Mathematics Commons^™

Quantum Econometrics: How To Explain Its Quantitative Successes And How The Resulting Formulas Are Related To Scale Invariance, Entropy, Fuzzy, And Copulas, Hung T. Nguyen, Kittawit Autchariyapanitkul, Olga Kosheleva, Vladik Kreinovich, Songsak Sriboonchitta

Departmental Technical Reports (CS)

Many aspects of human behavior seem to be well-described by formulas of quantum physics. In this paper, we explain this phenomenon by showing that the corresponding quantum-looking formulas can be derived from the general ideas of scale invariance, fuzziness, and copulas. We also use these ideas to derive a general family of formulas that include non-quantum and quantum probabilities as particular cases -- formulas that may be more adequate for describing human behavior than purely non-quantum or purely quantum ones.

Why Rectified Linear Neurons Are Efficient: Symmetry-Based, Complexity-Based, And Fuzzy-Based Explanations, Olac Fuentes, Justin Parra, Elizabeth Y. Anthony, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, neural networks used a sigmoid activation function. Recently, it turned out that piecewise linear activation functions are much more efficient -- especially in deep learning applications. However, so far, there have been no convincing theoretical explanation for this empirical efficiency. In this paper, we show that, by using different uncertainty techniques, we can come up with several explanations for the efficiency of piecewise linear neural networks. The existence of several different explanations makes us even more confident in our results -- and thus, in the efficiency of piecewise linear activation functions.

How To Make A Proof Of Halting Problem More Convincing: A Pedagogical Remark, Benjamin W. Robertson, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

As an example of an algorithmically undecidable problem, most textbooks list the impossibility to check whether a given program halts on given data. A usual proof of this result is based on the assumption that the hypothetical halt-checker works for all programs. To show that a halt-checker is impossible, we design an auxiliary program for which the existence of such a halt-checker leads to a contradiction. However, this auxiliary program is usually very artificial. So, a natural question arises: what if we only require that the halt-checker work for reasonable programs? In this paper, we show that even with such …

Do It Today Or Do It Tomorrow: Empirical Non-Exponential Discounting Explained By Symmetry And Fuzzy Ideas, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich, Thongchai Dumrongpokaphan

Departmental Technical Reports (CS)

At first glance, it seems to make sense to conclude that when a 1 dollar reward tomorrow is equivalent to a D < 1 dollar reward today, the day-after-tomorrow's 1 dollar reward would be equivalent to D * D = D² dollars today, and, in general, a reward after time t is equivalent to D(t) = D^t dollars today. This exponential discounting function D(t) was indeed proposed by the economists, but it does not reflect the actual human behavior. Indeed, according to this formula, the effect of distant future events is negligible, and thus, it would be reasonable for a person to take on huge loans or get engaged in unhealthy behavior even when the long-term consequences …

An Ancient Bankruptcy Solution Makes Economic Sense, Anh H. Ly, Michael Zakharevich, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While econometrics is a reasonable recent discipline, quantitative solutions to economic problem have been proposed since the ancient times. In particular, solutions have been proposed for the bankruptcy problem: how to divide the assets between the claimants? One of the challenges of analyzing ancient solutions to economics problems is that these solutions are often presented not as a general algorithm, but as a sequence of examples. When there are only a few such example, it is often difficult to convincingly extract a general algorithm from them. This was the case, for example, for the supposedly fairness-motivated Talmudic solution to the …

Maximum Entropy Beyond Selecting Probability Distributions, Thach N. Nguyen, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, the Maximum Entropy technique is used to select a probability distribution in situations when several different probability distributions are consistent with our knowledge. In this paper, we show that this technique can be extended beyond selecting probability distributions, to explain facts, numerical values, and even types of functional dependence.

Almost All Diophantine Sets Are Undecidable, Vladik Kreinovich

Departmental Technical Reports (CS)

The known 1970 solution to the 10th Hilbert problem says that no algorithm is possible that would decide whether a given Diophantine equation has a solution. In set terms, this means that not all Diophantine sets are decidable. In a posting to the Foundations of Mathematica mailing list, Timothy Y. Chow asked for possible formal justification for his impression that most Diophantine equations are not decidable. One such possible justification is presented in this paper.

Efficient Parameter-Estimating Algorithms For Symmetry-Motivated Models: Econometrics And Beyond, Vladik Kreinovich, Anh H. Ly, Olga Kosheleva, Songsak Sriboonchitta

Departmental Technical Reports (CS)

It is known that symmetry ideas can explain the empirical success of many non-linear models. This explanation makes these models theoretically justified and thus, more reliable. However, the models remain non-linear and thus, identification or the model's parameters based on the observations remains a computationally expensive nonlinear optimization problem. In this paper, we show that symmetry ideas can not only help to select and justify a nonlinear model, they can also help us design computationally efficient almost-linear algorithms for identifying the model's parameters.

