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A well-conditioned estimator for large-dimensional covariance matrices
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“Generalizing shrinkage estimation strategy for high-dimensional covariance matrix [Ledoit and Wolf, 2004], B ¨ ohm and von Sachs [2009] proposed shrinking averaged periodogram to estimate spectral density in high-dimension.”
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“63,129 To mitigate this issue, we built on top of prior work on covariance shrinkage operators, which are known to outperform ad hoc approaches, especially in the EEG context.”
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“In recent years, development of the shrinkage methods (Ledoit & Wolf 2004; Pope & Szapudi 2008; Ledoit & Wolf 2012; Joachimi 2016; Simpson et al. 2016) has been proved promising for alleviating the requirement on the number of mocks.”
MethodMeinel, A., Castaño-Candamil, S., Blankertz, B., Lotte, F., & Tangermann, M. (2018) · Neuroinformatics · doi:10.1007/s12021-018-9396-7 · Cited by 17
Evaluating prior predictions of production and seismic data
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“We compute an approximation of the Mahalanobis distance [2] using a regularized estimate of the covariance [26, 39] and in particular, we will use Target B of [39] to shrink the sample covariance towards a diagonal matrix with a constant value equal to the average variance, ν.”
MethodNonparametric Spectral Analysis of Multivariate Time Series
von Sachs, R. (2019) · Annual Review of Statistics and Its Application · doi:10.1146/annurev-statistics-031219-041138 · Cited by 17
“Shrinkage-based methods, which essentially amount to adding a ridge (e.g., a multiple of the identity matrix) to the covariance matrix, were first developed in the seminal work of Ledoit & Wolf (2004).”
Kang, X., Deng, X., Tsui, K., & Pourahmadi, M. (2019) · International Statistical Review · doi:10.1111/insr.12357 · Cited by 17
“The former stands for DCC-GARCH based on the linear shrinkage of Ledoit and Wolf (2004a) and Ledoit and Wolf (2004b).”
MethodShin, J., & Im, C. (2020) · Frontiers in Neuroscience · doi:10.3389/fnins.2020.00168 · Cited by 18
“The optimal γ between 0 and 1 was determined individually based on the Ledoit and Wolf (2004), Schäfer and Strimmer (2005), Blankertz et al. (2011), Lemm et al. (2011).”
MethodFormulating Divergence Framework for Multiclass Motor Imagery EEG Brain Computer Interface
Kumar, S., Reddy, T. K., Arora, V., & Behera, L. (2020) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp40776.2020.9053168 · Cited by 17
“All the implementations in this paper use shrinkage based estimator for estimation of sample co-variance matrices [15].”
MethodNowcasting GDP and its components in a data-rich environment: The merits of the indirect approach
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Regularized Covariance Matrix Estimation via Empirical Bayes
Coluccia, A. (2015) · IEEE Signal Processing Letters · doi:10.1109/lsp.2015.2462724 · Cited by 17
Shrinkage Estimation for Multivariate Hidden Markov Models
Fiecas, M. J. A., Franke, J., von Sachs, R., & Kamgaing, J. T. (2016) · Journal of the American Statistical Association · doi:10.1080/01621459.2016.1148608 · Cited by 17
“In the language of penalized maximum likelihood, this means that the regularization parameter λ is chosen adaptively to the underlying distribution as Ledoit and Wolf (2004) derived an explicit formula for the optimal shrink-age weight W 0 , and they discussed how to estimate the unknown quantities…”
BackgroundMethodInfluentialEstimating principal components of large covariance matrices using the Nyström method
Arcolano, N., & Wolfe, P. J. (2011) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.2011.5947175 · Cited by 17
Covariance-based clustering in multivariate and functional data analysis
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Linear programing models for portfolio optimization using a benchmark
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“Moreover, as in Becker, Gürtler, and Hibbeln (2015), if the variance of estimators is large, for example, for short observation periods or large samples, it is recommendable to additionally implement constraints or to use the estimator of Ledoit and Wolf (2004a, 2004b).”
