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PowerSpectralDensity

PowerSpectralDensity[data,ω]

估计 data 的功率谱密度.

PowerSpectralDensity[data,ω,sspec]

估计平滑规范 sspec 下 data 的功率谱密度.

PowerSpectralDensity[tproc,ω]

表示时间序列过程 tproc 的功率谱密度.

范例

打开所有单元关闭所有单元

基本范例 (3)

估计某些数据的功率谱密度：

Wolfram Language code: PowerSpectralDensity[Range[10], ω]

计算单变量时间序列的功率谱密度：

Wolfram Language code: PowerSpectralDensity[ARProcess[{a}, σ^2], ω]

来自自回归时间序列的随机样本的样本功率谱密度：

Wolfram Language code: data = RandomFunction[ARProcess[{.2}, .1], {1, 1000}];

计算给定截至点时的功率谱密度：

Wolfram Language code:

cutoffs = {5, 10, 17};
spec = Table[PowerSpectralDensity[data, ω, i], {i, cutoffs}];

Wolfram Language code: Plot[Evaluate@spec, {ω, -π, π}, PlotLegends -> cutoffs]

范围 (14)

经验估计 (4)

估计一个单变量时间序列的功率谱密度：

Wolfram Language code:

data = TemporalData[TimeSeries, {{{-0.0614065187043199, 2.6306604608936857, 3.9465133828074777, 
    3.745705410210661, 3.049768958525431, 1.4003183889805526, -0.755044374280679, 
    -2.0834316975526366, -1.3583280202011117, 0.14932145182161072, -0.8588219015885377}}, 
  {{0, 10, 1}}, 1, {"Continuous", 1}, {"Discrete", 1}, 1, 
  {ValueDimensions -> 1, ResamplingMethod -> None}}, False, 10.1];

Wolfram Language code: PowerSpectralDensity[data, ω]

Wolfram Language code: Plot[%, {ω, -π, π}, Filling -> Axis, PlotRange -> All]

向量时间序列的功率谱密度：

Wolfram Language code:

data = TemporalData[TimeSeries, {{{{-0.8805412843977076, 0.5330873177260956}, 
    {-1.091904325250007, 2.8824771870568764}, {0.4712280848587277, -2.563788099997642}, 
    {-1.0458566604266817, 1.2863067750997783}, {0.15937341588902115, 0.0582751907559111 ... 0441, -1.4624411116452787}, 
    {-1.2038000803608786, 0.6266434204415561}, {-1.5532848855025663, -1.4784686973103551}}}, 
  {{0, 10, 1}}, 1, {"Continuous", 1}, {"Discrete", 1}, 2, 
  {ValueDimensions -> 2, ResamplingMethod -> None}}, False, 10.1];

Wolfram Language code: psd = PowerSpectralDensity[data, ω]//Simplify

各个分量的功率谱密度：

Wolfram Language code: Table[Plot[psd[[i, i]], {ω, -π, π}], {i, 1, 2}]

各分量间的互功率谱密度：

Wolfram Language code:

Table[ParametricPlot[{Re[#], Im[#]}&@psd[[i, Mod[i, 2] + 1]], {ω, -π, π}, ColorFunction -> Function[{x, y, ω}, (ColorData["Rainbow"][ω])]], {i, 1, 2}]

估算一组路径的功率谱密度：

Wolfram Language code: data = RandomFunction[ARProcess[{.8}, 1], {0, 100}, 5];

Wolfram Language code: psd = PowerSpectralDensity[data, ω];

Wolfram Language code: Plot[psd, {ω, -π, π}, PlotRange -> All, Filling -> Axis]

比较经验和理论功率谱密度函数：

Wolfram Language code: proc = MAProcess[{.4, .3, .5, .6, .3}, 1.];

Wolfram Language code: data = RandomFunction[proc, {0, 10 ^ 2}];

Wolfram Language code:

Plot[#, {ω, -π, π}, Filling -> Axis, PlotRange -> All]& /@ {PowerSpectralDensity[proc, ω], PowerSpectralDensity[data, ω]}

平滑 (5)

截止于5，获取平滑处理过的估算功率谱密度：

Wolfram Language code: data = Range[10];

Wolfram Language code: sspec = PowerSpectralDensity[data, ω, 5]

将平滑处理过的功率谱密度和原来的进行比较：

Wolfram Language code: Plot[{Evaluate@PowerSpectralDensity[data, ω], sspec}, {ω, -π, π}, Filling -> Axis, PlotRange -> All]

使用 NuttallWindow 计算功率谱密度：

Wolfram Language code: data = Range[10];

