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A well-conditioned estimator for large-dimensional covariance matrices
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“The optimal value of α is computed by finding a bias-variance trade-off with no assumption on the data distribution.”
From Signal Estimation and Uncertainties Extraction in Terahertz Time-Domain Spectroscopy (2025) · cites this paper as [55]
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“This is due to the fact that high-dimensional data analysis problems have become increasingly common in a wide spectrum of fields, such as in finance [2], bioinformatics, and classification [10].”
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“It is also similar to performing covariance shrinkage [32, 33] in finance, where the extreme values are pulled towards more central values.”
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“A further possible issue would be the stability of the error covariance matrices, whether because of insufficient sampling (e.g. Ledoit and Wolf, 2004) or because the error characteristics are not stationary: they could for example vary with model version or between land and sea, as examined here…”
BackgroundShrinkage estimation of large covariance matrices: Keep it simple, statistician?
Ledoit, O., & Wolf, M. (2021) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2021.104796 · Cited by 44
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“Additionally, the pooled (global) covariance matrix Σ on combined data of both classes was estimated using shrinkage-regularization as initially proposed by Ledoit&Wolf [35] and first applied in BCI by Vidaurre et al. [36], see also [37].”
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“The weights of both matrices were automatically determined via the Ledoit–Wolf criterion (Ledoit and Wolf 2004, Geirnaert et al 2021).”
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“As a remark, Proposition 1 can be viewed as a generalization of Theorem 2.1 shown in Ledoit and Wolf (2004), where they studied the optimal linear combination of the sample covariance matrix and the identity matrix to estimate the covariance matrix for the complete data.”
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“This metric has been previously examined as a goodness-of-fit estimator in the context of structural equation modeling [Bentler and Bonett, 1980; MacCallum, 1996] and economic forecasting [Ledoit and Wolf, 2004], but has not yet been applied to functional neuroimaging.”
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“An alternative might be to use the covariance matrix calculated from the particles, but the full covariance matrix may occupy large memory and is difficult to estimate from a small number of samples [6].”
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“Many algorithms exist to estimate the real co-variance Σ of a set of measurements, such as the Ledoit-Wolf estimator[64].”
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“…a large variety of approaches for covariance regularization to improve the quality of large-scale covariance estimation (e.g., Laloux et al., 1999; Ledoit & Wolf, 2003, 2004; Bajeux-Besnainou et al., 2012; Bodnar et al., 2018b; Plachel, 2019; Nguyen et al., 2022; Mynbayeva et al., 2022; Costola…”
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“The asymptotic analysis-based methods [2], [3] may not perform well when the sample support is very low.”
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“Please refer to [12] for more details.”
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“where Cov 1 is the inverse feature covariance estimated on all training data from both classes (per time point, across trials) using a shrinkage estimator (Ledoit and Wolf, 2004).”
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“To tackle this inversion problem, we apply a transformation method of the empirical covariance matrix named shrinkage [16] to estimate the covariance.”
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“compute_covariance function, which uses four different estimators (the Ledoit-Wolf estimator (Ledoit and Wolf, 2004) with cross-validation for optimizing alpha, diagonal regularization, sample covariance, and factor analysis with low-rank (Barber, 2012)) and chooses the optimal solution based on log-likelihood estimat…”
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“Ledoit-Wolf Ledoit and Wolf (2004) proposed using the following shrinking coefficient: ρLW = min ( ∑N i=1 ‖xi x i − S‖F/D N 2(tr(S2) − tr2(S)/D) , 1 )”
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“In this case, ridge‐type regularization is closely related to shrinkage of eigenvalues.(11) Ridge‐type estimators, however, allow for other targets and therefore allow for improving the estimates of the eigenvectors as well.”
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“[30] Olivier Ledoit and Michael Wolf, “A well-conditioned estimator for large-dimensional covariance matrices,” Journal of Multivariate Analysis 88, 365 – 411 (2004).”
