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WatsonUSquareTest[data]

使用Watson 检验判断 data 是否服从正态分布.

WatsonUSquareTest[data,dist]

使用 Watson 检验判断 data 是否服从 dist 分布.

WatsonUSquareTest[data,dist,"property"]

返回 "property" 的值.

更多信息和选项
Details and Options Details and Options
范例  
基本范例  
范围  
检验  
报告  
选项  
Method  
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
按以下格式引用：

WatsonUSquareTest

WatsonUSquareTest[data]

使用Watson 检验判断 data 是否服从正态分布.

WatsonUSquareTest[data,dist]

使用 Watson 检验判断 data 是否服从 dist 分布.

WatsonUSquareTest[data,dist,"property"]

返回 "property" 的值.

更多信息和选项

  • WatsonUSquareTest 执行 Watson 拟合优度检验，其中零假设 为 data 取自分布为 dist 的总体，而备择假设 则对此否定.
  • 默认情况下，返回一个概率值或者 值.
  • 一个较小的 值表明 data 不可能服从 dist.
  • dist 可以是任意具有数值和符号参数或者数据集的符号分布.
  • data 可以是单变量 {x1,x2,…} 或多变量 {{x1,y1,…},{x2,y2,…},…}.
  • Watson 检验假设数据服从一个连续分布.
  • Watson 检验实际上使用基于 的检验统计量，其中 ，且 ， 是 data 的经验累积分布函数，而 是 dist 的累积分布函数.
  • 对于多变量检验，使用单变量边缘 值的总和并假设遵循 下的 UniformSumDistribution.
  • WatsonUSquareTest[data,dist,"HypothesisTestData"] 返回一个 HypothesisTestData 对象 htd，通过使用 htd["property"] 的形式，它可以用来提取额外的检验结果和属性.
  • WatsonUSquareTest[data,dist,"property"] 可以用来直接给出 "property" 的值.
  • 与检验报告相关的属性包括：
  • "PValue" 值
    "PValueTable""PValue" 的格式化版本
    "ShortTestConclusion"一个检验结论的简短描述
    "TestConclusion"一个检验结论的描述
    "TestData"检验统计量和 值
    "TestDataTable""TestData" 的格式化版本
    "TestStatistic"检验统计量
    "TestStatisticTable"格式化的 "TestStatistic"
  • 下列属性与所执行的检验类型无关.
  • 与数据分布相关的属性包括：
  • "FittedDistribution"数据的拟合分布
    "FittedDistributionParameters"数据的分布参数
  • 可以给出下列选项：
  • Method Automatic计算 值所用的方法
    SignificanceLevel0.05诊断和报告的分界点
  • 对于一个拟合优度检验，选择一个临界值 ，以使得只有当 时，否定 . 用于 "TestConclusion" 和 "ShortTestConclusion" 属性的 值由 SignificanceLevel 选项控制. 默认情况下， 设为 0.05.
  • 在设置 Method->"MonteCarlo" 下，在 下使用拟合分布，生成 个与输入 具有相同长度的数据集. 来自WatsonUSquareTest[si,dist,"TestStatistic"] 的 EmpiricalDistribution 用于估计 值.

范例

打开所有单元 关闭所有单元

基本范例  (4)

执行 Watson 检验对正态性进行校验：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 10^4];
Wolfram Language code: WatsonUSquareTest[data]

检验一些数据对一个特定分布的拟合情况：

Wolfram Language code: data = RandomVariate[LaplaceDistribution[1, 2], 10^3];
Wolfram Language code: WatsonUSquareTest[data, LaplaceDistribution[1, 2]]

比较两个数据集的分布：

Wolfram Language code: data1 = RandomVariate[NormalDistribution[], 100];
Wolfram Language code: data2 = RandomVariate[NormalDistribution[], 150];
Wolfram Language code: WatsonUSquareTest[data1, data2]

从 Watson 检验提取检验统计量：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 10^3];
Wolfram Language code: WatsonUSquareTest[data, NormalDistribution[], "TestStatistic"]

范围  (9)

检验  (6)

执行 Watson 检验对正态性进行校验：

Wolfram Language code: data1 = RandomVariate[NormalDistribution[], 10^4]; data2 = RandomVariate[StudentTDistribution[3], 10^4];

正态数据的 值相对于非正态数据的 值来说，显得比较大：

Wolfram Language code: WatsonUSquareTest[data1]
Wolfram Language code: WatsonUSquareTest[data2]

检验对一个特定分布的拟合优度：

Wolfram Language code: data1 = RandomVariate[NormalDistribution[], 10^3]; data2 = RandomVariate[CauchyDistribution[0, 1], 10^3];
Wolfram Language code: WatsonUSquareTest[data1, CauchyDistribution[0, 1]]
Wolfram Language code: WatsonUSquareTest[data2, CauchyDistribution[0, 1]]

比较两个数据集合的分布：

Wolfram Language code: data1 = RandomVariate[NormalDistribution[], 10^3]; data2 = RandomVariate[NormalDistribution[], 10^3];
Wolfram Language code: WatsonUSquareTest[data1, data2]

