A new approach to variable selection in least squares problems

@article{Osborne2000ANA,
  title={A new approach to variable selection in least squares problems},
  author={Michael R. Osborne and Brett Presnell and Berwin A. Turlach},
  journal={Ima Journal of Numerical Analysis},
  year={2000},
  volume={20},
  pages={389-403}
}
The title Lasso has been suggested by Tibshirani (1996) as a colourful name for a technique of variable selection which requires the minimization of a sum of squares subject to an l 1 bound κ on the solution. This forces zero components in the minimizing solution for small values of κ. Thus this bound can function as a selection parameter. This paper makes two contributions to computational problems associated with implementing the Lasso: (1) a compact descent method for solving the… 

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References

SHOWING 1-7 OF 7 REFERENCES

Regression Shrinkage and Selection via the Lasso

A new method for estimation in linear models called the lasso, which minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant, is proposed.

Resolving degeneracy in quadratic programming

A technique for the resolution of degeneracy in an Active Set Method for Quadratic Programming is described, which generalises Fletcher's method [2] which applies to the LP case and gives stronger guarantees than are available with other popular methods.

The Levenberg-Marquardt algo-rithm: Implementation and theory

A conduit arrangement for a tilt cylinder of a bulldozer comprises a trunnion having a hole to be connected to the hole in a truck frame and a conduit adapted to be connected through the frame to the

On Linear Restricted and Interval Least-Squares Problems

On etudie deux classes d'algorithmes pour la resolution du probleme des moindres carres lineaire sous contraintes

An effective method for computing regression quantiles

Regression quantiles were introduced in Koenker & Bassett [7] as quantities of interest in developing robust estimation procedures. They can be computed by linear programming combined with post

Applied Regression Analysis

  • R. Gunst
  • Computer Science
    Technometrics
  • 1999