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The *mean*, *median*, and *mode* are usually calculated from univariate observations as the most basic representative values of a random variable. To measure the spread of the distribution, the *standard deviation*, *interquartile range*, and *modal interval* are also calculated. When we analyze continuous relations between a pair of random variables from bivariate observations, *regression analysis* is often used. By minimizing appropriate costs evaluating regression errors, we estimate the conditional mean, median, and mode. The conditional standard deviation can be estimated if the bivariate observations are obtained from a Gaussian process. Moreover, the conditional interquartile range can be calculated for various distributions by the *quantile regression* that estimates any conditional quantile (percentile). Meanwhile, the study of the modal interval regression is relatively new, and *spline regression models*, known as flexible models having the optimality on the smoothness for bivariate data, are not yet used. In this paper, we propose a modal interval regression method based on spline quantile regression. The proposed method consists of two steps. In the first step, we divide the bivariate observations into bins for one random variable, then detect the modal interval for the other random variable as the lower and upper quantiles in each bin. In the second step, we estimate the conditional modal interval by constructing both lower and upper quantile curves as spline functions. By using the spline quantile regression, the proposed method is widely applicable to various distributions and formulated as a *convex optimization problem* on the coefficient vectors of the lower and upper spline functions. Extensive experiments, including settings of the bin width, the smoothing parameter and weights in the cost function, show the effectiveness of the proposed modal interval regression in terms of accuracy and visual shape for synthetic data generated from various distributions. Experiments for real-world meteorological data also demonstrate a good performance of the proposed method.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E106-A No.2 pp.106-123

- Publication Date
- 2023/02/01

- Publicized
- 2022/07/26

- Online ISSN
- 1745-1337

- DOI
- 10.1587/transfun.2022EAP1031

- Type of Manuscript
- PAPER

- Category
- Numerical Analysis and Optimization

Sai YAO

Ritsumeikan University

Daichi KITAHARA

Osaka University

Hiroki KURODA

Ritsumeikan University

Akira HIRABAYASHI

Ritsumeikan University

The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.

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Sai YAO, Daichi KITAHARA, Hiroki KURODA, Akira HIRABAYASHI, "Modal Interval Regression Based on Spline Quantile Regression" in IEICE TRANSACTIONS on Fundamentals,
vol. E106-A, no. 2, pp. 106-123, February 2023, doi: 10.1587/transfun.2022EAP1031.

Abstract: The *mean*, *median*, and *mode* are usually calculated from univariate observations as the most basic representative values of a random variable. To measure the spread of the distribution, the *standard deviation*, *interquartile range*, and *modal interval* are also calculated. When we analyze continuous relations between a pair of random variables from bivariate observations, *regression analysis* is often used. By minimizing appropriate costs evaluating regression errors, we estimate the conditional mean, median, and mode. The conditional standard deviation can be estimated if the bivariate observations are obtained from a Gaussian process. Moreover, the conditional interquartile range can be calculated for various distributions by the *quantile regression* that estimates any conditional quantile (percentile). Meanwhile, the study of the modal interval regression is relatively new, and *spline regression models*, known as flexible models having the optimality on the smoothness for bivariate data, are not yet used. In this paper, we propose a modal interval regression method based on spline quantile regression. The proposed method consists of two steps. In the first step, we divide the bivariate observations into bins for one random variable, then detect the modal interval for the other random variable as the lower and upper quantiles in each bin. In the second step, we estimate the conditional modal interval by constructing both lower and upper quantile curves as spline functions. By using the spline quantile regression, the proposed method is widely applicable to various distributions and formulated as a *convex optimization problem* on the coefficient vectors of the lower and upper spline functions. Extensive experiments, including settings of the bin width, the smoothing parameter and weights in the cost function, show the effectiveness of the proposed modal interval regression in terms of accuracy and visual shape for synthetic data generated from various distributions. Experiments for real-world meteorological data also demonstrate a good performance of the proposed method.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2022EAP1031/_p

