B - TAIDE Knowledge Graph

Okay, thank you very much for introduction and giving me a chance to talk a little about the Fuzzy System. So, in this case, after the very nice and intriguing presentation of Giovanni, we can look and say okay. Now, we have to build all the algorithms and we have to have the tools to really do it, and one of the things which also Giovanni mentioned is using the Fuzzy-based Control. And, the purpose of that short introduction is not by any means the full introduction to Fuzzy Systems, but the basic thing which allows you to at least start thinking and maybe going deeper into some aspects and potentially build the controlled Fuzzy Systems. The ones which you can use in the robotics. So, this is the reason why I will talk about the introduction to the Fuzzy Systems, and from that perspective, I emphasize the Human-Centric Approach because the Fuzzy Systems are something which is also meant to be an interface between human and machine. So, I will talk a little bit about what the fuzziness is. So, in this case, I will mention you the definition of the fuzzy sets, what are the membership functions, and what are the fuzzy operations. We'll talk about the fuzzy relations and eventually we'll talk about the fuzzy if-then rules which constitute a fuzzy decision system or fuzzy control systems. So, all the aspects which I will cover here will eventually allows you to understand how and the components and the process of building a control system and how eventually it works. So, before we start a little bit of a benefits. I think that you will hopefully see even more and understand more of the things what is beneficial when you use the fuzziness down at the end of the presentation. But, I would like to mention that they come from the aspect of looking and kind of able to express some of the things in the way how we do it. So,  it means that we look at the human intuition, human judgments in many cases, and many of the statements which we make are not very precise. We say that for example, with something like something is large, or something is expensive. This is all imprecise statements or imprecise terms, but this is something where the fuzziness will help us to translate this fuzzy imprecise linguistic terms, or phrases which we human use into the mathematics and into eventually numbers which you can apply and use in your computers or also in the Quantum Computing. So, we will talk from the  benefits from the aspect that we can eventually represent this linguistic terms into numbers and back. So, it also talks about imprecision so anything which is not precise not always as everything is, for example, in the case of the robots, you can say that's the robot is close to some object or it's far from the object. All of the terms, close, far would be nicely represented as fuzzy terms or fuzzy numbers and in this case could be used in the process. So, it's also relatively simple. It's not complicated story. You will see that the concept of fuzzy is kind of a straightforward intuitive, but at the same time quite powerful when we talk about interface between the human and the machine. Multiple different applications. I don't want to spend a lot of time, but in the control systems, in the case of the clustering data analysis,  decision making, pattern recognitions anything where the data are and some of them are involved in the imprecision or maybe missing data or imprecise values this is where we can use fuzziness. So, what it is ? So, let's start with the definition what the fuzzy sets are. So, it was introduced by L. A. Zadeh in 1965 as a method, an approach originally done for or in the kind of the thought idea how the human interacts with the machine. So, it was mostly demand to be used in describing imprecise concept, imprecise terms. So, what did it really mean? So, normally we know the concept of sets. So, in this particular case, if we talk about the crisp sets, kind of a something in the contrast with the fuzzy is something which we know that is a crisp border between concepts or between elements. So, in this particular case, let me just read and explain it. In a crisp set, the element fully belongs to given set, so let's assume that we have a temperature and let me just do it in this way. And, we have three sets of the temperature. One set which is cool so it's kind of represented in this way that anything which is in this range between 0 degree and 15. This is the representation of the set and as you can see here is the membership function because that mean this is a degree to which element belongs to a set. So, if I have a temperature, let's say 10 Celsius. I would say that it belongs to the level of 1.0 so it's fully belonging because that's the maximum degree of belonging to the set cool. I can have also another one which I, for   example, have a hot. Maybe, before we go there. One more thing any temperature in a range above 15 like any temperature here does not belong to the set cool and its membership is zero. So, degree to which this point belongs to the set cool is zero. In other words, it does not belong to the set cool. In the case of the hot, we have a little bit different scenario. In this case, opposite any temperature above, let's say 25 degree, is fully belonging to this set hot. So, if I have a temperature 30° degree, it belongs to 1.0 so that means it's to a degree of one meaning that it's fully belong to the set hot. On the other hand, any temperature below that like 9 or maybe 8 here or 10 or 20 is zero that means that the points don't belong. So, in other words if you think about it, it's kind of like a set which have a hot temperatures, I have a cool, and then if I look at another one which we have here nice which is between 15 and 25. It's kind of like this. So, nice meaning that we have three sets totally separate from each other. But, what the problem is here is that if we look at the temperatures around this area, it's kind of a little bit of a strange to say that if you have a temperature kind of on that side which is let me just go back 14.9, we treat it as a cool but when we have temperature 15.1, we can treat it as a nice. So, the kind of a transition between from one set to another is very crisp so the idea with the fuzzy sets as the name fuzzy in the case which means it's not crisp and is that we have really instead of that have situation like this. So, fuzzy sets that means that the given element belongs to a set to a degree. So let me explain. In the range, here if we look at the set cool, this is very similar story what we had before any temperature below 