BackgroundMethodPeriodogram Connectivity of EEG Signals for the Detection of Dyslexia
Martínez-Murcia, F. J., Ortíz, A., Morales-Ortega, R. C., López, P. J., Luque, J. L., Castillo-Barnés, D., Segovia, F., Illán, I. A., Ortega, J., Ramı́rez, J., et al. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-19591-5_36 · Cited by 17
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Covariance Estimation in Dynamic Portfolio Optimization: A Realized Single Factor ModelPreprint
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Transfer Learning Gaussian Anomaly Detection by Fine-tuning Representations
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“The shrinkage approach (see [41]) provides another potential method of estimating the covariance matrix.”
MethodXiong, W. H., & Wei, Q. (2022) · PLoS ONE · doi:10.1371/journal.pone.0263641 · Cited by 16
“To address the problem, researchers have presented four main types of approaches that are based on regularization [11, 12], user-to-user transfer [13–16], semi-supervised learning [17–19] and a prior physiological information [20, 21].”
BackgroundFast and accurate inference of gene regulatory networks through robust precision matrix estimation
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“We did not infer a using the approach from Ledoit–Wolf (Ledoit and Wolf, 2004), since our goal is to minimize quadratic risk on the precision matrix instead of the covariance matrix.”
InfluentialDetection of Multiple Structural Breaks in Large Covariance Matrices
Yu-ning, L., Li, D., & Fryźlewicz, P. (2022) · Journal of Business and Economic Statistics · doi:10.1080/07350015.2022.2076686 · Cited by 16
“…introduced by Fama and French (1993, 2015); the Momentum factor (MOM) proposed by Carhart (1997) in a four-factor model; and the short term reversal factor (STRev) and the long-term reversal factor (LTRev) which are less famous but equally important (see De Bondt and Thaler 1987; Lehmann 1990).”
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“…separation was performed by a regularized Linear Discriminant Analysis (Friedman, 1989; Lemm et al., 2011) with shrinkage of the covariance matrix (Ledoit and Wolf, 2004; Schäfer and Strimmer, 2005; Vidaurre et al., 2009; Blankertz et al., 2011), which proved to be successful for this type of…”
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“If a practitioner is still concerned about the singularity of vt, a well-conditioned estimator for covariance matrices proposed by [35] can be applied, where the estimator is the weighted sum of vt and Ip.”
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“The covariance matrix from each epoch is then estimated using the well-conditioned Ledoit-Wolf estimator (Ledoit and Wolf, 2004).”
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“MVP with the Elastic Net regularization EN Yen and Yen [34] (2) Benchmarks portfolio MVP with shortsale-constrained SC Jagannathan and Ma [17] MVP with shortsale-unconstrained SU Jagannathan and Ma [17] Equally-weighted (1/N) portfolio EW DeMiguel et al. [9] (3) Shrinkage of covariance Mixture of sample covariance and…”
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“This approach is similar to the approach that was proposed by Ledoit and Wolf (2004b) and Sch € afer and Strimmer (2005).”
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“Several of these estimators shrink sample quantities to a desired target; see Haff (1991), Daniels & Kass (2001), Ledoit & Wolf (2004), Khare & Rajaratnam (2011) or Won et al. (2013), for example.”
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“The optimal shrinkage parameter was calculated analytically using the closed form equation derived in [44], which is computationally less expensive than choosing the optimal parameter by cross-validation.”
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“ased on residuals from a fitted stochastic volatility model. Noisiness of sample correlation matrices is well documented. Several strategies have been proposed to remedy for it, most notably shrinkage [30, 46]. Alternative procedures developed in the context of graphical models consist of decomposing a noisy inverse co…”
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“Ledoit and Wolf (2004b) define QuEST functions (q1(λ), . . . , qN(λ), such that λ̃ minimizes the Euclidean distance between the QuEST functions and the sample eigenvalues, i.e., given by: λ̃ = arg min λ∈[0,∞)N N ∑ i=1 [qi(λ)− λ̂i]2.”
InfluentialAsymptotic inference for high-dimensional data
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Regularized parameter estimation of high dimensional t distribution
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NETS: Network Estimation for Time Series
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State-of-the-Art Versus Deep Learning: A Comparative Study of Motor Imagery Decoding Techniques
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Bongiorno, C., & Challet, D. (2022) · Finance research letters · doi:10.1016/j.frl.2022.103383 · Cited by 15
“The best known RIE is linear shrinkage [3].”
