Applied Mathematics Commons^™

Grading Homeworks, Verifying Code: How Thorough Should The Feedback Be?, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, we should assign many homeworks and give a thorough feedback for each homework. However, in reality, the instructor's time is limited, so we can either assign few homeworks and give a detailed feed back for all of them, or we can assign many homeworks, but give a less thorough feedback. What is the optimal thoroughness? A similar question can be raised for code verification: what is the optimal amount of feedback that should be provided to each programmer? In this paper, we provide answers to these questions.

Let Us Use Negative Examples In Regression-Type Problems Too, Jonatan Contreras, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich, Martine Ceberio

Departmental Technical Reports (CS)

In many practical situations, we need to reconstruct the dependence between quantities x and y based on several situations in which we know both x and y values. Such problems are known as regression problems. Usually, this reconstruction is based on positive examples, when we know y -- at least, with some accuracy. However, in addition, we often also know some examples in which we have negative information about y -- e.g., we know that y does not belong to a certain interval. In this paper, we show how such negative examples can be used to make the solution …

Why Quadratic Log-Log Dependence Is Ubiquitous And What Next, Sean R. Aguilar, Vladik Kreinovich, Uyen Pham

Departmental Technical Reports (CS)

In many real-life situations ranging from financial to volcanic data, growth is described either by a power law -- which is linear in log-log scale, or by a quadratic dependence in the log-log scale. In this paper, we use natural scale invariance requirement to explain the ubiquity of such dependencies. We also explain what should be a reasonable choice of the next model, if quadratic turns out to be not too accurate: it turns out that under scale invariance, the next class of models are cubic dependencies in the log-log scale, then fourth order dependencies, etc.

It Is Important To Take All Available Information Into Account When Making A Decision: Case Of The Two Envelopes Problem, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In situations when we know the probabilities of all possible consequences, traditional decision theory recommends selecting the action that maximizes expected utility. In many practical situations, however, we only have partial information about the corresponding probabilities. In this case, for different possible probability distributions, we get different values of expected utility. In general, possible values of expected utility form an interval. One way to approach this situation is to use the optimism-pessimism approach proposed by Nobelist Leo Hurwicz. Another approach is to select one of the possible probability distributions -- e.g., the one that has the largest possible entropy. Both …

How To Decide Which Cracks Should Be Repaired First: Theoretical Explanation Of Empirical Formulas, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Due to stress, cracks appear in constructions: cracks appear in buildings, bridges, pavements, among other structures, cracks appear in pavements, etc. In the long run, cracks need to be repaired. However, our resources are limited, so we need to decide which cracks are more dangerous. For this, we need to be able to predict how different cracks will grow. There are several empirical formulas describing crack growth. In this paper, we show that by using scale invariance, we can provide a theoretical explanation for these empirical formulas.

Euclidean Distance Between Intervals Is The Only Representation-Invariant One, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

An interval can be represented as a point in a plane, e.g., as a point with its endpoints as coordinates. We can thus define distance between intervals as the Euclidean distance between the corresponding points. Alternatively, we can describe an interval by its center and radius, which leads to a different definition of distance. Interestingly, these two definitions lead, in effect, to the same distance -- to be more precise, these two distances differ by a multiplicative constant. In principle, we can have more general distances on the plane. In this paper, we show that only for Euclidean distance, the …

The Less We Love A Woman, The More She Likes Us: Pushkin's Observation Explained, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Alexander Pushkin, the most famous Russian poet, made this observation in "Eugene Onegin", his novel in verse which is most known to non-Russian readers via Tchaikovsky's opera. This observation may not be an absolute truth -- there are counterexamples -- but the fact that it is still widely cited shows that there is some truth in this statement. In this paper, we recall the usual utility-based explanation for a similar statement, and propose a new explanation, which is even more fundamental -- it is on the biological level.

