Fit a generalized linear model regularized with the SLOPE (Sorted L-One Penalized Estimation) norm, which applies a decreasing \(\lambda\) (penalty sequence) to the coefficient vector (\(\beta\)) after having sorted it in decreasing order according to its absolute values.
owl( x, y, family = c("gaussian", "binomial", "multinomial", "poisson"), intercept = TRUE, standardize_features = TRUE, sigma = NULL, lambda = c("gaussian", "bh", "oscar"), lambda_min_ratio = if (n < p) 0.01 else 1e-04, n_sigma = 100, q = 0.1 * min(1, n/p), screening = TRUE, tol_dev_change = 1e-05, tol_dev_ratio = 0.995, max_variables = n * m, max_passes = 1e+06, tol_rel_gap = 1e-05, tol_infeas = 1e-04, diagnostics = FALSE, verbosity = 0 )
| x | the feature matrix, which can be either a dense matrix of the standard matrix class, or a sparse matrix inheriting from Matrix::sparseMatrix Data frames will be converted to matrices internally. |
|---|---|
| y | the response. For Gaussian models this must be numeric; for binomial models, it can be a factor. |
| family | response type. See Families for details. |
| intercept | whether to fit an intercept |
| standardize_features | whether to standardize features (predictors) |
| sigma | scale of lambda sequence |
| lambda | either a character vector indicating the method used to construct the lambda path or a vector with length equal to the number of coefficients in the model |
| lambda_min_ratio | smallest value for |
| n_sigma | length of regularization path |
| q | shape of lambda sequence |
| screening | whether the strong rule for SLOPE be used to screen variables for inclusion |
| tol_dev_change | the regularization path is stopped if the fractional change in deviance falls below this value. Note that this is automatically set to 0 if a sigma is manually entered |
| tol_dev_ratio | the regularization path is stopped if the deviance ratio \(1 - \mathrm{deviance}/\mathrm{(null-deviance)}\) is above this threshold |
| max_variables | criterion for stopping the path in terms of the maximum number of unique, nonzero coefficients in absolute value in model |
| max_passes | maximum number of passes for optimizer |
| tol_rel_gap | stopping criterion for the duality gap |
| tol_infeas | stopping criterion for the level of infeasibility |
| diagnostics | should diagnostics be saved for the model fit (timings, primal and dual objectives, and infeasibility) |
| verbosity | level of verbosity for displaying output from the program. Setting this to 1 displays basic information on the path level, 2 a little bit more information on the path level, and 3 displays information from the solver. |
An object of class "Owl" with the following slots:
a three-dimensional array of the coefficients from the model fit, including the intercept if it was fit. There is one row for each coefficient, one column for each target (dependent variable), and one slice for each penalty.
a three-dimensional boolean array indicating whether a coefficient was zero or not
the lambda vector that when multiplied by a value in sigma
gives the penalty vector at that point along the regularization
path
the vector of sigma, indicating the scale of the lambda vector
a character vector giving the names of the classes for binomial and multinomial families
the number of passes the solver took at each path
the number of violations of the screening rule
a list where each element indicates the indices of the coefficients that were active at that point in the regularization path
the number of unique predictors (in absolute value)
the deviance ratio (as a fraction of 1)
the deviance of the null (intercept-only) model
the name of the family used in the model fit
a data.frame of objective values for the primal and dual problems, as
well as a measure of the infeasibility, time, and iteration. Only
available if diagnostics = TRUE in the call to owl().
the call used for fitting the model
owl() tries to minimize the following composite objective, given
in Lagrangian form.
$$
f(\beta) + \sigma \sum_{i=j}^p \lambda_j |\beta|_{(j)},
$$
where \(f(\beta)\) is a smooth, convex function of \(\beta\), whereas
the second part is the SLOPE norm, which is convex but non-smooth.
In ordinary least-squares regression, for instance,
\(f(\beta)\) is simply the squared norm of the least-squares residuals.
See section Families for specifics regarding the various types of
\(f(\beta)\) (model families) that are allowed in owl().
By default, owl() fits a path of lambda sequences, starting from
the null (intercept-only) model to an almost completely unregularized
model. The path will end prematurely, however, if the criteria
related to any of the
arguments tol_dev_change, tol_dev_ratio, or max_variables
are reached before the path is complete. This means that unless these
arguments are modified, the path is not guaranteed to be of
length n_sigma.