BackgroundRácz, F. S., Kumar, S., Kaposzta, Z., Alawieh, H., Liu, D. H., Liu, R., Czoch, A., Mukli, P., & Millán, J. D. R. (2024) · Frontiers in Neuroscience · doi:10.3389/fnins.2024.1271831 · Cited by 14
Zhao, C., Yu, H., & Jiang, M. (2024) · IEEE Transactions on Geoscience and Remote Sensing · doi:10.1109/tgrs.2024.3404091 · Cited by 15
Light scalar meson and decay constant in SU(3) gauge theory with eight dynamical flavors
Brower, R. C., Owen, E. K., Claudio Rebbi, B., Culver, C., Schaich, D., Cushman, K. K., Fleming, G., Gasbarro, A., Anna Hasenfratz, A., Neil, E. T., et al. (2024) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.110.054501 · Cited by 15
Enhancing Markowitz's portfolio selection paradigm with machine learning
de Prado, M. L., Simonian, J., Fabozzi, F. A., & Fabozzi, F. J. (2024) · Annals of Operations Research · doi:10.1007/s10479-024-06257-1 · Cited by 18
“Ledoit and Wolf (2004a, 2004b) introduced a popular method to shrink a numerically ill-conditioned covariance matrix.”
An improved quadratic inference function for parameter estimation in the analysis of correlated data
Westgate, P. M., & Braun, T. (2012) · Statistics in Medicine · doi:10.1002/sim.5715 · Cited by 15
Sparse precision matrices for minimum variance portfolios
Torri, G., Giacometti, R., & Paterlini, S. (2019) · Computational Management Science · doi:10.1007/s10287-019-00344-6 · Cited by 15
“…to the traditional sample covariance matrix, as well as to the naive equally weighted portfolio strategy (EW) and to two state-of-art covariance estimation methods: randommatrix theory filtering (RMT) (Bouchaud and Potters 2009) and Ledoit–Wolf shrinkage estimation (LW) (Ledoit and Wolf 2004a).”
InfluentialPortfolio Construction by Mitigating Error Amplification: The Bounded-Noise Portfolio
Zhao, L., Chakrabarti, D., & Muthuraman, K. (2017) · Operations Research · doi:10.1287/opre.2019.1858 · Cited by 15
“One can also shrink the eigenvalues of the sample covariance matrix linearly (Ledoit and Wolf 2004) or nonlinearly (Ledoit and Wolf 2012, 2017).”
Modified Sparse Linear-Discriminant Analysis via Nonconvex Penalties
Cai, J., & Huang, X. (2018) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2017.2785324 · Cited by 15
Low-Rank Structured Covariance Matrix Estimation
Shikhaliev, A. P., Potter, L. C., & Chi, Y. (2019) · IEEE Signal Processing Letters · doi:10.1109/lsp.2019.2906405 · Cited by 15
Multivariate Heteroscedasticity Models for Functional Brain Connectivity
Seiler, C., & Holmes, S. (2017) · Frontiers in Neuroscience · doi:10.3389/fnins.2017.00696 · Cited by 17
“Therefore more elaborate estimators need to be employed, such as the Graphical Lasso (Friedman et al., 2008) estimator for inverse-covariance matrices or the Ledoit-Wolf shrinkage estimator (Ledoit and Wolf, 2004) for correlation matrices.”
BackgroundMethodTensor canonical correlation analysis
Min, E. J., Chi, E. C., & Zhou, H. (2019) · Stat · doi:10.1002/sta4.253 · Cited by 15
“As a remedy, several regularized estimation methods for obtaining nonsingular sample covariance matrices (Vinod, 1976; Ledoit & Wolf, 2004; González, Déjean, Martin, & Baccini, 2008; Ledoit & Wolf, 2012; Cai & Yuan, 2012; Bickel & Levina, 2008; 2008; Kubokawa et al., 2013; Srivastava & Reid, 2012)…”
MethodSingle-Option P300-BCI Performance Is Affected by Visual Stimulation Conditions
Peguero, J. D. C., Mendoza-Montoya, O., & Antelis, J. M. (2020) · Sensors · doi:10.3390/s20247198 · Cited by 16
“Here, we applied the option in which the covariance matrix is regularized in a fully automatic way, so no hyper-parameter tuning is needed [54,55].”
