title: loess predict with new x values
body: <p>I am attempting to understand how the <code>predict.loess</code> function is able to compute new predicted values (<code>y_hat</code>) at points <code>x</code> that do not exist in the original data.  For example (this is a simple example and I realize loess is obviously not needed for an example of this sort but it illustrates the point):</p>
  
  <pre><code>x &lt;- 1:10
  y &lt;- x^2
  mdl &lt;- loess(y ~ x)
  predict(mdl, 1.5)
  [1] 2.25
  </code></pre>
  
  <p><code>loess</code> regression works by using polynomials at each <code>x</code> and thus it creates a predicted <code>y_hat</code> at each <code>y</code>.  However, because there are no coefficients being stored, the "model" in this case is simply the details of what was used to predict each <code>y_hat</code>, for example, the <code>span</code> or <code>degree</code>.  When I do <code>predict(mdl, 1.5)</code>, how is <code>predict</code> able to produce a value at this new <code>x</code>?  Is it interpolating between two nearest existing <code>x</code> values and their associated <code>y_hat</code>? If so, what are the details behind how it is doing this?</p>
  
  <p>I have read the <code>cloess</code> documentation online but am unable to find where it discusses this.</p>