Computer Engineering Commons^™

Interval Methods In Non-Destructive Testing Of Material Structures, Keith Worden, Roberto A. Osegueda, Carlos M. Ferregut, Soheil Nazarian, Debra L. George, Mary J. George, Vladik Kreinovich, Olga Kosheleva, Sergio Cabrera

Departmental Technical Reports (CS)

In many practical situations, e.g., in aerospace applications and in mammography, it is important to test the structural integrity of material structures. We show that interval methods can help.

From Fuzzy Values To Intuitionistic Fuzzy Values To Intuitionistic Fuzzy Intervals Etc.: Can We Get An Arbitrary Ordering?, Vladik Kreinovich, Masao Mukaidono, Krassimir Atanassov

Departmental Technical Reports (CS)

Traditional fuzzy logic uses real numbers as truth values. This description is not always adequate, so in intuitionistic fuzzy logic, we use pairs of real numbers to describe a truth value. Such pairs can be described either as pairs (t,f) for which t+f<=1, or, alternatively, as pairs (t,1-f) for which t<=1-f. To make this description even more adequate, instead of using real numbers to described each value t and f, we can use intervals, and thus get interval-valued intuitionistic fuzzy values which can be described by 4 real numbers each. We can iterate this procedure again and again. The question is: can we get an arbitrary partially ordered set in this manner? An arbitrary lattice? In this paper, we show that although we cannot thus generate arbitrary lattices, we can actually generate an arbitrary partially ordered set in this manner. In this sense, the "intervalization" operation which underlies the notion of an intuitionistic fuzzy set, is indeed universal.

From Interval Methods Of Representing Uncertainty To A General Description Of Uncertainty, Vladik Kreinovich, Scott Ferson, Lev Ginzburg, Harry Schulte, Matthew R. Barry, Hung T. Nguyen

Departmental Technical Reports (CS)

Measurements do not result in an exact value of the measured quantity; even after the most accurate measurement, there is still some uncertainty about the actual value of the measured quantity. Traditionally, in science and engineering, this uncertainty is characterized by a probability distribution; however, often, we do not know this probability distribution exactly. So, to get a more adequate description of this uncertainty, we must consider classes of possible probability distributions. A natural question is: Are all possible classes needed for this description? In this paper, we show that even for simple situations, we indeed need arbitrary closed convex …

On Combining Statistical And Fuzzy Techniques: Detection Of Business Cycles From Uncertain Data, Hung T. Nguyen, Berlin Wu, Vladik Kreinovich

Departmental Technical Reports (CS)

Detecting the beginning and the end of the business cycle is an important and difficult economic problem. One of the reasons why this problem is difficult is that for each year, we have only expert estimates (subjective probabilities) indicating to what extent the economy was in growth or recession. In our previous papers, we used fuzzy techniques to process this uncertain information; namely, we used the operation min(a,b) to combine the subjective probabilities (expert estimates) of two events into a probability that both events happen. This function corresponds to the most optimistic estimate of the joint probability. In this paper, …

Chu Spaces - A New Approach To Describing Uncertainty In Systems, Vladik Kreinovich, Guoqing Liu, Hung T. Nguyen

Departmental Technical Reports (CS)

This paper proposes the use of a specific type of categories for modeling and fusing information in complex systems in which uncertainty of various types need to be taken into account.

Neural Network Approach To Speech Pathology, Antonio P. Salvatore, Nicole A. Thorne, Charlotte M. Gross

Departmental Technical Reports (CS)

A speech problem can be caused by different reasons, from psychological to organic. The existing diagnostic of speech pathologies relies on skilled doctors who can often diagnose by simply listening to the patient. We show that neural networks can simulate this ability and thus provide an automated (preliminary) diagnosis.

Kolmogorov Complexity-Based Ideas For Locating Text In Web Images, Martin Schmidt, Vladik Kreinovich, Luc Longpre

Departmental Technical Reports (CS)

The gaining popularity of the World Wide Web increases security risks. Search tools monitor plain text in web pages, but search for text in graphical images is still difficult. For this search, we use the fact that the compressed images with text have different size than images without text.