10 is fully belonging if I can say to the set cool. However, the temperatures which are between 10 and 18, as you can see if this line connects all the points which represent degrees we see that for example temperature 15 belongs to that set cool maybe to a degree of 0.4 so that means that there is a area in this particular case, the area here is this between 10 and 18 in which the point somehow belong to two different sets at the same time but to different degree. So, that means that if we look and we go back to set nice. If you look at this temperature 15, it belongs also to a degree 0.6 to this set nice but it also belongs to the degree of 0.4 to the set cool. So, that means that now we have a point which can belong to different degrees to different sets. So as we had this this scenario before, now when we have this we will do it if this is nice we have some area , we have some area in which these points belong to two sets. Normally, you would you know from the set theory that in this case it would be intersection between both but in our case we have intersection but the degree to which a given point belongs to the nice or cool is different and that is where the fuzziness comes in the place. So, as you can see, these functions or this functions which are these curves which or in this case the lines which represent the membership. I really identify that in this areas, we have really fuzziness. So, the border is not crisp but it's fuzzy, the points belong to two. So, this is the idea of the fuzzy set. So, let me just go a little bit more just to clear some ideas of the things. So, how you represent that set. So, in a crisp set, if we have this in this particular case, it was nice. So, let me just go and do it this way so if you have this nice set, this like this, so that means that with these points are 0, these points belong 1, and again these points belong to the set nice to degree of 0 or 1. And, you can see this a little bit more in a mathematical way that if it's just 1 or just 0 depending if the t is in this particular range when we have eventually definition of our set nice. So, think about this way, set of the temperatures. This is set of the temperatures which we call nice. In the case of the fuzziness, it is a little bit different. So, in this particular case, as you can see we have the area when the membership is 1 so that means that the given temperature belong to the set nice to a full degree. We have temperatures which don't belong at all and we have temperatures, they belong to a degree and in this case we call these values, membership values. So, now these all red points when you push them on y, we call the membership values. Okay. So, that means that now when we have a point, we can say okay it belongs to a set to a degree 1 or belongs to a set to degree 0 or belongs to degree 0.7 or 0.3 or 0.2 and this is how we eventually do it. So, this is a little bit of a math behind it that represents this membership value for a given temperature t. We can represent this in a multiple different ways. We can kind of show it in this way, all these points that for example, 5°, which is here the membership is 0, 10° the membership is 0, 12° is a little bit more 0.25, then it's 15° which is this point, then we have two points which are degree 1 and one point which is degree 0.5. So, you see that we can always show it in a way that for a given temperature, we provide the degree to which that temperature belongs to a specific set which we and that's important which we can label human, we, can say okay this is set of temperatures which we represent as a nice for you. Maybe, it's a little bit different for me maybe it's a little bit different but that is also something which is very interesting in the fuzziness that these things you will see in a moment that we call linguistic labels. So, we label each of this set with some kind of a nice with some word which is a meaningful for us. There's also ability to represent this as a function. Here, we had just points but we can also represent it as a as a line in this particular case like this and each of this if you look at that. I will leave that with Professor Lee for you to have also the access to that slide. You will see that if you kind of look and each of these options that will really define each of this part of the piecewise function which has multiple pieces in this case. Okay. So, that's definition. Of course, the shape of that function could be different. You can see that we for example look at the the function like this which is called the trapezoidal one but we can have also ones which are just like a triangular shape, Z-shaped, s-shaped, sigmoid, and gaussian, multiple different things. Which to choose from is one of the things you will learn as one of the approaches when you build the system, you can play around with different shapes and different places where how you shift that function across for example temperatures should it be like that, or should it be like this. This is where the Evolutionary Computing will help you and you will see that on the next presentation. So, as I mentioned earlier, one of the important things was that now we can label each of that fuzzy sets. And, in this particular case, we as human are able to determine eventually and kind of define what does it mean for us to be to have a cool temperature, what is nice, and what is hot for each of us. This place is where the things go. So, maybe for someone the hot will be like this that you know sooner hot. For some of you, it be may be like this and that okay and that is very interesting because in this particular case, you can make this system more personalized, something which depending on the problem, of course something which is more suitable for maybe a different environment, or maybe for a different person who uses the system.  