Lasso and probabilistic inequalities for multivariate point processes
Lasso and probabilistic inequalities for multivariate point processes
复制标题
DOI：
10.3150/13-bej562
复制
发表时间：
2015-02-01
期刊：
影响因子：
1.5
通讯作者：
Rivoirard, Vincent
中科院分区：
文献类型：
--
作者：
Hansen, Niels Richard;Reynaud-Bouret, Patricia;Rivoirard, Vincent
由于其低计算成本，Lasso是一种有吸引力的高维统计设置的正则化方法。本文考虑依赖于未知函数参数的多元计数过程，该参数由固定字典的线性组合来估计。为了选择系数，我们提出了一种自适应l（1）-惩罚方法，其中惩罚的数据驱动权重来自新的伯恩斯坦型鞅不等式。在字典的Gram矩阵的假设下，建立了Oracle不等式.证明了多变量Hawkes过程的非渐近概率结果，这使我们能够通过考虑基于直方图，傅立叶或小波基的一般字典来检查这些假设。受神经元活动推断问题的启发，我们最终对多元霍克斯过程进行了仿真研究，并将我们的方法与Zou在（J. Amer Statist.）中提出的自适应Lasso过程进行了比较。101（2006）1418 - 1429）。我们观察到我们的程序表现出色。我们依赖于理论方面的调整我们的方法的基本问题。不像自适应套索（J。Assoc.101（2006）1418 - 1429），我们的调整过程被证明是相对于问题的所有参数是鲁棒的，揭示了其用于具体目的的潜力，特别是在神经科学中。
Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive l(1)-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Nonasymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activity inference, we finally carry out a simulation study for multivariate Hawkes processes and compare our methodology with the adaptive Lasso procedure proposed by Zou in (J. Amer Statist. Assoc. 101 (2006) 1418-1429). We observe an excellent behavior of our procedure. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of (J. Amer Statist. Assoc. 101 (2006) 1418-1429), our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.