Studentized Residuals结合High Leverage Points图形找出Outliers Points

An outlier is a point for which $y_i$ is far from the value predicted by the model. Outliers can arise for a variety of reasons, such as incorrect recording of an observation during data collection.

To address the outlier problem, instead of plotting the residuals, we can plot the studentized residuals, computed by dividing each residual $e_i$ by its estimated standard error. Observations whose studentized residuals are greater than 3 in absolute value are possible outliers.

在去除outliers后，要考察对于模型的RSE和$R^2$的改善性况是否显著。

另一方面，我们也可以通过考查outliers的leverage水平高低，来判断其对least square fit线（红线）的影响程度，即leverage水平越高，对least square fit线的影响越大。

$$h_i=\frac{1}{n}+\frac{(x_i-\bar{x})^2}{\sum^n_{i \prime =1}(x_{i\prime}-\bar{x})^2}, 1/n \le h_i \le 1$$

正常情况下the average leverage for all the observations $h_i = (p+1)/n$，如果大幅超越了$(p+1)/n$，则要怀疑这个点的high leverage的影响。如：

Left: observation 41 is a high leverage point, while 20 is not. The red line is the fit to all the data, and the blue line is the fit with observation 41 removed. Center: the read observation is not unusual in terms of its $X_1$ value or its $X_2$ value, but still falls outside the bulk of the data, and hence has high leverage. Right: observation 41 has a high leverage and a high residual.