Eventually, you will see that once you define these fuzzy sets or the fuzzy functions because that eventually is each of them could be represented as a function because that's kind of is like this. Then, you can build out of that rules and eventually system. So, let's talk a little bit about to prepare ourselves for building a rule what we can do regarding the fuzzy relations because now we will try to build the fuzzy relation. So, in a crisp world, the mapping is between two domains. So, we all usually say that we have a X and then from X, we go to the Y so that means that we do the mapping between X and Y. In the crisp world, it is the mapping. In the case of the fuzzy, we talk about proposition. So, we talk about the domain and we talk about eventually fuzzy set with the linguistic term. So, we say that U so another for example in our case temperature is and here we use one of the things which we defined earlier for example nice, cool, or hot because that represents three different fuzzy sets defined on temperature. So, in this particular case, we see that temperature is nice. So, that is the proposition, that is a statement which we can use in a fuzzy rule and a fuzzy inferior system and fuzzy control system that each of the values which we for example measure, we can identify where it belongs so temperature is nice, age of the car is new, or maybe Susan is tall or maybe this bag is heavy or something along this line. Of course, you have to know that in this paticular case, we have a temperatures and we have a different fuzzy membership functions for different as we had before different our understanding what does it mean temperature to be nice, cold, or hot. In the case of the age of the car, we can have a age here and we can say that some cars are very new, maybe a not so new, maybe I'm kind of not so old, and maybe one of them old number of these sets on which we define on our domain. It also depends on us. So, there is no a clear rule. There is also something which you can play around and also make it as a thing, as a parameter in your design of the system. But, that is the single proposition. We can combine these proposition positions and you can see that we can use the logic expression. There is some mathematics behind it. You will get that eventually when you go deeper into that. I will not mention to cover this but in this particular case, we have ability to combine two propositions. So, you see here the age is new and the price is high. So, you see that we have two domains. One is the age of the car and the other one is price. We can specify what does it mean price to be high, maybe, the membership function twill be like this, maybe that will be medium, and that would be cheap. we also have the again the aspect of the age, and some of the membership functions associate with that if we have for example new here and here we have a high price. So, that means that if anything is in this range then and that means also in this particular case that we have the price is high. So, now we can combine these. We can say that. That means that we talk about two aspects here and age of the car and price of the car. So two different domains in each of them we define different membership functions, fuzzy functions or fuzzy sets representing different concepts like a price high, medium, or low, or maybe cheap, or depending how you want to name it, or maybe new car, old car, and not so old car, something like that whatever we want. Once we have these, we can build a rule and that is something very very useful and very important in the case of building a system. So, in this particular case, you have a if some of the statements or proposition which we have, then another proposition so what we had here before we define. Okay, this is just a statement which is a conjunction or a logical and between two statements. Now, we can place the statement a little bit  differently. We can say that if something is satisfied then something else will be satisfied. So, in this case, we have the in the same form if the age of the car is new then the price is high. So, you see that now we have a kind of a consequence and we have a condition. So, we condition something if the age of the car is new then the price is high and this is our rule. So, now you see that we have the fuzzy sets defined in the domain, we can make a proposition which says the car is new or the price is high or maybe temperature is low or nice and then we can combine it through the rule. So, in other words, this is a simple if-then rule. Once we have this, now we have the rules, we can do the inference. So, that means that we can eventually induce something from the rules depending what information we have. So, in this case what you see here inference is to combine fuzzy rules and linguistic propositions. So, what does it mean? You have a rule so this is a little bit more on the math side but you see that we have a one proposition and another proposition which eventually leads to this. If we can have a more on the conditional part, we can combine multiple statements. So, we have some facts. Now, we know something about what are the values potentially in this domain. It means for example in A, you will see an example in the moment, it will be clear in this domain or this domain. This domain, we don't know. We just know something which is in the If part of the condition and then what we can do once we know what is happening here here, we can induce what should be our output and that is eventually the way how we can determine the conclusion. So, this is the mathematics behind it. I will just show you a few example or one example which is associated with the car. So, let's assume that this is our rule so we're saying if engine is medium and the size is small and the torque of the engine is small then the price is small. So, in other words, we define a rule. Of course, there is the issue in which we have to decide what does it mean medium for the engines, what does it mean small for the size, and what does it mean small torque, and what does it mean small price. And, we can do this for example with these rules so we see that for the engine medium, we have defined a membership function like this. It means membership function which represents medium, for the size, we have a small like that for the torque small and for the price small. So, these are our membership functions of fuzzy sets which represent each of these particular concepts or linguistic terms or membership functions with medium, small, and small for the engine size, and torque, and price. And, now we have a specific fact, this is our definition of the rule, and now we have a concrete information, the engine is 2,000 cubic centimetres (cc) so that's a 2L engine. In other words, it has four cylinders, and this has a specific torque and that is the issue. Based on that information, we want to identify what is the price and that's where the imprecision also comes into place because we have this process. And, first of all, we look at the engine, the size is two liters, so that means that we have this point so in other words 2,000 or 2L has a small membership into the concept of engine small. Let's say 0.3. In the case of the four cylinders, again the four cylinders for the concept small size has a membership over let's say 0.5. So, you see that each of