MethodSpectrally-Corrected Estimation for High-Dimensional Markowitz Mean-Variance Optimization
Li, H., Bai, Z., Wong, W., & McAleer, M. J. (2021) · Econometrics and Statistics · doi:10.1016/j.ecosta.2021.10.005 · Cited by 15
Neural Reinstatement Tracks Spread of Attention between Object Features in Working Memory
Printzlau, F. A. B., Myers, N. E., Manohar, S., & Stokes, M. G. (2022) · Journal of Cognitive Neuroscience · doi:10.1162/jocn_a_01879 · Cited by 15
“We computed the pairwise Mahalanobis distances between the test trials and each of the orientation bins using the covariance matrix estimated from trials in the seven training folds, using a shrinkage estimator (Ledoit & Wolf, 2004).”
MethodCHARACTERISTICS OF OMEGA-OPTIMIZED PORTFOLIOS AT DIFFERENT LEVELS OF THRESHOLD RETURNS
Vilkancas, R. (2014) · Business Management and Education · doi:10.3846/bme.2014.235 · Cited by 15
Outlier detection and robust covariance estimation using mathematical programming
Nguyen, T.-D., & Welsch, R. E. (2010) · Advances in Data Analysis and Classification · doi:10.1007/s11634-010-0070-7 · Cited by 15
Partial Coherence Estimation via Spectral Matrix Shrinkage under Quadratic Loss
Schneider-Luftman, D., & Walden, A. T. (2016) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2016.2582464 · Cited by 15
Exact hypothesis testing for shrinkage-based Gaussian graphical models
Bernal, V., Bischoff, R., Guryev, V., Grzegorczyk, M. A., & Horvatovich, P. L. (2019) · Bioinformatics · doi:10.1093/bioinformatics/btz357 · Cited by 17
“On the other hand, GeneNet (Schäfer and Strimmer, 2005a) estimates a modified covariance matrix by using an (invertible) estimator based on shrinkage (Ledoit and Wolf, 2004).”
BackgroundMethodDirect comparison of supervised and semi-supervised retraining approaches for co-adaptive BCIs
Schwarz, A., Brandstetter, J., Pereira, J., & Muller-Putz, G. R. (2019) · Medical & Biological Engineering & Computing · doi:10.1007/s11517-019-02047-1 · Cited by 15
“These features were classified incorporating a shrinkage regularized based linear discriminant analysis (sLDA) classifier [11, 35, 36].”
MethodCastaño-Candamil, S., Piroth, T., Reinacher, P. C., Sajonz, B., Coenen, V. A., & Tangermann, M. (2019) · IEEE Transactions on Neural Systems and Rehabilitation Engineering · doi:10.1109/tnsre.2019.2941453 · Cited by 15
“[32] O. Ledoit and M. Wolf, “A well-conditioned estimator for largedimensional covariance matrices,” J. Multivariate Anal., vol. 88, no. 2, pp. 365–411, 2004.”
MethodInfluentialShah, R. D., Frot, B., Thanei, G.-A., & Meinshausen, N. (2020) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/rssb.12359 · Cited by 16
“A second approach to precision matrix estimation exploits the fact that coefficients from regressions of each variable on all others, known as nodewise regressions, match the entries of the precision matrix up to scale factors (Lauritzen, 1996; Meinshausen and Bühlmann, 2006).”
BackgroundAbdallah, M., Farrugia, N., Chirokoff, V., & Chanraud, S. (2020) · Network Neuroscience · doi:10.1162/netn_a_00149 · Cited by 16
“Specifically, we estimated full and partial correlation matrices using the Ledoit-Wolf estimator implemented in the nilearn and scikit-learn python packages (Ledoit and Wolf, 2004; Abraham et al., 2014).”
MethodTail Risks in Large Portfolio Selection: Penalized Quantile and Expectile Minimum Deviation Models
Giacometti, R., Torri, G., & Paterlini, S. (2020) · Quantitative Finance · doi:10.1080/14697688.2020.1820072 · Cited by 15
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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