Why Use a Fuzzy Partition in F-Transform?

Abstract

In many application problems, F-transform algorithms are very efficient. In F-transform techniques, we replace the original signal or image with a finite number of weighted averages. The use of a weighted average can be naturally explained, e.g., by the fact that this is what we get anyway when we measure the signal. However, most successful applications of F-transform have an additional not-so-easy-to-explain feature: the fuzzy partition requirement that the sum of all the related weighting functions is a constant. In this paper, we show that this seemingly difficult-to-explain requirement can also be naturally explained in signal-measurement terms: namely, this requirement can be derived from the natural desire to have all the signal values at different moments of time estimated with the same accuracy. This explanation is the main contribution of this paper.

Bitcoin Analysis and Forecasting Through Fuzzy Transform

“…F-transform is thus used to model Bitcoin as a function of GT100. The L 1 -norm and L 2 -norm based inverse iF-transforms of the data-set (GT100 t , BitCoin t ), t = 1, 2, ..., M are computed; taking into account the sparsity of the values GT100 t (in particular above the threshold 40), we use a non-uniform 1-partition (P, A) of the range [0, 100] of the observed GT100 t , namely the set of 25 nodes {0, 2,3,4,5,6,7,8,10,12,14,16,18,20,22,24,26,28,30,35,40,45,50, 60, 100}, as pictured in Figure 5. The two curves give the predominant relationship between BitCoin and GT100.…”

“…The Bitcoin observed time series is modelled through fuzzy-valued functions, whose level-cuts can be interpreted in the setting of expectile and quantile fuzzy regressions; these last are introduced in ([1], [2]) as non-parametric smoothing methodologies and are constructed by defining fuzzy-valued expectile (L 2 -norm) and quantile (L 1 -norm) extensions of the F-transforms. We recall that F-transform has been introduced by [3] (see also [4], [5], [6], [7])…”

Bitcoin Analysis and Forecasting through Fuzzy Transform

“…The notion of a fuzzy transform (F-transform) as a tool for modeling with fuzzy rules as specific transformation and for general approximation of functions has been introduced by Perfilieva in [1] (see also [2]) and is now recognized as a powerful technique with important properties and potentials for various applications, as developed in several papers and special issues (see, e.g, [3][4][5] and the references therein).…”