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[[Image:Loess_curveLowess_curve.svg|thumb|300 px|LOESSLOWESS curve fitted to a population sampled from a [[sine wave]] with uniform noise added. The LOESS curve approximates the original sine wave. ]]
'''LOESSLOWESS''', or '''LOWESSLOESS''' ('''locally weighted scatterplot smoothing'''), is one of many "modern" [[statistical model|modeling methods]] that build on [[classical statistics|"classical" methods]], such as linear and nonlinear (The correct original term is Lowess. This was not recognized by a spellcheck program which later substituted Loess, a term for windblown dust.[[Regression analysis|least squares regression]]. Modern regression methods are designed to address situations in which the classical procedures do not perform well or cannot be effectively applied without undue labor. LOESS combines much of the simplicity of linear least squares regression with the flexibility of [[Non-linear regression|nonlinear regression]]. It does this by fitting simple models to localized subsets of the data to build up a function that describes the deterministic part of the variation in the data, point by point. In fact, one of the chief attractions of this method is that the data analyst is not required to specify a global function of any form to fit a model to the data, only to fit segments of the data.
The trade-off for these features is increased computation. Because it is so computationally intensive, LOESSLOWESS would have been practically impossible to use in the era when least squares regression was being developed. Most other modern methods for process modeling are similar to LOESS in this respect. These methods have been consciously designed to use our current computational ability to the fullest possible advantage to achieve goals not easily achieved by traditional approaches.
Plotting a smooth curve through a set of data points using this statistical technique is called a '''LoessLowess Curve''', particularly when each smoothed value is given by a weighted quadratic least squares regression over the span of values of the y-axis [[scattergram]] criterion variable. When each smoothed value is given by a weighted linear least squares regression over the span, this is known as a '''Lowess curve'''; however, some authorities treat '''Lowess''' and Loess as synonyms.
==Definition of a LOESSLOWESS model==
LOESSLOWESS, originally proposed by Cleveland (1979)<!-- Please list this in a "References" section below. --> and further developed by Cleveland and Devlin (1988)<!-- Please list this in a "References" section below. -->, specifically denotes a method that is (somewhat) more descriptively known as locally weighted polynomial regression. At each point in the [[data set]] a low-degree [[polynomial]] is fitted to a subset of the data, with [[explanatory variable]] values near the point whose [[response variable|response]] is being estimated. The polynomial is fitted using [[weighted least squares]], giving more weight to points near the point whose response is being estimated and less weight to points further away. The value of the regression function for the point is then obtained by evaluating the local polynomial using the explanatory variable values for that data point. The LOESSLOwESS fit is complete after regression function values have been computed for each of the <math>n</math> data points. Many of the details of this method, such as the degree of the polynomial model and the weights, are flexible. The range of choices for each part of the method and typical defaults are briefly discussed next.
==Localized subsets of data==
The '''subsets''' of data used for each weighted least squares fit in LOESSLOWESS are determined by a nearest neighbors algorithm. A user-specified input to the procedure called the "bandwidth" or "smoothing parameter" determines how much of the data is used to fit each local polynomial. The smoothing parameter, <math>\alpha</math>, is a number between <math>\left(\lambda+1\right)/n</math> and 1, with <math>\lambda</math> denoting the degree of the local polynomial. The value of <math>\alpha</math> is the proportion of data used in each fit. The subset of data used in each weighted least squares fit comprises the <math>n\alpha</math> (rounded to the next largest integer) points whose explanatory variables values are closest to the point at which the response is being estimated.
<math>\alpha</math> is called the smoothing parameter because it controls the flexibility of the LOESSLOWESS regression function. Large values of <math>\alpha</math> produce the smoothest functions that wiggle the least in response to fluctuations in the data. The smaller <math>\alpha</math> is, the closer the regression function will conform to the data. Using too small a value of the smoothing parameter is not desirable, however, since the regression function will eventually start to capture the random error in the data. Useful values of the smoothing parameter typically lie in the range 0.25 to 0.5 for most LOESSLOWESS applications.
==Degree of local polynomials==
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