Computer Sciences Commons^™

Interleaving Enhances Learning: A Possible Geometric Explanation, Octavio Lerma, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional approach to learning, if we want students to learn how to solve different types of problems, we first teach them how to solve problems of the first type, then how to solve problems of the second type, etc. It turns out that we can speed up learning if we interleave problems of different types. In particular, it has bene empirically shown that interleaving problems of four different types leads to a double speed-up. In this paper, we provide a possible geometric explanation for this empirical fact.

From Global To Local Constraints: A Constructive Version Of Bloch's Principle, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Generalizing several results from complex analysis, A. Bloch formulated an informal principle -- that for every global implication there is a stronger local implication. This principle has been formalized for complex analysis, but is has been successfully used in other areas as well. In this paper, we propose a new formalization of Bloch's Principle, and we show that in general, the corresponding localized version can be obtained algorithmically.

How To Fully Represent Expert Information About Imprecise Properties In A Computer System -- Random Sets, Fuzzy Sets, And Beyond: An Overview, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

To help computers make better decisions, it is desirable to describe all our knowledge in computer-understandable terms. This is easy for knowledge described in terms on numerical values: we simply store the corresponding numbers in the computer. This is also easy for knowledge about precise (well-defined) properties which are either true or false for each object: we simply store the corresponding "true" and "false" values in the computer. The challenge is how to store information about imprecise properties. In this paper, we overview different ways to fully store the expert information about imprecise properties. We show that in the simplest …

A Feasible Algorithm For Checking N-Scissors Congruence Of Polyhedra In Rd, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While in R², every two polygons of the same area are scissors congruent (i.e., they can be both decomposed into the same finite number of pair-wise congruent polygonal pieces), in R³, there are polyhedra P and P' of the same volume which are not scissors-congruent. It is therefore necessary, given two polyhedra, to check whether they are scissors-congruent (and if yes -- to find the corresponding decompositions). It is known that while there are algorithms for performing this checking-and-finding task, no such algorithm can be feasible -- their worst-case computation time grows (at least) exponentially, so …

Why Injecting Fine Dust Into A Tornado Is More Promising Than Injecting Coarse Dust: A Geometric Explanation, Octavio Lerma, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the promising ways to tame a tornado is to inject dust into it. Somewhat counter-intuitively, injecting coarse dust only makes the tornado stronger, while injecting fine dust can indeed help in the taming. This difference has been explained by a mathematical analysis of the corresponding equations, but (in contrast to the usual physics practice) this mathematical analysis has not yet been accompanied by a simple qualitative physical explanation. We show that such a simple explanation can be obtained if we analyze the problem of taming tornados from the geometric viewpoint.

A Simple Geometric Model Provides A Possible Quantitative Explanation Of The Advantages Of Immediate Feedback In Student Learning, Octavio Lerma, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Calculus is a known bottleneck for many students studying science and engineering. Various techniques have been developed to enhance the students' success. A recent study published in the Notices of American Mathematical Society showed that only one factor determines the success of a technique: the presence of immediate feedback. On average, students who receive immediate feedback learn twice faster than students who are taught in a more traditional way, with a serious feedback only once or twice a semester (after a test).

The very fact that immediate feedback is helpful is not surprising: it helps the student clear misconceptions and …

Diversity Is Beneficial For A Research Group: One More Quantitative Argument, Komsan Suriya, Tatcha Sudtasan, Tonghui Wang, Octavio Lerma, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we propose a natural model describing competition between two research groups of the same average research strength. The analysis of this model enables us to conclude that a more diverse group has an advantage: namely, the more diverse the group, the higher the average quality of its publications.

Zipf's Law And 7 Plus Minus 2 Principle Lead To A Possible Explanation Of Daniel's Law, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In 1961, D. R. Daniel observed that the success of a company is usually determined by three to six major factors. This observation has led to many successful management ideas, but they leave one puzzled: why three to six? why not two or seven? In this paper, we provide a possible explanation to this puzzle; namely, we show that these numbers of factors can be derived from Zipf's Law and from the 7 plus minus 2 principle.