Theoretical Explanation For The Empirical Probability Of Detection (Pod) Curve: A Neural Network-Motivated Approach, Yohans Mendoza, Roberto Osegueda

Departmental Technical Reports (CS)

For non-destructive testing of aerospace structures, it is extremely important to know how the probability of detecting a fault depends on its size. Recently, an empirical formula has been found which described this dependence. In this paper, we provide the theoretical justification for this formula by using methods motivated by the neural network approach.

Fuzzy Logic In Non-Destructive Testing Of Aerospace Structures, Murali Krishna, Vladik Kreinovich, Roberto Osegueda

Departmental Technical Reports (CS)

In nondestructive testing, to locate the faults, we send an ultrasonic signal and measure the resulting vibration at different points. To describe and combine the uncertainty corresponding to different measurements and fuzzy estimates, we used fuzzy logic. As a result, we get reasonably simple computational models which lead to as good fault detection as the known more complicated models.

Decision Making Under Interval Probabilities, Ronald R. Yager, Vladik Kreinovich

Departmental Technical Reports (CS)

If we know the probabilities p(1),...,p(n) of different situations s1,...,sn, then we can choose a decision Ai for which the expected benefit C(i)=p(1)*c(i,1)+...+p(n)*c(i,n) takes the largest possible value, where c(i,j) denotes the benefit of decision Ai in situation sj. In many real life situations, however, we do not know the exact values of the probabilities p(j); we only know the intervals [p-(j),p+(j)] of possible values of these probabilities. In order to make decisions under such interval probabilities, we would like to generalize the notion of expected benefits to interval probabilities. In this paper, we show that natural requirements lead to …

Fuzzy/Probability ~ Fractal/Smooth, Hung T. Nguyen, Vladik Kreinovich, Berlin Wu

Departmental Technical Reports (CS)

Many applications of probability theory are based on the assumption that, as the number of cases increase, the relative frequency of cases with a certain property tends to a number - probability that this property is true. L. Zadeh has shown that in many real-life situations, the frequency oscillates and does not converge at all. It is very difficult to describe such situations by using methods from traditional probability theory. Fuzzy logic is not based on any convergence assumptions and therefore, provides a natural description of such situations. However, a natural next question arises: how can we describe this oscillating …

Multi-Resolution Techniques In The Rules-Based Intelligent Control Systems: A Universal Approximation Result, Yeung Yam, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

Intelligent control is a very successful method of transforming expert knowledge of control rules (formulated in terms of natural language, like "small") into a precise control strategy. It has led to many spectacular applications, ranging from appliances to automatic subway control to super-precise temperature control on a Space Shuttle mission.

It is known that fuzzy control is a universal approximator, i.e., that it can approximate every possible control strategy within an arbitrary accuracy. One of the main problems of fuzzy control is that the number of rules which are necessary to represent a given control strategy with a given accuracy, …

Multi-Resolution Methods In Non-Destructive Testing Of Aerospace Structures And In Medicine, Roberto Osegueda, Yohans Mendoza, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A fault in an aerospace structure can lead to catastrophic consequences; therefore, it is extremely important to test these structures regularly. Thorough testing of a huge aerospace structures results in a large amount of data, and processing this data takes a lot of time. To decrease the processing time, we use a "multi-resolution" technique, in which we first separate the data into data corresponding to different vibration modes, and then combine these data together. There are many possible ways to transform each mode's data into the probability of a fault, and many possible way of combining these mode-based probabilities; different …

Chu Spaces - A New Approach To Diagnostic Information Fusion, Hung T. Nguyen, Vladik Kreinovich, Berlin Wu

Departmental Technical Reports (CS)

This paper is rather theoretical. Its aim is to describe a general algebraic framework, known as Chu spaces, in which different type of information can be transformed into the same form, so that fusion procedures can be investigated in a single general framework.

We Live In The Best Of Possible Worlds: A Proof, Guoqing Liu, Vladik Kreinovich

Departmental Technical Reports (CS)

It is well known that equations of motions and equations which describe the dynamics of physical fields can be deduced from the condition the action S (determined by the corresponding Lagrange function) is optimal. In other words, there is an optimality criterion on the set of all trajectories, and the actual trajectory is optimal with respect to this criterion.

