tfHuber

Python implementation of Tuning-Free Huber Estimation and Regression

Description

This package implements the Huber mean estimator, Huber covariance matrix estimation and adaptive Huber regression estimators efficiently. For all these methods, the robustification parameter τ is calibrated via a tuning-free principle.

Specifically, for Huber regression, assume the observed data vectors (Y, X) follow a linear model Y = θ₀ + X θ + ε, where Y is an n-dimensional response vector, X is an n × d design matrix, and ε is an n-vector of noise variables whose distributions can be asymmetric and/or heavy-tailed. The package computes the standard Huber's M-estimator when d < n and the Huber-Lasso estimator when d > n. The vector of coefficients θ and the intercept term θ₀ are estimated successively via a two-step procedure. See Wang et al., 2020 for more details of the two-step tuning-free framework.

Requirement

numpy
setuptools
wheel

Functions

There are four functions in this package:

• mean(X, grad=True, tol=1e-5, max_iter=500): Huber mean estimation. Return a tuple of mean, tau and the number of iteration.
  X: A 1-d array.
  grad: Using gradient descent or weighed least square to optimize the mean, default True
  tol: Tolerance of the error, default 1e-5.
  max_iter: Maximum times of iteration, default 500.
• cov(X, type="element", pairwise=False, tol=1e-5, max_iter=500): Huber covariance matrix estimation. Return a 2d covariance matrix.
  X: A 2-d array.
  type: If set to "element", apply adaptive huber M-estimation; or if set to "spectrum", apply spectrum-wise truncated estimation. Default "element"
  pairwise: Pairwise covariance or difference based covariance. Default false.
  tol: Tolerance of the error, default 1e-5.
  max_iter: Maximum times of iteration, default 500.
• one_step_reg(X, Y, grad=True, tol=1e-5, max_iter=500 two_step_reg(X, Y, grad=True, tol=1e-5, constTau=1.345, max_iter=500)
  One or two step adaptive Huber regression. Return a tuple of coefficients, &tau and the number of iteration.
  X, Y: Arrays of data.
  grad: Using gradient descent or weighed least square to optimize the mean, default True.
  tol: Tolerance of the error, default 1e-5.
  constTau: Default 1.345. Used only in two-step method.
  max_iter: Maximum times of iteration, default 500.
• cvlasso(X, Y, lSeq=0, nlambda=30, constTau=2.5, phi0=0.001, gamma=1.5, tol=0.001, nfolds=3): K-fold cross validated Huber-lasso regression. Return a tuple of coefficients, tau, the number of iteration and minimun of lambda.
  X, Y: Arrays of data.
  lSeq: A list of Lasso parameter lambda . If not set, automatically find a range of &lambda to be cross validated.
  nlambda: The number of lambda used for validation.
  constTau, phi0, gamma: Some parameters.
  tol: Tolerance of the error, default 0.001.
  nfolds: Number of folds to be cross validated.

Examples

We present an example of adaptive Huber methods. Here we generate data from a linear model Y = X θ + ε, where ε follows a normal distribution, and estimate the intercept and coefficients by tuning-free Huber regression.

import numpy
import tfhuber
X = np.random.uniform(-1.5, 1.5, (10000, 10))
Y = intercept + np.dot(X, beta) + np.random.normal(0, 1, 10000)

mu, tau, iteration = tf.mean(Y, grad=True, tol=1e-5, max_iter=500)
cov = tf.cov(X, method=1, tol=1e-5, max_iter=500)

theta, tau, iteration = tf.one_step_reg(X, Y, grad=True, tol=1e-5, max_iter=500)
theta, tau, iteration = tf.two_step_reg(X, Y, grad=True, tol=1e-5, consTau=1.345, max_iter=500)

theta, tau, iteration, lam = tf.cvlasso(X, Y) 

Simulation result can be viewed in this colab notebook.

License

GPL (>= 3)

Author(s)

Yifan Dai yifandai@yeah.net, Qiang Sun qsun.ustc@gmail.com

Description and algorithms refer to Xiaoou Pan's page.

References

Guennebaud, G. and Jacob B. and others. (2010). Eigen v3. Website

Ke, Y., Minsker, S., Ren, Z., Sun, Q. and Zhou, W.-X. (2019). User-friendly covariance estimation for heavy-tailed distributions. Statis. Sci. 34 454-471, Paper

Pan, X., Sun, Q. and Zhou, W.-X. (2019). Nonconvex regularized robust regression with oracle properties in polynomial time. Preprint. Paper

Sanderson, C. and Curtin, R. (2016). Armadillo: A template-based C++ library for linear algebra. J. Open Source Softw. 1 26. Paper

Sun, Q., Zhou, W.-X. and Fan, J. (2020). Adaptive Huber regression. J. Amer. Stat. Assoc. 115 254-265. Paper

Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B. Stat. Methodol. 58 267–288. Paper

Wang, L., Zheng, C., Zhou, W. and Zhou, W.-X. (2020). A new principle for tuning-free Huber regression. Stat. Sinica to appear. Paper