Practical Need For Algebraic (Equality-Type) Solutions Of Interval Equations And For Extended-Zero Solutions, Ludmila Dymova, Pavel Sevastjanov, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main problems in interval computations is solving systems of equations under interval uncertainty. Usually, interval computation packages consider united, tolerance, and control solutions. In this paper, we explain the practical need for algebraic (equality-type) solutions, when we look for solutions for which both sides are equal. In situations when such a solution is not possible, we provide a justification for extended-zero solutions, in which we ignore intervals of the type [−a, a].

What Is The Optimal Bin Size Of A Histogram: An Informal Description, Afshin Gholamy, Vladik Kreinovich

Departmental Technical Reports (CS)

A natural way to estimate the probability density function of an unknown distribution from the sample of data points is to use histograms. The accuracy of the estimate depends on the size of the histogram's bins. There exist heuristic rules for selecting the bin size. In this paper, we show that these rules indeed provide the optimal value of the bin size.

How To Use Absolute-Error-Minimizing Software To Minimize Relative Error: Practitioner's Guide, Afshin Gholamy, Vladik Kreinovich

Departmental Technical Reports (CS)

In many engineering and scientific problems, there is a need to find the parameters of a dependence from the experimental data. There exist several software packages that find the values for these parameters -- values for which the mean square value of the absolute approximation error is the smallest. In practice, however, we are often interested in minimizing the mean square value of the relative approximation error. In this paper, we show how we can use the absolute-error-minimizing software to minimize the relative error.

Granular Approach To Data Processing Under Probabilistic Uncertainty, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, uncertainty can be naturally described as a combination of several components, components which are described by probabilistic, fuzzy, interval, etc. granules. In such situations, to process this uncertainty, it is often beneficial to take this granularity into account by processing these granules separately and then combining the results.

In this paper, we show that granular computing can help even in situations when there is no such natural decomposition into granules: namely, we can often speed up processing of uncertainty if we first (artificially) decompose the original uncertainty into appropriate granules.

How To Get Beyond Uniform When Applying Maxent To Interval Uncertainty, Songsak Sriboonchitta, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, the Maximum Entropy (MaxEnt) approach leads to reasonable distributions. However, in an important case when all we know is that the value of a random variable is somewhere within the interval, this approach leads to a uniform distribution on this interval -- while our intuition says that we should have a distribution whose probability density tends to 0 when we approach the interval's endpoints. In this paper, we show that in most cases of interval uncertainty, we have additional information, and if we account for this additional information when applying MaxEnt, we get distributions which are …

Quantitative Justification For The Gravity Model In Economics, Vladik Kreinovich, Songsak Sriboonchitta

Departmental Technical Reports (CS)

The gravity model in economics describes the trade flow between two countries as a function of their Gross Domestic Products (GDPs) and the distance between them. This model is motivated by the qualitative similarity between the desired dependence and the dependence of the gravity force (or potential energy) between the two bodies on their masses and on the distance between them. In this paper, we provide a quantitative justification for this economic formula.

In Education, Delayed Feedback Is Often More Efficient Than Immediate Feedback: A Geometric Explanation, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Feedback is important in education. It is commonly believed that immediate feedback is very important. That is why instructors stay often late at night grading students' assignments -- to make sure that the students get their feedback as early as possible. However, surprisingly, experiments show that in many cases, delayed feedback is more efficient that the immediate one. In this paper, we provide a simple geometric explanation of this seemingly counter-intuitive empirical phenomenon.

Are Permanent Or Temporary Teams More Efficient: A Possible Explanation Of The Empirical Data, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that in education, stable (long-term) student teams are more effective than temporary (short-term) ones. It turned out that the same phenomenon is true for workers working on a long-term project. However, somewhat surprisingly, for small-scale projects, the opposite is true: teams without any prior collaboration experience are more successful. Moreover, it turns out that if combine in a team members with prior collaboration experience and members without such experience, the efficiency of the team gets even lower. In this paper, we provide a possible explanation for this strange empirical phenomenon.

A Bad Plan Is Better Than No Plan: A Theoretical Justification Of An Empirical Observation, Songsak Sriboonchitta, Vladik Kreinovich

Departmental Technical Reports (CS)

In his 2014 book "Zero to One", a software mogul Peter Thiel lists the lessons he learned from his business practice. Most of these lessons make intuitive sense, with one exception -- his observation that "a bad plan is better than no plan" seems to be counterintuitive. In this paper, we provide a possible theoretical explanation for this somewhat counterintuitive empirical observation.

How Accurate Are Expert Estimations Of Correlation?, Michael Beer, Zitong Gong, Francisco Alejandro Diaz De La O, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, it is important to know the correlation between different quantities -- finding correlations helps find the causes of different phenomena, and helps to find way to improve the situation. Often, there is not enough empirical data to experimentally determine all possible correlation. In such cases, a natural idea is to supplement this situation with expert estimates. Expert estimates are rather crude. So, to decide whether to act based on these estimates, it is desirable to know how accurate are expert estimates. In this paper, we propose several techniques for gauging this accuracy.

