Physical Sciences and Mathematics Commons^™

Among Several Successful Algorithms, Simpler Ones Usually Work Better: A Possible Explanation Of An Empirical Observation, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Often, several different algorithms can solve a certain practical problem. Sometimes, algorithms which are successful in solving one problem can solve other problems as well. How can we decide which of the original algorithms is the most promising -- i.e., which is more probable to be able to solve other problem? In many cases, the simplest algorithms turns out to be the most successful. In this paper, we provide a possible explanation for this empirical observation.

Examining The Consistence Of Futures Margin Levels Using Bivariate Extreme Value Copulas, X. Gong, Hung T. Nguyen, Vladik Kreinovich, Songsak Sriboonchitta

Departmental Technical Reports (CS)

This study examines the consistence of the futures margin levels of different commodities and combinations in the CME group by Extreme Value Copula (EVC). We find that if we ignore the co-movements of the commodities, the margins become consistent with each other, and the margin violation rates hover around 0.5%. However, if we consider the co-movement of the related commodities using EVC, the margin levels are found to be not consistent anymore, especially in the combinations of strongly related commodities which are in the same category. Therefore, we suggest that the CME group should try to harmonize the margins policy …

Granularity Explains Empirical Factor-Of-Three Relation Between Probabilities Of Pulmonary Embolism In Different Patient Categories, Beverly Rivera, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

Pulmonary embolism is a very dangerous difficult-to-detect medical condition. To diagnose pulmonary embolism, medical practitioners combine indirect signs of this condition into a single score, and then classify patients into low-probability, intermediate-probability, and high-probability categories. Empirical analysis shows that, when we move from each category to the next one, the probability of pulmonary embolism increases by a factor of three. In this paper, we provide a theoretical explanation for this empirical relation between probabilities.

How To Gauge Unknown Unknowns: A Possible Theoretical Explanation Of The Usual Safety Factor Of 2, Joe Lorkowski, Vladik Kreinovich

Departmental Technical Reports (CS)

To gauge the accuracy of a measuring instrument, engineers analyze possible factors contributing to the instrument's inaccuracy. In addition to known factors, however, there are usually unknown factors which also contribute to the instrument's inaccuracy. To properly gauge the instrument's accuracy -- and thus, to make sure that we do not compromise our safety by underestimating the inaccuracy -- we need to also take these "unknown unknowns" into account. In practice, this is usually done by multiplying the original estimate for inaccuracy by a "safety" factor of 2. In this paper, we provide a possible theoretical explanation for this empirical …

Fuzzy Intervals As Foundation Of Metrological Support For Computations With Inaccurate Data, Konstantin K. Semenov, Gennady N. Solopchenko, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we discuss the possibility of using the formalism of fuzzy intervals as a basis for computational metrology. We consider advantages of using fuzzy intervals instead of the traditional intervals as a characteristic of uncertainty of the results of computations with inaccurate data.

Formalizing The Informal, Precisiating The Imprecise: How Fuzzy Logic Can Help Mathematicians And Physicists By Formalizing Their Intuitive Ideas, Olga Kosheleva, Renata Reiser, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy methodology transforms expert ideas -- formulated in terms of words from natural language -- into precise rules and formulas. In this paper, we show that by applying this methodology to intuitive physical and mathematical ideas, we can get known fundamental physical equations and known mathematical techniques for solving these equations. This fact makes us confident that in the future, fuzzy techniques will help physicists and mathematicians to transform their imprecise ideas into new physical equations and new techniques for solving these equations.

A Catalog Of While Loop Specification Patterns, Aditi Barua, Yoonsik Cheon

Departmental Technical Reports (CS)

This document provides a catalog of while loop patterns along with their skeletal specifications. The specifications are written in a functional form known as intended functions. The catalog can be used to derive specifications of while loops by first matching the loops to the cataloged patterns and then instantiating the skeletal specifications of the matched patterns. Once their specifications are formulated and written, the correctness of while loops can be proved rigorously or formally using the functional program verification technique in which a program is viewed as a mathematical function from one program state to another.

