Physical Sciences and Mathematics Commons^™

How To Explain The Results Of The Richard Thaler's 1997 Financial Times Contest, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In 1997, by using a letter published in Financial Times, Richard H. Thaler, the 2017 Nobel Prize winner in Economics, performed the following experiment: he asked readers to submit numbers from 0 to 100, so that the person whose number is the closest to 2/3 of the average will be the winner. An intuitive answer is to submit 2/3 of the average (50), i.e., 33 1/3. A logical answer, as can be explained, is to submit 0. The actual winning submission was -- depending on how we count -- 12 or 13. In this paper, we propose a possible explanation …

How Intelligence Community Interprets Imprecise (Fuzzy) Words, And How To Justify This Empirical-Based Interpretation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To provide a more precise meaning to imprecise (fuzzy) words like "probable" or "almost certain", researchers analyzed how often intelligence predictions hedged by each corresponding word turned out to be true. In this paper, we provide a theoretical explanation for the resulting empirical frequencies.

Gartner's Hype Cycle: A Simple Explanation, Jose Perez, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, any innovation should be gradually accepted. It is natural that initially some people are reluctant to adopt a new largely un-tested idea, but as more and more evidence appears that this new idea works, we should see a gradual increase in number of adoptees -- until the idea becomes universally accepted.

In real life, the adoption process is not that smooth. Usually, after the few first successes, the idea is over-hyped, it is adopted in situations way beyond the inventors' intent. In these remote areas, the new idea does not work well, so we have a …

How To Explain Empirical Distribution Of Software Defects By Severity, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the last decades, several tools have appeared that, given a software package, mark possible defects of different potential severity. Our empirical analysis has shown that in most situations, we observe the same distribution or software defects by severity. In this paper, we present this empirical distribution, and we use interval-related ideas to provide an explanation for this empirical distribution.

Working On One Part At A Time Is The Best Strategy For Software Production A Proof, Francisco Zapata, Maliheh Zargaran, Vladik Kreinovich

Departmental Technical Reports (CS)

When a company works on a large software project, it can often start recouping its investments by selling intermediate products with partial functionality. With this possibility in mind, it is important to schedule work on different software parts so as to maximize the profit. These exist several algorithms for solving the corresponding optimization problem, and in all the resulting plans, at each moment of time, we work on one part of software at a time. In this paper, we prove that this one-part-at-a-time property holds for all optimal plans.

Why Zipf's Law: A Symmetry-Based Explanation, Daniel Cervantes, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we have probability distributions for which, for large values of the corresponding quantity x, the probability density has the form ρ(x) ~ x^−αfor some α > 0. While, in principle, we have laws corresponding to different α, most frequently, we encounter situations -- first described by Zipf for linguistics -- when α is close to 1. The fact that Zipf's has appeared frequently in many different situations seems to indicate that there must be some fundamental reason behind this law. In this paper, we provide a possible explanation.

Virtual Agent Interaction Framework (Vaif): A Tool For Rapid Development Of Social Agents, Ivan Gris, David G. Novick

Departmental Technical Reports (CS)

Creating an embodied virtual agent is often a complex process. It involves 3D modeling and animation skills, advanced programming knowledge, and in some cases artificial intelligence or the integration of complex interaction models. Features like lip-syncing to an audio file, recognizing the users’ speech, or having the character move at certain times in certain ways, are inaccessible to researchers that want to build and use these agents for education, research, or industrial uses. VAIF, the Virtual Agent Interaction Framework, is an extensively documented system that attempts to bridge that gap and provide inexperienced researchers the tools and means to develop …

How To Gauge Repair Risk?, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

At present, there exist several automatic tools that, given a software, find locations of possible defects. A general tool does not take into account a specificity of a given program. As a result, while many defects discovered by this tool can be truly harmful, many uncovered alleged defects are, for this particular software, reasonably (or even fully) harmless. A natural reaction is to repair all the alleged defects, but the problem is that every time we correct a program, we risk introducing new faults. From this viewpoint, it is desirable to be able to gauge the repair risk. This will …

Why Encubation?, Vladik Kreinovich, Rohan Baingolkar, Swapnil S. Chauhan, Ishtjot S. Kamboj

Departmental Technical Reports (CS)

