Engineering Commons^™

Towards Optimal Few-Parametric Representation Of Spatial Variation: Geometric Approach And Environmental Applications, Misha Kosheleva, Octavio Lerma, Craig Tweedie

Departmental Technical Reports (CS)

In this paper, we use geometric approach to showthat under reasonable assumption, the spatialvariability of a field f(x), i.e., the expectedvalue F(z)=E[(f(x+z)-f(x))²], has the formF(z)=|Σ g_ij*z_i*z_j|^α.We explain how to find g_ij and αfrom the observations, and how to optimally place sensorsin view of this spatial variability.

Measures Of Deviation (And Dependence) For Heavy-Tailed Distributions And Heir Estimation Under Interval And Fuzzy Uncertainty, Nitaya Buntao, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, in science and engineering, most statistical techniques are based on the assumption that the random variables are normally distributed. For such distributions, a natural characteristic of the "average" value is the mean, and a natural characteristic of the deviation from the average is the variance. However, in many practical situations, e.g., in economics and finance, we encounter probability distributions for which the variance is infinite; such distributions are called heavy-tailed. For such distributions, we describe which characteristics can be used to describe the average and the deviation from the average, and how to estimate these characteristics under interval …

Towards A General Description Of Translation-Invariant And Translation-Covariant Linear Transformations: A Natural Justification Of Fourier Transforms And Fuzzy Transforms, Irina Perfilieva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we are interested in the dependencies that do not change with time, i.e., that do not change when we change the origin of the time axis. The corresponding translation-invariant transformations are easy to describe: they correspond to convolutions, or, equivalently, to fuzzy transforms.

It turns out that if we relax the invariance condition and require only that the transformation be translation-covariant (i.e., that it appropriately changes under translation), we get exactly two classes of transformations: Fourier transforms and fuzzy transforms. This result explain why both transforms have been successfully used in data processing.

Designing, Understanding, And Analyzing Unconventional Computation: The Important Role Of Logic And Constructive Mathematics, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we explain why, in our opinion, logic and constructive mathematics are playing -- and should play -- an important role in the design, understanding, and analysis of unconventional computation.