Interpretable Approximation of High-Dimensional Data

Potts, Daniel; Schmischke, Michael

doi:10.1137/21m1407707

Cited by 10 publications

(16 citation statements)

References 25 publications

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“…With the interpretation capabilities of the ANOVA method, c.f. [4], we are additionally able to explain the importance of our features and give reasonable explanation for the predictions.…”

Section: Introductionsupporting

confidence: 70%

“…In this section, we briefly summarize the interpretable ANOVA (analysis of variance) approximation method and the idea of grouped transformations, see [2,11]. The approach was considered for periodic functions, but has been expanded to nonperiodic functions in [3,4]. In this paper, we focus on functions f : R d → R from L 2 (R d , ω) with probability density ω from Equation (2).…”

Section: Interpretable Anova Approximationmentioning

confidence: 99%

“…. , d which was introduced in [4]. Note that this score has order-dependent weight and is normalized such that…”

Section: Now Taking the Unionmentioning

confidence: 99%

“…From the approximation S(X , I(U(d, d s )))f we can then calculate the global sensitivity indices ̺(u, S(X , I(U(d, d s )))f ), u ∈ U(d, d s ), and an attribute ranking r(i), i ∈ [d], see Equation (15). Then we are able to apply the strategies proposed in [4] to truncate terms from the set U(d, d s ).…”

Section: Active Setmentioning

confidence: 99%

“…The explainable ANOVA approximation method introduced in [2][3][4] is based on the wellknown multi-variate analysis of variance (ANOVA) decomposition, see, e.g,., [5][6][7][8][9][10], and relies on the existence of a complete orthonormal system in the space which is suitable for fast matrix-vector multiplication algorithms in grouped transformations, c.f. [11].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Interpretable Transformed ANOVA Approximation on the Example of the Prevention of Forest Fires

Potts

Schmischke

2022

Front. Appl. Math. Stat.

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The distribution of data points is a key component in machine learning. In most cases, one uses min-max-normalization to obtain nodes in [0, 1] or Z-score normalization for standard normal distributed data. In this paper, we apply transformation ideas in order to design a complete orthonormal system in the L2 space of functions with the standard normal distribution as integration weight. Subsequently, we are able to apply the explainable ANOVA approximation for this basis and use Z-score transformed data in the method. We demonstrate the applicability of this procedure on the well-known forest fires dataset from the UCI machine learning repository. The attribute ranking obtained from the ANOVA approximation provides us with crucial information about which variables in the dataset are the most important for the detection of fires.

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“…With the interpretation capabilities of the ANOVA method, c.f. [4], we are additionally able to explain the importance of our features and give reasonable explanation for the predictions.…”

Section: Introductionsupporting

confidence: 70%

Section: Interpretable Anova Approximationmentioning

confidence: 99%

“…. , d which was introduced in [4]. Note that this score has order-dependent weight and is normalized such that…”

Section: Now Taking the Unionmentioning

confidence: 99%

Section: Active Setmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Interpretable Transformed ANOVA Approximation on the Example of the Prevention of Forest Fires

Potts

Schmischke

2022

Front. Appl. Math. Stat.

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Fast hyperbolic wavelet regression meets ANOVA

2023

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We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low dimensional variable interactions. Compactly supported periodic Chui–Wang wavelets are used for the tensorized hyperbolic wavelet basis. In a first step we give a self-contained characterization of tensor product Sobolev–Besov spaces on the d-torus with arbitrary smoothness in terms of the decay of such wavelet coefficients. In the second part we perform and analyze scattered-data approximation using a hyperbolic cross type truncation of the basis expansion for the associated least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. In case of i.i.d. samples we can even bound the approximation error with high probability by loosing only $$\log $$ log -terms that do not depend on d compared to the best approximation. In addition, if the function has low effective dimension (i.e. only interactions of few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to increase the accuracy. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.

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Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation

Potts,

Weidensager

2024

Adv Comput Math

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We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low-dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis on the torus. With a variable transformation, we are able to transform the approximation rates and fast algorithms from the torus to other domains. We perform and analyze scattered data approximation for smooth but arbitrary density functions by using a least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. For non-periodic functions, we propose a new extension method. A proper choice of the extension parameter together with the piecewise polynomial Chui-Wang wavelets extends the functions appropriately. In every case, we are able to bound the approximation error with high probability. Additionally, if the function has a low effective dimension (i.e., only interactions of a few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to decrease the approximation error. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.

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Interpretable Approximation of High-Dimensional Data

Cited by 10 publications

References 25 publications

Interpretable Transformed ANOVA Approximation on the Example of the Prevention of Forest Fires

Interpretable Transformed ANOVA Approximation on the Example of the Prevention of Forest Fires

Fast hyperbolic wavelet regression meets ANOVA

Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation

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