Mathematics Commons^™

Linear Regression Under Partial Information, Tho M. Nguyen, Saeid Tizpaz-Niari, Vladik Kreinovich

Departmental Technical Reports (CS)

Often, we need to know how to estimate the value of a difficult-to-directly estimate quantity y -- e.g., tomorrow's temperature -- based on the known values of several quantities x1, ..., xn. In many practical situations, we know that the relation between y and xi can be accurately described by a linear function. So, to find this dependence, we need to estimate the coefficients of this linear dependence based on the known cases in which we know both y and xi; this is known as linear regression. In the ideal situation, when in each case, we know all the inputs …

When Is It Beneficial To Merge Two Companies? When Is It Beneficial To Start A Research Collaboration?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Merging two companies or splitting a company into two, teaming of two researchers or two research groups -- or splitting a research group into two -- these are frequent occurrences. Sometimes these actions lead to increased effectiveness, but sometimes, contrary to the optimistic expectations, the overall effectiveness decreases. To minimize the possibility of such failures, it is desirable to replace the current semi-intuitive way of making the corresponding decisions with a more objective approach. In this paper, we propose such an approach.

Algebraic Product Is The Only "And-Like"-Operation For Which Normalized Intersection Is Associative: A Proof, Thierry Denœx, Vladik Kreinovich

Departmental Technical Reports (CS)

For normalized fuzzy sets, intersection is, in general, not normalized. So, if we want to limit ourselves to normalized fuzzy sets, we need to normalize the intersection. It is known that for algebraic product, the normalized intersection is associative, and that for many other "and"-operations (t-norms), normalized intersection is not associative. In this paper, we prove that algebraic product is the only "and"-operation (even the only "and-like" operation) for which normalized intersection is associative.

Why Unit Two-Variable-Per-Inequality (Utvpi) Constraints Are So Efficient To Handle: Intuitive Explanation, Saeid Tizpaz-Niari, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, integer linear programming is NP-hard. However, there exists a class of integer linear programming problems for which an efficient algorithm is possible: the class of so-called unit two-variable-per-inequality (UTVPI) constraints. In this paper, we provide an intuitive explanation for why an efficient algorithm turned out to be possible for this class. Namely, the smaller the class, the more probable it is that a feasible algorithm is possible for this class, and the UTVPI class is indeed the smallest -- in some reasonable sense described in this paper.

Industry-Academia Collaboration: Main Challenges And What Can We Do, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

How can we bridge the gap between industry and academia? How can we make them collaborate more effectively? In this essay, we try to come up with answers to these important questions.

Towards A Psychologically Natural Relation Between Colors And Fuzzy Degrees, Victor L. Timchenko, Yuriy P. Kondratenko, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

A natural way to speed up computations -- in particular, computations that involve processing fuzzy data -- is to use the fastest possible communication medium: light. Light consists of components of different color. So, if we use optical color computations to process fuzzy data, we need to associate fuzzy degrees with colors. One of the main features -- and of the main advantages -- of fuzzy technique is that the corresponding data has intuitive natural meaning: this data comes from words from natural language. It is desirable to preserve this naturalness as much as possible. In particular, it is desirable …

Why Attitudes Are Usually Mutual: A Possible Mathematical Explanation, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we provide a possible mathematical explanation of why people's attitude to each other is usually mutual: we usually have good attitude who those who have good feelings towards us, and we usually have negative attitudes towards those who have negative feelings towards, Several mathematical explanations of this mutuality have been proposed, but they are based on specific approximate mathematical models of human (and animal) interaction. It is desirable to have a solid mathematical explanation that would not depend on such approximate models. In this paper, we show that a recent mathematical result about relation algebras can lead …

How To Select A Model If We Know Probabilities With Interval Uncertainty, Vladik Kreinovich

Departmental Technical Reports (CS)

Purpose: When we know the probability of each model, a natural idea is to select the most probable model. However, in many practical situations, we do not know the exact values of these probabilities, we only know intervals that contain these values. In such situations, a natural idea is to select some probabilities from these intervals and to select a model with the largest selected probabilities. The purpose of this study is to decide how to most adequately select these probabilities.