Maximum Entropy As A Feasible Way To Describe Joint Distributions In Expert Systems, Thongchai Dumrongpokaphan, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

In expert systems, we elicit the probabilities of different statements from the experts. However, to adequately use the expert system, we also need to know the probabilities of different propositional combinations of the experts' statements -- i.e., we need to know the corresponding joint distribution. The problem is that there are exponentially many such combinations, and it is not practically possible to elicit all their probabilities from the experts. So, we need to estimate this joint distribution based on the available information. For this purpose, many practitioners use heuristic approaches -- e.g., the t-norm approach of fuzzy logic. However, this …

Possible Explanation Of Empirical Values Of The Matern Smoothness Parameter For The Temporal Covariance Of Gps Measurements, Gaël Kermarrec, Steffen Schön, Vladik Kreinovich

Departmental Technical Reports (CS)

The measurement errors of GPS measurements are largely due to the atmosphere, and the unpredictable part of these errors are due to the unpredictable (random) atmospheric phenomena, i.e., to turbulence. Turbulence-generated measurement errors should correspond to the smoothness parameter ν = 5/6 in the Matern covariance model. Because of this, we expected the empirical values of this smoothness parameter to be close to 5/6. When we estimated ν based on measurement results, we indeed got values close to 5/6, but interestingly, all our estimates were actually close to 1 (and slightly larger than 1). In this paper, we provide a …

Efficient Algorithms For Synchroning Localization Sensors Under Interval Uncertainty, Raphael Voges, Bernardo Wagner, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that a practical need for synchronization of localization sensors leads to an interval-uncertainty problem. In principle, this problem can be solved by using the general linear programming algorithms, but this would take a long time -- and this time is not easy to decrease, e.g., by parallelization since linear programming is known to be provably hard to parallelize. To solve the corresponding problem, we propose more efficient and easy-to-parallelize algorithms.

No Idea Is A Bad Idea: A Theoretical Explanation, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

Many business publications state that no idea is a bad idea, that even if the idea is, at first glance, not helpful, there are usually some aspects of this idea which are helpful Usually, this statement is based on the experience of the author, and it is given without any theoretical explanation. In this paper, we provide a theoretical explanation for this statement.

Maybe The Usual Students' Practice Of Cramming For A Test Makes Sense: A Mathematical Analysis, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

We always teach students that cramming for a test is a bad idea, that they should study at the same speed throughout the semester – but many still cram. We ourselves are not that different: when we prepare papers for a conference, we often “cram” in the last days before the deadline instead of working with a regular speed for the whole time before the conference. The ubiquity of cramming makes us think that maybe it is not necessarily always a bad idea. And indeed, a simple model of a study process shows that an optimal solution often involve some …

How To Teach Implication, Martha Osegueda Escobar, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Logical implication is a somewhat counter-intuitive notion. For students, it is difficult to understand why a false statement implies everything. In this paper, we present a simple pedagogical way to make logical implication more intuitive.

Attraction-Repulsion Forces Between Biological Cells: A Theoretical Explanation Of Empirical Formulas, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Biological calls attract and repulse each other: if they get too close to each other, they repulse, and if they get too far away from each other, they attract. There are empirical formulas that describe the dependence of the corresponding forces on the distance between the cells. In this paper, we provide a theoretical explanation for these empirical formulas.

Towards Decision Making Under General Uncertainty, Andrzej Pownuk, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

There exist techniques for decision making under specific types of uncertainty, such as probabilistic, fuzzy, etc. Each of the corresponding ways of describing uncertainty has its advantages and limitations. As a result, new techniques for describing uncertainty appear all the time. Instead of trying to extend the existing decision making idea to each of these new techniques one by one, we attempt to develop a general approach that would cover all possible uncertainty techniques.

Can We Detect Crisp Sets Based Only On The Subsethood Ordering Of Fuzzy Sets? Fuzzy Sets And/Or Crisp Sets Based On Subsethood Of Interval-Valued Fuzzy Sets?, Christian Servin, Gerardo Muela, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy sets are naturally ordered by the subsethood relation. If we only know which set which fuzzy set is a subset of which -- and have no access to the actual values of the corresponding membership functions -- can we detect which fuzzy sets are crisp? In this paper, we show that this is indeed possible. We also show that if we start with interval-valued fuzzy sets, then we can similarly detect type-1 fuzzy sets and crisp sets.

Derivation Of Gross-Pitaevskii Version Of Nonlinear Schroedinger Equation From Scale Invariance, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be different from 3, it is desirable to analyze what happens in other spatial dimensions D. It turns out that while for D ≥ 3 we still get only the Schroedinger's equation, for D = 2, we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for D = 1, we also get a new nonlinear …

Why Stable Teams Are More Efficient In Education, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that study groups speed up learning. Recent studies have shown that stable study groups are more efficient than shifting-membership groups. In this paper, we provide a theoretical explanation for this empirical observation.

Why Linear Interpolation?, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

Linear interpolation is the computationally simplest of all possible interpolation techniques. Interestingly, it works reasonably well in many practical situations, even in situations when the corresponding computational models are rather complex. In this paper, we explain this empirical fact by showing that linear interpolation is the only interpolation procedure that satisfies several reasonable properties such as consistency and scale-invariance.