2. The Loess Method or Locally Weighted Regression Scatter Plot Smoothing Method

This smoothing technique is relatively more complex than band regression. The Loess method helps reveal the shape of the regression relationship by creating a smoothed curve from connecting each smoothed Y value that is generated from a local least squares regression function. We will clearly explain this method step by step:

1. Arrange pairs of X and Y values from the smallest value to the largest value based on the X value. For instance, let (X₁,Y₁) be a pair of X and Y values with the smallest X value, and (X₂,Y₂) be the second pair of X and Y values with the next smallest X value and so on.  The concept of arranging and grouping local X values is pretty much the same as in the band regression method. 
2. Each group of local X values will have its own fitted local regression function.  For example, the first least squares regression function can be obtained from a group of (X₁,Y₁), (X₂,Y₂), and (X₃,Y₃).  In practice, users have to decide the size of the group. Certainly, the bigger the size, the smoother the curve. However, you have to sacrifice some important features of the regression relationship.
3. So, each smoothed value of Y at a certain X value is actually the fitted value calculated by plugging in the value of a given X in its locally fitted least squares regression function.   For example, the locally fitted regression function in step 2 will provide the  smoothed value of Y at X₂

In addition, the Loess method has many other features to make smoothed curves smoother and less sensitive to extreme or outlying observations. These special features are:

1. The weight function offered by Cleveland is utilized in fitting linear regression. The Loess method assigns different weights to different X values in each local group. Within a certain group, the further apart of a X value is from the center X value, the less weight it will be given.
2. To reduce effects of outlying observations, the weights given to X values in each local group will be adjusted after the first fitting of the linear regression function. Of course, cases that produced huge residuals will receive smaller weights in the second fitting. Repeat this step at least two times to assure the robustness of the procedure for outlying observations.

As we mentioned earlier, smoothed curves are also very helpful in assuring our fitted linear regression function. The idea is the following if your smoothed curve appears to be within the confidence band generated from your fitted linear regression function, that means your linear regression function is valid.

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