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@ARTICLE{e106-a_2_106,

author={Sai YAO, Daichi KITAHARA, Hiroki KURODA, Akira HIRABAYASHI, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={Modal Interval Regression Based on Spline Quantile Regression},

year={2023},

volume={E106-A},

number={2},

pages={106-123},

abstract={The *mean*, *median*, and *mode* are usually calculated from univariate observations as the most basic representative values of a random variable. To measure the spread of the distribution, the *standard deviation*, *interquartile range*, and *modal interval* are also calculated. When we analyze continuous relations between a pair of random variables from bivariate observations, *regression analysis* is often used. By minimizing appropriate costs evaluating regression errors, we estimate the conditional mean, median, and mode. The conditional standard deviation can be estimated if the bivariate observations are obtained from a Gaussian process. Moreover, the conditional interquartile range can be calculated for various distributions by the *quantile regression* that estimates any conditional quantile (percentile). Meanwhile, the study of the modal interval regression is relatively new, and *spline regression models*, known as flexible models having the optimality on the smoothness for bivariate data, are not yet used. In this paper, we propose a modal interval regression method based on spline quantile regression. The proposed method consists of two steps. In the first step, we divide the bivariate observations into bins for one random variable, then detect the modal interval for the other random variable as the lower and upper quantiles in each bin. In the second step, we estimate the conditional modal interval by constructing both lower and upper quantile curves as spline functions. By using the spline quantile regression, the proposed method is widely applicable to various distributions and formulated as a *convex optimization problem* on the coefficient vectors of the lower and upper spline functions. Extensive experiments, including settings of the bin width, the smoothing parameter and weights in the cost function, show the effectiveness of the proposed modal interval regression in terms of accuracy and visual shape for synthetic data generated from various distributions. Experiments for real-world meteorological data also demonstrate a good performance of the proposed method.},

keywords={},

doi={10.1587/transfun.2022EAP1031},

ISSN={1745-1337},

month={February},}

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TY - JOUR

TI - Modal Interval Regression Based on Spline Quantile Regression

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 106

EP - 123

AU - Sai YAO

AU - Daichi KITAHARA

AU - Hiroki KURODA

AU - Akira HIRABAYASHI

PY - 2023

DO - 10.1587/transfun.2022EAP1031

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E106-A

IS - 2

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - February 2023

AB - The *mean*, *median*, and *mode* are usually calculated from univariate observations as the most basic representative values of a random variable. To measure the spread of the distribution, the *standard deviation*, *interquartile range*, and *modal interval* are also calculated. When we analyze continuous relations between a pair of random variables from bivariate observations, *regression analysis* is often used. By minimizing appropriate costs evaluating regression errors, we estimate the conditional mean, median, and mode. The conditional standard deviation can be estimated if the bivariate observations are obtained from a Gaussian process. Moreover, the conditional interquartile range can be calculated for various distributions by the *quantile regression* that estimates any conditional quantile (percentile). Meanwhile, the study of the modal interval regression is relatively new, and *spline regression models*, known as flexible models having the optimality on the smoothness for bivariate data, are not yet used. In this paper, we propose a modal interval regression method based on spline quantile regression. The proposed method consists of two steps. In the first step, we divide the bivariate observations into bins for one random variable, then detect the modal interval for the other random variable as the lower and upper quantiles in each bin. In the second step, we estimate the conditional modal interval by constructing both lower and upper quantile curves as spline functions. By using the spline quantile regression, the proposed method is widely applicable to various distributions and formulated as a *convex optimization problem* on the coefficient vectors of the lower and upper spline functions. Extensive experiments, including settings of the bin width, the smoothing parameter and weights in the cost function, show the effectiveness of the proposed modal interval regression in terms of accuracy and visual shape for synthetic data generated from various distributions. Experiments for real-world meteorological data also demonstrate a good performance of the proposed method.

ER -