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==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 LOESSLOWESS into a weighted [[moving average]]. Such a simple local model might work well for some situations, but may not always approximate the underlying function well enough. Higher-degree polynomials would work in theory, but yield models that are not really in the spirit of LOESS. LOESSLOWESS 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#statistics|parameter]] [[Statistical estimation|estimates]].
The traditional weight function used for LOESSLOWESS is the tri-cube weight function,
:<math>w(x) = (1 - |x|^3)^3 \operatorname{I}\left[\left| x\right| < 1\right] </math>
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.
==Advantages of LOESSLOWESS==
As discussed above, the biggest advantage LOESSLOWESS has over many other methods is the fact that it does not require the specification of a function to fit a model to all of the data in the sample. Instead the analyst only has to provide a smoothing parameter value and the degree of the local polynomial. In addition, LOESS is very flexible, making it ideal for modeling complex processes for which no theoretical models exist. These two advantages, combined with the simplicity of the method, make LOESS one of the most attractive of the modern regression methods for applications that fit the general framework of least squares regression but which have a complex deterministic structure.
Although it is less obvious than for some of the other methods related to linear least squares regression, LOESSLOWESS also accrues most of the benefits typically shared by those procedures. The most important of those is the theory for computing uncertainties for prediction and calibration. Many other tests and procedures used for validation of least squares models can also be extended to LOESSLOWESS models.
==Disadvantages of LOESSLOWESS==
LOESSLOWESS makes less efficient use of data than other least squares methods. It requires fairly large, densely sampled data sets in order to produce good models. This is not really surprising, however, since LOESSLOWESS needs good empirical information on the local structure of the process in order to perform the local fitting. In fact, given the results it provides, LOESSLOWESS could arguably be more efficient overall than other methods like nonlinear least squares. It may simply frontload the costs of an experiment in data collection but then reduce analysis costs.
Another disadvantage of LOESSLOWESS is the fact that it does not produce a regression function that is easily represented by a mathematical formula. This can make it difficult to transfer the results of an analysis to other people. In order to transfer the regression function to another person, they would need the data set and software for LOESSLOWESS calculations. In [[nonlinear regression]], on the other hand, it is only necessary to write down a functional form in order to provide estimates of the unknown parameters and the estimated uncertainty. Depending on the application, this could be either a major or a minor drawback to using LOESS.
Finally, as discussed above, LOESSLOWESS is a computationally intensive method. This is not usually a problem in our current computing environment, however, unless the data sets being used are very large. LOESSLOWESS is also prone to the effects of outliers in the data set, like other least squares methods. There is an iterative, [[robust statistics|robust]] version of LOESSLOWESS [Cleveland (1979)] that can be used to reduce LOESSLOWESS' sensitivity to [[outliers]], but extreme outliers can still overcome even the robust method.
==See also==
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