Engineering Commons^™

Homotopy Techniques In Solving Systems Of Nonlinear Equations: A Theoretical Justification Of Convex Combinations, Nicholas Sun

Departmental Technical Reports (CS)

One of the techniques for solving systems of non-linear equations F₁(x₁,...,x_n) = 0, ..., F_n(x₁,...,x_n) = 0, (F(x) = 0 in vector notations) is a homotopy method, when we start with a solution of a simplified (and thus easier-to-solve) approximate system G_i(x₁,...,x_n) = 0, and then gradually adjust this solution by solving intermediate systems of equation H_i(x₁,...,x_n) = 0 for an appropriate "transition" function H(x) = f(λ,F(x),G(x)). The success of this method depends on the …

Towards A Formal Description Of Understandability (Causality, Pre-Requisites): From Prosorov's Phonocentric Topology To More General Interior (Closure) Structures, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real life situations, a text consists of related parts; so, to understand a part, we need to first understand some (or all) preceding parts: e.g., to understand Chapter 3, we first need to understand Chapters 1 and 2. In many cases, this dependence is described by a partial order. For this case, O.~Prosorov proposed a natural description of the dependence structure as a topology (satisfying the separation axiom T₀).

In some practical situations, dependence is more general than partial order: e.g., to understand Chapter 3, we may need to understand either Chapter 1 or Chapter 2, …

From Numerical Probabilities To Linguistic Probabilities: A Theoretical Justification Of Empirical Granules Used In Risk Management, Beverly Rivera, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

In many risk management situations, instead of the exact probability values, specialists use a granule to which this probability belongs. Specifically, they use five granules, corresponding to thresholds 10%, 40%, 60%, and 90%. In this paper, we provide an explanation for such non-uniform granulation.

If Many Physicists Are Right And No Physical Theory Is Perfect, Then By Using Physical Observations, We Can Feasibly Solve Almost All Instances Of Each Np-Complete Problem, Olga Kosheleva, Michael Zakharevich, Vladik Kreinovich

Departmental Technical Reports (CS)

Many real-life problems are, in general, NP-complete, i.e., informally speaking, are difficult to solve -- at least on computers based on the usual physical techniques. A natural question is: can the use of non-standard physics speed up the solution of these problems? This question has been analyzed for several specific physical theories, e.g., for quantum field theory, for cosmological solutions with wormholes and/or casual anomalies, etc. However, many physicists believe that no physical theory is perfect, i.e., that no matter how many observations support a physical theory, inevitably, new observations will come which will require this theory to be updated. …

Security Risk Assessment: Towards A Justification For The Security Risk Factor Table Model, Beverly Rivera, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the widely used methods to gauge risk is the Security Risk Factor Table (SRFT) model. While this model has been empirically successful, its use is limited by the fact that its formulas do not have a theoretical explanation -- and thus, there is no guarantee that these formulas will work in other situations as well. In this paper, we provide a theoretical explanation for the SFRT formulas.

Log-Periodic Power Law As A Predictor Of Catastrophic Events: A New Mathematical Justification, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta

Departmental Technical Reports (CS)

To decrease the damage caused by meteorological disasters, it is important to be able to predict these disasters as accurately as possible. One of the most promising ways of achieving such a prediction comes from the observation that in the vicinity of a catastrophic event, many parameters exhibit log-periodic power behavior, with oscillations of increasing frequency. By fitting the corresponding formula to the observations, it is often possible to predict the catastrophic event. Such successful predictions were made in many application areas ranging from ruptures of fuel tanks to earthquakes to stock market disruptions. The fact that similar formulas can …

Increased Climate Variability Is More Visible Than Global Warming: A General System-Theory Explanation, Octavio Lerma, Craig Tweedie, Vladik Kreinovich

Departmental Technical Reports (CS)

While global warming is a statistically confirmed long-term phenomenon, its most visible consequence is not the warming itself but, somewhat surprisingly, the increased climate variability. In this paper, we use the general system theory ideas to explain why increased climate variability is more visible than the global warming itself.

How To Estimate Forecasting Quality: A System-Motivated Derivation Of Symmetric Mean Absolute Percentage Error (Smape) And Other Similar Characteristics, Vladik Kreinovich, Hung T. Nguyen, Rujira Ouncharoen

Departmental Technical Reports (CS)

When comparing how well different algorithms forecast time series, researchers use an average value of the ratio |x-y|/(|x|+|y|)/2), known as the Symmetric Mean Absolute Percentage Error (SMAPE). In this paper, we provide a system-motivated explanation for this formula. We also explain how this formula explains the use of geometric mean to combine different forecasts.

