Physical Sciences and 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.

Z-Numbers: How They Describe Student Confidence And How They Can Explain (And Improve) Laplacian And Schroedinger Eigenmap Dimension Reduction In Data Analysis, Vladik Kreinovich, Olga Kosheleva, Michael Zakharevich

Departmental Technical Reports (CS)

Experts have different degrees of confidence in their statements. To describe these different degrees of confidence, Lotfi A. Zadeh proposed the notion of a Z-number: a fuzzy set (or other type of uncertainty) supplemented by a degree of confidence in the statement corresponding to fuzzy sets. In this chapter, we show that Z-numbers provide a natural formalization of the competence-vs-confidence dichotomy, which is especially important for educating low-income students. We also show that Z-numbers provide a natural theoretical explanation for several empirically heuristic techniques of dimension reduction in data analysis, such as Laplacian and Schroedinger eigenmaps, and, moreover, show how …

Why Deep Learning Methods Use Kl Divergence Instead Of Least Squares: A Possible Pedagogical Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In most applications of data processing, we select the parameters that minimize the mean square approximation error. The same Least Squares approach has been used in the traditional neural networks. However, for deep learning, it turns out that an alternative idea works better -- namely, minimizing the Kullback-Leibler (KL) divergence. The use of KL divergence is justified if we predict probabilities, but the use of this divergence has been successful in other situations as well. In this paper, we provide a possible explanation for this empirical success. Namely, the Least Square approach is optimal when the approximation error is normally …

Why Triangular Membership Functions Are Often Efficient In F-Transform Applications: Relation To Interval Uncertainty\\ And Haar Wavelets, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy techniques describe expert opinions. At first glance, we would therefore expect that the more accurately the corresponding membership functions describe the expert's opinions, the better the corresponding results. In practice, however, contrary to these expectations, the simplest -- and not very accurate -- triangular membership functions often work the best. In this paper, on the example of the use of membership functions in F-transform techniques, we provide a possible theoretical explanation for this surprising empirical phenomenon.

Beyond Integration: A Symmetry-Based Approach To Reaching Stationarity In Economic Time Series, Songsak Sriboonchitta, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many efficient data processing techniques assume that the corresponding process is stationary. However, in areas like economics, most processes are not stationery: with the exception of stagnation periods, economies usually grow. A known way to apply stationarity-based methods to such processes -- integration -- is based on the fact that often, while the process itself is not stationary, its first or second differences are stationary. This idea works when the trend polynomially depends on time. In practice, the trend is usually non-polynomial: it is often exponentially growing, with cycles added. In this paper, we shod how integration techniques can be …

Why Sparse?, Thongchai Dumrongpokaphan, Olga Kosheleva, Vladik Kreinovich, Aleksandra Belina

Departmental Technical Reports (CS)

In many situations, a solution to a practical problem is sparse, i.e., corresponds to the case when most of the parameters describing the solution are zeros, and only a few attain non-zero values. This surprising empirical phenomenon helps solve the corresponding problems -- but it remains unclear why this phenomenon happens. In this paper, we provide a possible theoretical explanation for this mysterious phenomenon.

How To Best Apply Neural Networks In Geosciences: Towards Optimal "Averaging" In Dropout Training, Afshin Gholamy, Justin Parra, Vladik Kreinovich, Olac Fuentes, Elizabeth Y. Anthony

Departmental Technical Reports (CS)

The main objectives of geosciences is to find the current state of the Earth -- i.e., solve the corresponding inverse problems -- and to use this knowledge for predicting the future events, such as earthquakes and volcanic eruptions. In both inverse and prediction problems, often, machine learning techniques are very efficient, and at present, the most efficient machine learning technique is deep neural training. To speed up this training, the current learning algorithms use dropout techniques: they train several sub-networks on different portions of data, and then "average" the results. A natural idea is to use arithmetic mean for this …

Why Taylor Models And Modified Taylor Models Are Empirically Successful: A Symmetry-Based Explanation, Mioara Joldes, Christoph Lauter, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that symmetry-based ideas can explain the empirical success of Taylor models and modified Taylor models in representing uncertainty.

How To Store Tensors In Computer Memory: An Observation, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, after explaining the need to use tensors in computing, we analyze the question of how to best store tensors in computer memory. Somewhat surprisingly, with respect to a natural optimality criterion, the standard way of storing tensors turns out to be one of the optimal ones.

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 …

Propagation Of Probabilistic Uncertainty: The Simplest Case (A Brief Pedagogical Introduction), Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The main objective of this text is to provide a brief introduction to formulas describing the simplest case of propagation of probabilistic uncertainty -- for students who have not yet taken a probability course.