It is known that some algorithms are feasible, and some take too long to be practical/ For example, if the running time of an algorithm is 2ⁿ, where n = len(x) is the bit size of the input x, then already for n = 500, the computation time exceeds the lifetime of the Universe. In computer science, it is usually assumed that an algorithm A is feasible if and only if it is polynomial-time, i.e., if its number of computational steps t_A(x) on any input x is bounded by a polynomial P(n) of the input …

Why Burgers Equation: Symmetry-Based Approach, Leobardo Valera, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many application areas ranging from shock waves to acoustics, we encounter the same partial differential equation known as the Burgers' equation. The fact that the same equation appears in different application domains, with different physics, makes us conjecture that it can be derived from the fundamental principles. Indeed, in this paper, we show that this equation can be uniquely determined by the corresponding symmetries.

Lotfi Zadeh: A Pioneer In Ai, A Pioneer In Statistical Analysis, A Pioneer In Foundations Of Mathematics, And A True Citizen Of The World, Vladik Kreinovich

Departmental Technical Reports (CS)

Everyone knows Lotfi Zadeh as the Father of Fuzzy Logic. There have been -- and will be -- many papers on this important topic. What I want to emphasize in this paper is that his ideas go way beyond fuzzy logic:

• he was a pioneer in AI;
• he was a pioneer in statistical analysis; and
• he was a pioneer in foundations of mathematics.

My goal is to explain these ideas to non-fuzzy folks. I also want to emphasize that he was a true Citizen of the World.

Why 70/30 Or 80/20 Relation Between Training And Testing Sets: A Pedagogical Explanation, Afshin Gholamy, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

When learning a dependence from data, to avoid overfitting, it is important to divide the data into the training set and the testing set. We first train our model on the training set, and then we use the data from the testing set to gauge the accuracy of the resulting model. Empirical studies show that the best results are obtained if we use 20-30% of the data for testing, and the remaining 70-80% of the data for training. In this paper, we provide a possible explanation for this empirical result.

Why Skew Normal: A Simple Pedagogical Explanation, José Guadalupe Flores Muñiz, Vyacheslav Kalashnikov, Nataliya Kalashnykova, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we only know a few first moments of a random variable, and out of all probability distributions which are consistent with this information, we need to select one. When we know the first two moments, we can use the Maximum Entropy approach and get normal distribution. However, when we know the first three moments, the Maximum Entropy approach doe snot work. In such situations, a very efficient selection is a so-called skew normal distribution. However, it is not clear why this particular distribution should be selected. In this paper, we provide an explanation for this selection.

When Is Data Processing Under Interval And Fuzzy Uncertainty Feasible: What If Few Inputs Interact? Does Feasibility Depend On How We Describe Interaction?, Milan Hladík, Michal Čerńy, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that, in general, data processing under interval and fuzzy uncertainty is NP-hard -- which means that, unless P = NP, no feasible algorithm is possible for computing the accuracy of the result of data processing. It is also known that the corresponding problem becomes feasible if the inputs do not interact with each other, i.e., if the data processing algorithm computes the sum of n functions, each depending on only one of the $n$ inputs. In general, inputs x_i and x_j interact. If we take into account all possible interactions, and we use bilinear functions …

Which T-Norm Is Most Appropriate For Bellman-Zadeh Optimization, Vladik Kreinovich, Olga Kosheleva, Shahnaz Shahbazova

Departmental Technical Reports (CS)

In 1970, Richard Bellman and Lotfi Zadeh proposed a method for finding the maximum of a function under fuzzy constraints. The problem with this method is that it requires the knowledge of the minimum and the maximum of the objective function over the corresponding crisp set, and minor changes in this crisp set can lead to a drastic change in the resulting maximum. It is known that if we use a product "and"-operation (t-norm), the dependence on the maximum disappears. Natural questions are: what if we use other t-norms? Can we eliminate the dependence on the minimum? What if we …

From Traditional Neural Networks To Deep Learning: Towards Mathematical Foundations Of Empirical Successes, Vladik Kreinovich

Departmental Technical Reports (CS)

