Centroid . . . Geometric Meaning of . . . Mathematical Formula . . . Centroids Beyond First Meaning of This . . . Proof Defuzzification A Version of the First . . . ıa 1 , Christian Servin 2 , Juan Carlos Figueroa-Garc´ Second Meaning and Vladik Kreinovich 3 Home Page Title Page 1 Universidad Distrital Francisco Jos´ e de Caldas Bogot´ a, Colombia, jcfigueroag@udistrital.edu.co ◭◭ ◮◮ 2 Computer Science and Information Technology Systems Department ◭ ◮ El Paso Community College (EPCC), 919 Hunter Dr. Page 1 of 14 El Paso, TX 79915-1908, USA cservin1@epcc.edu Go Back 3 University of Texas at El Paso 500 W. University, El Paso, TX 79968, USA Full Screen vladik@utep.edu Close Quit
Centroid . . . 1. Centroid Defuzzification: A Brief Reminder Geometric Meaning of . . . • In fuzzy control, we start with the expert rules. Mathematical Formula . . . First Meaning of This . . . • These rules are formulated in terms in imprecise (“fuzzy”) Proof words from natural language. A Version of the First . . . • Given the inputs, we recommend what control value u Second Meaning to use. Home Page • This recommendation is also fuzzy: Title Page – for each possible value u , ◭◭ ◮◮ – we provides a degree µ ( u ) ∈ [0 , 1] to which u is a ◭ ◮ reasonable control. Page 2 of 14 • Such a fuzzy outcome is perfect if the main objective Go Back of the system is to advise a human controller. Full Screen • In many practical situations, however, we want this Close system to actually control. Quit
Centroid . . . 2. Centroid Defuzzification (cont-d) Geometric Meaning of . . . Mathematical Formula . . . • In such situations, it is important to transform: First Meaning of This . . . – the fuzzy recommendation – as expressed by the Proof function µ ( u ) (known as the membership function ) A Version of the First . . . – into a precise control value u that this system will Second Meaning apply. Home Page • Such a transformation is known as defuzzification . Title Page • The most widely used defuzzification procedure is cen- ◭◭ ◮◮ � u · µ ( u ) du troid defuzzification u = µ ( u ) du . ◭ ◮ � Page 3 of 14 Go Back Full Screen Close Quit
Centroid . . . 3. Geometric Meaning of Centroid Defuzzification Geometric Meaning of . . . • The name for this defuzzification procedure comes from Mathematical Formula . . . the fact that: First Meaning of This . . . Proof – if we take the subgraph of the function µ ( u ), i.e., A Version of the First . . . def the 2-D set S = { ( u, y ) : 0 ≤ y ≤ µ ( u ) } , Second Meaning – then the value u is actually the u -coordinate of this Home Page set’s center of mass (“centroid”) ( u, y ). Title Page • In fuzzy technique, we only use the u -coordinate of the ◭◭ ◮◮ center of mass. ◭ ◮ • A natural question is: is there a fuzzy-related meaning Page 4 of 14 of the y -coordinate y ? Go Back • In this talk, we describe such a meaning. Full Screen Close Quit
Centroid . . . 4. Mathematical Formula for the y -Component Geometric Meaning of . . . • In general, the y -component of the center of mass of a Mathematical Formula . . . � S y du dy First Meaning of This . . . 2-D body S has the form y = S du dy . � Proof A Version of the First . . . • The denominator is the same as for the u -component: � Second Meaning it is equal to µ ( u ) du . Home Page • The numerator can also be easily computed as Title Page � µ ( u ) � � y dy = 1 � µ 2 ( u ) du. y du dy = du · 2 · ◭◭ ◮◮ S u 0 u ◭ ◮ � µ 2 ( u ) du • Thus, we have y = 1 2 · µ ( u ) du . Page 5 of 14 � Go Back Full Screen Close Quit
Centroid . . . 5. First Meaning of This Formula Geometric Meaning of . . . • The u -component of the centroid is the weighted aver- Mathematical Formula . . . age value of u , with weights proportional to µ ( u ). First Meaning of This . . . Proof • Similarly, the expression y is the weighted average value A Version of the First . . . of µ ( u ). Second Meaning • Each value µ ( u ) is the degree of fuzziness of the sys- Home Page tem’s recommendation about the control value u . Title Page • Thus, the value y can be viewed with the weighted ◭◭ ◮◮ average value of the degree of fuzziness. ◭ ◮ • Let us show that this interpretation makes some sense. Page 6 of 14 • Proposition. Go Back – The value y is always between 0 and 1/2. Full Screen – For a measurable function µ ( u ) , y = 1 / 2 if and Close only if µ ( u ) is almost everywhere 0 or 1. Quit