Design/methodology/approach: We want the probability-selection method to preserve independence: If, according to the probability intervals, the two …

If Everything Is A Matter Of Degree, Why Do Crisp Techniques Often Work Better?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Numerous examples from different application domain confirm the statement of Lotfi Zadeh -- that everything is a matter of degree. Because of this, one would expect that in most -- if not all -- practical situations taking these degrees into account would lead to more effective control, more effective prediction, etc. In practice, while in many cases, this indeed happens, in many other cases, "crisp" methods -- methods that do not take these degrees into account -- work better. In this paper, we provide two possible explanations for this discrepancy: an objective one -- explaining that the optimal (best-fit) model …

How To Best Retrain A Neural Network If We Added One More Input Variable, Saeid Tizpaz-Niari, Vladik Kreinovich

Departmental Technical Reports (CS)

Often, once we have trained a neural network to estimate the value of a quantity y based on the available values of inputs x1, ..., xn, we learn to measure the values of an additional quantity that have some influence on y. In such situations, it is desirable to re-train the neural network, so that it will be able to take this extra value into account. A straightforward idea is to add a new input to the first layer and to update all the weights based on the patterns that include the values of the new input. The problem with …

Topological Explanation Of Why Complex Numbers Are Needed In Quantum Physics, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

In quantum computing, we only use states in which all amplitudes are real numbers. So why do we need complex numbers with non-zero imaginary part in quantum physics in general? In this paper, we provide a simple topological explanation for this need, explanation based on the Second Law of Thermodynamics.

How To Make Decision Under Interval Uncertainty: Description Of All Reasonable Partial Orders On The Set Of All Intervals, Tiago M. Costa, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to make a decision while for each alternative, we only know the corresponding value of the objective function with interval uncertainty. To help a decision maker in this situation, we need to know the (in general, partial) order on the set of all intervals that corresponds to the preferences of the decision maker. For this purpose, in this paper, we provide a description of all such partial orders -- under some reasonable conditions. It turns out that each such order is characterized by two linear inequalities relating the endpoints of the corresponding intervals, and …

Which Fuzzy Implications Operations Are Polynomial? A Theorem Proves That This Can Be Determined By A Finite Set Of Inequalities, Sebastia Massanet, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To adequately represent human reasoning in a computer-based systems, it is desirable to select fuzzy operations that are as close to human reasoning as possible. In general, every real-valued function can be approximated, with any desired accuracy, by polynomials; it is therefore reasonable to use polynomial fuzzy operations as the appropriate approximations. We thus need to select, among all polynomial operations that satisfy corresponding properties -- like associativity -- the ones that best fit the empirical data. The challenge here is that properties like associativity mean satisfying infinitely many constraints (corresponding to infinitely many possible triples of values), while most …

Fuzzy Mathematics Under Non-Minimal "And"-Operations (T-Norms): Equivalence Leads To Metric, Order Leads To Kinematic Metric, Topology Leads To Area Or Volume, Purbita Jana, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Most formulas analyzed in fuzzy mathematics assume -- explicitly or implicitly -- that the corresponding "and"-operation (t-norm) is the simplest minimum operation. In this paper, we analyze what happens if instead, we use other "and"-operations. It turns out that for such operations, a fuzzification of a mathematical theory naturally leads to a more complex mathematical setting: fuzzification of equivalence relation leads to metric, fuzzification of order leads to kinematic metric, and fuzzification of topology leads to area or volume.

Why Deep Learning Is Under-Determined? Why Usual Numerical Methods For Solving Partial Differential Equations Do Not Preserve Energy? The Answers May Be Related To Chevalley-Warning Theorem (And Thus To Fermat Last Theorem), Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we provide a possible explanation to two seemingly unrelated phenomena: (1) that in deep learning, under-determined systems of equations perform much better than the over-determined one -- which are typical in data processing, and that (2) usual numerical methods for solving partial differential equations do not preserve energy. Our explanation is related to the intuition of Fermat behind his Last Theorem and of Euler about more general statements, intuition that led to the proof of Chevalley-Warning Theorem in number theory.

