A - TAIDE Knowledge Graph

Hello everyone. My name is Marek Reformat. I am from University of Alberta, Canada. So, I will give you a little bit of a presentation about fuzzy set. So, what the fuzzy set is about? How you think about the fuzzy sets? Eventually, how you can use it to build the rule which you can apply to your applications. So, say things about what this fuzziness? The basic things so you know how to build and what the fuzzy sets is. I would say a few things about most propositions so the sentences which have the fuzzy sets so these sentences are used to build the if-then rules which you use in your fuzzy systems. So, what really is fuzziness about? It's close to the human judgment and intuition because many of the things which we say are not very precise and fuzzy sets are about not being very precise in the case of representing information. Of course, when it comes to the computations and calculations we have to be precise but the fuzzy sets allow us to bridge from very imprecise language which we human use to something which machines can understand and do the calculations. So, we use something which is called the linguistic terms so if you say that something is large or tall or it's cold or it's warm. That's the linguistic terms which we eventually translate into a fuzzy sets and then we use that to caculation. And, you see that this turns to numbers and back because we can also convert back the numbers into this linguistic terms so human can be easier to understand some of themselves. Concel is imprecision so it hides imprecision if we say that something is tall or something is hot you don't exactly specify what does it mean hot or tall. For us human, we kind of know what it is because from our experience, from our education, and whatever we were taught, we know what is tall, what is hot but that is a very imprecise statement which is hidden behind the fuzzy set. It's also relatively to be simple. You will see in a moment that it's not super complex. You can use it with more complex things down the road but the beginnings of the basics are personally. A lot of different applications so you know that fuzziness is really something which allows us to build interesting systems. And, we can do the things for the control for putting things together in the case of the clustering. That's kind of a more scientific term, decision making individual or with the groups different pattern recognition, and classification. This is all the places where the fuzziness and fuzzy sets could be used. so what really happened have it started? It started in 1965 close to sixty years ago when a lot designer introduced the concept of a fuzzy set and that is the the beginning of our kind of adventure of the fuzzy systems. So, what is really the fuzzy set? We know the concept of set. It's called the normal sets which we deal with. It is a crisp set so that means that the element fully belongs to a set or not. So, for example, if you have a set of all of us in the room, it's a crisp set because we can count each of us and all we are in the room or we are not in the room so there is no other option. We are in the room or outside room so that's a crisp set. Crisp set of all of us in the room And, in the case of the temperature, for example, we can say that anything from 0 to 15 is a cool, temperature between 15 and 25 is nice, temperature above 35 is hot. 000 It's a crisp. We have a crisp boundary between what they say as a cool versus what they say and that's a crisp set. They are crisp sets. So fuzzy sets present the state of element that belongs to a set to a degree so we cannot say that we present the fuzzy set because we do not belong to that room to a degree. So, the easiest or the nicest way to represent that is on the temperature. So, when you go back to the crisp situation, the interesting story happens kind of on the border between two sets, sets of the cool temperatures and sets of the nice temperatures, all the temperatures here belong to the set of cool but it really is very difficult to say that okay temperature is 15.5, it's already nice but if it's 14.5 it's cool. So, the idea is that we have the border which is not crisp but fuzzy. So, what really does mean that now we have different degrees of belonging to a set before we had just 001 something belong to the set or does not belong to the set. Right now, for example, looking at that visual representation of the processors you see that any temperatures between 0 and 10 if I look at this area here of the temperatures we see before we start I should say one more thing that the Y represents the degree to which element belongs to us so all the temperatures from 0 to 10 belong to the set cool, the degree of one. on the other hand, if we can look at this line here the temperatures between 10 to 18 belong to the set cool to a degree and that is the idea of fuzziness. Now we have some elements which belong to a set to a specific integrity. As you can see what is another interesting story is that if I look at another temperature nice, we look at the temperatures between 18 and 23 have fully belonged to the set nice but the temperatures between 10 and 18 not only belonged to the cold but they also belong these temperatures that we