Fuzziness And Bayesian Analysis In Engineering, Matthias Stein, Michael Beer, Vladik Kreinovich

Departmental Technical Reports (CS)

An engineering analysis requires a realistic quantification of all input information. The amount and quality of the available information dictates the uncertainty model and its associated quantification concept. For inconsistent information, a distinction between probabilistic and non-probabilistic characteristics is beneficial. In this distinction, uncertainty refers to probabilistic characteristics and non-probabilistic characteristics are summarized as imprecision. When uncertainty and imprecision occur simultaneously, the uncertainty model fuzzy randomness appears useful. In a Bayesian approach the fuzzy probabilistic model provides the opportunity to take account of imprecision in data and in prior expert knowledge. The Bayesian approach ex-tended to inconsistent information is demonstrated …

"And"- And "Or"-Operations For "Double", "Triple", Etc. Fuzzy Sets, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In the traditional fuzzy logic, the expert's degree of confidence d(A & B) in a complex statement A & B (or A\/B) is uniquely determined by his/her degrees of confidence d(A) and d(B) in the statements A and B, as f_&(d(A),d(B)) for an appropriate "and"-operation (t-norm). In practice, for the same degrees d(A) and d(B), we may have different degrees d(A & B) depending on the relation between A and B. The best way to take this relation into account is to explicitly elicit the corresponding degrees d(A & B) and d(A\/B), i.e., to come up with "double" …

Towards Designing Optimal Individualized Placement Tests, Octavio Lerma, Olga Kosheleva, Shahnaz Shahbazova, Vladik Kreinovich

Departmental Technical Reports (CS)

To find the current level of a student's knowledge, we use a sequence of problems of increasing complexity; if a student can solve a problem, the system generates a more complex one; if a student cannot solve a problem, the system generates an easier one. To find a proper testing scheme, we must take into account that every time a student cannot solve a problem, he/she gets discouraged. To take this into account, we define an overall effect on a student by combining "positive" and "negative" problems with different weights, and we design a testing scheme which minimizes this effect.

Why Triangular Membership Functions Work Well In F-Transform: A Theoretical Explanation, Jaime Nava, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, it is useful to represent a signal or an image by its average values on several fuzzy sets. The corresponding {\em F-transform} technique has many useful applications in signal and image processing. In principle, we can use different membership functions. Somewhat surprisingly, in many applications, the best results occur when we use triangular membership functions. In this paper, we provide a possible theoretical explanation for this empirical phenomenon.

Feasible Algorithms For Lattice And Directed Subspaces, Jennifer (Hamlyn) Del Valle, Vladik Kreinovich, Piotr Wojciechowski

Departmental Technical Reports (CS)

In some practical situations (e.g., in econometrics), it is important to check whether a given linear subspace of a space R^m with component-wise order is a lattice -- and if it is not, whether it is at least a directed ordered space. Because of the practical importance, it is desirable to have feasible algorithms for solving these problems -- which in Computer Science is usually interpreted as algorithms whose computation time does not exceed a polynomial of the length of the input. No such algorithms were previously known. In this paper, we present feasible algorithms for solving both problems.

Approximate Nature Of Traditional Fuzzy Methodology Naturally Leads To Complex-Valued Fuzzy Degrees, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional fuzzy logic, the experts' degrees of confidence in their statements is described by numbers from the interval [0,1]. These degree have a clear intuitive meaning. Somewhat surprisingly, in some applications, it turns out to be useful to also consider different numerical degrees -- e.g., complex-valued degrees. While these complex-valued degrees are helpful in solving practical problems, their intuitive meaning is not clear. In this paper, we provide a possible explanation for the success of complex-valued degrees which makes their use more intuitively understandable -- namely, we show that these degrees naturally appear due to the approximate nature …

Knowledge Geometry Is Similar To General Relativity: Both Mass And Knowledge Curve The Corresponding Spaces, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we explain and explore the idea that knowledge is similar to mass in physics: similarly to how mass curves space-time, knowledge curves the corresponding knowledge space.

Finding Specifications Of While Statements Using Patterns, Aditi Barua, Yoonsik Cheon

Departmental Technical Reports (CS)

A formal correctness proof of code containing loops such as while statements typically uses the technique of proof-by-induction, and often the most difficult part of carrying out an inductive proof is formulating a correct induction hypothesis, a specification for a loop statement. An incorrect induction hypothesis will surely lead to a proof failure. In this paper we propose a systematic way for identifying specifications of while statements. The key idea of our approach is to categorize and document common patterns of while statements along with their specifications. This is based on our observation that similarly-structured while statements frequently have similarly-structured …

Picture Fuzzy Sets - A New Concept For Computational Intelligence Problems, Bui Cong Cuong, Vladik Kreinovich

Departmental Technical Reports (CS)

Since Zadeh introduced fuzzy sets in 1965, a lot of new theories treating imprecision and uncertainty have been introduced. Some of these theories are extensions of fuzzy set theory, other try to handle imprecision and uncertainty in different way. In this paper, we introduce a new notion of picture fuzzy sets (PFS), which are directly extensions of fuzzy sets and of intuitonistic fuzzy sets (Atanassov). Then some operations on picture fuzzy sets are defined and some properties of these operations are considered. Here the basic preliminaries of PFS theory are presented.

