How does a LOESS model do its prediction? I understand the theory behind LOESS, but how does it do prediction without coefficients?
I'd like to use LOESS prediction, but need to be able to explain it.
 A: LOESS isn't really trained like some other models, you need to keep the entire training data around to do predictions.
So say your training data is $T$, and you want to make a prediction at a point $x$.  The general algorithm is this:


*

*Take the $k$ data points in $T$ whose  x-coordinate is closest to $x$, call this set $N_{x, k}$.  This gives you a collection of data points close to $x$, this is what the Local is about in the name LOESS.

*Fit a weighted linear (or polynomial) regression using the training set $N_{x, k}$.

*Use the resulting regression to make a prediction at $x$, this is the LOESS smoothed value at $x$.


It's important to realize that you have to do this entire procedure at each value of $x$ you want to get the smoothed value of $y$ for.  To get the nice curve you often see drawn through a scatterplot, you need to set down a grid of evenly spaced points to smooth, and then draw a piecewise linear interpolation through those smoothed values.  
If you would like to do predictions efficiently from LOESS, you should do much the same.  Set down a grid of evenly spaced x-coordinates and find the smoothed value of $y$ at each of these x-coordinates.  Then interpolate these smoothed points piecewise linearly, and remember the linear equations for each segment.  To get a prediction at $x$, figure out what segment it lies along, and use the linear equation you remembered for that segment.
I left the weighting unspecified, and you can do much of anything.  According to Hastie and Tibshirani "Generalized Additive Models" (1990) a popular choice is the funny looking
$$ w(x') = \left(1 - \left(\frac{|x - x'|}{\max_{x' \in N_{x, k}} |x - x'|}\right)^3 \right)^3 I\left(\frac{|x - x'|}{\max_{x' \in N_{x, k}} |x - x'|} < 1\right) $$
and Wikipedia lists the "traditional"
$$ w(x') = (1 - |x - x'|^3)^3 I(|x - x'| < 1) $$
I don't know where either of these expressions comes from, but they tend to weight points close to the "test point" $x'$ evenly, and then fall off rapidly after you deviate too far.  The $I$ is an indicator function which evaluates to 1 or 0 depending on whether the condition $|x - x'| < 1$ is true or false.  It has the effect of cutting off the weighting completely after you deviate away from $x$ too far.
