IDEAS home Printed from
MyIDEAS: Log in (now much improved!) to save this article

Best permutation analysis

Listed author(s):
  • Rajaratnam, Bala
  • Salzman, Julia
Registered author(s):

    High dimensional covariance estimation is an important topic in contemporary multivariate statistics and has recently received much attention in the mathematical statistics literature. The work of Bickel and Levina (2008) [2] introduces a general approach to such estimation problems in a large class of models: banding of the sample covariance matrix. Bickel and Levina show that banded estimators are consistent in the operator norm as the dimension of the covariance matrix, p, and the sample size, n, both go to infinity. Critically, these estimators rely on knowing the order of the covariates apriori before banding can be applied. A rigorous framework for order recovery is however not available in the literature. In this paper, we propose a novel framework and methodology that can be used to recover covariate order in general classes of banded models. Such models can also be framed as autoregressive processes, which in turn fall within the class of graphical models. We show that recovering covariate order is intimately related to minimizing functionals on the symmetric group. Indeed, an important contribution of the paper is a result showing that the natural time order in such an autoregressive process has the property that over all orderings of covariates, it minimizes the sum of the diagonals of the Cholesky decomposition, of both the covariance and the inverse covariance matrix. This result lays the foundation for the ensuing statistical methodology developed in this paper: an efficient algorithm called the Best Permutation Algorithm (BPA). The BPA can recover the natural order of variables in autoregressive models at the rate of Op((logp)/n), which is the same rate that the covariance matrix can be estimated if the natural time order were known. Hence the BPA yields the oracle rate. Moreover, the computational complexity of the BPA is proved to be polynomial in the number of variables, p, and hence allows for an efficient search over the full permutation group on p letters, a group whose size is super-exponential in p. The methodology is also successfully illustrated on numerical examples.

    If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

    File URL:
    Download Restriction: Full text for ScienceDirect subscribers only

    As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 121 (2013)
    Issue (Month): C ()
    Pages: 193-223

    in new window

    Handle: RePEc:eee:jmvana:v:121:y:2013:i:c:p:193-223
    DOI: 10.1016/j.jmva.2013.03.001
    Contact details of provider: Web page:

    Order Information: Postal:

    References listed on IDEAS
    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

    in new window

    1. Joong-Ho Won & Johan Lim & Seung-Jean Kim & Bala Rajaratnam, 2013. "Condition-number-regularized covariance estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(3), pages 427-450, June.
    2. Peng, Jie & Wang, Pei & Zhou, Nengfeng & Zhu, Ji, 2009. "Partial Correlation Estimation by Joint Sparse Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 735-746.
    3. Jianhua Z. Huang & Naiping Liu & Mohsen Pourahmadi & Linxu Liu, 2006. "Covariance matrix selection and estimation via penalised normal likelihood," Biometrika, Biometrika Trust, vol. 93(1), pages 85-98, March.
    4. Rothman, Adam J. & Levina, Elizaveta & Zhu, Ji, 2009. "Generalized Thresholding of Large Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 177-186.
    Full references (including those not matched with items on IDEAS)

    This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

    When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:121:y:2013:i:c:p:193-223. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu)

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If references are entirely missing, you can add them using this form.

    If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.