How do we make computers think? To make machines that fly, it is reasonable to look at the creatures that know how to fly: the birds. To make computers think, it is reasonable to analyze how we think -- this is the main origin of neural networks. At first, one of the main motivations was speed -- since even with slow biological neurons, we often process information fast. The need for speed motivated traditional 3-layer neural networks. At present, computer speed is rarely a problem, but accuracy is -- this motivated deep learning. In this paper, we concentrate on the …

A New Kalman Filter Model For Nonlinear Systems Based On Ellipsoidal Bounding, Ligang Sun, Hamza Alkhatib, Boris Kargoll, Vladik Kreinovich, Ingo Neumann

Departmental Technical Reports (CS)

In this paper, a new filter model called set-membership Kalman filter for nonlinear state estimation problems was designed, where both random and unknown but bounded uncertainties were considered simultaneously in the discrete-time system. The main loop of this algorithm includes one prediction step and one correction step with measurement information, and the key part in each loop is to solve an optimization problem. The solution of the optimization problem produces the optimal estimation for the state, which is bounded by ellipsoids. The new filter was applied on a highly nonlinear benchmark example and a two-dimensional simulated trajectory estimation problem, in …

Optimization Of Quadratic Forms And T-Norm Forms On Interval Domain And Computational Complexity, Milan Hladik, Michal Čerńy, Vladik Kreinovich

Departmental Technical Reports (CS)

We consider the problem of maximization of a quadratic form over a box. We identify the NP-hardness boundary for sparse quadratic forms: the problem is polynomially solvable for O(log n) nonzero entries, but it is NP-hard if the number of nonzero entries is of the order n^ε for an arbitrarily small ε > 0. Then we inspect further polynomially solvable cases. We define a sunflower graph over the quadratic form and study efficiently solvable cases according to the shape of this graph (e.g. the case with small sunflower leaves or the case with a restricted number of negative entries). Finally, …

Italian Folk Multiplication Algorithm Is Indeed Better: It Is More Parallelizable, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, many ethnic groups had their own versions of arithmetic algorithms. Nowadays, most of these algorithms are studied mostly as pedagogical curiosities, as an interesting way to make arithmetic more exciting to the kids: by applying to their patriotic feelings -- if they are studying the algorithms traditionally used by their ethic group -- or simply to their sense of curiosity. Somewhat surprisingly, we show that one of these algorithms -- a traditional Italian multiplication algorithm -- is actually in some reasonable sense better than the algorithm that we all normally use -- namely, it is easier to parallelize.

Reverse Mathematics Is Computable For Interval Computations, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

For systems of equations and/or inequalities under interval uncertainty, interval computations usually provide us with a box whose all points satisfy this system. Reverse mathematics means finding necessary and sufficient conditions, i.e., in this case, describing the set of {\it all} the points that satisfy the given system. In this paper, we show that while we cannot always exactly describe this set, it is possible to have a general algorithm that, given ε > 0, provides an ε-approximation to the desired solution set.

How To Monitor Possible Side Effects Of Enhanced Oil Recovery Process, Jose Manuel Dominguez Esquivel, Solymar Ayala Cortez, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

To extract all the oil from a well, petroleum engineers pump hot reactive chemicals into the well. This enhanced oil recovery process needs to be thoroughly monitored, since the aggressively hot liquid can seep out and, if unchecked, eventually pollute the sources of drinking water. At present, to monitor this process, engineers measure the seismic waves generated when the liquid fractures the minerals. However, the resulting seismic waves are weak in comparison with the background noise. Thus, the accuracy with which we can locate the spreading liquid based on these weak signals is low, and we get only a very …

Why Learning Has Aha-Moments And Why We Should Also Reward Effort, Not Just Results, Gerargo Uranga, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Traditionally, in machine learning, the quality of the result improves steadily with time (usually slowly but still steadily). However, as we start applying reinforcement learning techniques to solve complex tasks -- such as teaching a computer to play a complex game like Go -- we often encounter a situation in which for a long time, then is no improvement, and then suddenly, the system's efficiency jumps almost to its maximum. A similar phenomenon occurs in human learning, where it is known as the aha-moment. In this paper, we provide a possible explanation for this phenomenon, and show that this explanation …

Fuzzy Analogues Of Sets And Functions Can Be Uniquely Determined From The Corresponding Ordered Category: A Theorem, Christian Servin, Gerardo Muela, Vladik Kreinovich

Departmental Technical Reports (CS)

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms -- of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees.

