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PointCountDistribution

PointCountDistribution[pproc,reg]

表示区域 reg 中的点过程 pproc 的点数的分布.

PointCountDistribution[pproc,{reg₁,…,reg_n}]

表示区域 reg_i 中点数的联合分布.

范例

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基本范例 (3)

单位圆盘上的 PoissonPointProcess 的点数的分布：

Wolfram Language code: PointCountDistribution[PoissonPointProcess[μ, 2], Disk[]]

求点数的均值和方差：

Wolfram Language code: {Mean[%], Variance[%]}

单位球体内的 HardcorePointProcess 的 PointCountDistribution：

Wolfram Language code: dist = PointCountDistribution[HardcorePointProcess[10, .3, 3], Ball[]]

Wolfram Language code: DistributionParameterQ[dist]

模拟分布：

Wolfram Language code: sample = RandomVariate[dist, 10 ^ 3];

Wolfram Language code: Histogram[sample, Automatic, "PDF"]

GeoDisk 的 PointCountDistribution：

Wolfram Language code: proc = PoissonPointProcess[3 / Quantity[1, "Miles"] ^ 2, 2];

Wolfram Language code: PointCountDistribution[proc, GeoDisk[Here, Quantity[5, "Miles"]]]

范围 (4)

单位圆盘上的 BinomialPointProcess 的 PointCountDistribution：

Wolfram Language code: proc = BinomialPointProcess[10, Disk[]];

Wolfram Language code: PointCountDistribution[proc]

子区域中点数的分布：

Wolfram Language code: reg = ImplicitRegion[1 / 4 ≤ x ^ 2 + y ^ 2 ≤ 1∧1 ≥ y ≥ x ≥ 0, {x, y}];

Wolfram Language code: Region[reg]

Wolfram Language code: dist = PointCountDistribution[proc, reg]

计算平均点数：

Wolfram Language code: Mean[dist]//N

单位圆盘上的 InhomogeneousPoissonPointProcess 的 PointCountDistribution：

Wolfram Language code: proc = InhomogeneousPoissonPointProcess[Function[{x, y}, Exp[x + y]], 2];

Wolfram Language code: proc//PointProcessParameterQ

Wolfram Language code: PointCountDistribution[proc, Disk[]]

在两个不相交区域的并集上：

Wolfram Language code: PointCountDistribution[proc, {Rectangle[{-1, -1}, {1, 1}], Rectangle[{2, 0}, {3, 1}]}]

仿真：

Wolfram Language code: Histogram3D[RandomVariate[%, 10 ^ 6], 30, PDF]

在两个相交的区域上：

Wolfram Language code: PointCountDistribution[proc, {Rectangle[{-1, -1}, {1, 1}], Rectangle[{-2, 0}, {3, 1}]}]

仿真：

Wolfram Language code: Histogram3D[RandomVariate[%, 10 ^ 6], 30, PDF]

均值：

Wolfram Language code: Mean[%%]

Neyman–Scott 型的聚类过程的 PointCountDistribution 是 CompoundPoissonDistribution：

Wolfram Language code: reg = Disk[{0, 0}, r];

柯西点过程：

Wolfram Language code: PointCountDistribution[CauchyPointProcess[μ, λ, b, 2], reg]

Matérn 点过程：

Wolfram Language code: PointCountDistribution[MaternPointProcess[μ, λ, rad, 2], reg]

Thomas 点过程：

Wolfram Language code: PointCountDistribution[ThomasPointProcess[μ, λ, σ, 2], reg]

方差伽玛点过程：

Wolfram Language code: PointCountDistribution[VarianceGammaPointProcess[μ, λ, α, β, 2], reg]

通常，Neyman–Scott 点过程的点数分布不取决于聚类点的空间分布：

Wolfram Language code: PointCountDistribution[NeymanScottPointProcess[μ, λ, NormalDistribution[a, b], 2], reg]

Wolfram Language code: PointCountDistribution[NeymanScottPointProcess[μ, λ, BinormalDistribution[ρ], 2], reg]

在没有解析形式的 PointCountDistribution 的情况下计算概率和期望：

Wolfram Language code: dist = PointCountDistribution[HardcorePointProcess[10, .3, 3], Ball[]];

用仿真计算均值：

Wolfram Language code: NExpectation[n, ndist, Method -> {"MonteCarlo", "SamplingIncrement" -> 10 ^ 4}]

用仿真计算事件概率：

Wolfram Language code: NProbability[n > 20, ndist, Method -> {"MonteCarlo", "SamplingIncrement" -> 10 ^ 3}]