Observable Causality Implies Lorentz Group: Alexandrov-Zeeman-Type Theorem For Space-Time Regions, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The famous Alexandrov-Zeeman theorem proves that causality implies Lorentz group. The physical meaning of this result is that once we observe which event can causally affect which other events, then, using only this information, we can reconstruct the linear structure of the Minkowski space-time. The original Alexandrov-Zeeman theorem is based on the causality relation between events represented by points in space-time. Knowing such a point means that we know the exact moment of time and the exact location of the corresponding event - and that this event actually occurred at a single moment of time and at a single spatial …

Possible Geometric Explanations For Basic Empirical Dependencies Of Systems Engineering, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we provide possible geometric explanation for basic empirical dependencies of system engineering: that a properly designed system should have no more than 7 plus minus 2 elements reporting to it, and that the relative cost of correcting a defect on different stages of the system's life cycle is 3--6 on the second (design) stage, 20--100 on the third (development) stage, and 250--1000 on the fourth (production and testing) stage.

How To Explain The Definition Of Stochastic Affiliation To Economics Students, Tonghui Wang, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To formally describe the intuitive idea of "positive correlation" between two quantities, it is often helpful to use the notion of stochastic affiliation. While this notion is useful, its usual definition is not intuitively clear -- which make it difficult to explain this notion to, e.g., economics students. To help students understand this notion, in this paper, we show how the notion of stochastic affiliation can be explained in clear probabilistic terms.

Towards Fast And Reliable Localization Of An Underwater Object: An Interval Approach, Quentin Brefort, Luc Jaulin, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

To localize an underwater object, we measure the distance to this object from several sonar sensors with known locations. The problem is that the signal sent by some of the sonars is reflected not by the desired object(s), but by some auxiliary object and thus, the values measured by these sensors are drastically different from the distance to the desired object. To solve this problem, currently probabilistic methods are used; however, since we do not know the exact probability distributions, these methods may miss the actual location of the object. There exist interval-based methods which provide guaranteed (reliable) bounds on …

Protecting Patient Privacy While Preserving Medical Information For Research, Gang Xiang, Jason O'Rawe, Vladik Kreinovich, Janos Hajagos, Scott Ferson

Departmental Technical Reports (CS)

Patient health records possess a great deal of information that would be useful in medical research, but access to these data is impossible or severely limited because of the private nature of most personal health records. Anonymization strategies, to be effective, must usually go much further than simply omitting explicit identifiers because even statistics computed from groups of records can often be leveraged by hackers to re-identify individuals. Methods of balancing the informativeness of data for research with the information loss required to minimize disclosure risk are needed before these private data can be widely released to researchers who can …

F-Transform Enhancement Of The Sampling Theorem And Reconstruction Of Noisy Signals, Michal Holčapek, Irina Perfilieva, Vladik Kreinovich

Departmental Technical Reports (CS)

According to a sampling theorem, any band-limited and continuous signal can be uniquely reconstructed from certain of its values. We show that a reconstruction can be obtained from the set of F-transform components and moreover, the sampling theorem follows as a particular case. A special attention is paid to the case where sample values of a signal come with noise. We show that in the presence of noise, a more accurate reconstruction than that based on the sampling theorem can be obtained, if instead of noised sample values the F- transform components of the signal with respect to a generalized …

Imprecise Probabilities In Engineering Analyses, Michael Beer, Scott Ferson, Vladik Kreinovich

Departmental Technical Reports (CS)

Probabilistic uncertainty and imprecision in structural parameters and in environmental conditions and loads are challenging phenomena in engineering analyses. They require appropriate mathematical modeling and quantification to obtain realistic results when predicting the behavior and reliability of engineering structures and systems. But the modeling and quantification is complicated by the characteristics of the available information, which involves, for example, sparse data, poor measurements and subjective information. This raises the question whether the available information is sufficient for probabilistic modeling or rather suggests a set-theoretical approach. The framework of imprecise probabilities provides a mathematical basis to deal with these problems which …

Locating Local Extrema Under Interval Uncertainty: Multi-D Case, Karen Villaverde, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to locate local maxima and/or local minima of a function which is only know with interval uncertainty. For example, in radioastronomy, components of a radiosource are usually identified by locations at which the observed brightness reaches a local maximum. In clustering, different clusters are usually identified with local maxima of the probability density function (describing the relative frequency of different combinations of values). In the 1-D case, a feasible (polynomial-time) algorithm is known for locating local extrema under interval (and fuzzy) uncertainty. In this paper, we extend this result to the general multi-dimensional case.