Can Mass Be Negative?, Vladik Kreinovich, Sergei Soloviev

Departmental Technical Reports (CS)

Overcoming the force of gravity is an important part of space travel and a significant obstacle preventing many seemingly reasonable space travel schemes to become practical. Science fiction writers like to imagine materials that may help to make space travel easier. Negative mass -- supposedly causing anti-gravity -- is one of the popular ideas in this regard. But can mass be negative? In this paper, we show that negative masses are not possible -- their existence would enable us to create energy out of nothing, which contradicts to the energy conservation law.

How To Detect Crisp Sets Based On Subsethood Ordering Of Normalized Fuzzy Sets? How To Detect Type-1 Sets Based On Subsethood Ordering Of Normalized Interval-Valued Fuzzy Sets?, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

If all we know about normalized fuzzy sets is which set is a subset of which, will we be able to detect crisp sets? It is known that we can do it if we allow all possible fuzzy sets, including non-normalized ones. In this paper, we show that a similar detection is possible if we only allow normalized fuzzy sets. We also show that we can detect type-1 fuzzy sets based on the subsethood ordering of normalized interval-valued fuzzy sets.

How To Efficiently Compute Ranges Over A Difference Between Boxes, With Applications To Underwater Localization, Luc Jaulin, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When using underwater autonomous vehicles, it is important to localize them. Underwater localization is very approximate. As a result, instead of a single location x, we get a set X of possible locations of a vehicle. Based on this set of possible locations, we need to find the range of possible values of the corresponding objective function f(x). For missions on the ocean floor, it is beneficial to take into account that the vehicle is in the water, i.e., that the location of this vehicle is not in a set X' describing the under-floor matter. Thus, the actual set of …

How Many Monte-Carlo Simulations Are Needed To Adequately Process Interval Uncertainty: An Explanation Of The Smart Electric Grid-Related Simulation Results, Afshin Gholamy, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the possible ways of dealing with interval uncertainty is to use Monte-Carlo simulations. A recent study of using this technique for the analysis of different smart electric grid-related algorithms shows that we need approximately 500 simulations to compute the corresponding interval range with 5% accuracy. In this paper, we provide a theoretical explanation for these empirical results.

Measures Of Specificity Used In The Principle Of Justifiable Granularity: A Theoretical Explanation Of Empirically Optimal Selections, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To process huge amounts of data, one possibility is to combine some data points into granules, and then process the resulting granules. For each group of data points, if we try to include all data points into a granule, the resulting granule often becomes too wide and thus rather useless; on the other case, if the granule is too narrow, it includes only a few of the corresponding point -- and is, thus, also rather useless. The need for the trade-off between coverage and specificity is formalized as the principle of justified granularity. The specific form of this principle …

Geostatistical And Geospatial Distribution Of Multiple Metallic Elements In Agricultural Regions Of The Mid-Continental Usa, Najwah Alssaeidi Ahmed Almesleh

Open Access Theses & Dissertations

Regional geochemical mapping constitute the chemistry of the surface of the earth in a large scale and relatively low sample density. It is widely accepted scientific activity and man uses some cases to identify contamination. In the present study, data present in national geochemical survey database was used to investigate the chemical loading loading of the agricultural soil. In the mid-continent region of the United States. Several traditional parameters were used mainly the enrichment factor (EF), and geoaccumulation index (I geo). Total chemical influence (TCI) maps were used to guide sampling. In addition, regional trace element chemistry of bedrock was …

Forecasting Space Weather Using Deep Learning Techniques, Sumi None Dey

Open Access Theses & Dissertations

Solar activity gives rise to various kinds of space weather among which solar flares have serious detrimental eects on both near-Earth space and our upper atmosphere that will have consequent influence in our lives. For example, solar flares can damage satellite infrastructure, hinder power grids, disrupt Global Positioning Systems (GPS) and disrupt long-distance communication. Airplane pilots, cabin crew and astronauts can be aected by the harmful radiation released from the Sun. As a result, there is a need of a methodology to forecast space weather accurately. In this work, we have developed a deep learning architecture to do the short-range …