How To Detect Future Einsteins: Towards Systems Approach, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Talents are rare. It is therefore important to detect and nurture future talents as early as possible. In many disciplines, this is already being done -- via gifted and talented programs, Olympiads, and other ways to select kids with unusually high achievements. However, the current approach is not perfect: some of the kids are selected simply because they are early bloomers, they do not grow into unusually successful researchers; on the other hand, many of those who later become very successful are not selected since they are late bloomers. To avoid these problems, we propose to use systems approach: to …

Which Classes Of Bi-Intervals Are Closed Under Addition?, Olga Kosheleva, Vladik Kreinovich, Jonatan Contreras

Departmental Technical Reports (CS)

In many practical situations, uncertainty with which we know each quantity is described by an interval. In processing such data, it is useful to know that the sum of two intervals is always an interval. In some cases, however, the set of all possible value of a quantity is described by a bi-interval -- i.e., by a union of two intervals. It is known that the sum of two bi-intervals is not always a bi-interval. In this paper, we describe all the class of bi-intervals which are closed under addition -- i.e., for which the sum of bi-intervals is a …

Preference For Boys Does Not Necessarily Lead To A Gender Disbalance: A Realistic Example, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Intuitively, it seems that cultural preference for boys should lead to a gender disbalance -- more boys than girls. This disbalance is indeed what is often observed, and this disbalance is what many models predict. However, in this paper, we show, on a realistic example, that preference for boys does not necessarily lead to a gender disbalance: in our simplified example, boys are clearly preferred, but still there are exactly as many girls as there are boys.

Healthy Lifestyle Decreases The Risk Of Alzheimer Disease: A Possible Partial Explanation Of An Empirical Dependence, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent paper showed that for people who follow all five healthy lifestyle recommendations, the risk of Alzheimer disease is only 40% of the risk for those who do not follow any of these recommendations, and that for people two or three of these recommendations, the risk is 63% of the not-followers risk. In this paper, we show that a relation between the two numbers -- namely, that 0.40 is the square of 0.63 -- can be naturally explained by a simple model.

When Can We Be Sure That Measurement Results Are Consistent: 1-D Interval Case And Beyond, Hani Dbouk, Steffen Schön, Ingo Neumann, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, measurements are characterized by interval uncertainty -- namely, based on each measurement result, the only information that we have about the actual value of the measured quantity is that this value belongs to some interval. If several such intervals -- corresponding to measuring the same quantity -- have an empty intersection, this means that at least one of the corresponding measurement results is an outlier, caused by a malfunction of the measuring instrument. From the purely mathematical viewpoint, if the intersection is non-empty, there is no reason to be suspicious, but from the practical viewpoint, if …

What If Not All Interval-Valued Fuzzy Degrees Are Possible?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the applications of intervals is in describing experts' degrees of certainty in their statements. In this application, not all intervals are realistically possible. To describe all realistically possible degrees, we end up with a mathematical question of describing all topologically closed classes of intervals which are closed under the appropriate minimum and maximum operations. In this paper, we provide a full description of all such classes.

Common-Sense-Based Theoretical Explanation For An Empirical Formula Estimating Road Quality, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

The quality of a road is usually gauged by a group of trained raters; the resulting numerical value is known as the Present Serviceability Index (PSI). There are, however, two problems with this approach. First, while it is practical to use trained raters to gauge the quality of major highways, there are also numerous not-so-major roads, and there is not enough trained raters to gauge the quality of all of them. Second, even for skilled raters, their estimates are somewhat subjective: different groups of raters may estimate the quality of the same road segment somewhat differently. Because of these two …

How To Estimate The Stiffness Of The Multi-Layer Road Based On Properties Of Layers: Symmetry-Based Explanation For Odemark's Equation, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

When we design a road, we would like to check that the current design provides the pavement with sufficient stiffness to withstand traffic loads and climatic conditions. For this purpose, we need to estimate the stiffness of the road based on stiffness and thickness of its different layers. There exists a semi-empirical formula for this estimation. In this paper, we show that this formula can be explained by natural scale-invariance requirements.

