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EstimatedProcess[data,proc]

estimates the parametric process proc from data.

EstimatedProcess[data,proc,{{p,p0},{q,q0},…}]

estimates the parameters p, q, … with starting values p0, q0, ….

EstimatedProcess[data,proc,iproc]

estimates process proc with starting values taken from the instantiated process iproc.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Parametric Processes  
Time Series Processes  
Queueing Processes  
Finite Markov Processes  
Options  
ProcessEstimator  
WorkingPrecision  
Applications  
Properties & Relations  
See Also
Related Guides
History
Cite this Page

EstimatedProcess

EstimatedProcess[data,proc]

estimates the parametric process proc from data.

EstimatedProcess[data,proc,{{p,p0},{q,q0},…}]

estimates the parameters p, q, … with starting values p0, q0, ….

EstimatedProcess[data,proc,iproc]

estimates process proc with starting values taken from the instantiated process iproc.

Details and Options

  • EstimatedProcess returns the symbolic process proc with parameter estimates inserted for any non-numeric values.
  • The data can be given in the following forms:
  • {s0,…}a path with state si at time i
    {{t0,s0},…}a path with state si at time ti
    TemporalData[…]one or several paths
  • The times ti and states si must belong to the time and state domain of the process proc.
  • The process proc can be any parametric scalar- or vector-valued process.
  • The following options can be given:
  • AccuracyGoalAutomaticthe accuracy sought
    ProcessEstimator Automaticwhat process parameter estimator to use
    PrecisionGoalAutomaticthe precision sought
    WorkingPrecision Automaticthe precision used in internal computations
  • The following basic settings can be used for ProcessEstimator:
  • Automaticautomatically choose the parameter estimator
    "MaximumLikelihood"maximize the log likelihood directly
    "MethodOfMoments"match covariance
  • Special settings for ProcessEstimator are documented under the individual random process reference pages.
  • Additional settings for time series processes include "MaximumConditionalLikelihood" and "SpectralEstimator".
  • Additional settings for HiddenMarkovProcess include "BaumWelch" and "ViterbiTraining".

Examples

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Basic Examples  (2)

Estimate the parameter of a PoissonProcess:

Wolfram Language code: data = RandomFunction[PoissonProcess[.4], {0, 10 ^ 2}];
Wolfram Language code: eproc = EstimatedProcess[data, PoissonProcess[λ]]

Compare simulations of the estimated process to the original data:

Wolfram Language code: sims = RandomFunction[eproc, {0, 10 ^ 2}, 3];
Wolfram Language code: ListLinePlot[Join[data["Paths"], sims["Paths"]], PlotStyle -> {Thick, Dashed, Dashed, Dashed}, PlotLegends -> {"data", "simulations"}]

Find parameters for an ARProcess:

Wolfram Language code: data = RandomFunction[ARProcess[{.2, .1, -.3}, .1], {1, 100}];
Wolfram Language code: eproc = EstimatedProcess[data, ARProcess[{a, b, c}, v]]

Compare correlation functions for data and the estimated process:

Wolfram Language code: ρdata = CorrelationFunction[data, {25}];
Wolfram Language code: ρproc = CorrelationFunction[eproc, {25}];
Wolfram Language code: ListLinePlot[{ρdata, ρproc}, DataRange -> {0, 25}, PlotLegends -> {"data", "process"}]

Scope  (9)

Parametric Processes  (3)

Estimate the parameters for a RandomWalkProcess:

Wolfram Language code: SeedRandom[1234];data = RandomFunction[RandomWalkProcess[0.3], {0, 400}];
Wolfram Language code: ListLinePlot[data]
Wolfram Language code: EstimatedProcess[data, RandomWalkProcess[p]]

Use starting value for the parameter:

Wolfram Language code: EstimatedProcess[data, RandomWalkProcess[p], {{p, .5}}]

Estimate the parameters for a RenewalProcess:

Wolfram Language code: data = RandomFunction[RenewalProcess[GammaDistribution[2, 1]], {0, 10 ^ 3}];
Wolfram Language code: ListLinePlot[data]
Wolfram Language code: EstimatedProcess[data, RenewalProcess[GammaDistribution[a, b]]]

Estimate the parameters for a Wiener process:

Wolfram Language code: data = RandomFunction[WienerProcess[.4, .7], {0, 100, 0.01}];
Wolfram Language code: ListLinePlot[data]
Wolfram Language code: EstimatedProcess[data, WienerProcess[μ, σ]]

Time Series Processes  (3)

Estimate the parameters for an ARProcess:

Wolfram Language code: proc = ARProcess[{-.2, .3, -.3}, .1];
Wolfram Language code: data = RandomFunction[proc, {0, 10 ^ 3}];
Wolfram Language code: eproc = EstimatedProcess[data, ARProcess[3]]

Compare covariance functions:

Wolfram Language code: DiscretePlot[Evaluate@(CovarianceFunction[#, h]& /@ {proc, eproc}), {h, 0, 10}, ExtentSize -> 1 / 2]