Methodological Lesson Of Pythagorean Triples, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

There are many right triangles in which all three sides a, b, and c have integer lengths. The triples (a,b,c) formed by such lengths are known as Pythagorean triples. Since ancient times, it is known how to generate all Pythagorean triples: we can enumerate primitive Pythagorean triples -- in which the three numbers have no common divisors -- by considering all pairs of natural numbers m>n in which m and n have no common divisors, and taking a =m2 − n2, b = 2mn, and c = m2 + n2. Multiplying all elements of a triple by the same …

Why Resilient Modulus Is Proportional To The Square Root Of Unconfined Compressive Strength (Ucs): A Qualitative Explanation, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

The strength of the pavement is determine by its resilient modulus, i.e., by its ability to withstand (practically) instantaneous stresses caused by the passing traffic. However, the resilient modulus is not easy to measure: its measurement requires a special expensive equipment that many labs do not have. So, instead of measuring it, practitioners often measure easier-to-measure Unconfined Compressive Strength (UCS) -- that describes the effect of a continuously applied force -- and estimate the resilient modulus based on the result of this measurement. An empirical formula shows that the resilient modulus is proportional to the square root of the Unconfined …

How To Estimate Unknown Unknowns: From Cosmic Light To Election Polls, Talha Azfar, Vignesh Ponraj, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

In two different areas of research -- in the study of space light and in the study of voting -- the observed value of the quantity of interest is twice larger than what we would expect. That the observed value is larger makes perfect sense: there are phenomena that we do not take into account in our estimations. However, the fact that the observed value is exactly twice larger deserves explanation. In this paper, we show that Laplace Indeterminacy Principle leads to such an explanation.

How To Combine Probabilistic And Fuzzy Uncertainty: Theoretical Explanation Of Clustering-Related Empirical Result, Lázló Szilágyi, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In contrast to crisp clustering techniques that assign each object to a class, fuzzy clustering algorithms assign, to each object and to each class, a degree to which this object belongs to this class. In the most widely used fuzzy clustering algorithm -- fuzzy c-means -- for each object, degrees corresponding to different classes add up to 1. From this viewpoint, these degrees act as probabilities. There exist alternative fuzzy-based clustering techniques in which, in line with the general idea of the fuzzy set, the largest of the degrees is equal to 1. In some practical situations, the probability-type fuzzy …

Fuzzy Techniques Explain The Effectiveness Of Relu Activation Function In Deep Learning, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the last decades, deep learning has led to spectacular successes. One of the reasons for these successes was the fact that deep neural networks use a special Rectified Linear Unit (ReLU) activation function s(x) = max(0,x). Why this activation function is so successful is largely a mystery. In this paper, we show that common sense ideas -- as formalized by fuzzy logic -- can explain this mysterious effectiveness.

Why 6-Labels Uncertainty Scale In Geosciences: Probability-Based Explanation, Aaron Velasco, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To describe uncertainty in geosciences, several researchers have recently proposed a 6-labels uncertainty scale, in which one the labels corresponds to full certainty, one label to the absence of any knowledge, and the remaining four labels correspond to the degrees of confidence from the intervals [0,0.25], [0.25,0.5], [0.5,0.75], and [0.75,1]. Tests of this 6-labels scale indicate that it indeed conveys uncertainty information to geoscientists much more effectively than previously proposed uncertainty schemes. In this paper, we use probability-related techniques to explain this effectiveness.

Complex Numbers Explain Why In Chinese Tradition, 4 Is Bad But 8 Is Good, Luc Longpre, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional Chinese culture, 4 is considered to be an unlucky number, while the number 8 is considered to be very lucky. In this paper, we show that both "badness" and "goodness" can be explained if we take into account the role of complex numbers in the analysis of general dynamical systems.

Why Bump Reward Function Works Well In Training Insulin Delivery Systems, Lehel Dénes-Fazakas, Lásló Szilágyi, Gyorgy Eigner, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Diabetes is a disease when the body can no longer properly regulate blood glucose level, which can lead to life-threatening situations. To avoid such situations and regulate blood glucose level, patients with severe form of diabetes need insulin injections. Ideally, the system should automatically decide when best to inject insulin and how much to inject. To find the optimal control, researchers applied machine learning with different reward functions. It turns out that the most effective learning occurred when they used the so-called bump function. In this paper, we provide a possible explanation for this empirical result.