think about this points here as a degree to which element belongs to the set cool also to be nice. So, this is really where the fuzziness comes into the place but there are elements here that belong to a specific degree to one of the sets like cool as well as to another set of nice. And, that brings this imprecision and the fact that look everything is always 0 or 1. We have also degrees of belonging and that happens there will be example in the moments. I hope that would be you will see that again. So, in a crisp situation if you have a temperature nice, the set nice is that temperature, 15 or 18, this is just examples and that means that all we have belonging to one that means that we are here when the temperature is in that set if the temperature is not here or not between between 15 and 25 then the membership degree should be zero. The fuzziness and this is the situation yes the temperature here if we look at it just the nice, maybe I should forget about the cool and hot. If you don't get the nice, these temperatures do not belong to the set of the temperatures here. but they are here and then the membership means they belong to that set to the degree of one and the fuzziness situation like that the same temperatures if you take the same points and the same values now they look like this. These two they cannot belong to the set nice at all, these two belong to the fuzzy set nice but these two on that and this belong to a degree they belong to a degree to this positive nice. They also if you look at this if I look at this temperature and you see it crosses this line here for the fuzzy set nice. But if I go further it crosses here for the fuzzy set cool so that means that this temperature belongs to at the same time to two sets parts to a different degree lower degree to nice higher degree to cool and that is the essence of the whole fuzziness. So, then how we can represent it? Quite often we represent it when we have adjust specific temperatures like this. Now, the temperature 5 does not belong to the set nice. Temperature 10 also does not but temperature 12 belongs to the degree of 0.25 and this is the point to the set nice. Temperature 15 to the 0.65 then we have temperatures 18 and 23 are 1.0 and temperature 25 is here and it's 0.5. So, that represents the fuzzy set. Now when we talk about numbers. This is when we have discrete values, like 5, 10, 12, 15, and so on, but we can also have a function and quite often we talk about them function which we call them membership function. As the name in the gates it tells to what degree even element is a member of the function and this case we represent it as a function. Some kind of an indication that and the values between below 10, and the values between 10 and 18 is some kind of a function which in this case is just a simple linear function. Then, we have values between 18 and 23 always here one and then another part which is just increasing function of the slope 23 2o 22. Beyond that is again and that is also membership function but now we call it at the continuous membership function because if you're using your program some representation of temperature like in this case. Eventually, you will use some kind of function, I think that picture you should seen also in your own system. And, that could be a different shapes of this function depending on how we use these functions for what purpose. You have functions like this and you can have a function like that. Sometimes the function is one of the things which you should modify. We have some systems in which you cannot specify that the function is always like this but the function can change. We're going to shave its position because we want the function to represent information on one side. On the other hand, we want the functions to be such that our goal which is maybe the control system or maybe the decision support system, works at the best as possible. But, that's the beginning. Just one more thing, just to finish with the things which you'll eventually be using in the systems. Maybe, before that I can use that implicitly or a few times but one of the things which I mentioned is the concept of the linguistic terms. Because we are humans, we don't want to say this is the function, for example behaves like this, this is the function which have 0 here something between 0 and 1 here and 1 we want to label that. And, we call this linguistic terms. Linguistic is a language term, just a term award in this case, so we call it. But then when we talk about it, we call that the temperature is cool it's nice, it's hot, and that's all what we need. But then of course these terms are represented as functions fuzzy sets or fuzzy functions, or membership functions. So that's the beginning, then I have to mention a few words because I want to kind of end up with the rules. So, normally we also have propositions, it's a map from the logic point of view but kind of a statement. So, maybe the crisp statement, they would be like in shown this in Korea. That's the crisp state yes and we have on one side sets of cities, on another side set of example countries and we say that Incheon is in Korea. But we can also do that in the terms which are very useful when you use the linguistic