smoothing splines running mean target point interior knots smoothing parameter running-lines smooth equivalent kernels equivalent kernel basis functions running-lines smoother least-squares line kernel smoothers kernel smoother cubic smoothing spline target value symmetric nearest neighbourhood spaced data scatterplot smoother rigid form nearest neighbours nearest neighbourhood natural splines matrix locally-weighted running-lines kernel smooth data points weighted least-squares fit weight function unweighted running-lines smoother time series three interior knots third derivative symmetric nearest neighbours symmetric nearest neighbourhoods standard Gaussian density single predictor scatterplot smoothing scatterplot smoothers scatterplot smooth predictor space piecewise polynomials piecewise cubics piecewise cubic polynomials parametric fitting nearest neighbourhoods natural cubic splines natural cubic spline multiple regression multi-predictor smoothers moving average matrix containing interested reader fitted smooth fine grid evaluated-splines cubic-spline basis functions building block boundary knots bin smoother bibliographic notes Fig. shows Euclidean distance Smoothing What is a smoother? Scatterplot smoothing: definition Parametric Regression Bin smoothers Running-mean and running-lines smoothers Kernel smoothers Computational issues Running medians and enhancements Equivalent kernels Regression splines Computational aspects Cubic smoothing splines Computational aspects Locally weighted running-line smoothers Smoothers for multiple predictors {\sl smoother} {\sl nonparametric} nature {\sl smooth} {\sl scatterplot smoothing} {\sl categorical} {\sl smoothing} {\sl local averaging} {\sl neighbourhoods} around {\sl brand} {\sl smoothing parameter} {\sl fundamental tradeoff between bias and variance} {\em robustified} {\sl infinitely smooth} {\sl close} {\sl symmetric nearest neighbourhood} {\sl running mean} {\sl nearest neighbourhood} {\sl moving average} {\sl smoother} {\sl running lines smoother} {\sl weighted} least-squares {\sl loess} {\sl kernel} {\sl metric} distance {\sl metric} {\sl rank} distance {\em Hanning} {\em twicing} {\em regression smoothers} {\em linear} {\em degrees of freedom} {\sl equivalent kernels} {\sl linear} {\em equivalent kernel} {\em loess} smooth {\sl equivalent degrees of freedom} {\sl piecewise} {\sl knots} {\sl cubic} splines {\sl number} {\sl when the knots are given} {\sl natural cubic spline} {\sl effective} dimension {\sl sorted} values {\sl natural-spline} basis {\sl local averaging} {\sl kernel} {\sl loess} {\sl tri-cube} weight {\sl span} {\sl loess} {\sl nearest neighbours} {\sl Nearest} {\sl thin-plate spline} {\sl tensor product} {\sl per~se} {\sl Updating formula for running-line smooth} {\sl Basis for natural splines} {\sl Derivation of smoothing splines {\sl Semi-parametric regression {\sl estimating} equations {\sl Efficient kernel smoothing