How The Proportion Of People Who Agree To Perform A Task Depends On The Stimulus: A Theoretical Explanation Of The Empirical Formula, Laxman Bokati, Vladik Kreinovich, Doan Thanh Ha

Departmental Technical Reports (CS)

For each task, the larger the stimulus, the larger proportion of people agree to perform this task. In many economic situations, it is important to know how much stimulus we need to offer so that a sufficient proportion of the people will agree to perform the needed task. There is an empirical formula describing how this proportion increases as we increase the amount of stimulus. However, this empirical formula lacks a convincing theoretical explanation, as a result of which practitioners are somewhat reluctant to use it. In this paper, we provide a theoretical explanation for this empirical formula, thus making …

Reward For Good Performance Works Better Than Punishment For Mistakes: Economic Explanation, Olga Kosheleva, Julio Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

How should we stimulate people to make them perform better? How should we stimulate students to make them study better? Many experiments have shown that reward for good performance works better than punishment for mistakes. In this paper, we provide a possible theoretical explanation for this empirical fact.

Why Most Empirical Distributions Are Few-Modal, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In principle, any non-negative function can serve as a probability density function -- provided that it adds up to 1. All kinds of processes are possible, so it seems reasonable to expect that observed probability density functions are random with respect to some appropriate probability measure on the set of all such functions -- and for all such measures, similarly to the simplest case of random walk, almost all functions have infinitely many local maxima and minima. However, in practice, most empirical distributions have only a few local maxima and minima -- often one (unimodal distribution), sometimes two (bimodal), and, …

Economics Of Reciprocity And Temptation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Nguyen Ngoc Thach

Departmental Technical Reports (CS)

Behavioral economics has shown that in many situations, people's behavior differs from what is predicted by simple traditional utility-maximization economic models. It is therefore desirable to be able to accurately describe people's actual behavior. In some cases, the difference from the traditional models is caused by bounded rationality -- our limited ability to process information and to come up with a truly optimal solutions. In such cases, predicting people's behavior is difficult. In other cases, however, people actually optimize -- but the actual expression for utility is more complicated than in the traditional models. In such case, it is, in …

Scale-Invariance Ideas Explain The Empirical Soil-Water Characteristic Curve, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

The prediction of the road's properties under the influence of water infiltration is important for pavement design and management. Traditionally, this prediction heavily relied on expert estimates. In the last decades, complex empirical formulas have been proposed to capture the expert's intuition in estimating the effect of water infiltration on the stiffness of the pavement's payers. Of special importance is the effect of water intrusion on the pavement's foundation -- known as subgrade soil. In this paper, we show that natural scale-invariance ideas lead to a theoretical explanation for an empirical formula describing the dependence between soil suction and water …

Towards A Theoretical Explanation Of How Pavement Condition Index Deteriorates Over Time, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

To predict how the Pavement Condition Index will change over time, practitioners use a complex empirical formula derived in the 1980s. In this paper, we provide a possible theoretical explanation for this formula, an explanation based on general ideas of invariance. In general, the existence of a theoretical explanation makes a formula more reliable; thus, we hope that our explanation will make predictions of road quality more reliable.

New (Simplified) Derivation Of Nash's Bargaining Solution, Nguyen Hoang Phuong, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

According to the Nobelist John Nash, if a group of people wants to selects one of the alternatives in which all of them get a better deal than in a status quo situations, then they should select the alternative that maximizes the product of their utilities. In this paper, we provide a new (simplified) derivation of this result, a derivation which is not only simpler -- it also does not require that the preference relation between different alternatives be linear.