The next reasonable question is: where does this optimality criterion on the set of all trajectories (i.e., the corresponding Lagrange function) come from? It is reasonable to assume that (similarly) on the set of all Lagrange functions, there is an optimality …

Locating The Whole Pattern Is Better Than Locating Its Pieces: A Geometric Explanation Of An Empirical Phenomenon, Scott A. Starks, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical problems, we must find a pattern in an image. For situations in which the desired pattern consists of several simple components, the traditional approach is first to look for such components, and then to see whether the relative locations of these components are consistent with the pattern. Recent experiments have shown that a much more efficient pattern recognition can be achieved if we look for the whole pattern (without decomposing it first). In this paper, we give a simple geometric explanation of this empirical fact.

Arithmetic Of Complex Sets: Nickel's Classical Paper Revisited From A Geometric Viewpoint, Vladik Kreinovich, Juergen Wolff Von Gudenberg

Departmental Technical Reports (CS)

Due to measurement uncertainty, after measuring a value of a physical quantity (or quantities), we do not get its exact value, we only get a set of possible values of this quantity (quantities). In case of 1-D quantities, we get an interval of possible values. It is known that the family of all real intervals is closed under point-wise arithmetic operations (+,-,*) (i.e., this family forms an arithmetic). This closeness is efficiently used to estimate the set of possible values for y=f(x1,...,xn) from the known sets of possible values for xi.

In some practical problems, physical quantities are complex-valued; it …

Chu Spaces: Towards New Foundations For Fuzzy Logic And Fuzzy Control, With Applications To Information Flow On The World Wide Web, Hung T. Nguyen, Vladik Kreinovich, Guoqing Liu

Departmental Technical Reports (CS)

We show that Chu spaces, a new formalism used to describe parallelism and information flow, provide uniform explanations for different choices of fuzzy methodology, such as choices of fuzzy logical operations, of membership functions, of defuzzification, etc.

Towards Faster, Smoother, And More Compact Fuzzy Approximation, With An Application To Non-Destructive Evaluation Of Space Shuttle's Structural Integrity, Yeung Yam, Roberto Osegueda, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that fuzzy systems are universal approximators, i.e., any input-output system can be approximated, within any given accuracy, by a system described by fuzzy rules. Fuzzy rules work well in many practical applications. However, in some applications, the existing fuzzy rule approximation techniques are not sufficient:

First, in many practical problems (e.g., in many control applications), derivatives of the approximated function are very important, and so, we want not only the approximating function to be close to the approximated one, but we also want their derivatives to be close; however, standard fuzzy approximation techniques do not guarantee the …

A New Look At Fuzzy Theory Via Chu Spaces, Hung T. Nguyen, Berlin Wu, Vladik Kreinovich

Departmental Technical Reports (CS)

We propose to use Chu categories as a general framework for uncertainty analysis, with a special attention to fuzzy theory. We emphasize the fact that by viewing fuzzy concepts as Chu spaces, we can discover new aggregation operators, and model interactions and relationship between fuzzy data; these possibilities are due, in essence, to the category structure of Chu spaces, and especially to their morphisms. This paper is a tutorial introduction to the subject.

An Optimality Criterion For Arithmetic Of Complex Sets, Vladik Kreinovich, Juergen Wolff Von Gudenberg

Departmental Technical Reports (CS)

Uncertainty of measuring complex-valued physical quantities can be described by complex sets. These sets can have complicated shapes, so we would like to find a good approximating family of sets. Which approximating family is the best? We reduce the corresponding optimization problem to a geometric one: namely, we prove that, under some reasonable conditions, an optimal family must be shift-, rotation- and scale-invariant. We then use this geometric reduction to conclude that the best approximating low-dimensional families consist of sets with linear or circular boundaries. This result is consistent with the fact that such sets have indeed been successful in …

On Average Bit Complexity Of Interval Arithmetic, Chadi Hamzo, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we know only the intervals which contain the actual (unknown) values of physical quantities. If we know the intervals [x] for a quantity x and [y] for another quantity y, then, for every arithmetic operation *, the set of possible values of x*y also forms an interval; the operations leading from [x] and [y] to this new interval are called interval arithmetic operations. For addition and subtraction, corresponding interval operations consist of two corresponding operations with real numbers, so there is no hope of making them faster. The best known algorithms for interval multiplication consists of …

Towards Intelligent Virtual Environment For Training Medical Doctors In Surgical Pain Relief, Richard Alo, Kenneth Alo, Vladik Kreinovich

Departmental Technical Reports (CS)