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“In contrast, SRCSP does not need cross-validation or heuristics for parameter selection ( γ ) as it has a closed-form analytical solution obtained using Ledoit and Wolf ’ s method [59].”
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“Brown (1976), Bawa, Brown, and Klein (1979), Jorion (1986), MacKinlay and Pástor (2000), Jagannathan and Ma (2003), Ledoit and Wolf (2004), and Kan and Zhou (2007) are examples of early studies that provide various strategies to mitigate the estimation risk, but the performances of these strategies…”
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“-var + (1 ^)Wb 1 (Wickramasuriya et al., 2019), MinT-sam (cs-sam): W cs = Wb 1 (Wickramasuriya et al., 2019), where the symbol denotes the Hadamard product, ^ is an estimated shrinkage coefficient (Ledoit and Wolf, 2004, Sh¨afer and Strimmer, 2005), Wb 1 is the sample covariance matrix of the one-step ahead in-sample f…”
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“To avoid this problem, we implement a ridge-regularized covariance estimator ̂ε = (1−ε)̂+ε ·m̃Ip as in Ledoit and Wolf (2004), where m̃ = tr ̂/p and ε > 0 is a small constant.”
Estimation of covariance and precision matrix, network structure, and a view toward systems biology
Kuismin, M., & Sillanpää, M. J. (2017) · Wiley Interdisciplinary Reviews Computational Statistics · doi:10.1002/wics.1415 · Cited by 38
“The starting point of the shrinkage estimator is that the sample covariance matrix seems to overestimate the large eigenvalues and underestimate the small eigenvalues compared with the true covariance matrix eigenvalues.(18) The valuable property of the linear shrinkage estimator is that it will decrease the large eig…”
BackgroundTreating Sample Covariances for Use in Strongly Coupled Atmosphere‐Ocean Data Assimilation
Smith, P. J., Lawless, A. S., & Nichols, N. K. (2017) · Geophysical Research Letters · doi:10.1002/2017gl075534 · Cited by 35
“…eigenvalues closer together; the matrix is then reformed using the modified eigenvalues and original eigenvectors (see, e.g., Daniels & Kass, 2001; Ledoit & Wolf, 2004 for discussion of variations of this type of approach,andWestonetal.,2014andCampbelletal.,2017forrecentexamplesofitsapplication).”
MethodHuman influence on sub-regional surface air temperature change over India
Dileepkumar, R., AchutaRao, K. M., & Arulalan, T. (2018) · Scientific Reports · doi:10.1038/s41598-018-27185-8 · Cited by 35
Bank credit risk networks: Evidence from the Eurozone
Brownlees, C. T., Hans, C., & Nualart, E. (2020) · Journal of Monetary Economics · doi:10.1016/j.jmoneco.2020.03.014 · Cited by 35
Bauernfeind, G., Steyrl, D., Brunner, C., & Muller-Putz, G. R. (2014) · Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC) · doi:10.1109/embc.2014.6944008 · Cited by 35
Lecture notes on ridge regressionPreprint
van Wieringen (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1509.09169 · Cited by 164
Automatic diagonal loading for Tyler's robust covariance estimator
Zhang, T., & Wiesel, A. (2016) · IEEE Statistical Signal Processing Workshop (SSP) · doi:10.1109/ssp.2016.7551741 · Cited by 34
“Ledoit and Wolf proposed a simple approximation using the available data [4].”