We Can Always Reduce A Non-Linear Dynamical System To Linear -- At Least Locally -- But Does It Help?, Orsolya Csiszar, Gábor Csiszar, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Many real-life phenomena are described by dynamical systems. Sometimes, these dynamical systems are linear. For such systems, solutions are well known. In some cases, it is possible to transform a nonlinear system into a linear one by appropriately transforming its variables, and this helps to solve the original nonlinear system. For other nonlinear systems -- even for the simplest ones -- such transformation is not known. A natural question is: which nonlinear systems allow such transformations? In this paper, we show that we can always reduce a nonlinear system to a linear one -- but, in general, it does not …

What Was More Frequently Used -- "And" Or "Or": Based On Analysis Of European Languages, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional logic has two main connectives: "and" and "or". A natural question is: which of the two is more frequently used? This question is easy to answer for the current usage of these connectives -- we can simply analyze all the texts, but what can we say about the past usage? To answer this question, we use the known linguistics fact that, in general, notions that are more frequently used are described by shorter words. It turns out that in most European languages, the word for "and" is shorter -- or of the same length -- as the word for …

How To Propagate Interval (And Fuzzy) Uncertainty: Optimism-Pessimism Approach, Vinícius F. Wasques, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, inputs to a data processing algorithm are known with interval uncertainty, and we need to propagate this uncertainty through the algorithm, i.e., estimate the uncertainty of the result of data processing. Traditional interval computation techniques provide guaranteed estimates, but from the practical viewpoint, these bounds are too pessimistic: they take into account highly improbable worst-case situations when all the measurement and estimation errors happen to be strongly correlated. In this paper, we show that a natural idea of having more realistic estimates leads to the use of so-called interactive addition of intervals, techniques that has already …

Which Activation Function Works Best For Training Artificial Pancreas: Empirical Fact And Its Theoretical Explanation, Lehel Dénes-Fazakas, Lásló Szilágyi, György Eigner, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the most effective ways to help patients at the dangerous levels of diabetes is an artificial pancreas, a device that constantly monitors the patient's blood sugar level and injects insulin based on this level. Patient's reaction to insulin is highly individualized, so the artificial pancreas needs to be trained on each patient. It turns out that the best training results are attained when instead of the usual ReLU neurons, we use their minor modification known as Exponential Linear Units (ELU). In this paper, we provide a theoretical explanation for the empirically observed effectiveness of ELUs.

Dialogs Re-Enacted Across Languages, Version 2, Nigel G. Ward, Jonathan E. Avila, Emilia Rivas, Divette Marco

Departmental Technical Reports (CS)

To support machine learning of cross-language prosodic mappings and other ways to improve speech-to-speech translation, we present a protocol for collecting closely matched pairs of utterances across languages, a description of the resulting data collection and its public release, and some observations and musings. This report is intended for:

• people using this corpus
• people extending this corpus
• people designing similar collections of bilingual dialog data.

Change Notes. This version supersedes UTEP-CS-22-108. There is some new information and numerous clarifications, mostly arising from our experiences diversifying our corpus and helping a vendor to use this protocol.

Natural Color Interpretation Of Interval-Valued Fuzzy Degrees, Victor L. Timchenko, Yury P. Kondratenko, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Intuitively, interval-values fuzzy degrees are more adequate for representing expert uncertainty than the traditional [0,1]-based ones. Indeed, the very need for fuzzy degrees comes from the fact that experts often cannot describe their opinion not in terms of precise numbers, but by using imprecise ("fuzzy") words from natural language like "small". In such situations, it is strange to expect the same expert to be able to provide an exact number describing his/her degree of certainty; it is more natural to ask this expert to mark the whole interval (or even, more generally, a fuzzy set of possible degrees). In spite …

Logical Inference Inevitably Appears: Fuzzy-Based Explanation, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich, Orsolya Csiszar

Departmental Technical Reports (CS)

Many thousands years ago, our primitive ancestors did not have the ability to reason logically and to perform logical inference. This ability appeared later. A natural question is: was this appearance inevitable -- or was this a lucky incident that could have been missed? In this paper, we use fuzzy techniques to provide a possible answer to this question. Our answer is: yes, the appearance of logical inference in inevitable.