Physical Sciences and Mathematics Commons^™

How Resilient Modulus Of A Pavement Depends On Moisture Level: Towards A Theoretical Justification Of A Practically Important Empirical Formula, Pedro Barragan Olague, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Resilient modulus is a mechanical characteristic describing the stiffness of a pavement. Its value depends on the moisture level. In pavement construction, it is important to be able, knowing the resilient modulus corresponding to one moisture level, to predict resilient modulus corresponding to other moisture levels. There exists an empirical formula for this prediction. In this paper, we provide a possible theoretical explanation for this empirical formula.

What If We Use Different "And"-Operations In The Same Expert System, Mahdokhat Afravi, Vladik Kreinovich

Departmental Technical Reports (CS)

In expert systems, we often face a problem of estimating the expert's degree of confidence in a composite statement A & B based on the known expert's degrees of confidence a = d(A) and b = d(B) in individual statements A and B. The corresponding estimate f_&(a,b) is sometimes called an "and"-operation. Traditional fuzzy logic assumes that the same "and"-operation is applied to all pairs of statements. In this case, it is reasonable to justify that the "and"-operation be associative; such "and"-operations are known as t-norms. In practice, however, in different areas, different "and"-operations provide a good description …

Computers Of Generation Omega And The Future Of Computing, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

George Klir And The Great Chain Of Ideas, Vladik Kreinovich

Departmental Technical Reports (CS)

Earlier this year, I received sad news that my good friend and dear colleague George Klir is no more. George was a great scientist -- and, in my opinion, a greatly underestimated one. I therefore believe that it will be beneficial to explain his true role in science.

Why Compaction Meter Value (Cmv) Is A Good Measure Of Pavement Stiffness: Towards A Possible Theoretical Explanation, Andrzej Pownuk, Pedro Barragan Olague, Vladik Kreinovich

Departmental Technical Reports (CS)

To measure stiffness of the compacted pavement, practitioners use the Compaction Meter Value (CMV); a ratio between the amplitude for the first harmonic of the compactor's acceleration and the amplitude corresponding to the vibration frequency. Numerous experiments show that CMV is highly correlated with the pavement stiffness, but as of now, there is no convincing theoretical explanation for this correlation. In this paper, we provide a possible theoretical explanation for the empirical correlation. This explanation also explains why, the stiffer the material, the more higher-order harmonics we observe.

How To Transform Partial Order Between Degrees Into Numerical Values, Olga Kosheleva, Vladik Kreinovich, Joe Lorkowski, Martha Osegueda Escobar

Departmental Technical Reports (CS)

Fuzzy techniques are a successful way to handle expert knowledge, enabling us to capture different degrees of expert's certainty in their statements. To use fuzzy techniques, we need to describe expert's degree of certainty in numerical terms. Some experts can provide such numbers, but others can only describe their degrees by using natural-language words like "very", "somewhat", "to some extent", etc. In general, all we know about these word-valued degrees is that there is a natural partial order between these degrees: e.g., "very small" is clearly smaller than "somewhat small". In this paper, we propose a natural way to transform …

What Will Make Computers Faster: An Approach Based On Computational Complexity, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Which Point From An Interval Should We Choose?, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we know the exact form of the objective function, and we know the optimal decision corresponding to each value of the corresponding parameter x. What should we do if we do not know the exact value of x, and instead, we only know x with uncertainty -- e.g., with interval uncertainty? In this case, a reasonable idea is to select one value from the given interval, and to use the optimal decision corresponding to the selected value. But which value should we choose? In this paper, we provide a solution to this problem for …

Big Data: A Geometric Explanation Of A Seemingly Counterintuitive Strategy, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, the progress in science was usually achieved by gradually modifying known problem-solving techniques -- so that the modified techniques can solve problems similar to the already-solved ones. Recently, however, a different -- successful -- paradigm of big data appeared. In the big data paradigm, we, in contrast, look for problems which cannot be solved by gradual modifications of the existing methods. In this paper, we propose a geometric explanation for the empirical success of this new paradigm.

