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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 Loess

Source Notebook

Smooth noisy data using local regression

Contributed by: Jon McLoone

ResourceFunction["Loess"][data, bandwidth, x]

finds the interpolation of data at x by using local regression smoothing of bandwidth data points.

ResourceFunction["Loess"][data, Scaled[bandwidth], x]

finds the interpolation of data at x by using local regression smoothing of bandwidth fraction of data points.

ResourceFunction["Loess"][data,bandwidth,{x1,x2,…}]

finds the interpolation of data at each xi by using local regression smoothing of bandwidth data points.

Details and Options

Input data can be low- or higher-dimensional.

Examples

Basic Examples (3) 

Loess is useful when data is noisy but has an underlying trend:

In[1]:=
data = Table[{i, Sin[i] + RandomReal[0.3]}, {i, 0, 2 \[Pi], 0.05}]; ListPlot[data]
Out[2]=

Find an estimated value for the data at x=2 using the nearest 12 data points:

In[3]:=
ResourceFunction["Loess"][data, 12, 2]
Out[3]=

Find an estimated value for the data at x=2 using the nearest 10% of the data:

In[4]:=
ResourceFunction["Loess"][data, Scaled[0.10], 2]
Out[4]=
In[5]:=
ListPlot[ Table[{x, ResourceFunction["Loess"][data, 40, x]}, {x, 0, 2 \[Pi], 0.2}]]
Out[5]=

Scope (1) 

Loess can handle higher-dimensional data:

In[6]:=
data = Flatten[ Table[{i, j, Sin[i + Sin[j]] + RandomReal[0.3]}, {i, 0, 2 \[Pi], 0.1}, {j, 0, 2 \[Pi], 0.1}], {1, 2}]; ListPlot3D[data]
Out[7]=
In[8]:=
ResourceFunction["Loess"][data, 30, {4, 4}]
Out[8]=
In[9]:=
ListPlot3D[ Flatten[Table[{x, y, ResourceFunction["Loess"][data, 20, {x, y}]}, {x, 0, 6, 0.2}, {y, 0, 6, 0.2}], {1, 2}]]
Out[9]=

Options (2) 

By default, Loess fits straight lines to subsets of data, but you can increase the interpolation order to capture different detail:

In[10]:=
data = Table[{i, Sin[i] + RandomReal[0.3]}, {i, 0, 2 \[Pi], 0.2}]; ListPlot[data]
Out[11]=
In[12]:=
ListLinePlot[ Table[{x, ResourceFunction["Loess"][data, Scaled[0.1], x]}, {x, 0, 2 \[Pi], 0.2}]]
Out[12]=
In[13]:=
ListLinePlot[ Table[{x, ResourceFunction["Loess"][data, Scaled[0.1], x, InterpolationOrder -> 2]}, {x, 0, 2 \[Pi], 0.2}]]
Out[13]=

Loess fitting involves weighting the local data. Typically, points further from the estimated point are given less weight:

In[14]:=
ListLinePlot[ Table[{x, ResourceFunction["Loess"][data, Scaled[0.2], x, "WeightsFunction" -> (1/(0.1 + Norm[#]) &)]}, {x, 0, 2 \[Pi], 0.2}]]
Out[14]=

Possible Issues (3) 

If you intend to use Loess to predict many values from the same data, then it is more efficient to find all values in a single request:

In[15]:=
data = Table[{i, Sin[i] + RandomReal[0.3]}, {i, 0, 2 \[Pi], 0.05}]; ListPlot[data]
Out[16]=

This method:

In[17]:=
xvals = Range[0, 2 \[Pi], 0.2]; ListPlot[ Transpose[{xvals, ResourceFunction["Loess"][data, 40, xvals]}]]
Out[13]=

Is faster than this method:

In[18]:=
ListPlot[ Table[{x, ResourceFunction["Loess"][data, 40, x]}, {x, 0, 2 \[Pi], 0.2}]]
Out[18]=

Publisher

Jon McLoone

Requirements

Wolfram Language 11.3 (March 2018) or above

Version History

  • 1.0.0 – 27 February 2019

License Information