WOLFRAM

Wolfram Language & System Documentation Center

LocalModelFit[data]

creates a smooth approximation of data.

LocalModelFit[data,bw]

approximates data using the bandwidth bw.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Sparse Data  
See Also
Related Guides
History
Cite this Page

LocalModelFit

LocalModelFit[data]

creates a smooth approximation of data.

LocalModelFit[data,bw]

approximates data using the bandwidth bw.

Details and Options

  • LocalModelFit, also known as LOESS and adaptive smoothing, creates an approximation of a dataset using local polynomial fitting.
  • Local fitting is typically used for data smoothing in time series analysis, noise reduction in scientific measurements and trend analysis in economic data to reveal underlying patterns without overfitting.
  • For every sample point, the local fitting mechanism involves fitting a low-degree polynomial to a subset of data. Combining the fitted polynomials forms a continuous approximation of the entire dataset.
  • The bandwidth parameter bw determines the scale of locality, where smaller values emphasize local features and larger values smooth out broader trends.
  • For maximum precision, the fit can be computed at every sample point. For faster evaluation, the fit can be precomputed on a fixed number of samples, while the rest is obtained via interpolation.
  • Possible forms of data are:
  • {y1,y2,…}equivalent to the form {{1,y1},{2,y2},…}
    {{x11,x12,…,y1},…}a list of independent values xij and the responses yi
    {{x11,x12,…}y1,…}a list of rules between input values and responses
    {{x11,x12,…},…}{y1,y2,…}a rule between a list of input values and responses
    {{x11,…,y1,…},…}nfit the n^(th) column of a matrix
    Tabular[…]namefit the column name in a tabular object
  • Possible values for bw are:
  • dan absolute size d
    Scaled[f]a fraction f of the points in data
  • The following options can be given:
  • ComputeUncertaintyTruewhether to compute confidence bands
    DistanceFunctionEuclideanDistancethe distance metric to use
    FitDegree1degree of the polynomial interpolation
    InterpolationPointsAutomaticwhere the function should be interpolated
  • The value of FitDegree controls the degree of the polynomial interpolation.
  • The value of InterpolationPoints determines the number and placement of the interpolation points.
  • Possible settings for InterpolationPoints include:
  • Automaticautomatically determine the sampling points (default)
    Allsample on data
    nuse n equally spaced points
    {x1,…}explicit univariate values
    {{x11,…},…}explicit multivariate values
    Nonedo not interpolate
  • With InterpolationPointsNone, the actual fit is computed when evaluating the FittedModel.
    • Properties
  • The properties and diagnostics of the FittedModel can be obtained from model["property"].
  • Properties related to the fitted function include:
  • {"BestFit",vars}fitted function at vars
    {"BestFitAround",vars}fitted function with confidence intervals at vars
    {"BestFitDataAround",vars}fitted function with prediction intervals at vars
    "Function"best-fit interpolating function
  • Properties related to data include:
  • "Data"the input data or design matrix and response vector
    "DesignMatrix"design matrix for the model
    "Response"response values in the input data
  • Properties of predicted values include:
  • "FitResiduals"difference between actual and predicted responses
    {"MeanPredictionBands",vars}confidence bands for mean predictions at vars
    "MeanPredictions"confidence intervals for the mean predictions
    "PredictedResponse"fitted values for the data
    {"SinglePredictionBands",vars}confidence bands based on single observations at vars
    "SinglePredictions"confidence intervals for a single predicted response
  • Use model["Properties"] to obtain a list of all the supported properties.

Examples

open all close all

Basic Examples  (1)

Create a smooth interpolation from a dataset:

Wolfram Language code: data = {...};
Wolfram Language code: trend = LocalModelFit[data]
Wolfram Language code: Show[ListPlot[data], Plot[trend[x], {x, 0, 2Pi}, PlotRange -> All, PlotStyle -> Red], PlotRange -> All]

Scope  (5)

Specify different smoothing bandwidths:

Wolfram Language code: Plot[Evaluate@Table[LocalModelFit[{...}, bw][{"BestFit", x}], {bw, {.2, .5, 2}}], {x, 0, 2Pi}, PlotLegends -> {.2, .5, 2}]
Wolfram Language code: Plot[Evaluate@Table[LocalModelFit[{...}, Scaled[bw]][{"BestFit", x}], {bw, {.1, .3, .7}}], {x, 0, 2Pi}, PlotLegends -> {.2, .5, 2}]

Compute fit uncertainties:

Wolfram Language code: fit = LocalModelFit[{...}]
Wolfram Language code: ListPlot[Table[fit[{"BestFitAround", x}], {x, 0, 2Pi, .1}], IntervalMarkers -> "Bands"]

Compute an interpolation at specific points:

Wolfram Language code: data = {...}; smooth = LocalModelFit[data, InterpolationPoints -> Range[0, 2Pi, .2]];
Wolfram Language code: ListPlot[{data, Table[{x, smooth[x]}, {x, 0, 6, .1}]}, Joined -> {False, True}]

Compute a model that can be supports runtime fitting at specific points:

Wolfram Language code: data = {...};
Wolfram Language code: model = LocalModelFit[data, InterpolationPoints -> None]
Wolfram Language code: Show[ListPlot[data], Plot[model[x], {x, 0, 2Pi}, PlotPoints -> 20]]

Interpolate 2D points:

Wolfram Language code: data = Flatten[Table[{x, y, Sin[x]Sqrt@y Cos[y] + 0.1 RandomVariate[NormalDistribution[0, 5]]}, {x, 0, 6, 0.2}, {y, 0, 6, 0.2}], 1];
Wolfram Language code: ListPointPlot3D[data]

Build a smoothed version of the dataset on the sampling grid:

Wolfram Language code: smoothdata = LocalModelFit[data]
Wolfram Language code: Show[ListPointPlot3D[data], Plot3D[smoothdata[{x, y}], {x, 0, 6}, {y, 0, 6}]]

Applications  (1)

Sparse Data  (1)

Test the regression on a a sparse sub-sampling of a dataset:

Wolfram Language code: data = {...};
Wolfram Language code: ListPointPlot3D[data]
Wolfram Language code: sample = RandomSample[data, 50];
Wolfram Language code: ListPointPlot3D[sample]

Build a small, regular sampling grid:

Wolfram Language code: grid = Flatten[Table[{x, y}, {x, 0, 6, .2}, {y, 0, 6, .2}], 1];
Wolfram Language code: ListPlot[grid]

Approximate the data samples on the grid point:

Wolfram Language code: approx = LocalModelFit[sample, 2, InterpolationPoints -> grid];

Compare a contour plot of the original data, the sample and the sample approximation:

Wolfram Language code: ListContourPlot /@ {data, sample}
Wolfram Language code: ContourPlot[approx[x, y], {x, 0, 6}, {y, 0, 6}]
Wolfram Research (2025), LocalModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/LocalModelFit.html.

Text

Wolfram Research (2025), LocalModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/LocalModelFit.html.

CMS

Wolfram Language. 2025. "LocalModelFit." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LocalModelFit.html.

APA

Wolfram Language. (2025). LocalModelFit. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LocalModelFit.html

BibTeX

@misc{reference.wolfram_2026_localmodelfit, author="Wolfram Research", title="{LocalModelFit}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/LocalModelFit.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_localmodelfit, organization={Wolfram Research}, title={LocalModelFit}, year={2025}, url={https://reference.wolfram.com/language/ref/LocalModelFit.html}, note=[Accessed: 12-June-2026]}