Need to Fuse . . . Examples Fusing Expert . . . Example How to Fuse Expert Need to Consider the . . . Knowledge: Not Always In General, How . . . How to Define Degree . . . “And” but a Fuzzy Resulting Definition of . . . Discussion Combination of Home Page “And” and “Or” Title Page ◭◭ ◮◮ Christian Servin 1 , Olga Kosheleva 2 , and Vladik Kreinovich 3 1 Computer Science and Information Technology Systems Department ◭ ◮ El Paso Community College, 919 Hunter Page 1 of 28 El Paso, TX 79915, USA, cservin@gmail.com 2 , 3 Departments of 2 Teacher Education and 3 Computer Science Go Back University of Texas at El Paso, El Paso, Texas 79968, USA olgak@utep.edu, vladik@utep.edu Full Screen Close Quit
Need to Fuse . . . Examples 1. Need to Fuse Knowledge of Different Experts Fusing Expert . . . • Expert estimates of different quantities are usually not Example very accurate. Need to Consider the . . . In General, How . . . • In situations when measurements are possible, they are How to Define Degree . . . more accurate than expert estimates. Resulting Definition of . . . • When we can perform measurements: Discussion Home Page – we can further increase the measurement accuracy – if we use several different measuring instruments Title Page and then combine (“fuse”) their results. ◭◭ ◮◮ • It is known that such combinations are usually more ◭ ◮ accurate than all original measurement results. Page 2 of 28 Go Back Full Screen Close Quit
Need to Fuse . . . Examples 2. Need to Fuse Knowledge (cont-d) Fusing Expert . . . • In many situations, measurements are not realistically Example possible, so we have to rely on expert estimates only. Need to Consider the . . . In General, How . . . • In such situations: How to Define Degree . . . – we can increase the accuracy of the resulting esti- Resulting Definition of . . . mates the same way: Discussion – by combining (fusing) estimates of several experts. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 28 Go Back Full Screen Close Quit
Need to Fuse . . . Examples 3. Examples Fusing Expert . . . • To estimate the temperature, we can ask two experts. Example Need to Consider the . . . • Suppose that: In General, How . . . – one expert states that the temperature is between How to Define Degree . . . 22 and 25 degree C, and Resulting Definition of . . . – another expert states the temperature is in the low Discussion seventies, i.e., between 70 and 75 F; Home Page – this corresponds to between 21 and 24 C. Title Page • Then we can conclude that the actual temperature is ◭◭ ◮◮ larger than 22 C and smaller than 24 C – i.e., the actual ◭ ◮ temperature is between 22 and 24 C. Page 4 of 28 • If we only ask one expert, we get an interval of width 3 that contains the actual temperature. Go Back • But by fusing the opinions of the two experts, we get Full Screen a narrower interval [22 , 24] of width 2. Close Quit
Need to Fuse . . . Examples 4. Examples (cont-d) Fusing Expert . . . • So, we have indeed increased the accuracy. Example Need to Consider the . . . • Fusion is also possible on a non-quantitative level. In General, How . . . • For example, we can ask experts whether the wind is How to Define Degree . . . weak, moderate, or strong. Resulting Definition of . . . • Suppose that: Discussion Home Page – one expert says that the wind is not weak, while Title Page – another expert says that the wind is not strong. ◭◭ ◮◮ • By combining the opinions of both experts, we can ◭ ◮ conclude that the wind is moderate. Page 5 of 28 • On the other hand, if we only to one of the experts, we would not be able to come to this conclusion. Go Back Full Screen Close Quit
Need to Fuse . . . Examples 5. Fusing Expert Knowledge: Non-Fuzzy Case Fusing Expert . . . • Let us start with the case when expert estimates are Example crisp (non-fuzzy). Need to Consider the . . . In General, How . . . • So, for each possible value of the estimated quantity, How to Define Degree . . . the expert is: Resulting Definition of . . . – either absolutely sure that this value is possible Discussion – or is absolutely sure that the given value is not Home Page possible. Title Page • In this case, each expert estimate provides us with a ◭◭ ◮◮ set of possible values of the corresponding quantity. ◭ ◮ • In most practical cases, this set is an interval [ x, x ]. Page 6 of 28 Go Back Full Screen Close Quit
