Computer Sciences Commons^™

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.

What If We Do Not Know Correlations?, Michael Beer, Zitong Gong, Ingo Neumann, Songsak Sriboonchitta, Vladik Kreinovich

Departmental Technical Reports (CS)

It is well know how to estimate the uncertainty of the result y of data processing if we know the correlations between all the inputs. Sometimes, however, we have no information about the correlations. In this case, instead of a single value σ of the standard deviation of the result, we get a range [σ] of possible values. In this paper, we show how to compute this range.

Taking Into Account Interval (And Fuzzy) Uncertainty Can Lead To More Adequate Statistical Estimates, Ligang Sun, Hani Dbouk, Steffen Schön, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional statistical data processing techniques (such as Least Squares) assume that we know the probability distributions of measurement errors. Often, we do not have full information about these distributions. In some cases, all we know is the bound of the measurement error; in such cases, we can use known interval data processing techniques. Sometimes, this bound is fuzzy; in such cases, we can use known fuzzy data processing techniques.

However, in many practical situations, we know the probability distribution of the random component of the measurement error and we know the upper bound -- numerical or fuzzy -- on the …

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 …

Markowitz Portfolio Theory Helps Decrease Medicines' Side Effect And Speed Up Machine Learning, Thongchai Dumrongpokaphan, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that, similarly to the fact that distributing the investment between several independent financial instruments decreases the investment risk, using a combination of several medicines can decrease the medicines' side effects. Moreover, the formulas for optimal combinations of medicine are the same as the formulas for the optimal portfolio, formulas first derived by the Nobel-prize winning economist H. M. Markowitz. A similar application to machine learning explains a recent success of a modified neural network in which the input neurons are also directly connected to the output ones.

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 …

Kuznets Curve: A Simple Dynamical System-Based Explanation, Thongchai Dumrongpokaphan, Vladik Kreinovich

Departmental Technical Reports (CS)

In the 1950s, a future Nobelist Simon Kuznets discovered the following phenomenon: as a country's economy improves, inequality first grows but then decreases. In this paper, we provide a simple dynamical system-based explanation for this empirical phenomenon.

How To Gauge Accuracy Of Processing Big Data: Teaching Machine Learning Techniques To Gauge Their Own Accuracy, Vladik Kreinovich, Thongchai Dumrongpokaphan, Hung T. Nguyen, Olga Kosheleva

Departmental Technical Reports (CS)

When the amount of data is reasonably small, we can usually fit this data to a simple model and use the traditional statistical methods both to estimate the parameters of this model and to gauge this model's accuracy. For big data, it is often no longer possible to fit them by a simple model. Thus, we need to use generic machine learning techniques to find the corresponding model. The current machine learning techniques estimate the values of the corresponding parameters, but they usually do not gauge the accuracy of the corresponding general non-linear model. In this paper, we show how …

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 …

Why Student Distributions? Why Matern's Covariance Model? A Symmetry-Based Explanation, Steffen Schön, Gaël Kermarrec, Boris Kargoll, Ingo Neumann, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that empirical successes of Student distribution and of Matern's covariance models can be indirectly explained by a natural requirement of scale invariance -- that fundamental laws should not depend on the choice of physical units. Namely, while neither the Student distributions nor Matern's covariance models are themselves scale-invariant, they are the only one which can be obtained by applying a scale-invariant combination function to scale-invariant functions.

Entropy As A Measure Of Average Loss Of Privacy, Luc Longpre, Vladik Kreinovich, Thongchai Dumrongpokaphan

Departmental Technical Reports (CS)

Privacy means that not everything about a person is known, that we need to ask additional questions to get the full information about the person. It therefore seems to reasonable to gauge the degree of privacy in each situation by the average number of binary ("yes"-"no") questions that we need to ask to determine the full information -- which is exactly Shannon's entropy. The problem with this idea is that it is possible, by asking two binary questions -- and thus, strictly speaking, getting only two bits of information -- to sometimes learn a large amount of information. In this …