Which Robust Versions Of Sample Variance And Sample Covariance Are Most Appropriate For Econometrics: Symmetry-Based Analysis, Songsak Sriboonchitta, Ildar Batyrshin, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we do not know the shape of the corresponding probability distributions and therefore, we need to use robust statistical techniques, i.e., techniques that are applicable to all possible distributions. Empirically, it turns out the the most efficient robust version of sample variance is the average value of the p-th powers of the deviations |x_i- a| from the (estimated) mean a. In this paper, we use natural symmetries to provide a theoretical explanation for this empirical success, and to show how this optimal robust version of sample variance can be naturally extended to a robust …

Need For Most Accurate Discrete Approximations Explains Heavy-Tailed Distributions, Songsak Sriboonchitta, Vladik Kreinovich, Olga Kosheleva, Hung T. Nguyen

Departmental Technical Reports (CS)

In many practical situations, we encounter Gaussian distributions, for which the distribution tails are light -- in the sense that as the value increases, the corresponding probability density tends to 0 very fast. There are many theoretical explanations for the Gaussian distributions and for similar light-tail distributions. In practice, however, we often encounter heavy-tailed distributions, in which the probability density is asymptotically described, e.g., by a power law. In contrast to the light-tail distributions, there is no convincing theoretical explanation for the heavy-tailed ones. In this paper, we provide such a theoretical explanation. This explanation is based on the fact …

Empirically Successful Transformations From Non-Gaussian To Close-To-Gaussian Distributions: Theoretical Justification, Thongchai Dumrongpokaphan, Perdo Barragan, Vladik Kreinovich

Departmental Technical Reports (CS)

A large number of efficient statistical methods have been designed for a frequent case when the distributions are normal (Gaussian). In practice, many probability distributions are not normal. In this case, Gaussian-based techniques cannot directly applied. In many cases, however, we can apply these techniques indirectly -- by first applying an appropriate transformation to the original variables, after which their distribution becomes close to normal. Empirical analysis of different transformations has shown that the most successful are the power transformations X → X^h and their modifications. In this paper, we provide a symmetry-based explanation for this empirical success.

Bayesian Approach To Intelligent Control And Its Relation To Fuzzy Control, Kongliang Zhu, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In many application areas including economics, experts describe their knowledge by using imprecise ("fuzzy") words from natural language. To design an automatic control system, it is therefore necessary to translate this knowledge into precise computer-understandable terms. To perform such a translation, a special semi-heuristic fuzzy methodology was designed. This methodology has been successfully applied to many practical problem, but its semi-heuristic character is a big obstacle to its use: without a theoretical justification, we are never 100% sure that this methodology will be successful in other applications as well. It is therefore desirable to come up with either a theoretical …

Membership Functions Representing A Number Vs. Representing A Set: Proof Of Unique Reconstruction, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In some cases, a membership function m(x) represents an unknown number, but in many other cases, it represents an unknown crisp set. In this case, for each crisp set S, we can estimate the degree m(S) to which this set S is the desired one. A natural question is: once we know the values m(S) corresponding to all possible crisp sets S, can we reconstruct the original membership function? In this paper, we show that the original membership function m(x) can indeed be uniquely reconstructed from the values m(S).

Fuzzy Techniques Provide A Theoretical Explanation For The Heuristic L^P-Regularization Of Signals And Images, Fernando Cervantes, Bryan E. Usevitch, Leobardo Valera, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

One of the main techniques used to de-noise and de-blur signals and images is regularization, which is based on the fact that signals and images are usually smoother than noise. Traditional Tikhonov regularization assumes that signals and images are differentiable, but, as Mandelbrot has shown in his fractal theory, many signals and images are not differentiable. To de-noise and de-blur such images, researchers have designed a heuristic method of l^p-regularization.

l^p-regularization leads to good results, but it is not used as widely as should be, because it lacks a convincing theoretical explanation -- and thus, practitioners are often reluctant to …

Why Sparse? Fuzzy Techniques Explain Empirical Efficiency Of Sparsity-Based Data- And Image-Processing Algorithms, Fernando Cervantes, Bryan E. Usevitch, Leobardo Valera, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, it turned out to be efficient to assume that the signal or an image is sparse, i.e., that when we decompose it into appropriate basic functions (e.g., sinusoids or wavelets), most of the coefficients in this decomposition will be zeros. At present, the empirical efficiency of sparsity-based techniques remains somewhat a mystery. In this paper, we show that fuzzy-related techniques can explain this empirical efficiency. A similar explanation can be obtained by using probabilistic techniques; this fact increases our confidence that our explanation is correct.

