LOESS

LOESS locality¶

True or false: say you obtain an estimate of the regression function, $\hat{f}(x)$, using LOESS applied to a dataset $\left(\left(X_{1},Y_{1}\right),\ldots,\left(X_{n},Y_{n}\right)\right)$. Then, due to the locality property, for any point $(X_{i},Y_{i})$ from the training data itself, we have that $\hat{f}(X_{i})=Y_{i}$.

LOESS data gremlin¶

Say you collect a dataset with one predictor variable, one response variable, and 100 samples. Assume the predictor values are unique among the samples (i.e. there are no two samples that have the exact same value for their input). Then you use loess with a span of .8 to predict the response on a test point $x_{0}$, greater than the predictor value of any train point. Then you realize you made a mistake in data collection and the response for the point in your data with the very lowest predictor value must be changed. Finally, you recalculate the loess prediction on the test point $x_{0}$. True or false: the prediction is unchanged.

LOESS data gremlin, II¶

Say you collect a dataset with one predictor variable, one response variable, and 100 samples. Assume the predictor values are unique among the samples (i.e. there are no two samples that have the exact same value for their input). Then you use loess with a span of .8 to predict the response on a test point $x_{0}$, greater than the predictor value of any train point. Then you realize you made a mistake in data collection and the response for the point in your data with the median predictor value must be changed. Finally, you recalculate the loess prediction on the test point $x_{0}$. True or false: the two predictions must be the same.