Fuzzy Sets As Strongly Consistent Random Sets, Kittawit Autchariyapanitkul, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that from the purely mathematical viewpoint, fuzzy sets can be interpreted as equivalent classes of random sets. This interpretations helps to teach fuzzy techniques to statisticians and also enables us to apply results about random sets to fuzzy techniques. The problem with this interpretation is that it is too complicated: a random set is not an easy notion, and classes of random sets are even more complex. This complexity goes against the spirit of fuzzy sets, whose purpose was to be simple and intuitively clear. From this viewpoint, it is desirable to simplify this interpretation. In this …

Isn't Every Sufficiently Complex Logic Multi-Valued Already: Lindenbaum-Tarski Algebra And Fuzzy Logic Are Both Particular Cases Of The Same Idea, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

Usually, fuzzy logic (and multi-valued logics in general) are viewed as drastically different from the usual 2-valued logic. In this paper, we show that while on the surface, there indeed seems to be a major difference, a more detailed analysis shows that even in the theories based on the 2-valued logic, there naturally appear constructions which are, in effect, multi-valued, constructions which are very close to fuzzy logic.

From Fuzzy Universal Approximation To Fuzzy Universal Representation: It All Depends On The Continuum Hypothesis, Mahdokhat Michelle Afravi, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that fuzzy systems have a universal approximation property. A natural question is: can this property be extended to a universal representation property? Somewhat surprisingly, the answer to this question depends on whether the following Continuum Hypothesis: every infinite subset of the real line has either the same number of elements as the real line itself or as many elements as natural numbers.

A Natural Feasible Algorithm That Checks Satisfiability Of 2-Cnf Formulas And, If The Formulas Is Satisfiable, Finds A Satisfying Vector, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main results in Theory of Computation courses is the proof that propositional satisfiability is NP-complete. This means that, unless P = NP (which most computer scientists believe to be impossible), no feasible algorithm is possible for solving propositional satisfiability problems. This result is usually proved on the example of 3-CNF formulas, i.e., formulas of the type C₁ & ... & C_m, where each clause C_i has the form a \/ b or a \/ b \/ c, with no more than three literals -- i.e., propositional variables v_i or their negations ~v …

What Decision To Make In A Conflict Situation Under Interval Uncertainty: Efficient Algorithms For The Hurwzcz Approach, Bartłomiej Jacek Kubica, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show how to take interval uncertainty into account when solving conflict situations. Algorithms for conflict situations under interval uncertainty are know under the assumption that each side of the conflict maximizes its worst-case expected gain. However, it is known that a more general Hurwicz approach provides a more adequate description of decision making under uncertainty. In this approach, each side maximizes the convex combination of the worst-case and the best-case expected gains. In this paper, we describe how to resolve conflict situations under the general Hurwicz approach to interval uncertainty.

A Short Note On Pitch, Interval, And Melody Matching Assessment, Eric Hanson, Hannah Baslee, Eric Freudenthal

Departmental Technical Reports (CS)

This short note describes a metric and procedure for assessing an individual's overall simple pitch and interval matching proficiency when singing.

A Symmetry-Based Explanation For An Empirical Model Of Fatigue Damage Of Composite Materials, Pedro Barragan Olague, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we provide a symmetry-based explanation for an empirical formula that describes fatigue damage of composite materials.

It Is Possible To Determine Exact Fuzzy Values Based On An Ordering Of Interval-Valued Or Set-Valued Fuzzy Degrees, Gerardo Muela, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the usual [0,1]-based fuzzy logic, the actual numerical value of a fuzzy degree can be different depending on a scale, what is important -- and scale-independent -- is the order between different values. To make a description of fuzziness more adequate, it is reasonable to consider interval-valued degrees instead of numerical ones. Here also, what is most important is the order between the degrees. If we have only order between the intervals, can we, based on this order, reconstruct the original numerical values -- i.e., the degenerate intervals? In this paper, we show that such a reconstruction is indeed …

In System Identification, Interval (And Fuzzy) Estimates Can Lead To Much Better Accuracy Than The Traditional Statistical Ones: General Algorithm And Case Study, Sergey I. Kumkov, Vladik Kreinovich, Andrzej Pownuk

Departmental Technical Reports (CS)

In many real-life situations, we know the upper bound of the measurement errors, and we also know that the measurement error is the joint result of several independent small effects. In such cases, due to the Central Limit theorem, the corresponding probability distribution is close to Gaussian, so it seems reasonable to apply the standard Gaussian-based statistical techniques to process this data -- in particular, when we need to identify a system. Yes, in doing this, we ignore the information about the bounds, but since the probability of exceeding them is small, we do not expect this to make a …

