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An extended variable inclusion and shrinkage algorithm for correlated variables

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  • Mkhadri, Abdallah
  • Ouhourane, Mohamed
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    Abstract

    The problem of variable selection for linear regression in a high dimension model is considered. A new method, called Extended-VISA (Ext-VISA), is proposed to simultaneously select variables and encourage a grouping effect where strongly correlated predictors tend to be in or out of the model together. Moreover, Ext-VISA is capable of selecting a sparse model while avoiding the overshrinkage of a Lasso-type estimator. It combines the idea of the VISA algorithm which avoids the overshrinkage problem of regression coefficients and those of the Lasso-type estimators, based on ℓ1+ℓ2 penalty, that overcome the limitation of the grouping effect of Lasso in high dimension. It is based on a modified VISA algorithm, so it is also computationally efficient. Three interesting cases of Ext-VISA are examined. The first case is Smooth-VISA (SVISA), where the variations among successive regression coefficients are low. The second case is VISA-Net (VNET), where the correlations between predictors are taken into account. The third case is Laplacian-VISA (LVISA), where the predictors are measured on an undirected graph. A theoretical property on sparsity inequality of Ext-VISA is established. A detailed simulation study in small and high dimensional settings is performed, which illustrates the advantages of the new approach in relation to several other possible methods. Finally, we apply VNET, SVISA and LVISA to a GC-retention data set.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0167947312002976
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    Bibliographic Info

    Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

    Volume (Year): 57 (2013)
    Issue (Month): 1 ()
    Pages: 631-644

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    Handle: RePEc:eee:csdana:v:57:y:2013:i:1:p:631-644

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    Web page: http://www.elsevier.com/locate/csda

    Related research

    Keywords: Variable selection; VISA algorithm; Elastic-Net; LARS; Linear regression; Smooth Lasso;

    References

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    1. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108.
    2. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320.
    3. Daye, Z. John & Jeng, X. Jessie, 2009. "Shrinkage and model selection with correlated variables via weighted fusion," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1284-1298, February.
    4. Meinshausen, Nicolai, 2007. "Relaxed Lasso," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 374-393, September.
    5. Jerome H. Friedman & Trevor Hastie & Rob Tibshirani, . "Regularization Paths for Generalized Linear Models via Coordinate Descent," Journal of Statistical Software, American Statistical Association, vol. 33(i01).
    6. Gareth M. James & Peter Radchenko & Jinchi Lv, 2009. "DASSO: connections between the Dantzig selector and lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 127-142.
    7. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    8. Gareth M. James & Peter Radchenko, 2009. "A generalized Dantzig selector with shrinkage tuning," Biometrika, Biometrika Trust, vol. 96(2), pages 323-337.
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    Cited by:
    1. Korzeń, M. & Jaroszewicz, S. & Klęsk, P., 2013. "Logistic regression with weight grouping priors," Computational Statistics & Data Analysis, Elsevier, vol. 64(C), pages 281-298.

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