Use cubic instead of linear interpolation in predict

[andreasnoack]

The cubic interpolation is what is suggested in Cleveland and Grosee (1991) and also what R uses. It makes the prediction function once differentiable which is what I think most people would expect from LOESS.

Also, fix bug in use of partialsort!. Only the q'th element was ensured to be at the right localtion instead of all the first q elements.

Closes #61

[andreasnoack]

The first test failure here is a bit surprising. For some reason the KD tree ends up not using the x values in

Loess.jl/test/runtests.jl

Lines 21 to 24 in 31d924a

	x = [13.0,14.0,14.35,15.0,16.0]
	y = [0.369486, 0.355579, 0.3545, 0.356952, 0.36883]
	model = loess(x,y)
	@test Loess.predict(model,x) ≈ y

as vertices. It ends up with more vertices. Specifically

julia> model.verts
Dict{Vector{Float64}, Int64} with 6 entries:
  [14.0]  => 1
  [14.35] => 2
  [13.0]  => 3
  [16.0]  => 4
  [13.5]  => 5
  [15.5]  => 6

which means that there is no longer a perfect match for the x values. The match in the current master version seems to be somewhat coincidental. I'll try to figure out what is going on.

[andreasnoack]

I think I've finally managed to figure out how to compute the splits, see #64. It was pretty tedious to reverse engineer. When that one is in, this PR can be updated and the broken test I added in #64 should then pass.

I've also realized that with this PR, we shouldn't store the polynomial coefficients from the local regressions. Instead, we should just store the predictions and the derivatives of the predictions at all the vertices. That is all we need to construct the cubic interpolations. It took me way too long to realize that these quantities are what R stores in the $kd$vval field.

[codecov-commenter]

Codecov Report

Patch coverage: 96.66% and project coverage change: +1.08 🎉

Comparison is base (306d5de) 92.11% compared to head (1e4e6ac) 93.20%.

Additional details and impacted files

@@            Coverage Diff             @@
##           master      #63      +/-   ##
==========================================
+ Coverage   92.11%   93.20%   +1.08%     
==========================================
  Files           2        2              
  Lines         203      206       +3     
==========================================
+ Hits          187      192       +5     
+ Misses         16       14       -2     

Impacted Files	Coverage Δ
src/Loess.jl	88.23% <96.66%> (+1.36%)	⬆️

... and 1 file with indirect coverage changes

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[andreasnoack]

I think this one is ready now.

[devmotion]

I guess it would be good to add some test for #61.

[andreasnoack]

I guess it would be good to add some test for #61.

Please check the updated cars test. It should ensure that we interpolate exactly like the original implementation. Fr reference, I've recreated the plot from #61 with this branch

[andreasnoack]

@devmotion I reapplied the changes you suggested

[devmotion]

Looks good to me!