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InhomogeneousPoissonProcess[λ[t],t]

表示非齐次泊松过程，其中强度 λ[t] 是 t 的函数.

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范例  
基本范例  
范围  
基本用途  
过程切片性质  
应用  
属性和关系  
可能存在的问题  
巧妙范例  
参见
相关指南
历史
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InhomogeneousPoissonProcess

InhomogeneousPoissonProcess[λ[t],t]

表示非齐次泊松过程，其中强度 λ[t] 是 t 的函数.

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范例

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基本范例  (3)

模拟一个非齐次泊松过程：

Wolfram Language code: data = RandomFunction[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t], {0, 20}];
Wolfram Language code: ListStepPlot[data, Filling -> Axis]

均值和方差函数：

Wolfram Language code: Mean[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t][t]]
Wolfram Language code: Variance[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t][t]]

协方差函数：

Wolfram Language code: CovarianceFunction[InhomogeneousPoissonProcess[2 + t ^ 2, t], s, w]
Wolfram Language code: Plot3D[Evaluate@CovarianceFunction[InhomogeneousPoissonProcess[2 + t ^ 2, t], s, w], {s, 1, 10}, {w, 1, 10}, ColorFunction -> "Rainbow", PlotRange -> All, AxesLabel -> Automatic]

范围  (10)

基本用途  (5)

模拟一个轨道集成：

Wolfram Language code: data = RandomFunction[InhomogeneousPoissonProcess[(2 + Sin[t]) / 4, t], {0, 35}, 4]
Wolfram Language code: ListStepPlot[data, Filling -> Axis]

用任意精度进行模拟：

Wolfram Language code: RandomFunction[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t], {4}, WorkingPrecision -> 20]["Path"]

比较不同强度函数的轨道：

Wolfram Language code: f1[t_] = t; f2[t_] = 2 + 1 / (1 + t ^ 2); f3[t_] = (3 + Cos[t]) / 7;
Wolfram Language code: sample := (SeedRandom[3];RandomFunction[InhomogeneousPoissonProcess[#[t], t], {10}]&);
Wolfram Language code: ListStepPlot[sample[#], Filling -> Axis, PlotLabel -> #]& /@ {f1, f2, f3}

相关函数：

Wolfram Language code: CorrelationFunction[InhomogeneousPoissonProcess[2 + t, t], s1, s2]
Wolfram Language code: Simplify[%, s1 < s2]

绝对相关函数：

Wolfram Language code: AbsoluteCorrelationFunction[InhomogeneousPoissonProcess[Cos[t] + 2, t], s1, s2]
Wolfram Language code: Simplify[%, s1 < s2]

过程切片性质  (5)

单变量 SliceDistribution：

Wolfram Language code: DiscretePlot[Evaluate@Table[PDF[InhomogeneousPoissonProcess[2 + 1 / (1 + τ ^ 2), τ][t], x], {t, times = {1, 4, 8}}], {x, 0, 30}, PlotLegends -> ("t = " ~~ ToString[#]& /@ times)]
Wolfram Language code: InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t][t]

单变量概率密度：

Wolfram Language code: PDF[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t][t], x]

多时态切片分布：

Wolfram Language code: SliceDistribution[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t], {1, 3, 7}]//Mean

更高阶 PDF：

Wolfram Language code: PDF[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t][{1, 3, 7}], {x, y, z}]

计算一个表达式的期望：

Wolfram Language code: Expectation[x + E ^ (-x), xInhomogeneousPoissonProcess[2 + t, t][t]]

计算一个事件的概率：

Wolfram Language code: Probability[x[t] ^ 2 < t - 1, xInhomogeneousPoissonProcess[2 + t, t]]

Moment：

Wolfram Language code: 𝒫 = InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t];
Wolfram Language code: Moment[𝒫[t], r]

母函数：

Wolfram Language code: CharacteristicFunction[𝒫[t], w]
Wolfram Language code: MomentGeneratingFunction[𝒫[t], w]

CentralMoment 对于符号阶数没有解析式：

Wolfram Language code: CentralMoment[𝒫[t], r]
Wolfram Language code: CentralMoment[𝒫[t], 4]
Wolfram Language code: CentralMomentGeneratingFunction[𝒫[t], w]

