Time SeriesFiltering, Smoothing and Learning usingtheKernelKalman Filter

Inthis paper, wepropose anewmodel, theKernel KalmanFilter, toperform various nonlinear timeseries process- ing. ThismodelisbasedontheuseofMercerkernel functions intheframework oftheKalmanFilter orLinear Dynamical Systems. Thankstothekernel trick, alltheequations involved inourmodeltoperform filtering, smoothing andlearning tasks, onlyrequire matrix algebra calculus whilst providing theability tomodelcomplex timeseries. Inparticular, itispossible tolearn dynamics fromsomenonlinear noisy timeseries implementing anexact Expectation-Maximization I.INTRODUCTION Timeseries modeling hasbecomeanappealing field for machine learning approaches overthelast decade andinorder todealwithdatataking theformofnonlinear timeseries, there isaneedoftransversal andflexible tools that canbe engineered easily. Inthis paper, wepropose akernel-based approach totimeseries modeling enabling theimplementation ofprediction, denoising andlearning tasks. Ontheonehand, ourmethod presents theadvantage ofkeeping theframework oflinear dynamical systems usable, and, ontheother hand, theprocessing ofnonvectorial timeseries canbeconsidered. Thepaper isorganized asfollows. Insection II, wedescribe ourKernel Kalman Filter (KKF) model. Section IIIfocuses on afewworks which areclosely related toours, while section IV reports empirical results achieved byourmodelonvarious times series processing tasks. Eventually, section V gives hints about future developments ofKKF.