Optimizing Computer Representation And Computer Processing Of Epistemic Uncertainty For Risk-Informed Decision Making: Finances Etc., Vladik Kreinovich, Nitaya Buntao, Olga Kosheleva

Departmental Technical Reports (CS)

Uncertainty is usually gauged by using standard statistical characteristics: mean, variance, correlation, etc. Then, we use the known values of these characteristics (or the known bounds on these values) to select a decision. Sometimes, it becomes clear that the selected characteristics do not always describe a situation well; then other known (or new) characteristics are proposed. A good example is description of volatility in finance: it started with variance, and now many descriptions are competing, all with their own advantages and limitations.

In such situations, a natural idea is to come up with characteristics tailored to specific application areas: e.g., …

How To Describe And Propagate Uncertainty When Processing Time Series: Metrological And Computational Challenges, With Potential Applications To Environmental Studies, Christian Servin, Martine Ceberio, Aline Jaimes, Craig Tweedie, Vladik Kreinovich

Departmental Technical Reports (CS)

Time series comes from measurements, and measurements are never absolutely accurate. Traditionally, when we deal with an individual measurement or with a sample of measurement results, we subdivide a measurement error into random and systematic components: systematic error does not change from measurement to measurement which random errors corresponding to different measurements are independent. In time series, when we measure the same quantity at different times, we can also have correlation between measurement errors corresponding to nearby moments of time. To capture this correlation, environmental science researchers proposed to consider the third type of measurement errors: periodic. This extended classification …

Research-Related Projects For Graduate Students As A Tool To Motivate Graduate Students In Classes Outside Their Direct Interest Areas, Vladik Kreinovich

Departmental Technical Reports (CS)

In most graduate programs, students are required to take both "depth" classes -- classes in the areas of the student's direct interest -- and "breadth" classes, classes outside their direct interest areas. Naturally, the student's interest in "breadth" classes is often naturally lower than their interest in the "depth" classes. To enhance the students' interest in the "breadth" classes, a natural idea is to make research-related project an important part of the class, a project in which the student can apply the skills that he or she learns in the class to the research area of direct interest to this …

Towards Formalizing Non-Monotonic Reasoning In Physics: Logical Approach Based On Physical Induction And Its Relation To Kolmogorov Complexity, Vladik Kreinovich

Departmental Technical Reports (CS)

To formalize some types of non-monotonic reasoning in physics, researchers have proposed an approach based on Kolmogorov complexity. Inspired by Vladimir Lifschitz's belief that many features of reasoning can be described on a purely logical level, we show that an equivalent formalization can be described in purely logical terms: namely, in terms of physical induction.

One of the consequences of this formalization is that the set of not-abnormal states is (pre-)compact. We can therefore use Lifschitz's result that when there is only one state that satisfies a given equation (or system of equations), then we can algorithmically find this state. …

Negative Results Of Computable Analysis Disappear If We Restrict Ourselves To Random (Or, More Generally, Typical) Inputs, Vladik Kreinovich

Departmental Technical Reports (CS)

It is well known that many computational problems are, in general, not algorithmically solvable: e.g., it is not possible to algorithmically decide whether two computable real numbers are equal, and it is not possible to compute the roots of a computable function. We propose to constraint such operations to certain "sets of typical elements" or "sets of random elements".

In our previous papers, we proposed (and analyzed) physics-motivated definitions for these notions. In short, a set T is a set of typical elements if for every definable sequences of sets A_n for which each A_n is a subset …

Towards Optimizing Cloud Computing: An Example Of Optimization Under Uncertainty, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the most efficient way to store and process data is cloud computing. In cloud computing, instead of storing the data at the user-defined location (e.g., at the user's computer or at the centralized server), the computer system ("cloud") selects the location of the data storage that speeds up computations -- by minimizing the (average) communication time. In this chapter, we provide an analytical solution to the corresponding optimization problem.