Studying Volatility And Dependency Of Chinese Outbound Tourism Demand In Singapore, Malaysia, And Thailand: A Vine Copula Approach, Jianxu Liu, Songsak Sriboonchitta, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

This paper investigates the volatility and dependence of Chinese tourism demand for Singapore, Malaysia, and Thailand (SMT) destinations, using the vine copula based auto regression moving average-generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. It is found that a jolt to the tourist flow can have long-standing ramifications for the SMT countries. The estimation of the vine copulas among SMT show that the Survival Gumbel, Frank, and Gaussian copulas are the best copulas for Canonical vine (C-vine) or Drawable vine (D-vine) among the possible pair-copulas. In addition, this paper illustrates the making of time-varying Frank copulas for vine copulas. Finally, there is …

From Urysohn's Universal Metric Space To A Universal Space-Time, A. G. Aksoy, Z. Glassman, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A known Urysohn's result shows that there exists a universal} metric space, i.e., a metric space into every other (separable) metric space can be isomorphically embedded. Moreover, this universal metric space can be selected to be ultra-homogeneous -- every isomorphism of its two finite subsets can be extended to the isomorphism of the whole space.

Starting with Einstein's theories of Special and General relativity, space-times are described by a different type of structure -- a set (of events) equipped with the proper time t(a,b) between points a and b; such spaces are known as space-times with kinematic metric, or k-space-times. …

Algebraic Product Is The Only T-Norm For Which Optimization Under Fuzzy Constraints Is Scale-Invariant, Juan Carlos Figueroa Garcia, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to optimize under fuzzy constraints. There is a known Bellman-Zadeh approach for solving such problems, but the resulting solution, in general, depends on the choice of a not well-defined constant M. We show that this dependence disappears if we use an algebraic t-norm (and-operation) a * b, and we also prove that the algebraic product is the only t-norm for which the corresponding solution is independent on M.

A Vine Copula Approach For Analyzing Financial Risk And Co-Movement Of The Indonesian, Philippine And Thailand Stock Markets, Songsak Sriboonchitta, Jianxu Liu, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

This paper aims at analyzing the financial risk and co-movement of stock markets in three countries: Indonesia, Philippine and Thailand. It consists of analyzing the conditional volatility and test the leverage effect in the stock markets of the three countries. To capture the pairwise and conditional dependence between the variables, we use the method of vine copulas. In addition, we illustrate the computations of the value at risk and the expected shortfall using Monte Carlo simulation with copula based GJR-GARCH model. The empirical evidence shows that all the leverage effects add much to the capacity for explanation of the three …

Data Collection For The Similar Segments In Social Speech Task, Nigel G. Ward, Steven D. Werner

Departmental Technical Reports (CS)

Information retrieval systems rely heavily on models of similarity, but for spoken dialog such models currently use mostly standard textual-content similarity. As part of the MediaEval Benchmarking Initiative, we have created a new corpus to support development of similarity models for spoken dialog. This corpus includes 26 casual dialogs among members of two semi-cohesive groups, totaling about 5 hours, with 1889 labeled regions associated into 227 sets which annotators judged to be similar enough to share a tag. This technical report brings together information about this corpus and its intended uses.

