Bernstein conditional density estimation with application to conditional distribution and regression functions

Belalia, Mohamed; Bouezmarni, Taoufik; Leblanc, Alexandre

doi:10.1016/j.jkss.2019.05.005

Cited by 7 publications

(6 citation statements)

References 35 publications

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“…There are 8 layers AlexNet network structure, including 5 convolution layers, 3 fullyconnected layers. Each of them with node numbers of 128, 64, 32,16,6,192, 100 and 4 from the input end to the output end, respectively.The MMD distance is added to the 7-th layer (feature layer, the upper layer of softmax) to reduce the discrepancy between source and target domain.…”

Section: Ddcmentioning

confidence: 99%

“…Thus, feature distribution alignment is still a challenge for domain adaptation. Most of the existing methods try to align the marginal distribution [14], [15], or the conditional distribution [16], or assume that both distributions are equally important [17]. In the field of computer vision, the latest research has shown that perform dynamic distribution adaptation (DDA) can obtain better transfer performance [18].…”

Section: Introductionmentioning

confidence: 99%

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Deep Transfer Network With Multi-Kernel Dynamic Distribution Adaptation for Cross-Machine Fault Diagnosis

Liu

et al. 2021

IEEE Access

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Recently, various deep learning models, which are mainly based on data-driven algorithms, have received more and more attention in the field of intelligent fault diagnosis and prognostics. However, there are two major assumptions accepted by default in the existing studies: 1) The training (source domain) and testing (target domain) data sets obey the same feature distribution; 2) Sufficient labeled data with fault information is available for model training. In real industrial scenarios, especially for different machines, these assumptions are mostly invalid, which makes it a huge challenge to build reliable diagnostic model. Motivated by transfer learning, we present a novel intelligent method named deep transfer network (DTN) with multi-kernel dynamic distribution adaptation (MDDA) to address the problem of cross-machine fault diagnosis. In the proposed approach, the DTN has wide first-layer convolutional kernel and several small convolutional layers, which is utilized to extract transferable features across different machines and suppress high frequency noise. Then, the MDDA method constructs a weighted mixed kernel function to map different transferable features to a unified feature space, and the relative importance of the marginal and conditional distributions are also evaluated dynamically. The proposed method is verified by three transfer learning tasks of bearings, in which the health states of wind turbine bearings in real scenario are identified by using diagnosis knowledge from two different bearings in laboratories. The results show that the proposed method can achieve higher diagnosis accuracy and better transfer performance even under different noisy environment conditions than many other state-of-the-art methods. The presented framework offers a promising approach for cross-machine fault diagnosis. INDEX TERMS Deep transfer network, multi-kernel dynamic distribution adaptation, cross-machine fault diagnosis, transfer learning, bearings.

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Section: Ddcmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Deep Transfer Network With Multi-Kernel Dynamic Distribution Adaptation for Cross-Machine Fault Diagnosis

Liu

et al. 2021

IEEE Access

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“…Asymptotic properties of Bernstein estimators on compacts supports have been studied by many authors, see e.g. Vitale (1975); Gawronski & Stadtmüller (1981); Stadtmüller (1983); Tenbusch (1994); Babu et al (2002); Babu & Chaubey (2006); Bouezmarni & Rolin (2007); Leblanc (2010Leblanc ( , 2012a; Igarashi & Kakizawa (2014); Belalia (2016b); Belalia et al (2019); Ouimet (2020a,b), just to name a few. For the interested reader, there is an extensive review in Section 2 of Ouimet (2020a).…”

Section: Asymptotic Properties Of Bernstein Estimators With Poisson Weightsmentioning

confidence: 99%

On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators

Ouimet

2021

Journal of Mathematical Analysis and Applications

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In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard (2010), who uses a different approach. We also use the local limit theorem to derive the asymptotics of the variance for Bernstein c.d.f. and density estimators with Poisson weights on the positive half-line (also called Szasz estimators). The propagation of errors in the literature due to the incorrect estimate in Lemma 2 (iv) of Leblanc (2012a) is addressed in the Appendix.

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“…over the d-dimensional simplex. Some of their asymptotic properties were investigated by Vitale (1975), Stadtmüller (1986), Tenbusch (1997), , Ghosal (2001) Lu (2015), Guan (2016Guan ( , 2017 and Belalia et al (2017Belalia et al ( , 2019 when d = 1, by Tenbusch (1994) [31] when d = 2, and by Ouimet (2020) [32,33] for all d ≥ 1, using a local limit theorem from Ouimet (2020) [34] for the multinomial distribution (see also Arenbaev (1976) [35]). The estimator ( 5) is a discrete analogue of the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) [36] and studied theoretically in Brown and Chen (1999), Chen (1999Chen ( , 2000 and Bouezmarni and Rolin (2003) [37][38][39][40] when d = 1 (among others), and in Ouimet (2020) [41] for all d ≥ 1.…”

Section: Motivationmentioning

confidence: 99%

General Formulas for the Central and Non-Central Moments of the Multinomial Distribution

Ouimet

2021

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We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. These results expand significantly on those in Newcomer (2008) and Newcomer et al. (2008), where the non-central moments were calculated up to order 4.

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Bernstein conditional density estimation with application to conditional distribution and regression functions

Cited by 7 publications

References 35 publications

Deep Transfer Network With Multi-Kernel Dynamic Distribution Adaptation for Cross-Machine Fault Diagnosis

Deep Transfer Network With Multi-Kernel Dynamic Distribution Adaptation for Cross-Machine Fault Diagnosis

On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators

General Formulas for the Central and Non-Central Moments of the Multinomial Distribution

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