WhiteNoiseProcess
represents a Gaussian white noise process with mean 0 and standard deviation 1.
represents a Gaussian white noise process with mean 0 and standard deviation σ.
WhiteNoiseProcess[dist]
represents a white noise process based on the distribution dist.
Details
- WhiteNoiseProcess is also known as independent identically distributed (iid) process.
- WhiteNoiseProcess is a discrete-time random process.
- The slices of WhiteNoiseProcess are assumed to be independent and identically distributed random variables.
- The distribution dist can be any univariate distribution with mean 0 and finite variance.
- WhiteNoiseProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.
Examples
open all close allBasic Examples (1)
Scope (2)
Simulate a white noise process having normal slices with standard deviation 5:
𝒫 = WhiteNoiseProcess[5];data = RandomFunction[𝒫, {0, 30}];ListPlot[data, Filling -> Axis]𝒫 = WhiteNoiseProcess[UniformDistribution[{-1, 1}]];data = RandomFunction[𝒫, {0, 30}];ListPlot[data, Filling -> Axis]Discrete uniform distribution slices:
𝒫 = WhiteNoiseProcess[DiscreteUniformDistribution[{-1, 1}]];data = RandomFunction[𝒫, {0, 30}];ListPlot[data, Filling -> Axis]𝒫 = WhiteNoiseProcess[MixtureDistribution[{0.6, 0.4}, {NormalDistribution[], NormalDistribution[0, 3]}]];sample = RandomFunction[𝒫, {0, 100}, 2];ListPlot[sample, Filling -> Axis]Estimate the process parameters from sample data:
EstimatedProcess[sample, WhiteNoiseProcess[MixtureDistribution[{a, b}, {NormalDistribution[], NormalDistribution[0, 3]}]]]Applications (2)
Add white noise to a periodic signal:
𝒫 = TransformedProcess[Cos[t / 8] + noise[t], noiseWhiteNoiseProcess[UniformDistribution[{-1 / 5, 1 / 5}]], t];data = RandomFunction[𝒫, {0, 200}];ListPlot[data, Filling -> Axis]Define a moving-average process:
𝒫 = TransformedProcess[x[t] + 2 x[t - 1] + 7x[t - 2], xWhiteNoiseProcess[], t];data = RandomFunction[𝒫, {2, 50}, 3];ListLinePlot[data, PlotRange -> All]Mean, variance, and kurtosis for the process:
{Mean[𝒫[t]], Variance[𝒫[t]], Kurtosis[𝒫[t]]}Compare with the property values for the corresponding MAProcess:
𝒫1 = MAProcess[{2, 7}, 1];{Mean[𝒫1[t]], Variance[𝒫1[t]], Kurtosis[𝒫1[t]]}Properties & Relations (6)
WhiteNoiseProcess is a discrete-time process:
ProcessTimeDomain[WhiteNoiseProcess[]]The states may either be continuous or discrete:
ProcessStateDomain[WhiteNoiseProcess[UniformDistribution[{-1, 1}]]]ProcessStateDomain[WhiteNoiseProcess[DiscreteUniformDistribution[{-1, 1}]]]SliceDistribution[WhiteNoiseProcess[dist],t] is equal to dist:
dist = NormalDistribution[0, 5];𝒫 = WhiteNoiseProcess[dist];𝒫[t]Multivariate slices are products of dist with itself:
𝒫[{s, t}]The slice mean is always zero:
Mean[WhiteNoiseProcess[][t]]Mean[WhiteNoiseProcess[UniformDistribution[{-a, a}]][t]]WhiteNoiseProcess is uncorrelated according to the AutocorrelationTest:
data = RandomFunction[WhiteNoiseProcess[], {0, 10 ^ 5}];AutocorrelationTest[data, 1, "TestConclusion"]Gaussian white noise is a special case of an MAProcess:
Mean[#[t]]& /@ {MAProcess[{}, σ ^ 2], WhiteNoiseProcess[NormalDistribution[0, σ]]}CovarianceFunction[MAProcess[{}, σ ^ 2], h]CovarianceFunction[WhiteNoiseProcess[NormalDistribution[0, σ]], h]% - %%//SimplifyPossible Issues (1)
EstimatedProcess fails for this example involving white noise from a uniform distribution:
sample = RandomFunction[WhiteNoiseProcess[UniformDistribution[{-2, 2}]], {1, 10}];EstimatedProcess[sample, WhiteNoiseProcess[UniformDistribution[{a, b}]]]
Using a symmetric interval for UniformDistribution helps in this case:
EstimatedProcess[sample, WhiteNoiseProcess[UniformDistribution[{-a, a}]]]Neat Examples (1)
A family of white noise processes:
𝒫1 = WhiteNoiseProcess[];
𝒫2 = WhiteNoiseProcess[3];
𝒫3 = WhiteNoiseProcess[UniformDistribution[{-2, 2}]];
𝒫4 = WhiteNoiseProcess[DiscreteUniformDistribution[{-2, 2}]];
𝒫5 = WhiteNoiseProcess[MixtureDistribution[{3, 7}, {UniformDistribution[{-3, 3}], UniformDistribution[{-1, 1}]}]];
𝒫6 = WhiteNoiseProcess[MixtureDistribution[{2, 3}, {NormalDistribution[], NormalDistribution[0, 4]}]];data = RandomFunction[#, {0, 50}]& /@ {𝒫1, 𝒫2, 𝒫3, 𝒫4, 𝒫5, 𝒫6};Table[ListPlot[data[[i]], Filling -> Axis, PlotLabel -> StringJoin["𝒫", ToString[i]]], {i, 6}]Text
Wolfram Research (2014), WhiteNoiseProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/WhiteNoiseProcess.html.
CMS
Wolfram Language. 2014. "WhiteNoiseProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WhiteNoiseProcess.html.
APA
Wolfram Language. (2014). WhiteNoiseProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WhiteNoiseProcess.html