The demand for cloud computing is growing fast, and we expect that this demand -- and thus, the size of the resulting cloud -- will continue to grow. …

Do Constraints Facilitate Or Inhibit Creative Problem Solving: A Theoretical Explanation Of Two Seemingly Contradictory Experimental Studies, Karen Villaverde, Olga Kosheleva, Martine Ceberio

Departmental Technical Reports (CS)

Do constraints facilitate or inhibit creative problem solving? Recently, two experimental studies appeared, one showing that removing constraints may enhance creativity, another showing that adding constraints can facilitate creative problem solving. In this paper, we provide a theoretical explanation of these two seemingly contradictory experimental results.

Validated Templates For Specification Of Complex Ltl Formulas, Salamah Salamah, Ann Q. Gates, Vladik Kreinovich

Departmental Technical Reports (CS)

Formal verification approaches that check software correctness against formal specifications have been shown to improve program dependability. Tools such as Specification Pattern System (SPS) and Property Specification (Prospec) support the generation of formal specifications. SPS has defined a set of patterns (common recurring properties) and scopes (system states over which a pattern must hold) that allows a user to generate formal specifications by using direct substitution of propositions into parameters of selected patterns and scopes. Prospec extended SPS to support the definition of patterns and scopes that include the ability to specify parameters with multiple propositions (referred to as composite …

Towards Symmetry-Based Explanation Of (Approximate) Shapes Of Alpha-Helices And Beta-Sheets (And Beta-Barrels) In Protein Structure, Jaime Nava, Vladik Kreinovich

Departmental Technical Reports (CS)

Protein structure is invariably connected to protein function. There are two important secondary structure elements: alpha helices and beta-sheets (which sometimes come in a shape of beta-barrels). The actual shapes of these structures can be complicated, but in the first approximation, they are usually approximated by, correspondingly, cylindrical spirals and planes (and cylinders, for beta-barrels). In this paper, following the ideas pioneered by a renowned mathematician M. Gromov, we use natural symmetries to show that, under reasonable assumptions, these geometric shapes are indeed the best approximating families for secondary structures.

Estimating Statistical Characteristics Of Lognormal And Delta-Lognormal Distributions Under Interval Uncertainty: Algorithms And Computational Complexity, Nitaya Buntao, Sa-Aat Niwitpong, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional statistical estimates S(x₁, ..., x_n) for different statistical characteristics S (such as mean, variance, etc.) implicitly assume that we know the sample values x₁, ..., x_n exactly. In practice, the sample values Xi come from measurements and are, therefore, in general, different from the actual (unknown) values x_i of the corresponding quantities. Sometimes, we know the probabilities of different values of the measurement error d_i = X_i - x_i, but often, the only information that we have about the measurement error is the upper bound D_{i …}

A Framework To Create Ontologies For Scientific Data Management, Leonardo Salayandia, Paulo Pinheiro Da Silva, Ann Q. Gates

Departmental Technical Reports (CS)

Scientists often build and use highly customized systems to support observation and analysis efforts. Creating effective ontologies to manage and share data products created from those systems is a difficult task that requires collaboration among domain experts, e.g., scientists and knowledge representation experts. A framework is presented that scientists can use to create ontologies that describe how customized systems capture and transform data into products that support scientific findings. The framework establishes an abstraction that leverages knowledge representation expertise to describe data transformation processes in a consistent way that highlights properties relevant to data users. The intention is to create …

In The Beginning Was The Word, And The Word Was Fuzzy, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Why Bernstein Polynomials Are Better: Fuzzy-Inspired Justification, Jaime Nava, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is well known that an arbitrary continuous function on a bounded set -- e.g., on an interval [a,b] -- can be, with any given accuracy, approximated by a polynomial. Usually, polynomials are described as linear combinations of monomials. It turns out that in many computational problems, it is more efficient to represent a polynomial as Bernstein polynomials -- e.g., for functions of one variable, a linear combination of terms (x-a)^k * (b-x)^n-k. In this paper, we provide a simple fuzzy-based explanation of why Bernstein polynomials are often more efficient, and we show how this informal explanation …

Constraint Optimization: From Efficient Computation Of What Can Be Achieved To Efficient Computation Of A Way To Achieve The Corresponding Optimum, Ali Jalal-Kamali, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practically useful cases, we know how to efficiently compute the exact range of a function over given intervals (and, possibly, under additional constraints). In other words, we know how to efficiently compute the minimum and maximum of a given function f(x₁, ..., x_n) on any box. From the practical viewpoint, it is important not only to find the value of the corresponding maximum or minimum, but also to know for what values of the parameters x_i this optimum is attained. We prove a general result: that if we can efficiently compute the optimum, …