Miscellaneous information

On this page, miscellaneous information that for the moment doesn't fit elsewhere...

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About the concept of "firing strength"

This term can have two different meanings, we can have the firing strength of an input related to a membership function, or we can have the firing strength of a rule, given a set of input values. This latter meaning is used for Takagi-Sugeno inference.

Firing strength of an input value

This is about how much a given input value "triggers" a membership value. For example, if we have an input with an associated set of two triangular membership function such as the following:

If the input value is, for example 27.5, then the membership function mf1 will be triggered with a (fuzzy) value of 0.2, and the membership function mf2 will be triggered with a value of 0.8.

This can be used when plotting a 2-input rule base (class slifis_plot::PLOT_RB_2D), to visually show how much the input triggers the input functions.

Related functions:

Firing strength of a rule

This is about the firing strength of a rule, expressed as a fuzzy value, for some input values. It can be defined by the degree to which the antecedent part of a fuzzy rule is satisfied. It is also known as "degree of fulfillment" in some sources. Its computation depend on the rule operator:

if operator is "OR", then we use a s-norm (slifis::EN_SNORM), selected through RULE_IDX::s_RuleOrNorm
if operator is "AND", then we use a t-norm (slifis::EN_TNORM), selected through RULE_IDX::s_RuleAndNorm

It works as follow, for a rule r:

FOR each term ("a" is "b"):
- get input value for input a
- compute how much this value triggers membership function b
- take this fuzzy value into account to compute the final value, using the selected norm (see above).

For example, for a given "AND" rule, and using the (default) product t-norm

if input-1 is "blue" and input-2 is "hot" and input-3 is "nice", then output is ...

If input-1 triggers the "blue" membership function with a value of 0.9, input-2 triggers the "hot" membership function with a value of 0.8, and input-3 triggers the "nice" membership function with a value of 0.4, then the Firing strength of the rule will be $0.9 \times 0.8 \times 0.4 = 0.288$

Related functions:

RULE_IDX::GetRuleFiringStrength()

Enumeration to string functions

In order to print human readable strings from enumerations values, all the enum types have an associated GetString() function.

Usage: say you have an enum value enum_val of type EN_ENUMTYPE, and you want to print out its value in a human-readable form:

        EN_ENUMTYPE enum_val;
        enum_val = ...; // some value
        cout << "enum_val = " << GetString(enum_val) << endl;

Additionally, the following function are also related, although they have no argument:

Gaussian function approximation

The library provides an approximation of a Gaussian function using 6 line segments (3 per side). Each segment approximates a part of the Gaussian function $ f_G(x) $ . For a centered ( $\mu_0=0$ ) function, the segments are computed as follows:

The first segment goes from to , with $x_1=k_1 \cdot \sigma$ and
The second segment goes from to , with $x_2=k_2 \cdot \sigma$ and
The third segment goes from to , with $x_3=k_3 \cdot \sigma$

The values $ k_1, k_2, k_3 $ are at present arbitrary chosen to produce something that looks ok. The corresponding error can be plotted with the gnuplot script file misc/GaussianError.plt, that produces the following plots. Absolute error will be less than 3% most of the time.

Real Gaussian compared to approximation	Absolute error (adapted scale)