Estimate an ARMAProcess:

Wolfram Language code: proc = ARMAProcess[{.2, .3}, {0.26}, .1];
Wolfram Language code: data = RandomFunction[proc, {0, 1000}];
Wolfram Language code: eproc = EstimatedProcess[data, ARMAProcess[2, 1]]

Compare correlation functions:

Wolfram Language code: DiscretePlot[Evaluate@(CorrelationFunction[#, h]& /@ {proc, eproc}), {h, 0, 4}, ExtentSize -> 1 / 2]

Provide initial values for the estimation of an ARProcess:

Wolfram Language code: proc = ARProcess[{-.2, .3, -.3}, .1]; data = RandomFunction[proc, {0, 10 ^ 3}];
Wolfram Language code: EstimatedProcess[data, ARProcess[{a, b, c}, v], {{a, -.2}, {b, .3}, {c, -.3}, {v, .1}}]

The initial values can also be given as a corresponding model with numeric parameters:

Wolfram Language code: EstimatedProcess[data, ARProcess[{a, b, c}, v], proc]

Solve for repeated parameters:

Wolfram Language code: data = RandomFunction[ARProcess[{-.2, .3, -.3}, .1], {0, 10 ^ 3}];
Wolfram Language code: EstimatedProcess[data, ARProcess[{a, b, a}, v]]

Queueing Processes  (1)

Parameter estimation for an M/M/1 queue:

Wolfram Language code: 𝒬 = QueueingProcess[5., 13];
Wolfram Language code: data = RandomFunction[𝒬, {0, 200}];
Wolfram Language code: ListLinePlot[data]
Wolfram Language code: 𝒬1 = EstimatedProcess[data, QueueingProcess[λ, μ]]

Use the corresponding random path for the data:

Wolfram Language code: 𝒬2 = EstimatedProcess[data["Path"], QueueingProcess[λ, μ]]

Compare system sizes for the original and estimated processes:

Wolfram Language code: {QueueProperties[𝒬, "MeanSystemSize"], QueueProperties[𝒬2, "MeanSystemSize"]}

Finite Markov Processes  (2)

Estimate a four-state discrete Markov process:

Wolfram Language code: 𝒫 = DiscreteMarkovProcess[3, {{1 / 2, 1 / 2, 0, 0}, {1 / 2, 1 / 2, 0, 0}, {1 / 4, 1 / 4, 1 / 4, 1 / 4}, {0, 0, 0, 1}}];
Wolfram Language code: data = RandomFunction[𝒫, {0, 10 ^ 4}, 10];
Wolfram Language code: Map[MatrixForm, estproc = EstimatedProcess[data, DiscreteMarkovProcess[4], WorkingPrecision -> MachinePrecision]]

Estimate a four-state continuous Markov process:

Wolfram Language code: 𝒫 = ContinuousMarkovProcess[{1, 0, 0, 0}, (⁠| | | | | | -- | -- | -- | - | | -3 | 1 | 2 | 0 | | 3 | -6 | 2 | 1 | | 4 | 2 | -9 | 3 | | 0 | 0 | 0 | 0 |⁠)];
Wolfram Language code: data = RandomFunction[𝒫, {0, 10 ^ 2}, 10 ^ 3];
Wolfram Language code: Map[MatrixForm, estproc = EstimatedProcess[data, ContinuousMarkovProcess[4], WorkingPrecision -> MachinePrecision]]

Options  (5)

ProcessEstimator  (4)

Maximum likelihood for a parametric process:

Wolfram Language code: data = RandomFunction[WienerProcess[0, 1], {0, 100, .1}];
Wolfram Language code: EstimatedProcess[data, WienerProcess[a, b], ProcessEstimator -> "MaximumLikelihood"]

Time series process:

Wolfram Language code: data = RandomFunction[ARProcess[{.3, .2}, 1], {10 ^ 3}];
Wolfram Language code: EstimatedProcess[data, ARProcess[2], ProcessEstimator -> "MaximumLikelihood"]

Method of moments for a parametric process:

Wolfram Language code: data = RandomFunction[PoissonProcess[3], {0, 10 ^ 4}];
Wolfram Language code: EstimatedProcess[data, PoissonProcess[m], {{m, 3}}, ProcessEstimator -> "MethodOfMoments"]

Time series process:

Wolfram Language code: data = RandomFunction[ARMAProcess[{.3, .2}, {.5}, 1], {10 ^ 4}];
Wolfram Language code: EstimatedProcess[data, ARMAProcess[2, 1], ProcessEstimator -> "MethodOfMoments"]

Maximize conditional likelihood for a time series process:

Wolfram Language code: data = RandomFunction[ARMAProcess[{.3, .2}, {.5}, 1], {10 ^ 4}];
Wolfram Language code: EstimatedProcess[data, ARMAProcess[2, 1], ProcessEstimator -> "MaximumConditionalLikelihood"]

Spectral estimator for a time series process:

Wolfram Language code: data = RandomFunction[ARProcess[{.3, .2}, 1], {10 ^ 3}];
Wolfram Language code: EstimatedProcess[data, ARProcess[2], ProcessEstimator -> "SpectralEstimator"]

WorkingPrecision  (1)

Estimate a process using default machine precision:

Wolfram Language code: data = RandomFunction[QueueingProcess[5, 11], {0, 150}];
Wolfram Language code: EstimatedProcess[data, QueueingProcess[λ, μ]]

Use higher precision:

Wolfram Language code: data = RandomFunction[QueueingProcess[5, 11], {0, 150}, WorkingPrecision -> 20];
Wolfram Language code: EstimatedProcess[data, QueueingProcess[λ, μ], WorkingPrecision -> 20]

Applications  (2)

Model the mean daily temperature for Champaign in August 2012:

Wolfram Language code: temp = TemporalData[TimeSeries, {{{20.5, 20.89, 22.5, 27.44, 26.5, 20.33, 18.83, 23.06, 20.83, 19.72, 14.89, 15.28, 18.11, 18.72, 17.5, 20.83, 17.94, 13.56, 16.44, 14.33, 14.72, 15.83, 17.28, 19.89, 20.44, 21.39, 24.89, 22.78, 22.83, 20.22, 24.4 ... te", 1}, 1, {ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}, ValueDimensions -> 1, MetaInformation -> {"Source" -> HoldForm[WeatherData["Champaign", "MeanTemperature", {{2012, 8}, {2012, 8}, "Day"}]]}}}, True, 10.1];
Wolfram Language code: temp["Source"]
Wolfram Language code: DateListPlot[temp, Joined -> True, Filling -> Axis]

Find process parameters:

Wolfram Language code: eproc = EstimatedProcess[temp, ARProcess[20]]

Compare CorrelationFunction of the model and the data:

Wolfram Language code: ListPlot[TemporalData[CorrelationFunction[#, {30}]& /@ {eproc, temp}], Filling -> {1 -> {2}}, PlotStyle -> PointSize[Medium], PlotLegends -> {"Model", "Data"}]

Forecast the daily exchange rates of the euro to the dollar from May 2012 through September 2012:

Wolfram Language code: data = TemporalData[TimeSeries, {{{1.31, 1.31, 1.31, 1.3, 1.3, 1.3, 1.29, 1.29, 1.29, 1.27, 1.28, 1.28, 1.27, 1.26, 1.26, 1.26, 1.25, 1.24, 1.24, 1.23, 1.24, 1.24, 1.25, 1.26, 1.25, 1.26, 1.25, 1.25, 1.26, 1.26, 1.26, 1.26, 1.27, 1.27, 1.25, 1.2 ... yRange"]}, 1, {"Discrete", 1}, {"Discrete", 1}, 1, {ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}, MetaInformation -> {"Source" -> HoldForm[FinancialData]["EUR/USD", {{2012, 5, 1}, {2012, 9, 30}}]}}}, True, 10.1];
Wolfram Language code: data["Source"]
Wolfram Language code: DateListPlot[data, Joined -> True, Filling -> Bottom]

Fit an AR process to the exchange rates:

Wolfram Language code: eproc = EstimatedProcess[data, ARProcess[3]]

Forecast for 20 business days ahead:

Wolfram Language code: forecast = TimeSeriesForecast[eproc, data, {20}]

Plot the forecast with original data:

Wolfram Language code: DateListPlot[{data, forecast}, Joined -> True, Filling -> Bottom]

Properties & Relations  (2)

EstimatedProcess estimates a parametric process:

Wolfram Language code: data = RandomFunction[PoissonProcess[3], {0, 10 ^ 4}];
Wolfram Language code: EstimatedProcess[data, PoissonProcess[m]]

FindProcessParameters returns a list of parameter estimates for the process:

Wolfram Language code: FindProcessParameters[data, PoissonProcess[m]]

EstimatedProcess estimates a parametric process:

Wolfram Language code: data = RandomFunction[BernoulliProcess[0.4], {0, 10 ^ 5}];
Wolfram Language code: EstimatedProcess[data, BernoulliProcess[p]]

EstimatedDistribution estimates a parametric distribution:

Wolfram Language code: data = RandomVariate[BernoulliDistribution[0.4], 10 ^ 5];
Wolfram Language code: EstimatedDistribution[data, BernoulliDistribution[p]]
Wolfram Research (2012), EstimatedProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/EstimatedProcess.html.

Text

Wolfram Research (2012), EstimatedProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/EstimatedProcess.html.

CMS

Wolfram Language. 2012. "EstimatedProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/EstimatedProcess.html.

APA

Wolfram Language. (2012). EstimatedProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EstimatedProcess.html

BibTeX

@misc{reference.wolfram_2026_estimatedprocess, author="Wolfram Research", title="{EstimatedProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/EstimatedProcess.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_estimatedprocess, organization={Wolfram Research}, title={EstimatedProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/EstimatedProcess.html}, note=[Accessed: 13-June-2026]}