Loess (or lowess, Locally Weighted Scatterplot Smoothing) is a scatterplot smoother, which provides a flexible method for nonparametric regression. • blue points: data points under the smoother • black points: data points not under the smoother • black curve: the loess smoother • dark red dot: the loess fit at the current position • dark red line or curve: the loess curve with parameters fixed at those corresponding to the red dot • purple bars: window shape and the corresponding loess weights, with scale indicated on the right axis •  , the position at which the point on the loess curve is calculated. As you slide  , notice how the window width changes. The number of points in the window, whose positions are indicated by the green bars, remains constant. •  , the fraction of data under the smoother. As  increases, the window width increases and more smoothing is done.  indicates the degree of the local polynomial.
Given bivariate observations  ,  , the basic model that can be fitted may be written as  ,  , where  and  is a local polynomial of degree  , which may be written as  . The parameters  are estimated by weighted least squares for each value of  . The weight function weights the data,  , so that data values near to  have greater weight than those farther away from  . Following [1], we use the tricube weight function, with  to define the local neighborhood weights for the data at the point  . Here  and  controls the amount of smoothing (larger values of  result in more smoothing). As  ,  for each  , and the local linear model reduces to the standard parametric polynomial regression. For  ,  is the distance to the  nearest neighbor, where  (  is the integer part function). Hence,  , where  denotes the  largest value of  ,  . For  ,  . It follows that as  , the local linear model reduces to a parametric polynomial regression of degree  . In practice we work with local constant loess,  ; local linear,  , or local quadratic,  . [1] W. S. Cleveland, Visualizing Data, New Jersey: Summit, 1993.
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