Soft Computing Approach To Detecting Discontinuities: Seismic Analysis And Beyond, Solymar Ayala Cortez, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

Starting from Newton, the main equations of physics are differential equations -- which implicitly implies that all the corresponding processes are differentiable -- and thus, continuous. However, in practice, we often encounter processes or objects that change abruptly in time or in space. In physics, we have phase transitions when the properties change abruptly. In geosciences, we have sharp boundaries between different layers and discontinuing representing faults. In many such situations, it is important to detect these discontinuities. In some cases, we know the equations, but in many other cases, we do not know the equations, we only know that …

Fuzzy Techniques Explain Empirical Power Law Governing Wars And Terrorist Attacks, Hung T. Nguyen, Kittawit Autchariyapanitkul, Vladik Kreinovich

Departmental Technical Reports (CS)

The empirical distribution of the number of casualties in wars and terrorist attacks follows a power law with exponent 2.5. So far, there has not been a convincing explanation for this empirical fact. In this paper, we show that by using fuzzy techniques, we can explain this exponent. Interesting, we can also get a similar explanation if we use probabilistic techniques. The fact that two different techniques lead to the same explanation makes us reasonably confident that this explanation is correct.

Which Material Design Is Possible Under Additive Manufacturing: A Fuzzy Approach, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Additive manufacturing -- also known as 3-D printing -- is a very promising new way to generate complex material designs. However, even with the modern advanced techniques, some designs are too complex to be implemented. There exist an empirical formula that describes when the design is implementable. In this paper, we use fuzzy ideas to provide a theoretical justification for this empirical formula.

Uncertain Information Fusion And Knowledge Integration: How To Take Reliability Into Account, Hung T. Nguyen, Kittawit Autchariyapanitkul, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to fuse and integrate information and knowledge from different sources -- and do it under uncertainty. Most existing methods for information fusion and knowledge integration take into account uncertainty. In addition to uncertainty, we also face the problem of reliability: sensors may malfunction, experts can be wrong, etc. In this paper, we show how to take into account both uncertainty and reliability in information fusion and knowledge integration. We show this on the examples of probabilistic and fuzzy uncertainty.

Prediction Of Volcanic Eruptions As A Case Study Of Predicting Rare Events In Chaotic Systems With Delay, Justin Parra, Olac Fuentes, Elizabeth Y. Anthony, Vladik Kreinovich

Departmental Technical Reports (CS)

Volcanic eruptions can be disastrous; it is therefore important to be able to predict them as accurately as possible. Theoretically, we can use the general machine learning techniques for such predictions. However, in general, without any prior information, such methods require an unrealistic amount of computation time. It is therefore desirable to look for additional information that would enable us to speed up the corresponding computations. In this paper, we provide an empirical evidence that the volcanic system exhibit chaotic and delayed character. We also show that in general (and in volcanic predictions in particular), we can speed up the …

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.

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 …

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.

Physical Induction Explains Why Over-Realistic Animation Sometimes Feels Creepy, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the past, every progress of movie animation towards realism was viewed positively. However, recently, as computer animation is becoming more and more realistic, some people perceive the resulting realism negatively, as creepy. Similarly, everyone used to welcome robots that looked and behaved somewhat like humans; however, lately, too-human-like robots have started causing a similar negative feeling of creepiness. There exist complex psychology-based explanations for this phenomenon. In this paper, we show that this empirical phenomenon can be naturally explained simply by physical induction -- the main way we cognize the world.

Contradictions Do Not Necessarily Make A Theory Inconsistent, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Some religious scholars claim that while the corresponding holy texts may be contradictory, they lead to a consistent set of ethical and behavioral recommendations. Is this logically possible? In this paper, somewhat surprisingly, we kind of show that this is indeed possible: namely, we show that if we add, to statements about objects from a certain class, consequences of both contradictory abstract statements, we still retain a consistent theory. A more mundane example of the same phenomenon comes from mathematics: if we have a set-theoretical statement S which is independent from ZF and which is not equivalent to any arithmetic …