Wolfram Language code: sspec = PowerSpectralDensity[data, ω, NuttallWindow]

将平滑处理过的功率谱密度和原来的进行比较：

Wolfram Language code: Plot[{Evaluate@PowerSpectralDensity[data, ω], sspec}, {ω, -π, π}, Filling -> Axis, PlotRange -> All]

用纯函数定义窗函数：

Wolfram Language code: data = Range[10];

Wolfram Language code: sspec = PowerSpectralDensity[data, ω, HannPoissonWindow[#, (1/2)]&]

将平滑处理过的功率谱密度和原来的进行比较：

Wolfram Language code: Plot[{Evaluate@PowerSpectralDensity[data, ω], sspec}, {ω, -π, π}, Filling -> Axis, PlotRange -> All]

用指定窗函数值估计功率谱密度：

Wolfram Language code:

n = 10;
data = Range[n];
win = TukeyWindow[(#/2 * (n - 1)), (1/3)]& /@ Range[0, n - 1];

和用 TukeyWindow 作为窗函数的结果进行比较：

Wolfram Language code: PowerSpectralDensity[data, ω, win] === PowerSpectralDensity[data, ω, TukeyWindow]

将平滑处理过的功率谱密度和原来的进行比较：

Wolfram Language code:

Plot[{Evaluate@PowerSpectralDensity[data, ω], Evaluate@PowerSpectralDensity[data, ω, TukeyWindow]}, {ω, -π, π}, Filling -> Axis, PlotRange -> All]

给定截至点和窗函数，计算功率谱密度：

Wolfram Language code: data = Range[10];

Wolfram Language code: sspec = PowerSpectralDensity[data, ω, {5, BartlettWindow}]

将平滑处理过的功率谱密度和原来的进行比较：

Wolfram Language code: Plot[{Evaluate@PowerSpectralDensity[data, ω], sspec}, {ω, -π, π}, Filling -> Axis, PlotRange -> All]

随机过程 (5)

ARProcess 的功率谱密度：

Wolfram Language code: proc[a_] = ARProcess[{a}, 1];

Wolfram Language code: Plot[PowerSpectralDensity[#, w], {w, -π, π}, Filling -> Axis, PlotRange -> All]& /@ {proc[0.8], proc[-0.8]}

Wolfram Language code: PowerSpectralDensity[proc[a], w]

向量 ARProcess：

Wolfram Language code:

a = {{1 / 7, 0}, {1 / 3, 0}};
Σ = {{1, -1 / 3}, {-1 / 3, 1}};
proc = ARProcess[{a}, Σ];

Wolfram Language code:

psd = PowerSpectralDensity[proc, ω];
psd//MatrixForm

互谱密度：

Wolfram Language code:

ParametricPlot[{Re[#], Im[#]}&@psd[[1, 2]], {ω, -π, π}, ColorFunction -> Function[{x, y, ω}, (ColorData["Rainbow"][ω])]]

MAProcess 的功率谱密度：

Wolfram Language code: proc[b_] = MAProcess[{b}, 1];

Wolfram Language code: Plot[PowerSpectralDensity[#, w], {w, -π, π}, Filling -> Axis]& /@ {proc[0.9], proc[-0.9]}

Wolfram Language code: PowerSpectralDensity[proc[b], w]

向量 MAProcess：

Wolfram Language code:

a = {{1 / 10, 0}, {1 / 3, 0}};
Σ = {{1, 1 / 3}, {1 / 3, 1}};
proc = MAProcess[{a}, Σ];

Wolfram Language code:

psd = PowerSpectralDensity[proc, ω];
psd//MatrixForm

互谱密度：

Wolfram Language code:

ParametricPlot[{Re[#], Im[#]}&@psd[[1, 2]], {ω, -π, π}, ColorFunction -> Function[{x, y, ω}, (ColorData["Rainbow"][ω])], AspectRatio -> 1]

ARMAProcess 的功率谱密度：

Wolfram Language code: proc[a_, b_] = ARMAProcess[{a}, {b}, 1];

Wolfram Language code: Plot[PowerSpectralDensity[#, w], {w, -π, π}, Filling -> Axis, PlotRange -> All]& /@ {proc[0.3, 0.2], proc[-0.3, -0.2]}

Wolfram Language code: PowerSpectralDensity[proc[a, b], w]

向量 ARMAProcess：

Wolfram Language code:

a = {{1 / 9, 0}, {1 / 3, 1 / 2}};
b = {{1, 3 / 4}, {1 / 2, -1 / 4}};
Σ = {{1, 1 / 3}, {1 / 3, 1}};
proc = ARMAProcess[{a}, {b}, Σ];