Chemical Kinetics In Situations Intermediate Between Usual And High Concentrations: Fuzzy-Motivated Derivation Of The Formulas, Olga Kosheleva, Vladik Kreinovich, Laécio Carvalho Barros

Departmental Technical Reports (CS)

In the traditional chemical kinetics, the rate of each reaction A + ... + B --> ... is proportional to the product cA * ... * cB of the concentrations of all the input substances A, ..., B. For high concentrations cA, ..., cB, the reaction rate is known to be proportional to the minimum min(cA, ..., cB). In this paper, we use fuzzy-related ideas to derive the formula of the reaction rate for situations intermediate between usual and high concentrations.

How To Predict Nesting Sites And How To Measure Shoreline Erosion: Fuzzy And Probabilistic Techniques For Environment-Related Spatial Data Processing, Stephen Escarzaga, Craig Tweedie, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show how fuzzy and probabilistic techniques can be used in environment-related data processing. Specifically, we will show that these methods help in solving two environment-related problems: how to predict the birds' nesting sites and how to measure shoreline erosion.

How To Estimate Resilient Modulus For Unbound Aggregate Materials: A Theoretical Explanation Of An Empirical Formula, Pedro Barragan Olague, Soheil Nazarian, Vladik Kreinovich, Afshin Gholamy

Departmental Technical Reports (CS)

To ensure the quality of pavement, it is important to make sure that the resilient moduli -- that describe the stiffness of all the pavement layers -- exceed a certain threshold. From the mechanical viewpoint, pavement is a non-linear medium. Several empirical formulas have been proposed to describe this non-linearity. In this paper, we describe a theoretical explanation for the most accurate of these empirical formulas.

How To Describe Measurement Uncertainty And Uncertainty Of Expert Estimates?, Nicolas Madrid, Irina Perfilieva, Vladik Kreinovich

Departmental Technical Reports (CS)

Measurement and expert estimates are never absolutely accurate. Thus, when we know the result M(u) of measurement or expert estimate, the actual value A(u) of the corresponding quantity may be somewhat different from M(u). In practical applications, it is desirable to know how different it can be, i.e., what are the bounds f(M(u)) <= A(u) <= g(M(u)). Ideally, we would like to know the tightest bounds, i.e., the largest possible values f(x) and the smallest possible values g(x). In this paper, we analyze for which (partially ordered) sets of values such tightest bounds always exist: it turns out that they always exist only for complete lattices.

How To Make A Solution To A Territorial Dispute More Realistic: Taking Into Account Uncertainty, Emotions, And Step-By-Step Approach, Mahdokhat Afravi, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, it is necessary to divide a disputed territory between several interested parties. The usual way to perform this division is by using Nash's bargaining solution, i.e., by finding a partition that maximizes the product of the participants' utilities. However, this solution is based on several idealized assumptions: that we know the exact values of all the utilities, that division is performed on a purely rational basis, with no emotions involved, and that the entire decision is made once. In practice, we only know the utilities with some uncertainty, emotions are often involved, and the solution is …

How To Introduce Technical Details Of Quantum Computing In A Theory Of Computation Class: Using The Basic Case Of The Deutsch-Jozsa Algorithm, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many students taking the theory of computation class have heard about quantum computing and are curious about it. However, the usual technical description of quantum computing requires a large amount of preliminary information, too much to fit into an already packed class. In this paper, we propose a way to introduce technical details of quantum computing that does not require much time -- it can be described in less than an hour. As such an introduction, we use a simplified description of the basic case of one of the pioneering algorithms of quantum computing.