Need to Fuse . . . Examples 6. Non-Fuzzy Case (cont-d) Fusing Expert . . . • In these terms, when we have estimates of two different Example experts, this means that: Need to Consider the . . . In General, How . . . – based on the opinions of the first expert, we form How to Define Degree . . . a set S 1 of possible values; Resulting Definition of . . . – also, based on the opinions of the second expert, Discussion we form a set S 2 of possible values. Home Page • In general, different experts take into account different Title Page aspects of the situation. ◭◭ ◮◮ • For example, the first expert may know the upper bound ◭ ◮ x on the corresponding quantity. Page 7 of 28 • In this case, the set S 1 consists of all the numbers which are smaller than or equal to x , i.e., S 1 = ( −∞ , x ]. Go Back • The second expert may know the lower bound x , in Full Screen which case S 2 = [ x, ∞ ). Close Quit
Need to Fuse . . . Examples 7. Non-Fuzzy Case (cont-d) Fusing Expert . . . • A natural way to fuse the knowledge is to consider Example numbers which are possible according to both experts. Need to Consider the . . . In General, How . . . • In mathematical terms, we consider the intersection How to Define Degree . . . S 1 ∩ S 2 of the two sets S 1 and S 2 . Resulting Definition of . . . • A problem occurs when this intersection is empty, i.e., Discussion when the opinions of two experts are inconsistent. Home Page • This happens: experts are human and can thus make Title Page mistakes. ◭◭ ◮◮ • In this case, an extreme option is to say that: ◭ ◮ – since experts are not consistent with each other, Page 8 of 28 – this means that we do not trust what each of them Go Back says, Full Screen – so we can as well ignore both opinions; the result of fusion is then the whole real line. Close Quit
Need to Fuse . . . Examples 8. Non-Fuzzy Case (cont-d) Fusing Expert . . . • A more reasonable option is: Example Need to Consider the . . . – to conclude that, yes, both experts cannot be true, In General, How . . . but How to Define Degree . . . – we cannot conclude that both are wrong. Resulting Definition of . . . • They are experts after all, so it is reasonable to assume Discussion that one of them is right. Home Page • In this case, the result of the fusion is the union S 1 ∪ S 2 Title Page of the two sets. ◭◭ ◮◮ • In other words, the fusion S 1 f S 2 of the sets S 1 and S 2 ◭ ◮ has the following form; Page 9 of 28 – if S 1 ∩ S 2 � = ∅ , then S 1 f S 2 = S 1 ∩ S 2 ; Go Back – otherwise, if S 1 ∩ S 2 = ∅ , then S 1 f S 2 = S 1 ∪ S 2 . Full Screen Close Quit
Need to Fuse . . . Examples 9. Example Fusing Expert . . . • Suppose that: Example Need to Consider the . . . – one expert says that the temperature is between 22 In General, How . . . and 25, and How to Define Degree . . . – another one claims that it is between 18 and 21. Resulting Definition of . . . • In this case, the intersection of the corresponding in- Discussion tervals [22 , 25] and [18 , 21] is empty. Home Page • This means that the experts cannot be both right. Title Page ◭◭ ◮◮ • What we can conclude: ◭ ◮ – if we still believe that one of them is right Page 10 of 28 – is that the temperature is either between 22 and 25 or between 18 and 21. Go Back Full Screen Close Quit
Need to Fuse . . . Examples 10. Need to Consider the Fuzzy Case Fusing Expert . . . • In practice, experts are rarely absolutely confident about Example their opinions. Need to Consider the . . . In General, How . . . • Usually, they are only confident to a certain degree. How to Define Degree . . . • As a result, to adequately describe expert knowledge, Resulting Definition of . . . we need to describe: Discussion Home Page – for each number x , – the degree to which, according to this expert, the Title Page number x is possible. ◭◭ ◮◮ • This is the fuzzy logic approach; in the computer: ◭ ◮ – “true” (= “absolutely certain”) is usually repre- Page 11 of 28 sented as 1, and Go Back – “false” (= “absolutely certain this is false”) is rep- Full Screen resented as 0. Close Quit
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