应用 (5)

计算单位圆盘上的 Strauss 点过程的平均点数：

Wolfram Language code: 𝒟 = PointCountDistribution[StraussPointProcess[20, 1, .2, 2], Disk[]];

Wolfram Language code: NExpectation[n, n𝒟, Method -> {"MonteCarlo", "SamplingIncrement" -> 10 ^ 4}]

假设胶合板上平均每 50 平方英尺出现一个瑕疵. 模拟每平方英尺发现瑕疵的过程：

Wolfram Language code:

pp = PoissonPointProcess[1 / 50, 2];
reg1 = Rectangle[{0, 0}, {100, 100}];
sample = RandomPointConfiguration[pp, reg1]

Wolfram Language code: ListPlot[sample]

给定区域中瑕疵的数量：

Wolfram Language code: flawsDist1 = PointCountDistribution[pp, reg1]

求 4 英尺 x 8 英尺的板材没有瑕疵的概率：

Wolfram Language code: reg2 = Rectangle[{0, 0}, {4, 8}];

Wolfram Language code: flawsDist2 = PointCountDistribution[pp, reg2]

Wolfram Language code: Probability[x == 0, xflawsDist2]//N

对于面积为 7.54 cm 的圆镜，无瑕疵的概率为 0.91. 使用相同的抛光工艺，制造了另一个面积为 19.50 cm 的圆镜. 假设瑕疵是独立且随机分布的，求在较大的镜子上没有瑕疵的概率：

Wolfram Language code: flawProc[λ_] = PoissonPointProcess[λ, 2];

Wolfram Language code: mirror1 = Disk[{0, 0}, Sqrt[7.54 / π]];

镜子上瑕疵数量的分布：

Wolfram Language code: dist1 = PointCountDistribution[flawProc[λ], mirror1]

求瑕疵点过程的强度：

Wolfram Language code: sol = First[Solve[Probability[x == 0, xdist1] == 0.9100 && λ > 0, λ]]//Quiet

第二个镜子上瑕疵数量的分布：

Wolfram Language code: mirror2 = Disk[{0, 0}, Sqrt[19.50 / π]];

Wolfram Language code: dist2 = PointCountDistribution[flawProc[λ /. sol], mirror2]

在较大的镜子上没有瑕疵的概率：

Wolfram Language code: PDF[dist2, 0]

一个 LCD 显示器有 1920×1080 个像素. 如果有问题的像素少于或等于 15 个，则该显示器为合格品. 生产中像素出现故障的概率为，并且故障像素的位置是独立且随机的. 求合格显示器的比例：

Wolfram Language code: faultProcess = PoissonPointProcess[5 10^-6, 2];

Wolfram Language code: display = Rectangle[{0, 0}, {1920, 1080}];

模拟有问题像素的分布：

Wolfram Language code: samples = RandomPointConfiguration[faultProcess, display, 3]

Wolfram Language code: ListPlot[samples]

显示器上故障像素数的分布：

Wolfram Language code: dist = PointCountDistribution[faultProcess, display]

求显示器故障像素不超过 15 个的概率：

Wolfram Language code: NProbability[n ≤ 15, ndist]

求生产合格率至少为 90% 的 4000×2000 像素显示器所要求的像素故障率：

Wolfram Language code: display2 = Rectangle[{0, 0}, {4000, 2000}];

Wolfram Language code: dist2 = PointCountDistribution[PoissonPointProcess[λ, 2], display2]

Wolfram Language code: prob = Probability[n ≤ 15, ndist2]

绘制合格率与像素故障率的关系：

Wolfram Language code:

LogLogPlot[prob, {λ, 10^-7, 10^-5}, GridLines -> All, Frame -> True, FrameLabel -> {"Pixel Failure Rate", None}, PlotLabel -> "LCD Acceptance Rate"]

求可接受的最大像素故障率：

Wolfram Language code: Maximize[{λ, prob ≥ 0.90 && 10^-6 < λ < 2 10^-6}, λ, Reals]

检查结果：

Wolfram Language code: CDF[dist2, 15] /. Last[%]

芝加哥与汽车有关的盗窃事件：

Wolfram Language code:

geopos = GeoPosition[{{41.807586328, -87.615127788}, {41.961978526, -87.699709131}, 
  {41.9378739, -87.750043614}, {41.79137343, -87.657233828}, {41.869859534, -87.721345338}, 
  {41.903561493, -87.653509519}, {41.751579291, -87.578075178}, {41.813913673,  ... .771736514, -87.66400571}, {41.935270542, -87.740044903}, 
  {41.927801398, -87.754688495}, {41.770131721, -87.611528011}, {41.82467357, -87.626413225}, 
  {41.84418573, -87.701349795}, {41.782546496, -87.64088957}, {41.808896916, -87.611437797}}];