Chronic pain is a serious health problem affecting millions of people worldwide. Spinal cord stimulation is one of the most effective methods of easing the chronic pain. For most patients, a careful selection of weak electric currents drastically decreases the pain level. Engineering progress leads to more and more flexible devices that offer a wide variety of millions of possible simulation regimes. It is not possible to test all of them on each patient, we need an intelligent method of choosing an appropriate simulation regime. In this paper, we describe the need for an intelligent virtual environment for training medical …

System Reliability: A Case When Fuzzy Logic Enhances Probability Theory's Ability To Deal With Real-World Problems, Timothy J. Ross, Carlos M. Ferregut, Roberto Osegueda, Vladik Kreinovich

Departmental Technical Reports (CS)

In his recent paper "Probability theory needs an infusion of fuzzy logic to enhance its ability to deal with real-world problems", L. Zadeh explains that probability theory needs an infusion of fuzzy logic to enhance its ability to deal with real-world problems. In this talk, we give an example of a real-world problem for which such an infusion is indeed successful: the problem of system reliability.

From Fuzzy Models To Fuzzy Control, Chitta Baral, Vladik Kreinovich, Hung T. Nguyen, Yeung Yam

Departmental Technical Reports (CS)

Traditional (non-fuzzy) control methodology deals with situations when we know exactly how the system behaves and how it will react to different controls, and we want to choose an appropriate control strategy. This methodology enables us to transform the description of the plant's (system's) behavior into an appropriate control strategy. In many practical situations, we do not have the exact knowledge of the system's behavior, but we have expert-supplied fuzzy rules which describe this behavior. In such situations, it is desirable to transform these description rules into rules describing control. There exist several reasonable heuristics for such transformation; however, the …

Intervals Is All We Need: An Argument, Masao Mukaidono, Yeung Yam, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications of fuzzy methodology, it is desirable to go beyond the interval [0,1] and to consider more general fuzzy values: e.g., intervals, or more general sets of values. In this paper, we show that under some reasonable assumptions, there is no need to go beyond intervals.

Why Clustering In Function Approximation? Theoretical Explanation, Vladik Kreinovich, Yeung Yam

Departmental Technical Reports (CS)

Function approximation is a very important practical problem: in many practical applications, we know the exact form of the functional dependence y=f(x1,...,xn) between physical quantities, but this exact dependence is complicated, so we need a lot of computer space to store it, and a lot of time to process it, i.e., to predict y from the given xi. It is therefore necessary to find a simpler approximate expression g(x1,...,xn) for this same dependence. This problem has been analyzed in numerical mathematics for several centuries, and it is, therefore, one of the most thoroughly analyzed problems of applied mathematics. There are …

Beyond [0,1] To Intervals And Further: Do We Need All New Fuzzy Values?, Yeung Yam, Masao Mukaidono, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications of fuzzy methodology, it is desirable to go beyond the interval [0,1] and to consider more general fuzzy values: e.g., intervals, or real numbers outside the interval [0,1]. When we increase the set of possible fuzzy values, we thus increase the number of bits necessary to store each degree, and therefore, increase the computation time which is needed to process these degrees. Since in many applications, it is crucial to get the result on time, it is therefore desirable to make the smallest possible increase. In this paper, we describe such smallest possible increases.

Why Fundamental Physical Equations Are Of Second Order?, Takeshi Yamakawa, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we use a deep mathematical result (namely, a minor modification of Kolmogorov's solution to Hilbert's 13th problem) to explain why fundamental physical equations are of second order. This same result explain why all these fundamental equations naturally lead to non-smooth solutions like singularity.

Fuzzy Systems Are Universal Approximators For A Smooth Function And Its Derivatives, Vladik Kreinovich, Hung T. Nguyen, Yeung Yam

Departmental Technical Reports (CS)

One of the reasons why fuzzy methodology is successful is that fuzzy systems are universal approximators, i.e., that we can approximate an arbitrary continuous function within any given accuracy by a fuzzy system. In some practical applications (e.g., in control), it is desirable to approximate not only the original function, but also its derivatives (so that, e.g., a fuzzy control approximating a smooth control will also be smooth). In our paper, we show that for any given accuracy, we can approximate an arbitrary smooth function by a fuzzy systems so that not only the function is approximated within this accuracy, …