Wolfram Language code:

psd = PowerSpectralDensity[proc, ω];
psd//FullSimplify//MatrixForm

互谱密度：

Wolfram Language code:

ParametricPlot[{Re[#], Im[#]}&@psd[[1, 2]], {ω, -π, π}, ColorFunction -> Function[{x, y, ω}, (ColorData["Rainbow"][ω])], AspectRatio -> 1]

部份整合时间序列的功率谱密度：

Wolfram Language code: Plot[PowerSpectralDensity[FARIMAProcess[{.3, -.2}, -.4, {1}, 1], w], {w, -π, π}, Filling -> Axis, PlotRange -> All]

Wolfram Language code: PowerSpectralDensity[FARIMAProcess[{.3, .2}, .4, {1}, 1], w]

向量 FARIMAProcess：

Wolfram Language code:

a = {{1 / 9, 0}, {1 / 3, 1 / 2}};
b = {{1, 1 / 4}, {1 / 2, 0}};
Σ = {{1, 0}, {0, 1}};
proc = FARIMAProcess[{a}, {1 / 3, -1 / 4}, {b}, Σ];

Wolfram Language code:

psd = PowerSpectralDensity[proc, ω];
psd//FullSimplify//MatrixForm

互谱密度：

Wolfram Language code:

ParametricPlot[{Re[#], Im[#]}&@psd[[1, 2]], {ω, -π, π}, ColorFunction -> Function[{x, y, ω}, (ColorData["Rainbow"][ω])], AspectRatio -> 1]

季节性时间序列的功率谱密度：

Wolfram Language code: Plot[PowerSpectralDensity[SARMAProcess[{.2}, {.4}, {12, {.3}, {1}}, 1], w], {w, -π, π}, Filling -> Axis]

Wolfram Language code: PowerSpectralDensity[SARMAProcess[{a}, {b}, {12, {g}, {h}}, σ^2], ω]

向量 SARMAProcess：

Wolfram Language code:

a = {{1 / 9, 0}, {1 / 3, 1 / 2}};
b = {{1 / 3, 1 / 4}, {1 / 2, 0}};
Σ = {{1, 0}, {0, 1}};
proc = SARMAProcess[{}, {b}, {4, {b}, {}}, Σ];

Wolfram Language code:

psd = PowerSpectralDensity[proc, ω];
psd//FullSimplify//MatrixForm

互谱密度：

Wolfram Language code:

ParametricPlot[{Re[#], Im[#]}&@psd[[1, 2]], {ω, -π, π}, ColorFunction -> Function[{x, y, ω}, (ColorData["Rainbow"][ω])], AspectRatio -> 1]

选项 (2)

FourierParameters 的缺省值：

Wolfram Language code: PowerSpectralDensity[Range[10], ω, FourierParameters -> {1, 1}]

Wolfram Language code: PowerSpectralDensity[Range[10], ω]

Wolfram Language code: % - %%

改变 FourierParameters：

Wolfram Language code: PowerSpectralDensity[Range[10], ω, FourierParameters -> {-1, 1}]

缺省值被缩放了：

Wolfram Language code: PowerSpectralDensity[Range[10], ω]

Wolfram Language code: % / %%//Simplify

应用 (1)

用功率谱密度估计时间序列过程：

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

Wolfram Language code: EstimatedProcess[data, ARProcess[2], ProcessEstimator -> "SpectralEstimator"]

使用平滑窗：

Wolfram Language code: EstimatedProcess[data, ARProcess[2], ProcessEstimator -> {"SpectralEstimator", "Window" -> 5}]

属性和关系 (11)

时间序列的功率谱密度是 CovarianceFunction 的变换：

Wolfram Language code:

proc = ARProcess[c, {a}, σ^2];
cf = CovarianceFunction[proc, h]

使用 FourierSequenceTransform：

Wolfram Language code: ft = FourierSequenceTransform[cf, h, z]

和功率谱比较：

Wolfram Language code: PowerSpectralDensity[proc, z]

Wolfram Language code: FullSimplify[% - ft]

向量时间序列：

Wolfram Language code: proc = ARProcess[{{{1 / 3, 11 / 10}, {1 / 5, 1 / 10}}}, {{1, 0}, {0, 1}}];

Wolfram Language code: cov = CovarianceFunction[proc, h]//PiecewiseExpand;

Wolfram Language code: fst = FourierSequenceTransform[cov, h, w]

