We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions; the similarity dimension via the Moran equation (at least in the case of self-similar sets); the order of the (box-)counting function; the classic result on...

This paper looks at the extraction of nonlinear turbulence information of the speech signal using Multi-Scale Fractal Dimension (MFD) as the additional feature to the convectional MFCC features. The MFD is estimated using both Box-Counting and Minkowiski-Bouligand dimensions. The proposed framework uses this feature extraction together with sub-band based speaker modeling rather than the wide-b...

The object of this paper is to present the numerical algorithm to obtain a finite difference solution for an elliptic transmission problem by use of the direct method of lines ([Nak65],[KK98], [KK99]). Let Π be a rectangular domain in R, Ω1 be an open subset of Π and Ω2 = Π \ Ω1, Γ = ∂Ω1 (see Figure 1). Then the elliptic transmission problem is formulated as follows. And it is well known that (...

We consider a class of “box-like” statistically self-affine functions, and compute the almost-sure box-counting dimension their graphs. Furthermore, we differentiability our prove that, depending on an explicitly computable functional model, they are almost surely either differentiable everywhere or non-differentiable everywhere.

Validation is a process that determines the fitness of an assay, which has been properly developed, optimised and standardised, for an intended purpose. All diagnostic assays (laboratory and field assays) should be validated for the species in which they will be used. Validation includes estimates of the analytical and diagnostic performance characteristics of a test. In the context of this cha...

The motivation for this paper comes from the study of a simple model of bicycle motion. The bicycle is modeled as an oriented segment in the plane of fixed length `, the wheelbase of the bicycle. The motion is constrained so that the segment is always tangent to the path of the rear wheel; this non-holonomic constraint is due to the fact that the rear wheel is fixed on the frame, whereas the fr...

We find the almost sure Hausdorff and box-counting dimensions of random subsets of self-affine fractals obtained by selecting subsets at each stage of the hierarchical construction in a statistically self-similar manner.

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In many graph mining settings, measuring node proximity is a fundamental problem. While most of existing measurements are (implicitly or explicitly) designed for undirected graphs; edge directions in the graph provide a new perspective to proximity measurement: measuring the proximity from A to B; rather than between A and B. (See Figure 1 as an example). In this chapter, we study the role of e...