Why Lp-Methods In Signal And Image Processing: A Fuzzy-Based Explanation, Fernando Cervantes, Bryan E. Usevitch, Vladik Kreinovich

Departmental Technical Reports (CS)

In signal and image processing, it is often beneficial to use semi-heuristic Lp-methods, i.e., methods that minimize the sum of the p-th powers of the discrepancies. In this paper, we show that a fuzzy-based analysis of the corresponding intuitive idea leads exactly to the Lp-methods.

Model-Order Reduction Using Interval Constraint Solving Techniques, Leobardo Valera, Martine Ceberio

Departmental Technical Reports (CS)

Many natural phenomena can be modeled as ordinary or partial differential equations. A way to find solutions of such equations is to discretize them and to solve the corresponding (possibly) nonlinear large systems of equations.

Solving a large nonlinear system of equations is very computationally complex due to several numerical issues, such as high linear-algebra cost and large memory requirements. Model-Order Reduction (MOR) has been proposed as a way to overcome the issues associated with large dimensions, the most used approach for doing so being Proper Orthogonal Decomposition (POD). The key idea of POD is to reduce a large number …

How To Estimate Amount Of Useful Information, In Particular Under Imprecise Probability, Luc Longpre, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional Shannon's information theory describes the overall amount of information, without distinguishing between useful and unimportant information. Such a distinction is needed, e.g., in privacy protection, where it is crucial to protect important information while it is not that crucial to protect unimportant information. In this paper, we show how Shannon's definition can be modified so that it will describe only the amount of useful information.

Limitations Of Realistic Monte-Carlo Techniques, Andrzej Pownuk, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Because of the measurement errors, the result Y = f(X1, ..., Xn) of processing the measurement results X1, ..., Xn is, in general, different from the value y = f(x1, ..., xn) that we would obtain if we knew the exact values x1, ..., xn of all the inputs. In the linearized case, we can use numerical differentiation to estimate the resulting difference Y -- y; however, this requires >n calls to an algorithm computing f, and for complex algorithms and large $n$ this can take too long. In situations when for each input xi, we know the probability distribution …

Why Min-Based Conditioning, Salem Benferhat, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we do not have full information about which alternatives are possible and which are not. In such situations, an expert can estimate, for each alternative, the degree to which this alternative is possible. Sometimes, experts can produce numerical estimates of their degrees, but often, they can only provide us with qualitative estimates: they inform us which degrees are higher, but do not provide us with numerical values for these degrees. After we get these degrees from the experts, we often gain additional information, because of which some alternatives which were previously considered possible are now excluded. …

Why Locating Local Optima Is Sometimes More Complicated Than Locating Global Ones, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In most applications, practitioners are interested in locating global optima. In such applications, local optima that result from some optimization algorithms are an unnecessary side effect. In other words, in such applications, locating global optima is a much more computationally complex problem than locating local optima. In several practical applications, however, local optima themselves are of interest. Somewhat surprisingly, it turned out that in many such applications, locating all local optima is a much more computationally complex problem than locating all global optima. In this paper, we provide a theoretical explanation for this surprising empirical phenomenon.

On Geometry Of Finsler Causality: For Convex Cones, There Is No Affine-Invariant Linear Order (Similar To Comparing Volumes), Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Some physicists suggest that to more adequately describe the causal structure of space-time, it is necessary to go beyond the usual pseudo-Riemannian causality, to a more general Finsler causality. In this general case, the set of all the events which can be influenced by a given event is, locally, a generic convex cone, and not necessarily a pseudo-Reimannian-style quadratic cone. Since all current observations support pseudo-Riemannian causality, Finsler causality cones should be close to quadratic ones. It is therefore desirable to approximate a general convex cone by a quadratic one. This cane be done if we select a hyperplane, and …

Bell-Shaped Curve For Productivity Growth: An Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent analysis of the productivity growth data shows, somewhat surprisingly, that the dependence of the 20-century productivity growth on time can be reasonably well described by a Gaussian formula. In this paper, we provide a possible theoretical explanation for this observation.