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Fitting a local polynomial model to a noisy sequence of uniformly sampledobservations or measurements (i.e. regressing) by minimizing the sum ofweighted squared errors (i.e. residuals) may be used to design digital filtersfor a diverse range of signal-analysis problems, such as detection,classification and tracking, in biomedical, financial, and aerospaceapplications, for instance. Furthermore, the recursive realization of suchfilters, using a network of so-called leaky integrators, yields simple digitalcomponents with a low computational complexity and an infinite impulse response(IIR) that are ideal in embedded online sensing systems with high data rates.Target tracking, pulse-edge detection, peak detection and anomaly/changedetection are considered in this tutorial as illustrative examples.Erlang-weighted polynomial regression provides a design framework within whichthe various design trade-offs of state estimators (e.g. bias errors vs. randomerrors) and IIR smoothers (e.g. frequency isolation vs. time localization) maybe intuitively balanced. Erlang weights are configured using a smoothingparameter which determines the decay rate of the exponential tail; and a shapeparameter which may be used to discount more recent data, so that a greaterrelative emphasis is placed on a past time interval. In Morrison's 1969treatise on sequential smoothing and prediction, the exponential weight (i.e.the zero shape-parameter case) and the Laguerre polynomials that are orthogonalwith respect to this weight, are described in detail; however, more generalErlang weights and the resulting associated Laguerre polynomials are notconsidered there, nor have they been covered in detail elsewhere since. Thus,one of the purposes of this tutorial is to explain how Erlang weights may beused to shape and improve the response of recursive regression filters.

The leaky integrator that could: Or recursive polynomial regression for online signal analysis