FactorialMoment 与其母函数：

Wolfram Language code: FactorialMoment[𝒫[t], r]
Wolfram Language code: FactorialMomentGeneratingFunction[𝒫[t], w]

Cumulant 与其母函数：

Wolfram Language code: Cumulant[𝒫[t], r]
Wolfram Language code: CumulantGeneratingFunction[𝒫[t], w]

多时态切片分布的协方差：

Wolfram Language code: Covariance[InhomogeneousPoissonProcess[2 + t, t][{1, 2, 3}]]
Wolfram Language code: N[%]//MatrixForm

与模拟的结果比较：

Wolfram Language code: Covariance[RandomVariate[InhomogeneousPoissonProcess[2 + t, t][{1, 2, 3}], 10 ^ 4]]//N//MatrixForm

应用  (3)

模拟一个小快餐店的到达过程，如果客户的小时到达率为：

Wolfram Language code: arrivalrates = {{0, 8}, {1, 9}, {2, 7}, {4, 6}, {5, 12}, {6, 14}, {7, 11}, {8, 6}, {9, 4}, {10, 3}, {11, 8}, {12, 15}, {13, 12}, {14, 10}, {15, 8}, {16, 6}, {17, 12}, {18, 17}, {19, 15}, {20, 12}, {21, 6}, {22, 5}, {23, 7}};

使用线性插值获取到达过程的密度函数：

Wolfram Language code: λ[t_] = Interpolation[arrivalrates, InterpolationOrder -> 1][t];
Wolfram Language code: Plot[λ[t], {t, 0, 23}, Exclusions -> None, Filling -> Axis]

定义一个到达的非齐次泊松过程：

Wolfram Language code: 𝒫 = InhomogeneousPoissonProcess[λ[t], t];

每天有 200 多个客户光顾餐馆的概率：

Wolfram Language code: NProbability[x > 200, x𝒫[23]]

模拟一天的到达过程：

Wolfram Language code: data = RandomFunction[𝒫, {0, 23, 1}, 4];
Wolfram Language code: ListStepPlot[data]

使用模拟找到有效的每小时平均到达率：

Wolfram Language code: sim = RandomFunction[𝒫, {0., 23, 1}, 10 ^ 3]; slopes = Map[# / 23.&, sim["LastValues"]]; Mean[slopes]

定义非齐次泊松过程的平方：

Wolfram Language code: 𝒫 = TransformedProcess[p[t] ^ 2, pInhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t], t];

模拟此过程：

Wolfram Language code: data = RandomFunction[𝒫, {0, 30, 1}, 3]
Wolfram Language code: ListStepPlot[data]

过程切片的均值和方差：

Wolfram Language code: Mean[𝒫[t]]
Wolfram Language code: Variance[𝒫[t]]

带有韦伯失败率密度的非齐次泊松过程是众所周知的韦伯泊松过程：

Wolfram Language code: λ[α_, β_, t_] = Refine[HazardFunction[WeibullDistribution[α, β], t], t > 0]
Wolfram Language code: WeibullPoissonProcess = InhomogeneousPoissonProcess[λ[2 / 3, 1 / 2, t], t]

采样过程轨迹：

Wolfram Language code: paths = RandomFunction[WeibullPoissonProcess, {0, 30}, 6, Method -> "Direct"]
Wolfram Language code: ListStepPlot[paths]

使用模拟找到一天的有效平均强度比：

Wolfram Language code: sim = RandomFunction[WeibullPoissonProcess, {0, 30}, 10 ^ 3, Method -> "Direct"]; slopes = Map[# / 30.&, sim["LastValues"]]; Mean[slopes]

比较由比例函数积分给出的平均强度比：

Wolfram Language code: NIntegrate[λ[2 / 3, 1 / 2, t], {t, 0, 30}] / 30

属性和关系  (3)

InhomogeneousPoissonProcess 是一个跳跃过程：

Wolfram Language code: path = RandomFunction[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t], {0, 10}]; f = path["PathFunction"]; jumps = path["Times"];
Wolfram Language code: Plot[f[t], {t, 0, 10}, Exclusions -> jumps, ExclusionsStyle -> Red]