Centroid . . . 6. First Meaning of This Formula (cont-d) Geometric Meaning of . . . Mathematical Formula . . . • In other words, if we ignore sets of measure 0: First Meaning of This . . . – the value y is equal to 1/2 if and only if Proof – the corresponding fuzzy set is actually crisp. A Version of the First . . . • For all non-crisp fuzzy sets, we have y < 1 / 2. Second Meaning Home Page • For a triangular membership function, one can check that we always have y = 1 / 3. Title Page ◭◭ ◮◮ • For trapezoid membership functions, y can take any possible value between 1/3 and 1/2. ◭ ◮ • The larger the value-1 part, the larger y . Page 7 of 14 Go Back Full Screen Close Quit
Centroid . . . 7. Proof Geometric Meaning of . . . • Since µ ( u ) ∈ [0 , 1], we always have µ 2 ( u ) ≤ µ ( u ), thus Mathematical Formula . . . µ 2 ( u ) du � First Meaning of This . . . µ 2 ( u ) du ≤ � � µ ( u ) du , hence µ ( u ) du ≤ 1 . � Proof A Version of the First . . . • So, y ≤ 1 / 2. Second Meaning µ 2 ( u ) du � • Vice versa, if y = 1 / 2, then µ ( u ) du = 1 . Home Page � Title Page • Multiplying both sides of this equality by the denomi- � µ 2 ( u ) du = � nator, we conclude that µ ( u ) du , i.e.: ◭◭ ◮◮ � � ◭ ◮ µ ( u ) − µ 2 ( u ) � du = 0 . Page 8 of 14 • The difference µ ( u ) − µ 2 ( u ) is always non-negative. Go Back • Since its integral is 0, this means that almost always Full Screen µ ( u ) − µ 2 ( u ), i.e., µ ( u ) = 0 or µ ( u ) = 1. Close • The proposition is proven. Quit
Centroid . . . 8. A Version of the First Meaning Geometric Meaning of . . . • In general, in the fuzzy case, we have different values Mathematical Formula . . . of µ ( u ) for different u . First Meaning of This . . . Proof • We want to find a single degree µ 0 which best repre- A Version of the First . . . sents all these values. Second Meaning • This is natural to interpret as requiring that: Home Page – the mean square difference weighted by µ ( u ) – i.e., Title Page ( µ ( u ) − µ 0 ) 2 · µ ( u ) du � the value ◭◭ ◮◮ – attains its smallest possible value. ◭ ◮ • Differentiating the minimized expression with respect Page 9 of 14 to µ 0 and equating the derivative to 0, we get: � Go Back 2( µ 0 − µ ( u )) · µ ( u ) du = 0 . Full Screen � µ 2 ( u ) du Close • Hence µ 0 = µ ( u ) du , and y 0 = (1 / 2) · µ 0 . � Quit
Centroid . . . 9. Second Meaning Geometric Meaning of . . . • Membership functions µ ( u ) and probability density func- Mathematical Formula . . . tions ρ ( u ) differ by their normalization: First Meaning of This . . . Proof – for a membership function µ ( u ), we require that A Version of the First . . . max µ ( u ) = 1, while u Second Meaning – for a probability density function ρ ( u ), we require Home Page � that ρ ( u ) du = 1. Title Page • For every f ( u ) ≥ 0, we can divide it by an appropriate ◭◭ ◮◮ constant c and get µ ( u ) or ρ ( u ): ◭ ◮ – if we divide f ( u ) by c = max f ( v ), then we get a v Page 10 of 14 f ( u ) membership function µ ( u ) = f ( v ); max Go Back v Full Screen � – if we divide f ( u ) by c = µ ( v ) dv , we get a proba- f ( u ) Close bility density function ρ ( u ) = f ( v ) dv. � Quit
Centroid . . . 10. Second Meaning (cont-d) Geometric Meaning of . . . Mathematical Formula . . . • In particular, for each µ ( u ), we can construct the cor- responding probability density function First Meaning of This . . . Proof µ ( u ) ρ ( u ) = µ ( v ) dv. A Version of the First . . . � Second Meaning • In terms of this expression ρ ( u ), the formulas for both Home Page components of the center of mass are simplified. Title Page • The result u of centroid defuzzification takes the form ◭◭ ◮◮ � u = u · ρ ( u ) du. ◭ ◮ Page 11 of 14 • It is simply the expected value of control under this probability distribution. Go Back • Similarly, the value µ 0 = 2 y takes the form Full Screen � Close µ 0 = µ ( u ) · ρ ( u ) du. Quit
Centroid . . . 11. Second Meaning (cont-d) Geometric Meaning of . . . Mathematical Formula . . . • So, µ 0 is simply the expected value of the membership function. First Meaning of This . . . Proof � • Note: µ ( u ) · ρ ( u ) du is Zadeh’s formula for the prob- A Version of the First . . . ability of the fuzzy event. Second Meaning • Reminder: µ ( u ) characterizes to what extent a control Home Page value u is reasonable. Title Page • So, µ 0 is the probability that a control value selected ◭◭ ◮◮ by fuzzy control will be reasonable. ◭ ◮ Page 12 of 14 Go Back Full Screen Close Quit
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