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.. _sphx_glr_auto_gallery_1-glm_plot_a1_power_of_abess.py:


====================================================
Power of **abess** Library: Empirical Comparison
====================================================

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Introduction
^^^^^^^^^^^^
In this part, we are going to explore the power of the ``abess`` package
using simulated data. We compare the abess package with popular Python
packages:
`scikit-learn <https://scikit-learn.org/stable/supervised_learning.html#supervised-learning>`__
for linear and logistic regressions in the following section. (Actually,
we also compare with
`python-glmnet <https://github.com/civisanalytics/python-glmnet>`__,
`statsmodels <https://github.com/statsmodels/statsmodels>`__ and
`L0bnb <https://github.com/alisaab/l0bnb>`__, but the ``python-glmnet``
presents a poor prediction error, the ``statsmodels`` runs slow and the
``L0bnb`` cannot adaptively choose sparsity level. So their results are not
showed here.)


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Simulation Setting
^^^^^^^^^^^^^^^^^^

Both packages are compared in three aspects including the prediction
performance, the variable selection performance, and the computation
efficiency.

-  The prediction performance of the linear model is measured by
   :math:`||y−\hat{y}||_2` on a test set and for logistic regression
   this is measured by the area under the ROC curve (AUC).
-  The coefficient estimation performance are measured by the coefficient error :math:`||\beta - \hat{\beta}||_2`.
-  For the variable selection performance, we compute true positive rate (TPR,
   which is the proportion of variables in the active set that are
   correctly identified) and the false positive rate (FPR, which is the
   proportion of the variables in the inactive set that are falsely
   identified as a signal).
-  Timings of the CPU execution are recorded in seconds, and all of methods select the best model among 100 models
   under different regularization strength or support size.

The simulated data are made by ``abess.datasets.make_glm_data()``. The
number of predictors is :math:`p=8000` and the size of data is
:math:`n=500`. The true coefficient contains :math:`k=10` nonzero
entries uniformly distributed in :math:`[b,B]`:

-  For linear regression (``family = "gaussian"``), we set :math:`b = 5\sqrt{2\ln p / n}` and :math:`B = 100b`.
-  For logistic regression (``family = "binomial"``), we set :math:`b = 10\sqrt{2\ln p / n}` and :math:`B = 5b`.

In each regression, we test for both low
(:math:`\rho=0.1`) and high correlation (:math:`\rho=0.7`) scenarios.
What’s more, a random noise generated from a standard Gaussian
distribution is added to the linear predictor :math:`x^{\top} \beta` for linear
regression.

All the performances are averaged over 20 replications.
All experiments are evaluated on a Arch Linux platform with Intel(R)
Core(TM) i5-6500 CPU @ 3.20GHz and 16 RAM.

.. code:: bash

   $ python abess/docs/simulation/Python/plot_results_figure.py

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Numerical Results
^^^^^^^^^^^^^^^^^

For linear regression, we compare three methods in the two packages:
Lasso, OMP and abess. For logistic regression, we compare two
methods: lasso and abess.
The results are presented in the following pictures. The first column is
the result of linear regression and the second one is of logistic
regression.

-  Firstly, among all of the methods implemented in different packages,
   the estimator obtained by the abess package shows both the best
   prediction performance and the best coefficient error.

-  Secondly, the estimator obtained by the abess package can reasonably
   control FPR in a low level while the TPR stays at 1. (Since all
   methods’ TPR are 1, the figure is not plotted.)

-  Furthermore, our abess package is highly efficient compared with
   other packages, especially in the linear regression.

|image0|
|image1|

For ``abess`` R library's performance, please view
https://abess-team.github.io/abess/articles/v11-power-of-abess.html.

.. |image0| image:: ../../Tutorial/figure/perform.png
.. |image1| image:: ../../Tutorial/figure/timings.png

sphinx_gallery_thumbnail_path = 'Tutorial/figure/timings.png'



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