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Directional representation of data in Linear Discriminant Analysis

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Grala-Michalak J.

Biometrical Letters

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2015

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tom 52

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nr 2

55-74

EN

Sometimes feature representations of measured individuals are better described by spherical coordinates than Cartesian ones. The author proposes to introduce a preprocessing step in LDA based on the arctangent transformation of spherical coordinates. This nonlinear transformation does not change the dimension of the data, but in combination with LDA it leads to a dimension reduction if the raw data are not linearly separated. The method is presented using various examples of real and artificial data.

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On constructions of isometric embeddings of nonseparable \(L^p\) spaces, \(0 \lt p \leq 2\)

63%

Grala-Michalak J., Michalak A.

Commentationes Mathematicae

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2008

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tom 48

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nr 2

EN

Let \(J\) be an infinite set. Let \(X\) be a real or complex \(\sigma\)-order continuous rearrangement invariant quasi-Banach function space over \((\{0, 1\}^J,\ \mathcal{B}^J,\ \lambda_J)\), the product of \(J\) copies of the measure space \((\{0, 1\},\ 2^{0,1},\ \frac{1}{2} \delta_0 + \frac{1}{2}\delta_1)\). We show that if \(0 \lt p \lt 2\) and \(X\) contains a function \(f\) with the decreasing rearrangement \(f^∗\) such that \(f^∗(t) \gt t^{-\frac{1}{p}}\) for every \(t\in (0, 1)\), then it contains an isometric copy of the Lebesgue space \(L^p (\lambda_J)\). Moreover, if \(X\) contains a function \(f\) such that \(f^∗(t) \gt \sqrt{|\text{ln}(t)|}\) for every \(t\in (0, 1)\), then it contains an isometric copy of the Lebesgue space \(L^2(\lambda_J)\).

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On constructions of isometric copies of \(L^p (0, 1)\) spaces \((0 \lt p \leq 2)\) by stochastic \(p\)-stable processes

63%

Grala-Michalak J., Michalak A.

Commentationes Mathematicae

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2008

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tom 48

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nr 1

EN

Let \(S^p = \{S_t^p : t = \frac{k}{2^n},\ 0 \leq k \leq 2^n,\ n \in\mathbb{N}\}\) be a stochastic process on a probability space \((\Omega, \Sigma, P)\) with independent and time homogeneous increments such that \(S_t^p - S_u^p\) is identically distributed as \((t- u)^{1/p} Z_p\) for each \(0 \leq u \lt t \leq 1\) where \(Z_p\) is a given symmetric \(p\)-stable distribution. We show that the closed linear hull of \(S^p\) forms an isometric copy of the real Lebesgue space \(L^p (0, 1)\) in any quasi-Banach space \(X\) consisting of \(P\)-a.e. equivalence classes of \(\Sigma\)-measurable real functions on \(\Omega\) equipped with a rearrangement invariant quasi-norm which contains \(S^p\) as a subset. It is possible to construct processes \(S^p\) for \(0 \lt p \leq 2\) on \([0, 1]\) with the Lebesgue measure. We show also a complex version of the result.

Ograniczanie wyników

2 Commentationes Mathematicae

1 Biometrical Letters

3 Grala-Michalak J.

2 Michalak A.

1 2015

2 2008

Directional representation of data in Linear Discriminant Analysis

On constructions of isometric embeddings of nonseparable \(L^p\) spaces, \(0 \lt p \leq 2\)

On constructions of isometric copies of \(L^p (0, 1)\) spaces \((0 \lt p \leq 2)\) by stochastic \(p\)-stable processes