Kekule's Benzene Structure: A Case Study Of Teaching Usefulness Of Symmetry, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Benzene is one of the basic building blocks of organic molecules. One of the reasons for benzene's ubiquity is its unusual ring structure first discovered by Kekule in 1865. In this paper, we show that a simple symmetry-based analysis can narrow down possible benzene structures to three ring ones, including the Kekule's ring. Thus, Kekule's benzene structure provides a good pedagogical example on which one can explain usefulness of symmetries.

Construction Of Shear Wave Models By Applying Multi-Objective Optimization To Multiple Geophysical Data Sets, Lennox Thompson, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

For this work, our main purpose is to obtain a better understanding of the Earth's tectonic processes in the Texas region, which requires us to analyze the Earth structure. We expand on a constrained optimization approach for a joint inversion least-squares (LSQ) algorithm to characterize a one-dimensional Earth's structure of Texas with the use of multiple geophysical data sets. We employed a joint inversion scheme using multiple geophysical datasets for the sole purpose of obtaining a three-dimensional velocity structure of Texas in order to identify an ancient rift system within Texas. In particular, we use data from the USArray, which …

What If We Only Have Approximate Stochastic Dominance?, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta

Departmental Technical Reports (CS)

In many practical situations, we need to select one of the two alternatives, and we do not know the exact form of the user's utility function -- e.g., we only know that it is increasing. In this case, stochastic dominance result says that if the cumulative distribution function (cdf) corresponding to the first alternative is always smaller than or equal than the cdf corresponding to the second alternative, then the first alternative is better. This criterion works well in many practical situations, but often, we have situations when for most points, the first cdf is smaller but at some points, …

Constructive Mathematics In St. Petersburg, Russia: A (Somewhat Subjective) View From Within, Vladik Kreinovich

Departmental Technical Reports (CS)

In the 1970 and 1980s, logic and constructive mathematics were an important part of my life; it's what I defended in my Master's thesis, it was an important part of my PhD dissertation. I was privileged to work with the giants. I visited them in their homes. They were who I went to for advice. And this is my story.

Using Second-Order Probabilities To Make Maximum Entropy Approach To Copulas More Reasonable, Hung T. Nguyen, Vladik Kreinovich, Berlin Wu

Departmental Technical Reports (CS)

Copulas are a general way of describing dependence between two or more random variables. When we only have partial information about the dependence, i.e., when several different copulas are consistent with our knowledge, it is often necessary to select one of these copulas. A frequently used method of selecting this copula is the maximum entropy approach, when we select a copula with the largest entropy. However, in some cases, the maximum entropy approach leads to an unreasonable selection -- e.g., even if we know that the two random variables are positively correlated, the maximum entropy approach completely ignores this information. …

Dealing With Uncertainties In Computing: From Probabilistic And Interval Uncertainty To Combination Of Different Types Of Uncertainty, Vladik Kreinovich

Departmental Technical Reports (CS)

To predict values of future quantities, we apply algorithms to the current and past measurement results. Because of the measurement errors and model inaccuracy, the resulting estimates are, in general, different from the desired values of the corresponding quantities. There exist methods for estimating this difference, but these methods have been mainly developed for the two extreme cases: the case when we know the exact probability distributions of all the measurement errors and the interval case, when we only know the bounds on the measurement errors. In practice, we often have some partial information about the probability distributions which goes …

R-Bounded Fuzzy Measures Are Equivalent To Epsilon-Possibility Measures, Karen A. Richart-Ruiz, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional probabilistic description of uncertainty is based on additive probability measures. To describe non-probabilistic uncertainty, it is therefore reasonable to consider non-additive measures. An important class of non-additive measures are possibility measures, for which m(A union B) = max(m(A), m(B)). In this paper, we show that possibility measures are, in some sense, universal approximators: for every epsilon > 0, every non-additive measure which satisfies a certain reasonable boundedness property is equivalent to a measure which is epsilon-close to a possibility measure.