Wolfram Language code: fst - PowerSpectralDensity[proc, w]//Simplify

数据的功率谱密度是样本 CovarianceFunction 的变换：

Wolfram Language code:

sample = Range[20];
n = Length[sample];
cov = CovarianceFunction[sample, {-n + 1, n - 1}];

运用 ListFourierSequenceTransform：

Wolfram Language code: ListFourierSequenceTransform[cov, w, -n + 1]//Simplify

和 SamplePowerSpectralDensity 比较：

Wolfram Language code: PowerSpectralDensity[sample, w]//Simplify

Wolfram Language code: % - %%//FullSimplify

向量值时间序列：

Wolfram Language code:

data = RandomFunction[ARProcess[{{{.3, .1}, {.2, .1}}}, {{1, 0}, {0, 1}}], {10}];
n = data["PathLength"];

Wolfram Language code: cov = CovarianceFunction[data, {-n + 1, n - 1}]["Values"];

Wolfram Language code: lfst = Table[ListFourierSequenceTransform[cov[[All, i, j]], w, -n + 1], {i, 1, 2}, {j, 1, 2}]

Wolfram Language code: PowerSpectralDensity[data, w] - lfst//Simplify//Chop

白噪声的功率谱：

Wolfram Language code: FourierSequenceTransform[Piecewise[{{1, x == 0}}, 0], x, w]

和 MAProcess 的特例进行比较：

Wolfram Language code: AbsoluteCorrelationFunction[MAProcess[{}, 1], h]

Wolfram Language code: PowerSpectralDensity[MAProcess[{}, 1], w]

进行积分以求出方差：

Wolfram Language code:

Integrate[PowerSpectralDensity[ARProcess[{a}, σ^2], w, FourierParameters -> {-1, 1}], {w, -π, π}, Assumptions -> σ > 0 && -1 ≤ a ≤ 1]

和时间序列的方差进行比较：

Wolfram Language code: CovarianceFunction[ARProcess[{a}, σ^2], 0]

Wolfram Language code: % - %%//Simplify

进行积分以求出样本二阶矩：

Wolfram Language code:

data = TemporalData[TimeSeries, {{{0.30013118796154015, 0.4256944865636253, -1.2546039329698115, 
    0.09639437311894994, -1.9173223832884778, 0.16259596973060586, -0.001109688881539693, 
    0.6619148052033135, 2.232751727745077, 0.06794816837802453, 1. ... 5330770230747, 0.22924102933021667, 0.1509993303482705, 
    -0.4613641689767734}}, {{1, 100, 1}}, 1, {"Continuous", 1}, {"Discrete", 1}, 1, 
  {ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}, ValueDimensions -> 1}}, False, 
 10.1];

Wolfram Language code: NIntegrate[PowerSpectralDensity[data, w, FourierParameters -> {-1, 1}], {w, -π, π}]

和样本二阶矩进行比较：

Wolfram Language code: CovarianceFunction[data, 0]

Wolfram Language code: % - %%//Chop

谐波频率的功率谱密度和 PeriodogramArray 相关：

Wolfram Language code:

data = Range[20];
n = Length[data];
spec = Table[PowerSpectralDensity[data, 2π j / n], {j, 0, n - 1}]//N

和 PeriodogramArray 比较：

Wolfram Language code: periodogram = PeriodogramArray[data]

频率为0：

Wolfram Language code: First[periodogram] - n * Mean[data] ^ 2

非零频率：

Wolfram Language code: Rest[spec] - Rest[periodogram]//Chop

向量数据的功率谱密度的对角元素：

Wolfram Language code:

data = TemporalData[TimeSeries, {{{{0.22321490696028234, -0.09955096694103477}, 
    {-1.2036012283524826, 0.21356213785864084}, {-2.170758245872314, -2.172602353385546}, 
    {-2.0417769517615256, -0.9854153357651101}, {-1.8679940757580273, 0.11399938753 ... 3335557631977, 1.4574116828470514}, {-0.6330195762966424, -0.6361997810358189}}}, 
  {{0, 10, 1}}, 1, {"Continuous", 1}, {"Discrete", 1}, 2, 
  {ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}, ValueDimensions -> 2}}, False, 
 10.1];

Wolfram Language code: data["ValueDimensions"]

Wolfram Language code: psd = PowerSpectralDensity[data, ω];

将各数据分量的功率谱密度和单变量功率谱密度比较:

Wolfram Language code: dens = Table[PowerSpectralDensity[data["Values"][[All, i]], ω], {i, 2}];

Wolfram Language code:

