Abstract
Suppose that the random matrix \({\textbf{X}}_{p\times m}\) has a matrix variate normal distribution with the mean matrix \(\varvec{\Theta }\) and covariance matrix \({{\varvec{\Sigma }}} \otimes {\varvec{\Psi }}\), where \({{\varvec{\Sigma }}}\) and \({\varvec{\Psi }}\) are known positive definite covariance matrices. This paper studies the Bayesian shrinkage wavelet estimation of the mean matrix \(\varvec{\Theta }\) under the balanced loss function. Two soft Bayesian shrinkage wavelet estimators are proposed based on two prior distributions: the matrix variate normal \(N_{p,m}({\textbf{0}}, {\varvec{\Lambda }}\otimes {\varvec{\Psi }})\), where \({\varvec{\Lambda }}\) is a known positive definite covariance matrix, and the improper prior \(\pi (\varvec{\Theta })=1\). Using Bayes estimators as the target estimator and Stein’s unbiased risk estimate technique, the soft Bayesian shrinkage wavelet threshold and the soft generalized Bayesian shrinkage wavelet threshold are obtained. Based on the newly proposed thresholds, we derive the soft Bayesian shrinkage wavelet and the soft generalized Bayesian shrinkage wavelet estimators. The performance of the presented theoretical topics is measured through a simulation study and three real examples. The results show that the soft generalized Bayesian shrinkage wavelet estimator outperforms four classical soft shrinkage wavelet estimators and the soft Bayesian shrinkage wavelet estimator.
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Data availability
The datasets that support the findings of this study are openly available at the following link: https://archive.ics.uci.edu/dataset/464/superconductivty+data. https://archive.ics.uci.edu/dataset/165/concrete+compressive+strength. https://archive.ics.uci.edu/dataset/844/average+localization+error+(ale)+in+sensor +node+localization+process+in+wsns.
References
Afshari M, Lak F, Gholizadeh B (2017) A new Bayesian wavelet thresholding estimator of nonparametric regression. J Appl Stat 44(4):649–666
Antoniadis A (2007) Wavelet methods in statistics: Some recent developments and their applications.
Batvandi Z, Afshari M, Karamikabir H (2024) Bayesian estimation for mean vector of multivariate normal distribution on the linear and nonlinear exponential balanced loss based on wavelet decomposition. Int J Wavelets Multiresolution Inf Process. published online.
Batvandi Z, Afshari M, Karamikabir H (2023) Two new Bayesian-wavelet thresholds estimations of elliptical distribution parameters under non-linear exponential balanced loss. Communications in Statistics-Simulation and Computation. pp.1-21
Blyth CR (1951) On minimax statistical decision procedures and their admissibility. The Annals of Mathematical Statistics. pp.22-42
Chen X, Li S, Wang W (2015) New de-noising method for speech signal based on wavelet entropy and adaptive threshold. J Inf Comput Sci 12(3):1257–1265
Chipman HA, Kolaczyk ED, McCulloch RE (1997) Adaptive Bayesian wavelet shrinkage. J Am Stat Assoc 92(440):1413–1421
Donoho DL, Johnstone IM (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3):425–455
Donoho DL, Johnstone IM (1995) Adapting to unknown smoothness via wavelet shrinkage. J Am Stat Assoc 90(432):1200–1224
Fourdrinier D, Strawderman W (2015) Robust minimax Stein estimation under invariant data-based loss for spherically and elliptically symmetric distributions. Metrika 78:461–484
Gao HY (1998) Wavelet shrinkage denoising using the non-negative garrote. J Comput Graphical Stat 7(4):469–488
Ghosh M, Shieh G (1991) Empirical Bayes minimax estimators of matrix normal means. J Multivar Anal 38(2):306–318
Gupta AK, Nagar DK (2018) Matrix Variate Distrib. CRC, Chapman and Hall
Hamidieh K (2018) A data-driven statistical model for predicting the critical temperature of a superconductor. Comput Mater Sci 154:346–354
Karamikabir H, Afshari M (2020) Generalized Bayesian shrinkage and wavelet estimation of location parameter for spherical distribution under balance-type loss: Minimaxity and admissibility. J Multivar Anal 177:104583
Karamikabir H, Afshari M, Lak F (2021) Wavelet threshold based on Stein’s unbiased risk estimators of restricted location parameter in multivariate normal. J Appl Stat 48(10):1712–1729
Karamikabir H, Afshari M (2021) New wavelet SURE thresholds of elliptical distributions under the balance loss. Stat Sinica 31(4):1829–1852
Karamikabir H, Asghari AN, Salimi A (2023) Soft thresholding wavelet shrinkage estimation for mean matrix of matrix-variate normal distribution: low and high dimensional. Soft Comput 27(18):13527–13542
Karamikabir H, Sanati A, Hamedani GG (2024) Low and high dimensional wavelet thresholds for matrix-variate normal distribution. Communications in Statistics-Simulation and Computation. pp.1-20
Konno Y (1990) Families of minimax estimators of matrix of normal means with unknown covariance matrix. J Jpn Stat Soc Jpn Issue 20(2):191–201
Matsuda T, Komaki F (2015) Singular value shrinkage priors for Bayesian prediction. Biometrika 102(4):843–854
Lehmann EL, Casella G (1998) Theory of point estimation. Springer, Cham
Petersen KB, Pedersen MS (2008) The matrix cookbook. Tech Univ Denmark 7(15):510
Singh A, Kotiyal V, Sharma S, Nagar J, Lee CC (2020) A machine learning approach to predict the average localization error with applications to wireless sensor networks. IEEE Access 8:208253–208263
Tsukuma H (2008) Admissibility and minimaxity of Bayes estimators for a normal mean matrix. J Multivar Anal 99(10):2251–2264
Tsukuma H (2009) Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix. J Multivar Anal 100(10):2296–2304
Yeh IC (1998) Modeling of strength of high-performance concrete using artificial neural networks. Cement Concrete Res 28(12):1797–1808
Yuasa R, Kubokawa T (2023) Generalized Bayes estimators with closed forms for the normal mean and covariance matrices. J Stat Plan Inference 222:182–194
Zellner A (1994) Bayesian and non-Bayesian estimation using balanced loss functions. Statistical decision theory and related topics. Springer, New York, pp 377–390
Zinodiny S, Rezaei S, Nadarajah S (2017) Bayes minimax estimation of the mean matrix of matrix variate normal distribution under balanced loss function. Stat Prob Lett 125:110–120
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The authors would like to thank the Research Committee of Persian Gulf University.
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Batvandi, Z., Afshari, M. & Karamikabir, H. Bayesian shrinkage wavelet estimation of mean matrix of the matrix variate normal distribution with application in de-noising. Comp. Appl. Math. 44, 43 (2025). https://doi.org/10.1007/s40314-024-02997-9
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DOI: https://doi.org/10.1007/s40314-024-02997-9
Keywords
- Bayesian shrinkage wavelet
- Mean matrix
- Matrix variate normal distribution
- Stein’s unbiased risk estimate
- Threshold