Towards Making Fuzzy Techniques More Adequate For Combining Knowledge Of Several Experts, Nguyen Hoang Phuong, Vladik Kreinovich

Departmental Technical Reports (CS)

In medical and other applications, expert often use rules with several conditions, each of which involve a quantity within the domain of expertise of a different expert. In such situations, to estimate the degree of confidence that all these conditions are satisfied, we need to combine opinions of several experts -- i.e., in fuzzy techniques, combine membership functions corresponding to different experts. In each area of expertise, different experts may have somewhat different membership functions describing the same natural-language ("fuzzy") term like small. It is desirable to present the user with all possible conclusions corresponding to all these membership functions. …

Why Mean, Variance, Moments, Correlation, Skewness Etc. – Invariance-Based Explanations, Olga Kosheleva, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

In principle, we can use many different characteristics of a probability distribution. However, in practice, a few of such characteristics are mostly used: mean, variance, moments, correlation, etc. Why these characteristics and not others? The fact that these characteristics have been successfully used indicates that there must be some reason for their selection. In this paper, we show that the selection of these characteristics can be explained by the fact that these characteristics are invariant with respect to natural transformations -- while other possible characteristics are not invariant.

Strength Of Lime Stabilized Pavement Materials: Possible Theoretical Explanation Of Empirical Dependencies, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

When building a road, it is often necessary to strengthen the underlying soil layer. This strengthening is usually done by adding lime. There are empirical formulas that describe how the resulting strength depends on the amount of added lime. In this paper, we provide a theoretical explanation for these empirical formulas.

Fusion Of Probabilistic Knowledge As Foundation For Sliced-Normal Approach, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, it turns out to be efficient to use Sliced-Normal multi-D distributions, i.e., distributions for which the logarithm of the probability density function (pdf) is a polynomial -- -- to be more precise, it is a sum of squares of several polynomials. This class is a natural extension of normal distributions, i.e., distributions for which the logarithm of the pdf is a quadratic polynomial.

In this paper, we provide a possible theoretical explanation for this empirical success.

Toward Automated Region Detection & Parcellation Of Rat Brain Tissue Images, Alexandro Arnal

Open Access Theses & Dissertations

People who analyze images of biological tissue often rely on segmentation of structures as a preliminary step. In particular, laboratories studying the rat brain manually delineate brain regions to position scientific findings on a brain atlas to propose hypotheses about the rat brain, and ultimately, the human brain. Our work intersects with the preliminary step of delineating regions in images of brain tissue via computational methods.

We investigate pixel-wise classification or segmentation of brain regions using ten histological images of brain tissue sections stained for Nissl substance, and two deep learning models: U-Net and Tile2Vec. Our goal is to assess …

Predicting Stochastic Volatility For Extreme Fluctuations In High Frequency Time Series, Md Al Masum Bhuiyan

Open Access Theses & Dissertations

This work is devoted to the study of modeling high frequency time series including extreme fluctuations. As the high frequency data are collected at extremely fine scales, the fluctuations can capture the dynamics of data that evolve over time. A class of volatility models with time-varying parameters is used to forecast the volatility in a stationary condition at different lags. The modeling of stationary time series with consistent properties facilitates prediction with much certainty.

A large set of high frequency financial returns, closing prices of stock markets, high magnitudes of seismograms generated by the natural earthquakes, and the mining explosions …

Applications Of Ornstein-Uhlenbeck Type Stochastic Differential Equations, Osei Kofi Tweneboah

Open Access Theses & Dissertations

In this Dissertation, we show with plausible arguments that the Stochastic Differential Equations (SDEs) arising on the superposition and coupling system of independent Ornstein-Uhlenbeck process is a new method available in modern literature that takes the properties and behavior of the data into consideration when performing the statistical analysis of the time series.

The time series to be analyzed is thought of as a source of fluctuations, and thus we need a model that takes this behavior into consideration when performing such analysis. Most of the standard methods fail to take into account the physical behavior of the time series, …

Stochastic Modeling Of Earthquakes And Option Pricing Using Bns-Gamma-Ou Model, Mandela Bright Quashie

Open Access Theses & Dissertations

High frequency data are becoming increasingly popular these days. They are fundamental in basically every facet of people’s lives. They are the determining factors in hedging in the field of finance. In geology, they help in the accurate prediction of earthquakes’ magnitude which goes along way to help save lives and properties.

High frequency data are also used more and more frequently for speculations. For this reason, it is important not only for scientists to apply models allowing correct quantification of these data, but also to improve the eciency of these models.

The Black-Scholes model, which is widely used because …