a small measurement error or a large uncertainty requires separate interpretation; the rule does not make this distinction on its own.

The most common mistake is reporting only the peak (the mean) for an asymmetric Gaussian curve and ignoring the difference in spread; this produces the same outcome as applying the symmetric formula to asymmetric data. A second mistake is confusing the spread width (sigma) with the curve's support (the outer points where membership drops to zero); sigma alone does not affect a symmetric Gaussian's centre of mass, but the two concepts are frequently mixed up. A third mistake is applying the triangular fuzzy number's formula (the average of three corners) to Gaussian-type data; this is the wrong formula and does not give the correct result.

The governing principle is this:

The Gaussian centroid gives the bell curve's centre of mass; if the curve is symmetric this coincides with the peak, if not it shifts towards the more widely spread side, and the report must explain the reason for this shift.