Wolfram Language code: geoplot = GeoGraphics[GeoMarker[geopos, "Scale" -> Scaled[0.015]]]

Wolfram Language code: pts = Reverse /@ First[geopos];

Wolfram Language code: region = ConvexHullMesh[pts];

Wolfram Language code: f = HistogramPointDensity[pts]

定义具有此强度的非齐次泊松点过程：

Wolfram Language code: proc = InhomogeneousPoissonPointProcess[Function[{x, y}, f[{x, y}]], 2];

Wolfram Language code: dist = PointCountDistribution[proc, region]

求平均犯罪数量：

Wolfram Language code: Mean[dist]

计算犯罪数量小于 1200 的概率：

Wolfram Language code: Probability[n < 1200, ndist]

计算邮政编码为 60639 的区域内预期的犯罪数量：

Wolfram Language code:

reg = EntityValue[Entity["ZIPCode", "60639"], "Polygon"][[1, 1, 1, 1]];
reg = Polygon[Reverse /@ %];

Wolfram Language code: RegionQ[reg]

Wolfram Language code: RegionPlot[reg]

Wolfram Language code: Counts[RegionMember[reg, pts]][True]

Wolfram Language code: zip𝒟 = PointCountDistribution[proc, reg]

Wolfram Language code: Mean[zip𝒟]

属性和关系 (4)

一对不相连区域上的 PointCountDistribution 是 ProductDistribution：

Wolfram Language code:

reg1 = Disk[];
reg2 = Rectangle[{1, 1}, {3, 2.5}];
reg3 = Triangle[{{2, 3}, {4, 6}, {0, 3}}];

Wolfram Language code: Graphics[{EdgeForm[Gray], GrayLevel[.6], {reg1, reg2, reg3}}]

Wolfram Language code: proc = PoissonPointProcess[30, 2];

每个区域上的 PointCountDistribution：

Wolfram Language code: PointCountDistribution[proc, #]& /@ {reg1, reg2, reg3}

区域列表上的多元 PointCountDistribution：

Wolfram Language code: PointCountDistribution[proc, {reg1, reg2, reg3}]

重叠区域上的 PointCountDistribution：

Wolfram Language code:

reg1 = Disk[];
reg2 = Rectangle[{0, 0}, {2, 2}];
reg3 = Triangle[{{1, 1}, {4, 6}, {0, 3}}];

Wolfram Language code: Graphics[{EdgeForm[Gray], Opacity[.3], {reg1, reg2, reg3}}]

Wolfram Language code: proc = PoissonPointProcess[30, 2];

每个区域上的 PointCountDistribution：

Wolfram Language code: PointCountDistribution[proc, #]& /@ {reg1, reg2, reg3}

区域列表上的多元 PointCountDistribution：

Wolfram Language code: PointCountDistribution[proc, {reg1, reg2, reg3}]

区域相交的地方是相关的：

Wolfram Language code: Covariance[%]//MatrixForm

覆盖有 BinomialPointProcess 的区域上的 PointCountDistribution：

Wolfram Language code:

reg = Rectangle[];
proc = BinomialPointProcess[100, reg];

定义三种覆盖：

Wolfram Language code:

r1 = Rectangle[{0, 1 / 2}, {1 / 3, 1}];
r2 = RegionDifference[Disk[{0, 0}, 2 / 3, {0, π / 2}], r1];

Wolfram Language code: covering = {r1, r2, RegionDifference[reg, RegionUnion[r1, r2]]};

Wolfram Language code: Show[Table[Region[Style[covering[[i]], ColorData[97, i]]], {i, 3}]]

覆盖的点数分布：

Wolfram Language code: PointCountDistribution[proc, covering]

具有非数值参数的区域的 PointCountDistribution：

Wolfram Language code: PointCountDistribution[PoissonPointProcess[μ, 2], Disk[{0, 0}, r]]

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PointCountDistribution

更多信息

范例

基本范例 (3)

范围 (4)

应用 (5)

属性和关系 (4)

文本

CMS

APA

BibTeX

BibLaTeX

PointCountDistribution

更多信息

范例

基本范例 (3)

范围 (4)

应用 (5)

属性和关系 (4)

参见

相关指南

历史

文本

CMS

APA

BibTeX

BibLaTeX