Functional data analysis uses basis functions to construct a notion of continuum from the discrete data. Are wavelets used for this purpose or are they unsuitable.
This is a very good question with many possible answers.
A good place to start is the 2014 Ph.D. thesis:
http://www.math.ku.dk/noter/filer/phd15mmu.pdf
In Section 3.1, page 14, it is observed that wavelets are especially useful for data with sharp peaks. At the beginning of Chapter 4, p. 17, it is observed that waveles are used to convert discretely observed noisy spectral data to smooth functions. And that is the beginning of an answer to this question.
The basic approach is to represent a discrete sample by a square integrable function (Section 4.1.4, page 23).
To look more deeply into the relation between wavelets and discrete data, see
https://arxiv.org/pdf/1101.4744.pdf
See, for example, Section 2.1, starting on page 4.
Sir, it is very interesting. One of the central points of functional data analysis is the construction of functional data from discrete data. The method chosen at this point restricts our "world view" in some sense as it is more or less the function we assume to be at the heart of the problem.
Wavelets are assumed to be linked to the way how humans hears and see (aquire this information from the surroundings)
A more fundamental function can perhaps improve the idea.
@Muhammad Zubair Ahmad: A more fundamental function can perhaps improve the idea.
Exactly!
An example of such a function lives in a spatial fibre space $(E,\pi,B)$ in which $E,B$ are topological spaces and $\pi$ is surjective and continuous. The inverse $\pi^{-1}(b)$ for an element $b\in B$ is called a fibre. And $\pi^{-1}(B)$ on the whole space $B$ is a fibre bundle. The arrow diagram in the first attached image represents a spatial fibre space, where the mapping $f$ is called a projection mapping. This becomes richer, if we move to what is known as a descriptive fibre space.
Let $\re X\in \mathscr{R}_{\re A}$ and let $\Phi(\re X)$ be a feature vector that describes region $\re X$ in the class of regions $\mathscr{R}_{\re A}$. The continuous mapping $\Phi:\mathscr{R}_{\re A}\longrightarrow B$ is a \emph{projection} on the class of regions $\mathscr{R}_{\re A}$ onto $B$ such that $(\mathscr{R}_{\re A}, \Phi, B)$ is a \emph{descriptive fibre space} and the set of feature vectors $B\subset \mathbb{R}^n$ is the \emph{descriptive base space}. For $x$ in $B$, $\Phi^{-1}(x)\in \mathscr{R}_{\re A}$ is a fibre over $x$.
The leads to a descriptive fibre bundle which is a BreMiller-Sloyer sheaf for a class of regions. The arrow diagram in the second attached image represents a descriptive fibre space, in which $f$ is a projection mapping that maps a region $A$ (denoted by ReA) to a class of regions.
I believe it is safe to say to that fibre bundles offer more than wavelets, if we consider descriptive fibre spaces that characterize the shapes of space.
Sir,
Correct me if I am wrong
I had to sit through it pretty much the whole after noon. The following is what I understand.
We have a topological space and \reA are the regions defined in them
If we have have a mapping $\Phi$ that maps the regions to a feature vector then the inverse map $\Phi^-1$ would be a ring of continuous functions on the regions included in the class of the regions. Hence this would give us the continuous mapping instead of the interpolations.
Wavelets can be used to construct continuous signals from discrete measurement. This is a principle used in conversion of digital television and digital music to analog output devices. Quality derives from discrete sampling rate, data speed, and wavelet definitions.
Behind this practice there is a long history of debate about how continuous functions can derive from discrete sources. Example is how the concept of velocity can exist at a single point. After much argument mathematics has been modified to define continuous functions as a mapping of discrete points in domains.
With wavelets a convolution can be constructed in which a single discrete point can be promoted to represent one cycle of a vibration like a musical note of a particular pitch timbre and duration. Your question relates to how quality is defined and produced.
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