this concrete information, we call it activates a fuzzy set to a different degree because it is associated where it fits into our fuzzy set. For the torque, small torque, this torque is not a small one, so its membership in that is very small, for example 0.1. Now, what is really happening, we also have a definition of the concept small price. So, what we do is we take a minimum approach, this is one of the approaches, we take a minimum of these three elements and we see this here. And, this red spot or red part of that represents the activation of that function so that means that this is the activation. I don't want to say the level but kind of like a level, activation degree to which that particular concept small price was activated. And, then what we do is we do some process which is called defuzzication so from the point of view of something which is not very precise, we want eventually as a human number. We could leave it as to a price so the output as the indication that price is small to a degree of 0.1. But, that would not be maybe a very useful it could stay and be very valuable but we do this process of going back to the numbers. And, in this case, there are multiple different ways how you can do it that will be for you to kind of look at after. But, in this particular case, we look at the something which is called the center of gravity (COG) or Center of Maximum (CoM). So, in this case, we look at some place where it's more or less equally that it will be balanced and in this for example it will be 20,000$. So, that means that if we have a car which is two liters cc, four cylinders, and torque for 440Nm, it will cost 20,000$ based on that rule. Of course, in the system, you can have multiple rules so that is another thing that you can have multiple rules in one system not just one. Giovanni mentioned that many of the fuzzy systems when you have multiple dimensions, multiple inputs, and you provide quite a number of these fuzzy sets meaning that we have for example like what here. We could have for example three here, three here, but we could have maybe four and three that number of rules goes quickly, goes up very quickly. But, what is the idea when you have this, when you have this, each of this rule is activated to a degree, so I mentioned that this level of activation here is relatively small and we do this for each rule. So, for example, going back and this is a little bit of mathematics how it is done but the idea is that let me just go here. If we have the same thing, the rule which we just talked about here and you see that this is the same thing copied from the other slide. We have just this activation very small activation of the price is small. But, if the same values, now we have the other rule which says engine is large, size is small, and torque is medium, then price is medium. So, now this is the if part and this is then part, so now the same values which we used to determine these points use right now for another. Because this is the input to our system, now system has two rules. This one is we already talked about but now the new rule is also activated to a degree and it's done in the same way, so we look at this point here, this point here, and this point here and this again we take a minimum, now the price is medium is this membership function or this fuzzy set. And, we say okay now this activation is like that and then what is happening quite often the there are some variation but in the principle we take this element and this element and we combine it. As you can see here and we also take the Center of Gravity, sorry Center of Maximum and then we can try to identify that maybe in this particular case, it's about 30,000$. So, this is the way how if you build your system, you eventually define first what are different fuzzy sets in each domain. Domain mean engine, size, torque, and price whatever you are dealing with. And then, you try to identify if the engine is medium, and size is small, and small torque, then the price is small. This is what you set up. And, you can have multiple of these rules then when the input comes, in this particular case, the input will be this value here, this value here, and this value here so these values of course are also here, so then this value goes to each rule and each rule is activated to a different degree. Once this rule is activated to different degree, then we look at the consequence of that rule. And, we specify okay we have also definition of membership functions on the output if we can say output and then we can see to what degree each of the rule activates the output the consequence then we combine it and the you via defuzzification meaning that from that area because that will be very difficult for human to understand what is the meaning of that. We eventually one of the techniques is the center of maximum. We identify what is the precise number from the fuzzy fuzzy fuzzy everything so the input is crisp because we have specific values. We go through the fuzzy process of fuzzifying everything because these are eventually the points which are fuzzy number numbers representing different degrees from that particular fuzzy sets. We get to the output. We get the fuzzy elements from the output this and this we combine it, and eventually get the number at the end. That's in a super natural super short introduction from the introduction to fuzzy sets through the propositions to the rules and to the systems. So, now I hope that that will engage your in the interests how to really go into details because eventually you can build something along these slides. So, you have input, you have the rules, you have this inference engine which I talk about the process, the defuzzification, and then you can control your robots, you can control your decision making anything which involves some kind of a process of what we call fuzzy inference. I would like to emphasize that that's a fuzzy inference, this process which you see here. So, with that I hope that I will generate some interests in fuzziness and hopefully you will be able to get into the details some of the things regarding its mathematics. It is not very complicated but that allows you to and I like to kind of finish with this fuzziness you just learn a few things about it but it allows you eventually to built linguistic or intelligent system using linguistic labels. And with that, I finish and thank you for being letting me go through the whole process. I hope that that was useful to understand what the fuzziness is about in a very short more less half an hour presentation. Thank you very much.