How To Assign Weights To Different Factors In Vulnerability Analysis: Towards A Justification Of A Heuristic Technique, Beverly Rivera, Irbis Gallegos, Vladik Kreinovich

Departmental Technical Reports (CS)

The main objective of vulnerability analysis is to select the alternative which is the least vulnerable. To make this selection, we must describe the vulnerability of each alternative by a single number -- then we will select the alternative with the smallest value of this vulnerability index. Usually, there are many aspects of vulnerability: vulnerability of a certain asset to a storm, to a terrorist attack, to hackers' attack, etc. For each aspect, we can usually gauge the corresponding vulnerability, the difficulty is how to combine these partial vulnerabilities into a single weighted value. In our previous research, we proposed …

Observable Causality Implies Lorentz Group: Alexandrov-Zeeman-Type Theorem For Space-Time Regions, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The famous Alexandrov-Zeeman theorem proves that causality implies Lorentz group. The physical meaning of this result is that once we observe which event can causally affect which other events, then, using only this information, we can reconstruct the linear structure of the Minkowski space-time. The original Alexandrov-Zeeman theorem is based on the causality relation between events represented by points in space-time. Knowing such a point means that we know the exact moment of time and the exact location of the corresponding event - and that this event actually occurred at a single moment of time and at a single spatial …

How To Modify Grade Point Average (Gpa) To Make It More Adequate, Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At present, the amounts of knowledge acquired by different graduates of the same program are usually compared by comparing their GPAs. We argue that this is not always the most adequate description: for example, if, after completing all required classes with the highest grade of "excellent" (A), a student takes an additional challenging class and gets a "satisfactory" grade (C), the amount of her knowledge increases, but the GPA goes down. We propose a modification of the GPA which is free of this drawback and is, thus, more adequate for describing the student's knowledge. We also provide a psychological explanation …

How To Estimate Relative Spatial Resolution Of Different Maps Or Images Of The Same Area?, Christian Servin, A A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we describe how to estimate relative spatial resolution of different maps or images of the same area under uncertainty. We consider probabilistic and fuzzy approaches and we show that both approaches lead to the same estimates -- which makes us more confident that this joint result is reasonable.

Interval And Symmetry Approaches To Uncertainty -- Pioneered By Wiener -- Helps Explain Many Seemingly Irrational Human Behaviors: A Case Study, Joe Lorkowski, Vladik Kreinovich

Departmental Technical Reports (CS)

It has been observed that in many cases, when we present a user with three selections od different price (and, correspondingly, different quality), then the user selects the middle selection. This empirical fact -- known as a compromise effect -- seems to contradicts common sense. Indeed, when a rational decision-maker selects one of the two alternatives, and then we add an additional option, then the user will either keep the previous selection or switch to a new option, but he/she will not select a previously rejected option. However, this is exactly what happens under the compromise effect. If we present …

Wiener's Conjecture About Transformation Groups Helps Predict Which Fuzzy Techniques Work Better, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Often, application success only comes when we select specific fuzzy techniques (t-norm, membership function, etc.) -- and in different applications, different techniques are the best. How to find the best technique? Exhaustive search of all techniques is not an option: there are too many of them. We need to come up with a narrow class of promising techniques, so that trying them all is realistic. In this paper, we show that such a narrowing can be obtained from transformation groups techniques motivated by N. Wiener's conjecture -- which was, in its turn, motivated by observations about human vision.

A Simple Probabilistic Explanation Of Term Frequency-Inverse Document Frequency (Tf-Idf) Heuristic (And Variations Motivated By This Explanation), Lukas Havrlant, Vladik Kreinovich

Departmental Technical Reports (CS)

In document analysis, an important task is to automatically find keywords which best describe the subject of the document. One of the most widely used techniques for keyword detection is a technique based on the term frequency-inverse document frequency (tf-idf) heuristic. This techniques has some explanations, but these explanations are somewhat too complex to be fully convincing. In this paper, we provide a simple probabilistic explanation for the tf-idf heuristic. We also show that the ideas behind explanation can help us come up with more complex formulas which will hopefully lead to a more adequate detection of keywords.