Similarity Approach To Defining Basic Level Of Concepts Explained From The Utility Viewpoint, Joe Lorkowski, Martin Trnecka

Departmental Technical Reports (CS)

In many practical situations, it is necessary to describe an image in words. From the purely logical viewpoint, to describe the same object, we can use concepts of different levels of abstraction: e.g., when the image includes a dog, we can say that it is a dog, or that it is a mammal, or that it is a German Shepherd. In such situations, humans usually select a concept which, to them, in the most natural; this concept is called the basic level concept. However, the notion of a basic level concept is difficult to describe in precise terms; as a …

Complete Description Of Idempotent Hedges In Fuzzy Logic, Jaime Nava

Departmental Technical Reports (CS)

In describing expert knowledge, it is often important to properly take into account hedges} like "very", "somewhat", etc. In particular, fuzzy logic provides a consistent way of describing hedges. For some of the hedges, a repetition changes the meaning: e.g., "very very small" is smaller than "very small". However, other hedges -- like "somewhat" -- are idempotent, in the sense that repeating this hedge twice does not change the meaning. In this paper, we provide a complete description of such idempotent hedges.

On Early Stages Of Idea Propagation, The Number Of Adopters Grows As N(T) ~ C * Ta: Theoretical Explanation Of The Empirical Observation, L. Octavio Lerma, Deana Pennington, Vladik Kreinovich

Departmental Technical Reports (CS)

New good ideas sometimes propagate too slowly. To speed up their propagation, we need to have a quantitative understanding of how ideas propagate. An intuitive understanding of ideas propagation has led to several reasonable first-approximation mathematical models. These models provide a good description of idea propagation on the later stages, when the ideas have already been adopted by a reasonably large number of people. However, at the critically important early stages, these models are not perfect: these models predict a linear growth with time, while empirical growth data is often better described by a power law. In this paper, we …

Vine Copulas As A Way To Describe And Analyze Multi-Variate Dependence In Econometrics: Computational Motivation And Comparison With Bayesian Networks And Fuzzy Approaches, Songsak Sriboonchitta, Jianxu Liu, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

In the last decade, vine copulas emerged as a new efficient techniques for describing and analyzing multi-variate dependence in econometrics. Our experience has shown, however, that while these techniques have been successfully applied to many practical problems of econometrics, there is still a lot of confusion and misunderstanding related to vine copulas. In this paper, we provide a motivation for this new technique from the computational viewpoint. We show that other techniques used to described dependence -- Bayesian networks and fuzzy techniques -- can be viewed as a particular case of vine copulas.

How To Distinguish True Dependence From Varying Independence?, Marketa Krmelova, Martin Trnecka, Vladik Kreinovich, Berlin Wu

Departmental Technical Reports (CS)

A usual statistical criterion for the quantities X and Y to be independent is that the corresponding distribution function F(x,y) is equal to the product of the corresponding marginal distribution functions. If this equality is violated, this is usually taken to mean that X and Y are dependent. In practice, however, the inequality may be caused by the fact that we have a mixture of several populations, in each of which X and Y are independent. In this paper, we show how we can distinguish true dependence from such varying independence. This can also lead to new measures to degree …

How To Faster Test A Device For Different Combinations Of Parameters, Francisco Zapata, Luis C. Gutierrez, Vladik Kreinovich

Departmental Technical Reports (CS)

A device has to function properly under all possible conditions: e.g., for all temperatures within a given range, for all possible humidity values within a given range, etc. Ideally, it would be nice to be able to test a device for all possible combinations of these parameters, but the number of such combinations is often so huge that such an exhaustive testing is not possible. Instead, it is reasonable to check the device for all possible values of each parameter, for each possible pairs of values of two parameters, and, in general, for all possible combinations of values of k …

Why Trapezoidal And Triangular Membership Functions Work So Well: Towards A Theoretical Explanation, Aditi Barua, Lalitha Snigdha Mudunuri, Olga Kosheleva

Departmental Technical Reports (CS)

In fuzzy logic, an imprecise ("fuzzy") property is described by its membership function μ(x), i.e., by a function which describes, for each real number x, to what degree this real number satisfies the desired property. In principle, membership functions can be of different shape, but in practice, trapezoidal and triangular membership functions are most frequently used. In this paper, we provide an interval-based theoretical explanation for this empirical fact.

Note On Fair Price Under Interval Uncertainty, Joshua Mckee, Joe Lorkowski, Thavatchai Ngamsantivong

Departmental Technical Reports (CS)

Often, in decision making situations, we do not know the exact value of a gain resulting from making each decision, we only know the bounds on this gain. To make a reasonable decision under such interval uncertainty, it makes sense to estimate the fair price of each alternative, and then to select the alternative with the highest price. In this paper, we show that the value of the fair price can be uniquely determined from some reasonable requirements: e.g., the additivity requirement, that the fair price of two objects together should be equal to the sum of the fair prices …