Table[Plot[{psd[[i, i]], dens[[i]]}, {ω, -π, π}, PlotStyle -> {Automatic, Dotted}, PlotLegends -> {"Vector Component", "Univariate"}], {i, 2}]

向量过程的功率谱密度关于零点共轭对称：

Wolfram Language code: proc = ARProcess[{{{1 / 3, 1 / 10}, {1 / 5, 1 / 10}}}, {{1, 0}, {0, 1}}];

Wolfram Language code: Conjugate /@ PowerSpectralDensity[proc, -2]

Wolfram Language code: PowerSpectralDensity[proc, 2]

Wolfram Language code: % - %%

单变量过程的功率谱密度关于零点对称：

Wolfram Language code: proc = ARMAProcess[{a}, {b}, σ^2];

Wolfram Language code: PowerSpectralDensity[proc, w] == PowerSpectralDensity[proc, -w]

向量过程的功率谱密度是厄尔米特矩阵：

Wolfram Language code: proc = ARMAProcess[{{{1 / 3, 1 / 10}, {1 / 5, 1 / 10}}}, {{{1 / 8, -3 / 10}, {3 / 5, 1 / 6}}}, {{1, -.3}, {-.3, 1}}];

Wolfram Language code: HermitianMatrixQ[PowerSpectralDensity[proc, RandomReal[{-π, π}]]]

而且还是非负定矩阵：

Wolfram Language code: PositiveSemidefiniteMatrixQ[PowerSpectralDensity[proc, RandomReal[{-π, π}]]]

样本互谱密度的幅值由每个分量给出：

Wolfram Language code:

data = TemporalData[EventSeries, 
 {{{{0.4294925809742698255912415200470240013516218314553046886407`30., 
     -1.7225955472058514396692544902832573500729728703324221048`30.}, 
    {0.0548388212316276096199767991813010875210668838810634829247`30., 
     0 ... 255531611908244000251642161167860416322414`30., 
     0.0404300321804073415502129760700924797739015008346027682413`30.}}}, {{0, 20, 1}}, 1, 
  {"Continuous", 1}, {"Discrete", 1}, 2, {ResamplingMethod -> None, ValueDimensions -> 2}}, False, 
 10.1];

Wolfram Language code: data["ValueDimensions"]

Wolfram Language code: psd = PowerSpectralDensity[data, 3]//Chop

Wolfram Language code: Abs[psd[[1, 2]]] ^ 2 - Times@@Diagonal[psd]

样本功率谱密度的行列式总是等于0：

Wolfram Language code: FindMaximum[Abs@Det[PowerSpectralDensity[data, w]], {w, -3, 3}]

用 TransferFunctionModel 来计算时间序列的 PowerSpectralDensity：

Wolfram Language code:

proc = ARMAProcess[{a}, {b}, σ^2];
psd[ω_] = PowerSpectralDensity[proc, ω]

定义传递函数：

Wolfram Language code: g[z_] = TransferFunctionModel[proc][z][[1, 1]]

计算谱密度：

Wolfram Language code: σ^2g[Exp[I ω]]g[Exp[-I ω]]//ComplexExpand//TrigExpand//Simplify

检验：

Wolfram Language code: % - psd[ω]//Simplify

巧妙范例 (1)

绘制两个功率谱密度乘积的三维图形：

Wolfram Language code:

proc = SARMAProcess[{.2}, {.4}, {24, {.3}, {.1}}, 1];
Plot3D[Evaluate[PowerSpectralDensity[proc, w] * PowerSpectralDensity[proc, z]], {w, -π, π}, {z, -π, π}, PlotRange -> All, ColorFunction -> "IslandColors", ViewPoint -> {-3, 3, .5}]

Top

	c	使用 c 作为截止点
	w	使用窗函数 w
	{c,w}	同时使用截止点和窗函数

	{1,1}		缺省设置
	{-1,1}		常用于时间序列
	{a,b}		通用设置

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PowerSpectralDensity

更多信息和选项

范例

基本范例 (3)

范围 (14)

经验估计 (4)

平滑 (5)

随机过程 (5)

选项 (2)

应用 (1)

属性和关系 (11)

巧妙范例 (1)

文本

CMS

APA

BibTeX

BibLaTeX

PowerSpectralDensity

更多信息和选项

范例

基本范例 (3)

范围 (14)

经验估计 (4)

平滑 (5)

随机过程 (5)

选项 (2)

应用 (1)

属性和关系 (11)

巧妙范例 (1)

参见

相关指南

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历史

文本

CMS

APA

BibTeX

BibLaTeX