Gaussian
The Gaussian model (Ordinary Least Squares) minimizes the following objective. $$ ||y - X\beta||_2^2 $$
Binomial
The binomial model (logistic regression) has the following objective. $$ \sum_{i=1}^n \log\left(1+ \exp\left(- y_i \left(x_i^T\beta + \alpha \right) \right) \right) $$ with \(y \in \{-1, 1\}\).
Poisson
In poisson regression, we use the following objective.
$$ -\sum_{i=1}^n \left(y_i\left(x_i^T\beta + \alpha\right) - \exp\left(x_i^T\beta + \alpha\right)\right) $$
Multinomial
In multinomial regression, we minimize the full-rank objective $$ -\sum_{i=1}^n\left( \sum_{k=1}^{m-1} y_{ik}(x_i^T\beta_k + \alpha_k) - \log\sum_{k=1}^{m-1} \exp(x_i^T\beta_k + \alpha_k) \right) $$ with \(y_{ik}\) being the element in a \(n\) by \((m-1)\) matrix, where \(m\) is the number of classes in the response.
There are multiple ways of specifying the lambda sequence
in owl(). It is, first of all, possible to select the sequence manually by
using a non-increasing
numeric vector as argument instead of a character.
If all lambda are the same value, this will
lead to the ordinary lasso penalty. The greater the differences are between
consecutive values along the sequence, the more clustering behavior
will the model exhibit. Note, also, that the scale of the \(\lambda\)
vector makes no difference if sigma = NULL, since sigma will be
selected automatically to ensure that the model is completely sparse at the
beginning and almost unregularized at the end. If, however, both
sigma and lambda are manually specified, both of the scales will
matter.
Instead of choosing the sequence manually, one of the following automatically generated sequences may be chosen.
BH (Benjamini--Hochberg)
If lambda = "bh", the sequence used is that referred to
as \(\lambda^{(\mathrm{BH})}\) by Bogdan et al, which sets
\(\lambda\) according to
$$
\lambda_i = \Phi^{-1}(1 - iq/(2p)),
$$
where \(\Phi^{-1}\) is the quantile function for the standard
normal distribution and \(q\) is a parameter that can be
set by the user in the call to owl().
Gaussian
This penalty sequence is related to BH, such that $$ \lambda_i = \lambda^{(\mathrm{BH})}_i \sqrt{1 + w(i-1)\cdot \mathrm{cumsum}(\lambda^2)_i}, $$ where \(w(k) = 1/(n-k-1)\). We let \(\lambda_1 = \lambda^{(\mathrm{BH})}_1\) and adjust the sequence to make sure that it's non-increasing. Note that if \(p\) is large relative to \(n\), this option will result in a constant sequence, which is usually not what you would want.
OSCAR
This sequence comes from Bondell and Reich and is a linearly non-increasing sequence such that $$ \lambda_i = q(p - i) + 1. $$
Bogdan, M., van den Berg, E., Sabatti, C., Su, W., & Candès, E. J. (2015). SLOPE -- adaptive variable selection via convex optimization. The Annals of Applied Statistics, 9(3), 1103–1140. https://doi.org/10/gfgwzt
Bondell, H. D., & Reich, B. J. (2008). Simultaneous Regression Shrinkage, Variable Selection, and Supervised Clustering of Predictors with OSCAR. Biometrics, 64(1), 115–123. JSTOR. https://doi.org/10.1111/j.1541-0420.2007.00843.x
# Gaussian response, default lambda sequence fit <- owl(bodyfat$x, bodyfat$y) # Binomial response, BH-type lambda sequence fit <- owl(heart$x, heart$y, family = "binomial", lambda = "bh") # Poisson response, OSCAR-type lambda sequence fit <- owl(abalone$x, abalone$y, family = "poisson", lambda = "oscar", q = 0.4) # Multinomial response, custom sigma and lambda m <- length(unique(wine$y)) - 1 p <- ncol(wine$x) sigma <- 0.005 lambda <- exp(seq(log(2), log(1.8), length.out = p*m)) fit <- owl(wine$x, wine$y, family = "multinomial", lambda = lambda, sigma = sigma)