非齐次泊松过程不是弱平稳的：

Wolfram Language code: WeakStationarity[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t]]

平均函数不是常数：

Wolfram Language code: Mean[InhomogeneousPoissonProcess[2 + 1 / (1 + t ^ 2), t][t]]

带有常密度的非齐次泊松过程是 PoissonProcess：

Wolfram Language code: ipp = InhomogeneousPoissonProcess[λ, t];
Wolfram Language code: pp = PoissonProcess[λ];

比较单变量切片分布：

Wolfram Language code: Mean /@ {ipp[t], pp[t]}
Wolfram Language code: Variance /@ {ipp[t], pp[t]}
Wolfram Language code: PDF[#, x]& /@ {ipp[t], pp[t]}

多切片属性：

Wolfram Language code: CovarianceFunction[#, s, t]& /@ {ipp, pp}
Wolfram Language code: PDF[#, {x, y}]& /@ {ipp[{s, t}], pp[{s, t}]}

可能存在的问题  (1)

某些模拟方法需要有界密度函数：

Wolfram Language code: λ[t_] = Abs[Tan[t]] + 1; Plot[λ[t], {t, 0, 3}, PlotRange -> All]
Wolfram Language code: 𝒫 = InhomogeneousPoissonProcess[λ[t], t];

默认的 "Thinning" 方法失败：

Wolfram Language code: TimeConstrained[RandomFunction[𝒫, {0, 3}], 1]
Wolfram Language code: TimeConstrained[RandomFunction[𝒫, {0, 3}, Method -> "Thinning"], 1]

对于该密度函数，反演方法也会失败：

Wolfram Language code: RandomFunction[𝒫, {0, 3}, Method -> "Inversion"]

直接方法适于无界密度函数，虽然可能会慢：

Wolfram Language code: RandomFunction[𝒫, {0, 3}, Method -> "Direct"]
Wolfram Language code: ListLinePlot[%, PlotRange -> All]

巧妙范例  (1)

模拟非齐次泊松过程的轨道：

Wolfram Language code: SeedRandom[34];𝒫 = InhomogeneousPoissonProcess[1.1 + Sin[t] / (1 + t ^ 2), t]; data = RandomFunction[𝒫, {0, 50}, 300];

在 50 取切片并可视化其分布：

Wolfram Language code: sd = data["SliceData", 50];
Wolfram Language code: cf = ColorData["Rainbow"]; sliced = BarChart[Last[#], Axes -> False, BarOrigin -> Left, AspectRatio -> 4, ChartStyle -> (cf /@ Rescale[MovingAverage[First[#], 2], {Min[sd], Max[sd]}, {0, 1}]), ImageSize -> 43]&[HistogramList[sd, {Range[Min[sd], Max[sd], (Max[sd] - Min[sd]) / 20]}]];

绘制在 50 的切片分布的轨道和直方图分布：

Wolfram Language code: ListLinePlot[data, ImageSize -> 400, PlotRange -> All, AspectRatio -> 3 / 4, Epilog -> Inset[sliced, {50.5, 57}, {0, 10.5}], PlotStyle -> (cf /@ Rescale[sd]), BaseStyle -> Thin, PlotRangePadding -> {{0, 20}, {.5, 7}}]
Wolfram Research (2015)，InhomogeneousPoissonProcess，Wolfram 语言函数，https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html.

文本

Wolfram Research (2015)，InhomogeneousPoissonProcess，Wolfram 语言函数，https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html.

CMS

Wolfram 语言. 2015. "InhomogeneousPoissonProcess." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html.

APA

Wolfram 语言. (2015). InhomogeneousPoissonProcess. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html 年

BibTeX

@misc{reference.wolfram_2026_inhomogeneouspoissonprocess, author="Wolfram Research", title="{InhomogeneousPoissonProcess}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html}", note=[Accessed: 04-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_inhomogeneouspoissonprocess, organization={Wolfram Research}, title={InhomogeneousPoissonProcess}, year={2015}, url={https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html}, note=[Accessed: 04-November-2026]}