In modern time series problems, one aims at forecasting multiple time series with possible missing and noisy values. In this paper, we introduce the Sliding Mask Method (SMM) for forecasting multiple nonnegative time series by means of nonnegative matrix completion: observed noisy values and forecast/missing values are collected into matrix form, and learning is achieved by representing its rows as a convex combination of a small number of nonnegative vectors, referred to as the archetypes. We introduce two estimates, the mask Archetypal Matrix factorization (mAMF) and the mask normalized Nonnegative Matrix Factorization (mNMF) which can be combined with the SMM method. We prove that these estimates recover the true archetypes with an error proportional to the noise. We use a proximal alternating linearized method (PALM) to compute the archetypes and the convex combination weights. We compared our estimators with state-of-the-art methods (Transformers, LSTM, SARIMAX...) in multiple time series forecasting on real data and obtain that our method outperforms them in most of the experiments.

翻译：在现代时间序列问题中，我们旨在预测可能存在缺失值和噪声的多变量时间序列。本文提出滑动掩码方法（SMM），通过非负矩阵补全技术对多个非负时间序列进行预测：将观测到的含噪值与预测/缺失值整合为矩阵形式，通过将矩阵行表示为少量非负向量（称为原型）的凸组合来实现学习。我们引入两种估计方法：掩码原型矩阵分解（mAMF）和掩码归一化非负矩阵分解（mNMF），二者均可与SMM方法结合使用。我们证明这些估计量能以与噪声幅度成比例的误差恢复真实原型。采用近端交替线性化方法（PALM）计算原型及其凸组合权重。在真实数据上的多时间序列预测实验中，我们将所提估计器与当前最优方法（Transformer、LSTM、SARIMAX等）进行对比，结果表明我们的方法在大多数实验场景中表现更优。