Fuzzy systems

There are two main categories of fuzzy systems - Mamdani and Sugeno systems. In Mamdani system everything is encoded using fuzzy variables - fuzzy output forces us to perform defuzzification (with may be quite expansive). In Sugeno system implication rule have form of crisp function with simplifies calculus and gives more flexibility for price of intuitiveness.

Immortal example of fuzzy system usage is calculating tip % based on food and service quality. We can create empty Mamdani system using:

from yvain.fuzzy_system import MamdaniSystem
system = MamdaniSystem.empty()

Fuzzy variable

Fuzzy variable allow us to describe linguistic terms. In classical system we would simply describe that service can be rated in range of [0, 10]. Fuzzy logic allow us to describe service using fuzzy terms as “poor”, “good or “excellent”

from yvain.membership_functions import Gaussian
system.add_input("service", {
    "poor": Gaussian(0, 1.5),
    "good": Gaussian(5, 1.5),
    "excellent": Gaussian(10, 1.5),
})

(Source code, png, hires.png, pdf)

For purpose of example we will also add second input variable, with allows us to rate food:

from yvain.membership_functions import Trapezoid
system.add_input("food", {
    "rancid": Trapezoid(-2, 0, 2, 4),
    "delicious": Trapezoid(7, 9, 11, 13),
})

(Source code, png, hires.png, pdf)

As we mentioned before Mamdani system will also feature fuzzy output. We can add fuzzy outputs to our system in similar way as adding inputs:

from yvain.membership_functions import Triangle
system.add_output("tip", {
    "cheap": Triangle(0, 5, 10),
    "average": Triangle(7.5, 12.5, 17.5),
    "generous": Triangle(15, 20, 25),
})

(Source code, png, hires.png, pdf)

Fuzzy rules

Another important thing in fuzzy system is to describe relationships between inputs and outputs. Such relationships are called rules. Fuzzy rules in Mamdani system are formulated as IF service is good THEN tip is average, we are also allowed to use operators known from boolean logic like AND, OR. In this library fuzzy rule can be build via fluent api starting from when function:

from yvain.fuzzy_system import when
system.add_rule(when("service", "good").then("tip", "average"))

Mamdani system rule always have to end with .then method call and there can be only one such call per rule. OR and AND operators are chained via .or_is and .and_is methods. Those methods do always take two arguments - variable and state.

system.add_rule(when("service", "poor").or_is("food", "rancid").then("tip", "cheap"))

system.add_rule(when("service", "excellent").or_is("food", "delicious").then("tip", "generous"))

Having all variables and rules added to the system we can use it by simply calling .run method. This method will require dictionary mapping variable to crisp value. This value will be fuzzified, rules are going to be applied, and we will get mapping from output variable to crisp value (our result).

print(system.run({"service": 3, "food": 8}, (0, 25)))