4. Process Modeling 4.1. Introduction to Process Modeling 4.1.4. What are the some of the different statistical methods for model building? 4.1.4.3. Weighted Least Sum of Squares Regression
Handles Cases Where Data Quality Varies	One of the common assumptions underlying most process modeling methods, including linear and nonlinear least squares regression, is that each data point provides equally accurate information about the deterministic part of the total process variation. This assumption, however, clearly does not hold in every modeling application. For example, in the semiconductor photomask linespacing data shown below, it appears that the precision of the linespacing measurements decreases as the line spacing increases. In situations like this, where it may not reasonable to assume that every observation should be treated equally, weighted least squares can often be used to maximize the efficiency of parameter estimation. This is done by attempting to give each data point the proper amount of influence over the parameter estimates rather than giving some points more influence than they should have and giving others less by using a procedure that treats all of the data equally.
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Definition of a Weighted Least Squares Model	Unlike linear and nonlinear least squares regression, weighted least squares regression is not associated with a particular type of function used to describe the relationship between the process variables. Instead, as suggested above, it is used to describe the behavior of the random errors in the model. Weighted least squares can be used with functions that are either linear or nonlinear in the parameters. It works by incorporating an extra nonnegative constant, or weight, associated with each data point, into the fitting criterion. The size of the weight indicates the accuracy of the information contained in the associated observation. The weight for each observation is given relative to the other observations and their weights. Then, when the weighted fitting criterion is minimized to find the parameter estimates, the weights determine the contribution of each observation to the final parameter estimates.
Advantages of Weighted Least Squares	Like all of the least squares methods discussed so far, weighted least squares is an efficient method that makes good use of small data sets. It also shares the ability to provide different types of easily interpretable statistical intervals for prediction, calibration and optimization. In addition, as discussed above, the main advantage that weighted least squares enjoys over other methods is the ability handle regression situations where the data points are of varying quality. For example, if the standard deviation of the random errors in the data is not constant across all levels of the explanatory variables using weighted least squares with weights that are inversely proportional to the variance at each level of the explanatory variables yields the most precise parameter estimates possible. On the other hand, if the random errrors in the data do have constant standard devation, but the data contains a few outliers, weighted least squares can be conveniently used to eliminate the outliers from an analysis. This is done by setting the weights for the points to be eliminated equal to zero.
Disadvantages of Weighted Least Squares	The biggest disadvantage of weighted least squares, which many people choose to ignore or are not aware of, is probably the fact that the theory behind this method is based on the assumption that the weights are known exactly. This is almost never the case in real applications, of course, so estimated weights must be used instead. The effect of using estimated weights is difficult to assess in general, but experience indicates that the results of most regression analyses are not very sensitive to the weights used. The benefits of a weighted analysis are often obtained in large degree, though not fully, with approximate weights. It is important to remain aware of this problem, however, and to only use weights when they can be estimated precisely relative to one another.
	Weighted least squares regression, like the other least squares methods, is also sensitive to to the effects of outliers. Though it provides a convenient way to elminate outliers, it is not a robust procedure itself. If potential outliers are not investigated and dealt with, they will likely have a negative influence on the parameter estimation and other aspects of a weighted least squares analysis.