R: Compute the generalized lasso solution path for arbitrary...
genlasso
R Documentation
Compute the generalized lasso solution path for arbitrary penalty matrix
Description
This function computes the solution path of the generalized lasso
problem for an arbitrary penalty matrix. Speciality functions exist
for the trend filtering and fused lasso problems; see
trendfilter and fusedlasso.
an optional matrix of predictor variables, with observations along
the rows, and variables along the columns. If missing, X is
assumed to be the identity matrix. If the passed X does not
have full column rank, then a warning is given, and a small ridge
penalty is added to the generalized lasso criterion before the path
is computed.
D
a penalty matrix. Its number of columns must be equal to the number
of columns of X, or if no X is given, the length of
y. This can be a sparse matrix from Matrix package,
but this will be ignored (converted to a dense matrix) if D
is row rank deficient or if X is specified. See "Details"
below.
approx
a logical variable indicating if the approximate solution path
should be used (with no dual coordinates leaving the boundary).
Default is FALSE.
maxsteps
an integer specifying the maximum number of steps for the algorithm
to take before termination. Default is 2000.
minlam
a numeric variable indicating the value of lambda at which the path
should terminate. Default is 0.
rtol
a numeric variable giving the tolerance for determining the rank of
a matrix: if a diagonal value in the R factor of a QR decomposition
is less than R, in absolute value, then it is considered zero. Hence
making rtol larger means being less stringent with determination of
matrix rank. In general, do not change this unless you know what you
are getting into! Default is 1e-7.
btol
a numeric variable giving the tolerance for accepting "late" hitting
and leaving times: future hitting times and leaving times should always
be less than the current knot in the path, but sometimes for numerical
reasons they are larger; any computed hitting or leaving time larger
than the current knot + btol is thrown away. Hence making btol larger
means being less stringent withthe determination of hitting and leaving
times. Again, in general, do not change this unless you know what you
are getting into! Default is 1e-7.
eps
a numeric variable indicating the multiplier for the ridge penalty,
in the case that X is column rank deficient. Default is
1e-4.
verbose
a logical variable indicating if progress should be reported after
each knot in the path.
svd
a logical variable indicating if the genlasso function should use
singular value decomposition to solve least squares problems at each
path step, which is slower, but should be more stable.
Details
The generalized lasso estimate minimizes the criterion
1/2 |y - X β|_2^2 + λ |D β|_1.
The solution hat{β} is computed as a function of
the regularization parameter λ. The advantage of the
genlasso function lies in its flexibility, i.e., the user can
specify any penalty matrix D of their choosing. However, for a
trend filtering problem or a fused lasso problem, it is strongly
recommended to use one of the speciality functions,
trendfilter or fusedlasso. When compared
to these functions, genlasso is not as numerically stable and
much less efficient.
Note that, when D is passed as a sparse matrix, the linear
systems that arise at each step of the path algorithm are solved
separately via a sparse solver. The usual strategy (when D is
simply a matrix) is to maintain a matrix factorization of D,
and solve these systems by (or downdating) this factorization, as
these linear systems are highly related. Therefore,
when D is sufficiently sparse and structured, it can be
advantageous to pass it as a sparse matrix; but if D is truly
dense, passing it as a sparse matrix will be highly inefficient.
Value
Returns an object of class "genlasso", a list with at least following
components:
lambda
values of lambda at which the solution path changes slope,
i.e., kinks or knots.
beta
a matrix of primal coefficients, each column corresponding to a knot
in the solution path.
fit
a matrix of fitted values, each column corresponding to a knot in
the solution path.
u
a matrix of dual coefficients, each column corresponding to a knot
in the solution path.
hit
a vector of logical values indicating if a new variable in the dual
solution hit the box contraint boundary. A value of FALSE
indicates a variable leaving the boundary.
df
a vector giving an unbiased estimate of the degrees of freedom of
the fit at each knot in the solution path.
y
the observed response vector. Useful for plotting and other
methods.
completepath
a logical variable indicating whether the complete path was
computed (terminating the path early with the maxsteps or
minlam options results in a value of FALSE).
bls
the least squares solution, i.e., the solution at lambda = 0.
call
the matched call.
Author(s)
Taylor B. Arnold and Ryan J. Tibshirani
References
Tibshirani, R. J. and Taylor, J. (2011), "The solution path of the
generalized lasso", Annals of Statistics 39 (3) 1335–1371.
Arnold, T. B. and Tibshirani, R. J. (2014), "Efficient implementations
of the generalized lasso dual path algorithm", arXiv: 1405.3222.
# Using the generalized lasso to run a standard lasso regression
# (for example purposes only! for pure lasso problems, use LARS
# instead)
set.seed(1)
n = 100
p = 10
X = matrix(rnorm(n*p),nrow=n)
y = 3*X[,1] + rnorm(n)
D = diag(1,p)
out = genlasso(y,X,D)
coef(out, lambda=sqrt(n*log(p)))