terms. So, in this particular case, we have a phrases like this. This is the one of the most simplests and in this case, the example of the temperature is nice, and age of the car is new, so we have a statement which tells something from a given domain so that in this case, domain of temperatures or domian of cars and here we have information about how we describe after some degree to this domain. And, you see that in this particular case, if this is clear, the indication what we talk about is something less precise because this is something which we have defined in our concept so with that being nice without being new. That is, for example, John or Susan is tall that would be statements that we describe. But, we use the fuzzy linguistic terms which they have a presentation with functions because the membership functions. When we have that we can combine. We have a logical statement, a logical operator in this case is 'and' so you can say that the age of the car is new and price is high and again here we talk about the car and we describe it with the fuzzy statement. Here, we have another thing which is the price of the car and we also use this fuzzy representation in this case. What does the mean price high? We have got probably have to define our function but for each of us, it will be a little bit different, some person will be high price. I know 20,000 or something 10,000 depending what is the currency and what we or what the person thinks and how much money he has. Using the same kind of approach means that we can have the same statements about some kind of element using describing it with the function, the fuzzy set. We can put them together and create something which is called the if-then roots so you see that in the red that we specify condition if the age of the car is new then once this condition is satisfied you're saying that the price is high. This is our things which you use very often to build a system. That means that you try to describe your input and then depending if the input satisfies that condition then we can say that this has happened. So, in this particular case if the ages of the car is new then the price is high. Something like that is used to build the if-then rule. As you can see if and then, called the if-then rule when you have a few of these rules, you build a system. And, that's kind of what is the last thing I want to show you but let's keep that part. This is the most interesting thing So, I have a few rules. Each of you is given me some outlook. But, each of the rule is satisfied to a different condition because as we had before stand up this one. If you think about this new I could have a membership function. Sometimes if the price if the example is a brand new car probably this level of degree to which this statement satisfied the rule is 1. But, if the car is, for example, five years old, it's not really new now that's all at old, maybe the degree to which this is satisfied is 0.25 or maybe 0.4 depending what you think is the new car. But, in the case of the car which is 15 years old that would be 0 but that means that in this case the age is 15 that means that the car is not new anymore. This rule we call it not fired. When we call that the rule is satisfied that condition, we call that the rule is fired. If this condition is not satisfied, the rule is not fired. So, it means that there's no fuzzy at all because this condition is not satisfied. And, then because we have the fuzziness, then this rule could be satisfied to a degree or 0 or 1. One will be when this is really new. Zero will be the car is not new. It's something between would be also indication that the rule is fired to a degree. Therefore, if you have a few of these rules, and maybe I will jump here that would be because indication. So very time specifically I went with the cars. I have a three inputs, engine, size of the car, and what does the part how engine is the strong. I can have some functions which I haven't showed you here but I showed you a few things because I have a rule. I have a rule A. It says that the engine is medium if it sizes very small and the torque is small, then its price is small. And, I have another rule which says rule B that if the engine is large, the size is small, and torque is medium, then its price is medium. Now, each of these terms which is linguistic term. It is represented by a function so medium engine is represented by this function, engine large is represented by this function, and small size by this, very small is by that, small torque is by this, and medium is like that. Now, I have some data. I have this engine, size of the engine is here, this is my value, this is my different values of the size, size of the car is here, the torque of the engine is here, and what happens. This line meets somewhere in our fuzzy set. Now this one is here, this meets here, this size, specific size, which is here meets this line of this model of this membership function of this fuzzy set here. This one here, this one on the other hand is here, and here, so why these two are red not the best because that is the issue that this rule indicates that if the engine is medium and the size is very small and the torque is small then the price is small. So, that