In many engineering situations, we are interested in finding the correlation ρ between different quantities x and y based on the values xi and yi of these quantities measured in different situations i. Measurements are never absolutely accurate; it is therefore necessary to take this inaccuracy into account when estimating the correlation ρ. Sometimes, we know the probabilities of different values of measurement errors, but in many cases, we only know the upper bounds ∆xi and ∆yi on the corresponding measurement errors. In such situations, after we get the measurement results xi and yi, the only information that we have about the actual (unknown) values xi and yi is that they belong to the corresponding intervals [ xi − ∆xi, xi + ∆xi] and [ yi − ∆yi, yi + ∆yi]. Different values from these intervals lead, in general, to different values of the correlation ρ. It is therefore desirable to find the range [ρ, ρ] of possible values of the correlation when xi and yi take values from the corresponding intervals. In general, the problem of computing this range is NP-hard. In this paper, we provide a feasible (= polynomial-time) algorithm for computing at least one of the endpoints of this interval: for computing ρ when ρ > 0 and for computing ρ when ρ < 0.
Abstract-Some probability distributions (e.g., Gaussian) are symmetric, some (e.g., lognormal) are non-symmetric (skewed). How can we gauge the skeweness? For symmetric distributions, the third central moment C3 def
Abstract-In many situations, we are interested in finding the correlation ρ between different quantities x and y based on the values xi and yi of these quantities measured in different situations i. The correlation is easy to compute when we know the exact sample values xi and yi. In practice, the sample values come from measurements or from expert estimates; in both cases, the values are not exact. Sometimes, we know the probabilities of different values of measurement errors, but in many cases, we only know the upper bounds ∆xi and ∆yi on the corresponding measurement errors. In such situations, after we get the measurement results xi and yi, the only information that we have about the actual (unknown) values xi and yi is that they belong to the corresponding intervals [ xi − ∆xi, xi + ∆xi] and [ yi−∆yi, yi+∆yi]. For expert estimates, we get different intervals corresponding to different degrees of certainty -i.e., fuzzy sets. Different values of xi and yi lead, in general, to different values of the correlation ρ. It is therefore desirable to find the range [ρ, ρ] of possible values of the correlation when xi and yi take values from the corresponding intervals. In general, the problem of computing this range is NP-hard. In this paper, we provide a feasible (= polynomial-time) algorithm for computing at least one of the endpoints of this interval: for computing ρ when ρ > 0 and for computing ρ when ρ < 0.
One of the main objectives of science and engineering is to predict the future state of the world -and to come up with actions which will lead to the most favorable outcome. To be able to do that, we need to have a quantitative model describing how the values of the desired quantities change -and for that, we need to know which factors influence this change. Usually, these factors are selected by using traditional statistical techniques, but with the current drastic increase in the amount of available data -known as the advent of big data -the traditional techniques are no longer feasible. A successful semi-heuristic method has been proposed to detect true connections in the presence of big data. However, this method has its limitations. The first limitation is that this method is heuristic -its main justifications are common sense and the fact that in several practical problems, this method was reasonably successful. The second limitation is that this heuristic method is based on using "crisp" granules (clusters), while in reality, the corresponding granules are flexible ("fuzzy"). In this chapter, we explain how the known semi-heuristic method can be justified in statistical terms, and we also show how the ideas behind this justification enable us to improve the known method by taking granule flexibility into account.
Computer Programming for Science/ Engineering" (CPSE) is an introductory programming course for STEM students other than Computer Science (CS) major. Typical attendees are required to attend either CPSE or the Java-based first programming course of the CS majors' sequence (CS1). The previous curriculum for CPSE was a traditional introductory programming course with chronically low enrollment despite less its substantially relaxed learning outcomes.Post-reform, CPSE has become a popular alternative to CS1. Pass rates in CPSE are substantially higher than both local and national pass rates for CS1 and equivalent courses, and in-class tests of CPSE students indicate that 70% satisfy the majority of CS1 outcomes, which exceeds the pass rate for most CS1 sections at our institution.While both CPSE and CS1 teach the Java language, CPSE integrates inductive teaching strategies developed by the second author's iMPaCT program that exploit the relaxed syntax of the Jython language. Like immersive foreign language programs, lessons during the first half of the course exploit the relaxed grammatical requirements of a simpler programming language (Jython) to introduce programming concepts (semantics) incrementally in a conversational manner. These early lessons are motivated by accessible graphical problems that incidentally review foundational math concepts.
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