# Performs the lasso on the cross product matrices X'X and X'y

### Description

Perform L1-regularized regression of y onto X using only the cross-product matrices X'X and X'y. In the case of covariance-regularized regression, this is useful if you would like to try out something other than L1 or L2 regularization of the inverse covariance matrix.

Suppose you use your own method to regularize X'X. Then let Sigma denote your estimate of the population covariance matrix. Now say you want to minimize beta' Sigma beta - 2 beta' X'y + lambda ||beta||_1 in order to get the regression estimate beta, which maximizes the second scout criterion when an L_1 penalty is used. You can do this by calling crossProdLasso(Sigma, X'y,rho).

If you run crossProdLasso(X'X,X'y,rho) then it should give the same result as lars(X,y)

Notice that the xtx that you pass into this function must be POSITIVE SEMI DEFINITE (or positive definite) or the problem is not convex and the algorithm will not converge.

### Usage

1 | ```
crossProdLasso(xtx,xty,rho,thr=1e-4,maxit=100,beta.init=NULL)
``` |

### Arguments

`xtx` |
A pxp matrix, which should be an estimate of a covariance matrix. This matrix must be POSITIVE SEMI DEFINITE (or positive definite) or the problem is not convex and the algorithm will not converge. |

`xty` |
A px1 vector, which is generally obtained via X'y. |

`rho` |
Must be non-negative; the regularization parameter you are using. |

`thr` |
Convergence threshold. |

`maxit` |
How many iterations to perform? |

`beta.init` |
If you're running this over a range of rho values, then set beta.init equal to the solution you got for a previous rho value. It will speed things up. |

### Details

If your xtx is simply X'X for some X, and your xty is simple X'y with some y, then the results will be the same as running lars on data (X,y) for a single shrinkage parameter value.

Note that when you use the scout function with p2=1, the crossProdLasso function is called internally to give the regression coefficients, after the regularized inverse covariance matrix is estimated. It is provided here in case it is useful to the user in other settings.

### Value

`beta` |
A px1 vector with the regression coefficients. |

### Note

The FORTRAN code that this function links to was kindly written and provided by Jerry Friedman.

### Author(s)

FORTRAN code by Jerry Friedman. R interface by Daniela M. Witten and Robert Tibshirani

### References

Witten, DM and Tibshirani, R (2008) Covariance-regularized regression and classification for high-dimensional problems. Journal of the Royal Statistical Society, Series B 71(3): 615-636. <http://www-stat.stanford.edu/~dwitten>

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | ```
set.seed(1)
#data(diabetes)
#attach(diabetes)
x2 <- matrix(rnorm(10*20),ncol=20)
y <- rnorm(10)
# First, let's do scout(2,1) the usual way).
scout.out <- scout(x2,y,p1=2,p2=1)
print(scout.out)
# Now, suppose I want to do develop a covariance-regularized regression
# method as in Section 3.2 of Witten and Tibshirani (2008). It will work
# like this:
# 1. Develop some positive definite estimate of Sigma
# 2. Find \beta by minimize \beta^T \Sigma \beta - 2 \beta^T X^T y +
# \lamda ||\beta||_1
# 3. Re-scale \beta.
# Step 1:
regcovx <- cov(x2)*(abs(cov(x2))>.005) + diag(ncol(x2))*.01
# Step 2:
betahat <- crossProdLasso(regcovx, cov(x2,y), rho=.02)$beta
# Step 3:
betahat.sc <- betahat*lsfit(x2%*%betahat, y, intercept=FALSE)$coef
print(betahat.sc)
# Try a different value of rho:
betahat2 <- crossProdLasso(regcovx,cov(x2,y),rho=.04,beta.init=betahat)$beta
plot(betahat,betahat2, xlab="rho=.02",ylab="rho=.04")
#detach(diabetes)
``` |

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