Adaptive Nonparametric Density Estimation with B-Spline Bases
Learning density estimation is important in probabilistic modeling and reasoning with uncertainty. Since B-spline basis functions are piecewise polynomials with local support, density estimation with B-splines shows its advantages when intensive numerical computations are involved in the subsequent...
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MDPI AG
2023-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/2/291 |
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| author | Yanchun Zhao Mengzhu Zhang Qian Ni Xuhui Wang |
| author_facet | Yanchun Zhao Mengzhu Zhang Qian Ni Xuhui Wang |
| author_sort | Yanchun Zhao |
| collection | DOAJ |
| description | Learning density estimation is important in probabilistic modeling and reasoning with uncertainty. Since B-spline basis functions are piecewise polynomials with local support, density estimation with B-splines shows its advantages when intensive numerical computations are involved in the subsequent applications. To obtain an optimal local density estimation with B-splines, we need to select the bandwidth (i.e., the distance of two adjacent knots) for uniform B-splines. However, the selection of bandwidth is challenging, and the computation is costly. On the other hand, nonuniform B-splines can improve on the approximation capability of uniform B-splines. Based on this observation, we perform density estimation with nonuniform B-splines. By introducing the error indicator attached to each interval, we propose an adaptive strategy to generate the nonuniform knot vector. The error indicator is an approximation of the information entropy locally, which is closely related to the number of kernels when we construct the nonuniform estimator. The numerical experiments show that, compared with the uniform B-spline, the local density estimation with nonuniform B-splines not only achieves better estimation results but also effectively alleviates the overfitting phenomenon caused by the uniform B-splines. The comparison with the existing estimation procedures, including the state-of-the-art kernel estimators, demonstrates the accuracy of our new method. |
| first_indexed | 2024-03-09T11:46:53Z |
| format | Article |
| id | doaj.art-cf3afce04f254f549727d7edcf1c9f82 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:46:53Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-cf3afce04f254f549727d7edcf1c9f822023-11-30T23:20:08ZengMDPI AGMathematics2227-73902023-01-0111229110.3390/math11020291Adaptive Nonparametric Density Estimation with B-Spline BasesYanchun Zhao0Mengzhu Zhang1Qian Ni2Xuhui Wang3School of Mathematics, Hefei University of Technology, Hefei 230009, ChinaSchool of Mathematics, Hefei University of Technology, Hefei 230009, ChinaSchool of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing 211816, ChinaDepartment of Mathematics, Hohai University, Nanjing 211100, ChinaLearning density estimation is important in probabilistic modeling and reasoning with uncertainty. Since B-spline basis functions are piecewise polynomials with local support, density estimation with B-splines shows its advantages when intensive numerical computations are involved in the subsequent applications. To obtain an optimal local density estimation with B-splines, we need to select the bandwidth (i.e., the distance of two adjacent knots) for uniform B-splines. However, the selection of bandwidth is challenging, and the computation is costly. On the other hand, nonuniform B-splines can improve on the approximation capability of uniform B-splines. Based on this observation, we perform density estimation with nonuniform B-splines. By introducing the error indicator attached to each interval, we propose an adaptive strategy to generate the nonuniform knot vector. The error indicator is an approximation of the information entropy locally, which is closely related to the number of kernels when we construct the nonuniform estimator. The numerical experiments show that, compared with the uniform B-spline, the local density estimation with nonuniform B-splines not only achieves better estimation results but also effectively alleviates the overfitting phenomenon caused by the uniform B-splines. The comparison with the existing estimation procedures, including the state-of-the-art kernel estimators, demonstrates the accuracy of our new method.https://www.mdpi.com/2227-7390/11/2/291adaptive strategyB-splinenonparametric density estimationerror indictor |
| spellingShingle | Yanchun Zhao Mengzhu Zhang Qian Ni Xuhui Wang Adaptive Nonparametric Density Estimation with B-Spline Bases Mathematics adaptive strategy B-spline nonparametric density estimation error indictor |
| title | Adaptive Nonparametric Density Estimation with B-Spline Bases |
| title_full | Adaptive Nonparametric Density Estimation with B-Spline Bases |
| title_fullStr | Adaptive Nonparametric Density Estimation with B-Spline Bases |
| title_full_unstemmed | Adaptive Nonparametric Density Estimation with B-Spline Bases |
| title_short | Adaptive Nonparametric Density Estimation with B-Spline Bases |
| title_sort | adaptive nonparametric density estimation with b spline bases |
| topic | adaptive strategy B-spline nonparametric density estimation error indictor |
| url | https://www.mdpi.com/2227-7390/11/2/291 |
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