is the issue to what degree each of this input satisfies each of these parts of our statement. This one is here, this one is here, this one is here and usually what we look upon here is what is the meaning. This is the reason why this has pointed out because if I put the line here this line is below this point and also is below this one. So, this rule even if this size satisfies that very small group higher degree, we only look at the minimum level. So, in this particular case, this rule is satisfied to that degree. In this particular case, perhaps scenario that it happened that this is also the new because if I cross this line with this, it is above this dashed one. Here is also about that. So, now what happens is I'm taking this point and this point as my indication to what degree rule A is satisfied that will be just not very low level and it will be satisfied without that. There is the old output so now is the price small and the price is impacted so what really happens is that this is the membership function representing this small price. This is the membership function representing medium price. Now, this rule was satisfied with that so what really happens is kind of push this line further to the output and this is the area which I'm taking to build my output because only that level of this price is small is taken into consideration. Here, this level is higher so I'm taking this one because this is really indication that this rule is fired to that degree and now this output price is medium. It's kind of true. so I'm taking the whole area because it is application So, then what happens I'm taking this together and really I have a kind of a strange kind of a curve of almost like this because I combine this together. But, then what this is happening is we have a process which is called the defuzzification. So, that means that from this fuzzy sets we want to get the real life. And, in this case, this is the indication that we have a specific value of the engine the size, and the torque, we kept that our system we push the data through the system. We obtain eventually a value which says if this system or if this kind of a way how we present this information what is the input if we get this satisfied to a specific degree and eventually that would be about and that is the basic of many things which we can use. so jumping jumping Wait for a moment because this afternoon, we will have the competition. For, the elementary-school students, at least two inputs and one output. For the high-school and the undergraduate students, three or four inputs. I explain in Chinese. I say in Chinese. 今天下午比賽結束，我們會馬上評分 我們昨天已經討論出評分的標準 所以等一下下午小學生有4組 至少要兩個輸入變數 一個輸出變數 然後解釋 你的技術深度 等一下，你要想一下你的題目 然後下午馬上做 完成後就會馬上評分 我們會有很多教授在下面評分 問問題 完成評分後，名字次會出來 所以，小學生至少要有兩個輸入 兩個輸入 如果有三個更好 最多四個 會及時收資料 會透過這個工具 So, the students will collect data by this one. 所以下午你們要自己訓練模型 等一下，Yusuke老師還會再介紹 So, later Prof. Yusuke Nojima will introduce how to collect the data and construct the model using PSO. So, for the elementary-school students should have at least two inputs and you can explain why, then I think is good. We hope you know what, what to do, and how to do. 我們知道 我們希望小朋友能夠知道你要做什麼 你要解決什麼問題 比如說預測系統 或者是同學要做智慧馬桶 比如說那個就想那些很有趣的題目 如果能夠解釋的話 那個Marek老師在解釋這一個 就是你們去接下來學物 理就會學到重心 就找到這兩個的重心 你們到高中的時候就學物理 這就是解模糊化 那我們明年在日本就是要用量子 現在還沒有量子電腦 現在有模擬器我們用歐盟 就是未來Quantum AI跟Quantum CI 那計算智慧會快很多 就不用再用GPU 現在用GPU要耗費很多能源 So, in the future, next year, we hope we have the quantum computational intelligence. Maybe, five years later or ten years later, we will have quantum computer. I think the world will change very very faster. I think this is very important for you so this afternoon, one team will present five minutes and 2-min Q&A. 下午每組報告只有5分鐘 然後再有2到3分鐘Q&A 然後我們進行評分 所以你們需要了解這個原理 如果不了解的話 等一下要討論 我們希望你們除了會使用那些工具 So, later you will use the tool to construct the fuzzy system. I think this is very important for the young students. 以人類的human language 人類表達現在很冷或很熱 今天有點冷或有點熱 人類不會使用非常明確的數字 不是25度、26度、27度 這樣的想法有了量子電腦 量子技術使我們認為未來可能會出現這種情況 這個理論應該很快就會實現 出來 I think human language 這是在1965年 美國加州柏克萊大學的 Zadeh教授 提出來的一個想法和理論 在我們這個領域已經很久 因為最近的電腦速度快很多 So, I think engine engine is large and size is small and torque is medium then price is medium we give. That's the language. But the language is fuzzy, and from the next presentation, the adjustment where was essential look like where should be, could be done. okay so thank you okay. I think that's more or less and then based on that, we can build the whole fuzzy system in which we can have a different input, the process of making from the crisp to the fuzziness, we have different rules, and then we put them together, and we get the defuzzication. Then, we have a certainty in the case of the control and then again and again. So, these fuzzy sets and different rules are making elements for any quite a complex fuzzy system. What is the fuzziness next? Just to send you the fuzziness. Next, you just learn a few things about it so far but this enables you to build comparative systems with linguistic terms. And, another thing is important that we had the human language, human phrases, human statements, and human description of things, we can use together and eventually beyond the intelligence. Okay, so that's all. Thank you very much.