Fitting a local polynomial model to a noisy sequence of uniformly sampled observations or measurements (i.e. regressing) by minimizing the sum of weighted squared errors (i.e. residuals) may be used to design digital filters for a diverse range of signal-analysis problems, such as detection, classification and tracking, in biomedical, financial, and aerospace applications, for instance. Furthermore, the recursive realization of such filters, using a network of so-called leaky integrators, yields simple digital components 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/change detection are considered in this tutorial as illustrative examples. Erlang-weighted polynomial regression provides a design framework within which the various design trade-offs of state estimators (e.g. bias errors vs. random errors) and IIR smoothers (e.g. frequency isolation vs. time localization) may be intuitively balanced. Erlang weights are configured using a smoothing parameter which determines the decay rate of the exponential tail; and a shape parameter which may be used to discount more recent data, so that a greater relative emphasis is placed on a past time interval. In Morrison's 1969 treatise on sequential smoothing and prediction, the exponential weight (i.e. the zero shape-parameter case) and the Laguerre polynomials that are orthogonal with respect to this weight, are described in detail; however, more general Erlang weights and the resulting associated Laguerre polynomials are not considered 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 be used to shape and improve the response of recursive regression filters.
翻译:将本地的多元模型适用于统一抽样观测或测量的噪音序列( 即递减), 最大限度地减少加权正方差( 剩余值)的总和, 以将加权正方差( 即残余值) 用于设计数字过滤器, 解决各种信号分析问题, 例如生物医学、 金融和航空航天应用中的检测、 分类和跟踪。 此外, 这种过滤器的循环实现, 使用所谓的漏泄整合器网络, 产生简单的数字组件, 其计算复杂度低, 以及无限的脉冲反应( IIR), 这些组件在嵌入的在线感测系统中是理想的, 且具有高数据比重。 目标跟踪、 脉冲检测、 峰值检测和异常/ 变异形检测, 可以作为说明性实例。 欧朗加权 提供了一个设计框架, 其中, 州估量器的各种设计交易误差( 如误差与随机误差) 和 IIR 平流( 如: 频率隔离值与时间反应) 可能是直径直径直径, 。 直径比重( 或直径比重) 。