Sudoku App: Model-Driven Development Of Android Apps Using Ocl?, Yoonsik Cheon, Aditi Barua

Departmental Technical Reports (CS)

Model driven development (MDD) shifts the focus of software development from writing code to building models by developing an application as a series of transformations on models including eventual code generation. Can the key ideas of MDD be applied to the development of Android apps, one of the most popular mobile platforms of today? To answer this question, we perform a small case study of developing an Android app for playing Sudoku puzzles. We use the Object Constraint Language (OCL) as the notation for creating precise models and translate OCL constraints to Android Java code. Our findings are mixed in …

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 …

Impacts Of Java Language Features On The Memory Performances Of Android Apps, Yoonsik Cheon, Adriana Escobar De La Torre

Departmental Technical Reports (CS)

Android apps are written in Java, but unlike Java applications they are resource-constrained in storage capacity and battery lifetime. In this document, we perform an experiment to measure quantitatively the impact of Java language and standard API features on the memory efficiency of Android apps. We focus on garbage collection because it is a critical process for performance affecting user experience. We learned that even Java language constructs and standard application programming interfaces (APIs) may be a source of a performance problem causing a significant memory overhead for Android apps. Any critical section of code needs to be scrutinized on …

Need For A Large-N Array (And Wavelets And Differences) To Determine The Assumption-Free 3-D Earth Model, Solymar Ayala Cortez, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main objectives of geophysical seismic analysis is to determine the Earth's structure. Usually, to determine this structure, geophysicists supplement the measurement results with additional geophysical assumptions. An important question is: when is it possible to reconstruct the Earth's structure uniquely based on the measurement results only, without the need to use any additional assumptions? In this paper, we show that for this, one needs to use large-N arrays -- 2-D arrays of seismic sensors. To actually perform this reconstruction, we need to use differences between measurements by neighboring sensor and we need to apply wavelet analysis to …

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.

Is It Legitimate Statistics Or Is It Sexism: Why Discrimination Is Not Rational, Martha Osegueda Escobar, Vladik Kreinovich, Thach N. Nguyen

Departmental Technical Reports (CS)

While in the ideal world, everyone should have the same chance to succeed in a given profession, in reality, often the probability of success is different for people of different gender and/or ethnicity. For example, in the US, the probability of a female undergraduate student in computer science to get a PhD is lower than a similar probability for a male student. At first glance, it may seem that in such a situation, if we try to maximize our gain and we have a limited amount of resources, it is reasonable to concentrate on students with the higher probability of …

Does The Universe Really Expand Faster Than The Speed Of Light: Kinematic Analysis Based On Special Relativity And Copernican Principle, Reynaldo Martinez, Vladik Kreinovich

Departmental Technical Reports (CS)

In the first approximation, the Universe's expansion is described by the Hubble's law v = H * R, according to which the relative speed v of two objects in the expanding Universe grows linearly with the distance R between them. This law can be derived from the Copernican principle, according to which, cosmology-wise, there is no special location in the Universe, and thus, the expanding Universe should look the same from every starting point. The problem with the Hubble's formula is that for large distance, it leads to non-physical larger-than-speed-of-light velocities. Since the Universe's expansion is a consequence of Einstein's …

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.

A Thought On Refactoring Java Loops Using Java 8 Streams, Khandoker Rahad, Zejing Cao, Yoonsik Cheon

Departmental Technical Reports (CS)

Java 8 has introduced a new abstraction called a stream to represent an immutable sequence of elements and to provide a variety of operations to be executed on the elements in series or in parallel. By processing a collection of data in a declarative way, it enables one to write more concise and clean code that can also leverage multi-core architectures without needing a single line of multithread code to be written. In this document, we describe our preliminary work on systematically refactoring loops with Java 8 streams to produce more concise and clean code. Our idea is to adapt …

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 …

How Better Are Predictive Models: Analysis On The Practically Important Example Of Robust Interval Uncertainty, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta, Olga Kosheleva

Departmental Technical Reports (CS)

One of the main applications of science and engineering is to predict future value of different quantities of interest. In the traditional statistical approach, we first use observations to estimate the parameters of an appropriate model, and then use the resulting estimates to make predictions. Recently, a relatively new predictive approach has been actively promoted, the approach where we make predictions directly from observations. It is known that in general, while the predictive approach requires more computations, it leads to more accurate predictions. In this paper, on the practically important example of robust interval uncertainty, we analyze how more accurate …

How To Estimate Statistical Characteristics Based On A Sample: Nonparametric Maximum Likelihood Approach Leads To Sample Mean, Sample Variance, Etc., Vladik Kreinovich, Thongchai Dumrongpokaphan

Departmental Technical Reports (CS)

In many practical situations, we need to estimate different statistical characteristics based on a sample. In some cases, we know that the corresponding probability distribution belongs to a known finite-parametric family of distributions. In such cases, a reasonable idea is to use the Maximum Likelihood method to estimate the corresponding parameters, and then to compute the value of the desired statistical characteristic for the distribution with these parameters.

In some practical situations, we do not know any family containing the unknown distribution. We show that in such nonparametric cases, the Maximum Likelihood approach leads to the use of sample mean, …