| CNplot {rags2ridges} | R Documentation |
Visualize the spectral condition number against the regularization parameter
Description
Function that visualizes the spectral condition number of the regularized precision matrix against the domain of the regularization parameter. The function can be used to heuristically determine an acceptable (minimal) value for the penalty parameter.
Usage
CNplot(
S,
lambdaMin,
lambdaMax,
step,
type = "Alt",
target = default.target(S, type = "DUPV"),
norm = "2",
Iaids = FALSE,
vertical = FALSE,
value = 1e-100,
main = "",
nOutput = FALSE,
verbose = TRUE,
suppressChecks = FALSE
)
Arguments
S |
Sample covariance |
lambdaMin |
A |
lambdaMax |
A |
step |
An |
type |
A |
target |
A target |
norm |
A |
Iaids |
A |
vertical |
A |
value |
A |
main |
A |
nOutput |
A |
verbose |
A |
suppressChecks |
A |
Details
Under certain target choices the proposed ridge estimators (see
ridgeP) are rotation equivariant, i.e., the eigenvectors of
\mathbf{S} are left intact. Such rotation equivariant situations help
to understand the effect of the ridge penalty on the precision estimate: The
effect can be understood in terms of shrinkage of the eigenvalues of the
unpenalized precision estimate \mathbf{S}^{-1}. Maximum shrinkage
implies that all eigenvalues are forced to be equal (in the rotation
equivariant situation). The spectral condition number w.r.t. inversion
(ratio of maximum to minimum eigenvalue) of the regularized precision matrix
may function as a heuristic in determining the (minimal) value of the
penalty parameter. A matrix with a high condition number is near-singular
(the relative distance to the set of singular matrices equals the reciprocal
of the condition number; Demmel, 1987) and its inversion is numerically
unstable. Such a matrix is said to be ill-conditioned. Numerically,
ill-conditioning will mean that small changes in the penalty parameter lead
to dramatic changes in the condition number. From a numerical point of view
one can thus track the domain of the penalty parameter for which the
regularized precision matrix is ill-conditioned. When plotting the condition
number against the (domain of the) penalty parameter, there is a point of
relative stabilization (when working in the p > n situation) that can
be characterized by a leveling-off of the acceleration along the curve when
plotting the condition number against the (chosen) domain of the penalty
parameter. This suggest the following fast heuristic for determining the
(minimal) value of the penalty parameter: The value of the penalty parameter
for which the spectral condition number starts to stabilize may be termed an
acceptable (minimal) value.
The function outputs a graph of the (spectral) matrix condition number over
the domain [lambdaMin, lambdaMax]. When norm = "2" the
spectral condition number is calculated. It is determined by exact
calculation using the spectral decomposition. For most purposes this exact
calculation is fast enough, especially when considering rotation equivariant
situations (see ridgeP). For such situations the amenities for
fast eigenvalue calculation as provided by
RSpectra are used
internally. When exact computation of the spectral condition number is
deemed too costly one may approximate the computationally friendly
L1-condition number. This approximation is accessed through the
rcond function (Anderson et al. 1999).
When Iaids = TRUE the basic condition number plot is amended/enhanced
with two additional plots (over the same domain of the penalty parameter as
the basic plot): The approximate loss in digits of accuracy (for the
operation of inversion) and an approximation to the second-order derivative
of the curvature in the basic plot. These interpretational aids can enhance
interpretation of the basic condition number plot and may support choosing a
value for the penalty parameter (see Peeters, van de Wiel, & van Wieringen,
2016). When vertical = TRUE a vertical line is added at the constant
value. This option can be used to assess if the optimal penalty
obtained by, e.g., the routines optPenalty.LOOCV or
optPenalty.aLOOCV, has led to a precision estimate that is
well-conditioned.
Value
The function returns a graph. If nOutput = TRUE the function
also returns an object of class list:
lambdas |
A |
conditionNumbers |
A |
Note
The condition number of a (regularized) covariance matrix is
equivalent to the condition number of its corresponding inverse, the
(regularized) precision matrix. Please note that the target argument
(for Type I ridge estimators) is assumed to be specified in precision terms.
In case one is interested in the condition number of a Type I estimator
under a covariance target, say \mathbf{\Gamma}, then just specify
target = solve(\mathbf{\Gamma}).
Author(s)
Carel F.W. Peeters <carel.peeters@wur.nl>
References
Anderson, E, Bai, Z., ..., Sorenson, D. (1999). LAPACK Users' Guide (3rd ed.). Philadelphia, PA, USA: Society for Industrial and Applied Mathematics.
Demmel, J.W. (1987). On condition numbers and the distance to the nearest ill-posed problem. Numerische Mathematik, 51: 251–289.
Peeters, C.F.W., van de Wiel, M.A., & van Wieringen, W.N. (2020). The spectral condition number plot for regularization parameter evaluation. Computational Statistics, 35: 629–646.
See Also
covML, ridgeP,
optPenalty.LOOCV, optPenalty.aLOOCV,
default.target
Examples
## Obtain some (high-dimensional) data
p = 25
n = 10
set.seed(333)
X = matrix(rnorm(n*p), nrow = n, ncol = p)
colnames(X)[1:25] = letters[1:25]
Cx <- covML(X)
## Assess spectral condition number across grid of penalty parameter
CNplot(Cx, lambdaMin = .0001, lambdaMax = 50, step = 1000)
## Include interpretational aids
CNplot(Cx, lambdaMin = .0001, lambdaMax = 50, step = 1000, Iaids = TRUE)