Ensemble Kalman filter implementations based on shrinkage covariance matrix estimation
Niño-Ruiz, E. D., & Sandu, A. (2015) · Ocean Dynamics · doi:10.1007/s10236-015-0888-9 · Cited by 34
Selection bias in the reported performances of AD classification pipelines
Mendelson, A., Zuluaga, M. A., Lorenzi, M., Hutton, B. F., & Sebastien Ourselin, A. (2016) · NeuroImage Clinical · doi:10.1016/j.nicl.2016.12.018 · Cited by 34
Evaluation of a Compact Hybrid Brain-Computer Interface System
Shin, J., Müller, K., Schmitz, C., Kim, D.-W., & Hwang, H.-J. (2017) · BioMed Research International · doi:10.1155/2017/6820482 · Cited by 34
Behncke, J., Schirrmeister, R. T., Burgard, W., & Tonio Ball, A. (2018) · International Winter Conference on Brain-Computer Interface (BCI) · doi:10.1109/iww-bci.2018.8311531 · Cited by 34
Improving Power System State Estimation Based on Matrix-Level Cleaning
Yang, H., Qiu, R. C., Chu, L., Mi, T., Shi, X. C., & Liu, C. M. (2020) · IEEE Transactions on Power Systems · doi:10.1109/tpwrs.2020.2984926 · Cited by 34
“, [19]–[21]), aimed to clean measurement noise based on RMT, only involved eliminating noise of covariance matrices, but they did not intend to clean errors in original data of power systems.”
BackgroundBayesian Estimation of Graph Signals
Kroizer, A., Routtenberg, T., & Eldar, Y. C. (2022) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2022.3159393 · Cited by 39
“This is because the sample covariance matrix is not well-conditioned in the small sample size regime, and inverting it amplifies the estimation error [55].”
BackgroundFully automatic computation of diagonal loading levels for robust adaptive beamforming
Li, J., Du, L., & Stoica, P. (2008) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.2008.4518112 · Cited by 34
Fronto‐Parietal Subnetworks Flexibility Compensates For Cognitive Decline Due To Mental Fatigue
Taya, F., Dimitriadis, S. I., Dragomir, A., Lim, J., Sun, Y., Wong, K. F., Thakor, N. V., & Bezerianos, A. (2018) · Human Brain Mapping · doi:10.1002/hbm.24192 · Cited by 37
“…constructed with cutting-edge techniques including maximum overlap discrete wavelet transform (MODWT), Ledoit-Wolf shrinkage covariance estimator (Ledoit & Wolf, 2004) and a data-driven topological filtering scheme based on the orthogonal minimum spanning trees (OMSTs) (Dimitriadis, Antonakakis,…”
MethodIntegrating neurophysiologic relevance feedback in intent modeling for information retrieval
Jacucci, G., Barral, O., Daee, P., Wenzel, M. A., Serim, B., Ruotsalo, T., Pluchino, P., Freeman, J., Gamberini, L., Samuel Kaski, A., et al. (2019) · Journal of the Association for Information Science and Technology · doi:10.1002/asi.24161 · Cited by 33
Metadata
- Authors
- Ledoit, O., & Wolf, M.
- Affiliations
- University of California, Los Angeles (US)
- Universitat Pompeu Fabra (ES)
- Corresponding author
- Wolf, Michael
- Venue
- Journal of Multivariate Analysis
- Publisher
- Elsevier BV
- Published
- 2004-02
- Volume
- 88
- Issue
- 2
- Pages
- 365-411
- ISSN
- 0047-259X (ISSN-L)
- OpenAlex
- W2062125287
- Semantic Scholar
- b40890f3d917379ac4bd2e0933a134d45b4dc251
- Cited by
- 3,046
- References
- 24
- Citations by year
- 2026: 355 · 2025: 228 · 2024: 191 · 2023: 178 · 2022: 178 · 2021: 193 · 2020: 198 · 2019: 208 · 2018: 199 · 2017: 146 · 2016: 140 · 2015: 152 · 2014: 137 · 2013: 108 · 2012: 99
- FWCI
- 12.70
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Journal of Multivariate Analysis (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- e-Archivo (Carlos III University of Madrid) (submittedVersion, cc-by-nc-nd)
- Funders
- Yale University
- McDonnell Center for Systems Neuroscience
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Random Matrices and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Keywords
- covariance matrix estimation, sample covariance matrix, convex combination, asymptotic optimality, squared error loss
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