combss 1.0.2

A package implementation of COMBSS, a novel continuous optimisation method toward best subset selection

COMBSS

This is the package for COMBSS, a novel continuous optimisation method toward best subset selection, developed from the paper Moka et al. (2024).

For a more detaled overview of COMBSS, refer to https://link.springer.com/article/10.1007/s11222-024-10387-8.

Dependencies

This package relies on the following libraries:

• numpy (version 1.21.0 or later): Numerical computing.
• scipy (version 1.7.0 or later): Sparse matrix operations and linear algebra.
• scikit-learn (version 1.0.0 or later): Machine learning and evaluation metrics.

These will be installed automatically if you install the package via pip. Alternatively, they can also be installed manually.

COMBSS Installation and Usage Guide

Installation

Users can install COMBSS using the pip command-line tool:

pip install combss

Usage Guide

For demonstrative purposes, we apply COMBSS on a dataset created beforehand, with X_train, y_train, X_test, y_test generated from a 80-20 train-test split prior to this example.

Importing COMBSS

To import COMBSS after installation, use the following command:

import combss

COMBSS is implemented as a class named model within the linear module. Users can instantiate an instance of the model class to utilize its methods:

# Instantiating an instance of the combss class
optimiser = combss.linear.model()

Fitting the Model

To use COMBSS for best subset selection, call the fit method within the model class. Here are some commonly used arguments:

• q: Maximum subset size. Defaults to min(number of observations, number of predictors).
• nlam: Number of λ values in the dynamic grid. Default is 50.
• scaling: Boolean to enable feature scaling. Default is False.

Example usage 1:

# A sample usage of the commonly used arguments
optimiser.fit(X_train=X_train, y_train=y_train, X_test=X_test, y_test=y_test, q=8, nlam=20, scaling=True)

Additional Fitting Arguments

Other arguments include:

• t_init: Initial point for the vector t.
• tau: Threshold parameter for subset mapping.
• delta_frac: Value of δ/n in the objective function.
• eta: Truncation parameter during gradient descent.
• patience: Number of iterations before termination.
• gd_maxiter: Maximum iterations for gradient descent.
• gd_tol: Tolerance for gradient descent.
• cg_maxiter: Maximum iterations for the conjugate gradient algorithm.
• cg_tol: Tolerance for the conjugate gradient algorithm.

Modified usage example 2:

# A modified usage of the fit method
optimiser.fit(X_train=X_train, y_train=y_train, X_test=X_test, y_test=y_test, q=10, nlam=50, scaling=True, tau=0.9, delta_frac=20)

Model Attributes

After fitting, the following attributes can be accessed:

• subset: Indices of the optimal subset.
• mse: Mean squared error on test data.
• coef_: Coefficients of the linear model.
• lambda_: Optimal λ value.
• run_time: Time taken for fitting.
• lambda_list: List of λ values explored.
• subset_list: Subsets obtained for each λ.

Example:

optimiser.subset
# Output: array([0, 1, 2, 3, 4, 6, 7, 8])

optimiser.mse
# Output: 19.94

Illustrative Examples

Example 1

# A sample usage of the commonly used arguments
optimiser.fit(X_train=X_train, y_train=y_train, X_test=X_test, y_test=y_test, q = 8, nlam = 20, scaling=True)

optimiser.subset
# array([0, 1, 2, 3, 4, 6, 7, 8])

optimiser.mse
# 19.940929997277212

optimiser.coef_
# array([ 0.85215,  1.50009,  0.39557,  2.3919,  -0.56994,
#         0.     ,  2.6758 ,  0.72726,  1.70696,  0.        ,
#         0.     ,  0.     ,  0.     ,  0.     ,  0.        ,
#         0.     ,  0.     ,  0.     ,  0.     ,  0.        ])

optimiser.lambda_
# 0.6401161339265333

optimiser.run_time
# 2.591932

optimiser.lambda_list
# [65.54789211407702,
# 32.77394605703851,
# 16.386973028519254,
# .
# .
# 0.5120929071412267,
# 0.6401161339265333]

optimiser.subset_list
# [array([], dtype=int64),
# array([], dtype=int64),
# array([], dtype=int64),
# .
# .
# array([0, 1, 2, 3, 4, 6, 7, 8]),
# array([0, 1, 2, 3, 4, 6, 7, 8])]

One can observe that a model of size q = 8 was recovered from the training data after approximately 2.59 seconds. The recovered model with elements of indices in the optimiser.subset array achieved a mean squared error of approximately 19.94 on the test data, after a series of up to nlam = 50 values of λ were explored in the dynamic grid search, starting with an null model explored when COMBSS was initialised with λ approximately equal to 65.548.

One can additionally observe the following output after performing the fitting in the modified code example 2. In this setting, q is instead taken to equal 10, exploring 50 values of λ with feature scaling, a more stringent thresholding value of 𝜏 = 0.9, and taking the fraction delta/n for the objective function equal to 20. All other arguments take their default values.

Example 2

# A sample usage of additional arguments
combssOptimiser.fit(X_train=X_train, y_train=y_train, X_test=X_test, y_test=y_test, q = 10, nlam = 50, scaling=True, tau = 0.9, delta_frac = 20)

optimiser.subset
# array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])

optimiser.mse
# 19.50638319557191

optimiser.coef_
# array([ 0.76678,  1.51074,  0.49312,  2.45588,  -0.69150,
#         0.13782,  2.43072,  0.89641,  0.88130,  1.13421 ,  
#         0.     ,  0.     ,  0.     ,  0.     ,  0.        ,
#         0.     ,  0.     ,  0.     ,  0.     ,  0.        ])

optimiser.lambda_
# 0.022003992103724584

optimiser.run_time
# 5.400080000000001

optimiser.lambda_list
# [65.54789211407702,
# 32.77394605703851,
# 16.386973028519254,
# .
# .
# 0.020003629185204166,
# 0.016002903348163334]

optimiser.subset_list
# [array([], dtype=int64),
# array([], dtype=int64),
# array([], dtype=int64),
# .
# .
# array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10]),
# array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10])]

One can observe that the changes to tau, delta_frac and nlam result in different values of lambda being explored, with a different navigation of subsets as the threshold parameter 𝜏 is increased in the subset mapping process, and the landscape of the objective function is changed. Consequently, an additional predictor from the true model is recovered at the expense of a larger computational cost.