SLOPE

Installation

You can install the package using the Julia package manager:

using Pkg
Pkg.add("SLOPE")

Alternatively, you can also install the latest development version of the package from the source code on GitHub by calling

Pkg.add(url = "https://github.com/jolars/SLOPE.jl")

Getting Started

SLOPE is a Julia package for Sorted L1 Penalized Estimation (SLOPE), which is a type of regularized regression. SLOPE minimizes the following objective function:

\[\frac{1}{n} \sum_{i=1}^n f(y_i, x_i^\intercal \beta) + \alpha \sum_{j=1}^p \lambda_j |\beta_{(j)}|.\]

where $f(y,\eta)$ is the negative log-likelihood contribution of a single observation $(y, \eta)$. $\beta_{(j)}$ is the $j$-th coefficient, $\beta_0$ is the intercept, and $\lambda$ is a decreasing sequence of regularization weights. $x_i$ is the $i$th row of the design matrix, and $n$ is the number of observations.

SLOPE is a type of sparse regression, which means that it will, given high enough penalization, set some of the coefficients to zero, effectively removing the corresponding features from the model. If you are familiar with the lasso, then you should know that SLOPE is actually a generalization of the lasso (which you can see by setting all $\lambda_j$ to the same value). Unlike the lasso, however, SLOPE also clusters coefficients by settings them to the same magnitude. This helps remove some deficiencies of the lasso in highly correlated settings, and it also allows for selection of more features that is possible with the lasso.

Basic Usage

First, we'll load the package and fit a simple model:

using SLOPE
using Random
using Statistics

# Generate some sample data
n, p = 100, 20
Random.seed!(123)
X = randn(n, p)
β = vcat(fill(3.0, 5), fill(0.0, p-5))  # 5 non-zero coefficients
y = X * β + 0.5 * randn(n)

# Fit a SLOPE model
fit = slope(X, y)

Examining the Model

You can examine the results of the fitted model:

fit.coefficients

Cross-Validation

To determine the optimal regularization strength, you can use cross-validation:

# Perform cross-validation to find optimal parameters
cvfit = slopecv(X, y)

# View the optimal α value
cvfit.best_params

Contributing

The SLOPE.jl package is a thin wrapper around the C++ slope library, which provides all of the core functionality. Therefore, if you find any bugs or have feature requests, then it's likely that you should open a ticket in the slope repository rather than here.

That being said, if you find any bugs in the Julia wrapper or there are features in the C++ library that are yet to be implemented in the Julia wrapper, then please open an issue in this repository.