Nonnegative Matrix Factorization with Side Information for Time Series Recovery and Prediction

Abstract

Motivated by the reconstruction and the prediction of electricity consumption, we extend Nonnegative Matrix Factorization (NMF) to take into account side information (column or row features). We consider general linear measurement settings, and propose a framework which models non-linear relationships between features and the response variables. We extend previous theoretical results to obtain a sufficient condition on the identifiability of the NMF in this setting. Based the classical Hierarchical Alternating Least Squares (HALS) algorithm, we propose a new algorithm (HALSX, or Hierarchical Alternating Least Squares with eXogeneous variables) which estimates the factorization model. The algorithm is validated on both simulated and real electricity consumption datasets as well as a recommendation dataset, to show its performance in matrix recovery and prediction for new rows and columns. * J. Mei, Y. Goude and G. Hébrail are with EDF Lab Paris-Saclay,

Recovering Missing Values From Corrupted Historical Observations: Approaching the Limit of Predictability in Spectrum Prediction Tasks

“…Besides the data loss, the observed sensory data are also easily polluted by anomalies occur due to activity from malicious operations, or misconfigurations and failures of network equipments [11]. Previous studies [12]- [14] have demonstrated that the existence of missing values and anomalies in historical spectrum observations will compromise the prediction accuracy of several spectrum prediction algorithms. However, the algorithm-specific evaluations cannot reveal the inherent deficiency caused by missing values and anomalies for spectrum prediction, because the compromise could be incurred by either the structure of algorithms or the missing values and anomalies.…”

“…This appendix describes the derivation of A [t]. Defining F a f [t] as the inner objective to be minimized in(14), its derivative with regard to a f [t] is calculated as∂F (a f [t]) ∂(a f [t]) = (ωτ ) f,w (Zτ ) f,w − a f αw [τ ] αw[τ ] +µ h [τ ] W −1 w=1 λ t−τ (ωτ ) f,w a f βw[τ ] βw[τ ] +µr[t] a f .Then, by setting this derivative equal to zero and usinga f α w [τ ] α w [τ ] = α w [τ ] (α w [τ ]) a f , a f β w [τ ] β w [τ ] = β w [τ ] (β w [τ ]) a f , we get the following:…”

Tight Risk Bound for High Dimensional Time Series Completion

“…It is thus tempting to use low-rank matrix completion algorithms to recover partially observed highdimensional time series, and this was indeed done in many applications: [44,42,18] used low-rank matrix completion to reconstruct data from multiple sensors. Similar techniques were used by [35,34] to recover the electricity consumption of many households from partial observations, by [5] on panel data in economics, and by [38,6] for policy evaluation. Some algorithms were proposed to take into account the temporal updates of the observations (see [40]).…”

DNN-MVL: DNN-Multi-View-Learning-Based Recover Block Missing Data in a Dam Safety Monitoring System

“…Some global initialization methods have been proposed to avoid the unstable inputs for the matrix completion. For example, non-negative matrix factorization based on the multi-view [16,17], the user-based collaborative filtering [18], and matrix factorization and a salient version called empirical orthogonal functions model [19,20,21,22] can be applied to fill missing values. Although those methods consider the spatial-temporal views, they just adopt the linear fusion model with multiple views results, which cannot generate more accurate estimate for filling missing data.…”