Towards Efficient Algorithms For Approximating A Fuzzy Relation By Fuzzy Rules: Case When "And"- And "Or"-Operation Are Distributive, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

A generic fuzzy relation often requires too many parameters to represent -- especially when we have a relation between many different quantities x1, ..., xn. There is, however, a class of relations which require much fewer parameters to describe - namely, relations which come from fuzzy rules. It is therefore reasonable to approximate a given relation by fuzzy rules. In this paper, we explain how this can be done in an important case when "and"- and "or"-operation are distributive -- and we also explain why this case is important.

Writing Self-Testing Java Classes With Selftest, Yoonsik Cheon

Departmental Technical Reports (CS)

This document provides a tutorial introduction to Java annotations called SelfTest. The SelfTest annotations allow one to annotate Java classes with test data, and the SelfTest annotation processor generates executable JUnit test classes from annotated Java classes by translating test cases to executable JUnit tests. The SelfTest annotations not only automate unit testing of Java classes significantly but also provides a step toward writing self-testing Java classes by embedding test data in source code for both compile and runtime processing.

In Category Of Sets And Relations, It Is Possible To Describe Functions In Purely Category Terms, Vladik Kreinovich, Martine Ceberio, Quentin Brefort

Departmental Technical Reports (CS)

We prove that in the category of sets and relations, it is possible to describe functions in purely category terms.

How To Understand Connections Based On Big Data: From Cliques To Flexible Granules, Ali Jalal-Kamali, M. Shahriar Hossain, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main objectives of science and engineering is to predict the future state of the world -- and to come up with actions which will lead to the most favorable outcome. To be able to do that, we need to have a quantitative model describing how the values of the desired quantities change -- and for that, we need to know which factors influence this change. Usually, these factors are selected by using traditional statistical techniques, but with the current drastic increase in the amount of available data -- known as the advent of {\it big data} -- …

Towards Decision Making Under Interval, Set-Valued, Fuzzy, And Z-Number Uncertainty: A Fair Price Approach, Joe Lorkowski, Vladik Kreinovich, Rafik Aliev

Departmental Technical Reports (CS)

In this paper, we explore one of the possible ways to make decisions under uncertainty: namely, we explain how to define a fair price for a participation in such a decision, and then select an alternative for which the corresponding fair price is the largest. This idea is explained on the examples of interval uncertainty, set-valued, fuzzy, and Z-number uncertainty.

Simpler-To-Describe Cases Are Often More Difficult To Prove: A Possible Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many areas of mathematics, simpler-to-describe cases are often more difficult to prove. In this paper, we provide examples of such phenomena (Bieberbach's Conjecture, Poincar\'e Conjecture, Fermat's Last Theorem), and we provide a possible explanation for this empirical fact.

Logic Of Scientific Discovery: How Physical Induction Affects What Is Computable, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Most of our knowledge about a physical world comes from physical induction: if a hypothesis is confirmed by a sufficient number of observations, we conclude that this hypothesis is universally true. We show that a natural formalization of this property affects what is computable when processing measurement and observation results, and we explain how this formalization is related to Kolmogorov complexity and randomness. We also consider computational consequences of an alternative idea also coming form physics: that no physical law is absolutely true, that every physical law will sooner or later need to be corrected. It turns out that this …

Decision Making Under Interval Uncertainty: What Can And What Cannot Be Computed In Linear Time And In Real Time, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In engineering, we constantly need to make decisions: which design to select, which parameters to select for this design, etc.

The traditional approach to decision making is based on the assumption that we know all possible consequences of each alternative, and we know the probability of each such consequence. Under this assumption, we can describe a rational decision-making process: to each possible consequence, we assign a numerical values called its utility, and we select the alternative for which the expected value of the utility is the largest.

An important advantage of this approach is that it can be performed in …

Extending Ocl To Better Express Uml Qualified Associations, Alla Dove, Aditi Barua, Yoonsik Cheon

Departmental Technical Reports (CS)

A qualified association in the Unified Modeling Language (UML) is an association that allows one to restrict the objects referred in an association using a key called a qualifier. A qualified association can appear in a constraint written in the Object Constraint Language (OCL) to specify a precise UML model. However, the OCL notation fails to provide appropriate support for expressing certain types of constraints written using qualified associations. In this paper we first describe a deficiency of OCL in expressing qualified associations and then propose a small extension to OCL to make it more expressive. The key idea of …