A Technique for Integration by Parts

Kasube, Herbert E.

doi:10.1080/00029890.1983.11971195

Cited by 3 publications

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“…The second term is intractable due to the unknown derivative of the density difference. To make this term tractable, we use integration by parts (Kasube, 1983) to obtain the following:…”

Section: Direct Estimation Of the Derivative Of The Density Differencementioning

confidence: 99%

Direct Estimation of the Derivative of Quadratic Mutual Information with Application in Supervised Dimension Reduction

2017

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A typical goal of supervised dimension reduction is to find a low-dimensional subspace of the input space such that the projected input variables preserve maximal information about the output variables. The dependence maximization approach solves the supervised dimension reduction problem through maximizing a statistical dependence between projected input variables and output variables. A well-known statistical dependence measure is mutual information (MI) which is based on the Kullback-Leibler (KL) divergence. However, it is known that the KL divergence is sensitive to outliers. On the other hand, quadratic MI (QMI) is a variant of MI based on the L 2 distance which is more robust against outliers than the KL divergence, and a computationally efficient method to estimate QMI from data, called least-squares QMI (LSQMI), has been proposed recently. For these reasons, developing a supervised dimension reduction method based on LSQMI seems promising. However, not QMI itself, but the derivative of QMI is needed for subspace search in supervised dimension reduction, and the derivative of an accurate QMI estimator is not necessarily a good estimator of the derivative of QMI. In this paper, we propose to directly estimate the derivative of QMI without estimating QMI itself. We show that the direct estimation of the derivative of QMI is more accurate than the derivative of the estimated QMI. Finally, we develop a supervised dimension reduction algorithm which efficiently uses the proposed derivative estimator, and demonstrate through experiments that the proposed method is more robust against outliers than existing methods.

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“…The second term is intractable due to the unknown derivative of the density difference. To make this term tractable, we use integration by parts (Kasube, 1983) to obtain the following:…”

Section: Direct Estimation Of the Derivative Of The Density Differencementioning

confidence: 99%

Direct Estimation of the Derivative of Quadratic Mutual Information with Application in Supervised Dimension Reduction

2017

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The Five Columns Rule in Solving Definite Integration by Parts Through Transformation of Integral Limits

Jandja

Lutfi

2018

J. Phys.: Conf. Ser.

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A guide for using integration by parts: <i>Pet-LoPo-InPo</i>

Guo

2022

era

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<abstract> <p>Based on the tutorial cases accumulated in the past several years, by reclassifying the arithmetic functions (A) in LIATE into the polynomial function (<italic>P</italic>), the standard power function (<italic>Po</italic>), and the integer power function (<italic>nPo</italic>), a new guide, comprising three sub-guides, <italic>Pet</italic>, <italic>LoPo</italic> and <italic>InPo</italic>, or <italic>Pet-LoPo-InPo</italic>, is summarized in this note to guide practicing integration by parts. This new guide removes many incompatible combinations included in LIATE, rationalizes the relationship between the exponential and trigonometric functions in LIATE, and expands the coverage of the <italic>P</italic>-functions beyond the traditional definitions. Hence, the new guide can reduce potential confusions that students may experience in using LIATE for their practices of integration by parts. The advantages of this new guide are demonstrated by many worked examples in this note.</p> </abstract>

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A Technique for Integration by Parts

Cited by 3 publications

References 0 publications

Direct Estimation of the Derivative of Quadratic Mutual Information with Application in Supervised Dimension Reduction

Direct Estimation of the Derivative of Quadratic Mutual Information with Application in Supervised Dimension Reduction

The Five Columns Rule in Solving Definite Integration by Parts Through Transformation of Integral Limits

A guide for using integration by parts: <i>Pet-LoPo-InPo</i>

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