Local regression - Biblioteka.sk

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Local regression
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LOESS curve fitted to a population sampled from a sine wave with uniform noise added. The LOESS curve approximates the original sine wave.

Local regression or local polynomial regression,[1] also known as moving regression,[2] is a generalization of the moving average and polynomial regression.[3] Its most common methods, initially developed for scatterplot smoothing, are LOESS (locally estimated scatterplot smoothing) and LOWESS (locally weighted scatterplot smoothing), both pronounced /ˈlɛs/ LOH-ess. They are two strongly related non-parametric regression methods that combine multiple regression models in a k-nearest-neighbor-based meta-model. In some fields, LOESS is known and commonly referred to as Savitzky–Golay filter[4][5] (proposed 15 years before LOESS).

LOESS and LOWESS thus build on "classical" methods, such as linear and nonlinear least squares regression. They 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 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, LOESS 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.

A smooth curve through a set of data points obtained with this statistical technique is called a loess 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.[6][7]

Model definition

In 1964, Savitsky and Golay proposed a method equivalent to LOESS, which is commonly referred to as Savitzky–Golay filter. William S. Cleveland rediscovered the method in 1979 and gave it a distinct name. The method was further developed by Cleveland and Susan J. Devlin (1988). LOWESS is also known as locally weighted polynomial regression.

At each point in the range of the data set a low-degree polynomial is fitted to a subset of the data, with explanatory variable values near the point whose 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 LOESS fit is complete after regression function values have been computed for each of the 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 LOESS 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, , is the fraction of the total number n of data points that are used in each local fit. The subset of data used in each weighted least squares fit thus comprises the points (rounded to the next largest integer) whose explanatory variables' values are closest to the point at which the response is being estimated.[7]

Since a polynomial of degree k requires at least k + 1 points for a fit, the smoothing parameter must be between and 1, with denoting the degree of the local polynomial.

is called the smoothing parameter because it controls the flexibility of the LOESS regression function. Large values of produce the smoothest functions that wiggle the least in response to fluctuations in the data. The smaller 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.

Degree of local polynomials

The local polynomials fit to each subset of the data are almost always of first or second degree; that is, either locally linear (in the straight line sense) or locally quadratic. Using a zero degree polynomial turns LOESS into a weighted moving average. Higher-degree polynomials would work in theory, but yield models that are not really in the spirit of LOESS. LOESS is based on the ideas that any function can be well approximated in a small neighborhood by a low-order polynomial and that simple models can be fit to data easily. High-degree polynomials would tend to overfit the data in each subset and are numerically unstable, making accurate computations difficult.

Weight function

As mentioned above, the weight function gives the most weight to the data points nearest the point of estimation and the least weight to the data points that are furthest away. The use of the weights is based on the idea that points near each other in the explanatory variable space are more likely to be related to each other in a simple way than points that are further apart. Following this logic, points that are likely to follow the local model best influence the local model parameter estimates the most. Points that are less likely to actually conform to the local model have less influence on the local model parameter estimates.

The traditional weight function used for LOESS is the tri-cube weight function,

where d is the distance of a given data point from the point on the curve being fitted, scaled to lie in the range from 0 to 1.[7]

However, any other weight function that satisfies the properties listed in Cleveland (1979) could also be used. The weight for a specific point in any localized subset of data is obtained by evaluating the weight function at the distance between that point and the point of estimation, after scaling the distance so that the maximum absolute distance over all of the points in the subset of data is exactly one.

Consider the following generalisation of the linear regression model with a metric on the target space that depends on two parameters, . Assume that the linear hypothesis is based on input parameters and that, as customary in these cases, we embed the input space into as , and consider the following loss function

Here, is an real matrix of coefficients, and the subscript i enumerates input and output vectors from a training set. Since is a metric, it is a symmetric, positive-definite matrix and, as such, there is another symmetric matrix such that . The above loss function can be rearranged into a trace by observing that . By arranging the vectors and into the columns of a matrix and an matrix respectively, the above loss function can then be written as







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