这两个数据集合不具有相同的分布：

Wolfram Language code: data3 = RandomVariate[NormalDistribution[0, 1.25], 10^3];
Wolfram Language code: WatsonUSquareTest[data1, data3]

检验多元正态性：

Wolfram Language code: data1 = RandomVariate[BinormalDistribution[.5], 10^3]; data2 = RandomVariate[LaplaceDistribution[1, 2], {10^3, 2}];
Wolfram Language code: WatsonUSquareTest[data1]
Wolfram Language code: WatsonUSquareTest[data2]

检验任意多变量分布的拟合优度：

Wolfram Language code: data1 = RandomVariate[BinormalDistribution[.5], 10^3]; data2 = RandomVariate[𝒹 = LaplaceDistribution[1, 2], {10^3, 2}];
Wolfram Language code: 𝒟 = ProductDistribution[𝒹, 𝒹];
Wolfram Language code: WatsonUSquareTest[data1, 𝒟]
Wolfram Language code: WatsonUSquareTest[data2, 𝒟]

创建一个 HypothesisTestData 对象以进行重复属性提取：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 10^5];
Wolfram Language code: ℋ = WatsonUSquareTest[data, Automatic, "HypothesisTestData"]

可用于提取的属性：

Wolfram Language code: ℋ["Properties"]

报告  (3)

将 Watson 检验的结果制作成表格：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 100];
Wolfram Language code: ℋ = WatsonUSquareTest[data, Automatic, "HypothesisTestData"];

完全检验表：

Wolfram Language code: ℋ["TestDataTable"]

一个 值表：

Wolfram Language code: ℋ["PValueTable"]

检验统计量：

Wolfram Language code: ℋ["TestStatisticTable"]

从一个 Watson 检验表中提取项目，用于生成定制的报告：

Wolfram Language code: data1 = RandomVariate[NormalDistribution[], 100]; data2 = RandomVariate[NormalDistribution[], 100];
Wolfram Language code: ℋ1 = WatsonUSquareTest[data1, Automatic, "TestData"]
Wolfram Language code: ℋ2 = WatsonUSquareTest[data2, Automatic, "TestData"]
Wolfram Language code: BarChart[{Labeled[ℋ1, "Set 1"], Labeled[ℋ2, "Set 2"]}, ChartLabels -> {"SubscriptBox[D, n]", "p‐value"}]

使用 "ShortTestConclusion" 和 "TestConclusion" 报告检验结论：

Wolfram Language code: data = BlockRandom[SeedRandom[1];RandomVariate[ParetoDistribution[1.05, 2], 100]];
Wolfram Language code: ℋ = WatsonUSquareTest[data, ParetoDistribution[1, 2], "HypothesisTestData"];
Wolfram Language code: ℋ["ShortTestConclusion"]
Wolfram Language code: ℋ["TestConclusion"]//TraditionalForm

结论在不同的显著性水平上可能是不同的：

Wolfram Language code: ℋ = WatsonUSquareTest[data, ParetoDistribution[1, 2], "HypothesisTestData", SignificanceLevel -> .001];
Wolfram Language code: ℋ["ShortTestConclusion"]
Wolfram Language code: ℋ["TestConclusion"]//TraditionalForm

选项  (3)

Method  (3)

使用基于蒙特卡罗的方法或者一个可计算的公式：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 100];
Wolfram Language code: WatsonUSquareTest[data, NormalDistribution[], Method -> "MonteCarlo"]
Wolfram Language code: WatsonUSquareTest[data, NormalDistribution[], Method -> Automatic]

对于基于蒙特卡罗的方法，设置样本数：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 100];
Wolfram Language code: pts = Table[{i, WatsonUSquareTest[data, NormalDistribution[], Method -> {"MonteCarlo", "MonteCarloSamples" -> i}]}, {i, Range[5, 100, 5]}];

当样本数增加时，蒙特卡罗估计量收敛到真实的 值：

Wolfram Language code: pval = WatsonUSquareTest[data, NormalDistribution[]];
Wolfram Language code: Show[ListLinePlot[pts, PlotRange -> {0, 1}, FrameLabel -> {"Samples", "P-Value"}, Frame -> True, AxesOrigin -> {0, 0}], Graphics[{Dashed, Line[{{0, pval}, {100, pval}}]}]]

设置在蒙特卡罗方法中所使用的随机种子：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 100];
Wolfram Language code: pts = Table[{i, WatsonUSquareTest[data, NormalDistribution[], Method -> {"MonteCarlo", "RandomSeed" -> i, "MonteCarloSamples" -> 50}]}, {i, Range[1, 10]}];

种子对产生器的状态有影响，而且对所得的 值也有一定影响：

Wolfram Language code: pval = WatsonUSquareTest[data, NormalDistribution[]];
Wolfram Language code: Show[ListLinePlot[pts, PlotRange -> {Min[pts[[All, 2]]], Max[pts[[All, 2]]]}, FrameLabel -> {"Seed", "P-Value"}, Frame -> True, AxesOrigin -> {0, 0}], Graphics[{Dashed, Line[{{0, pval}, {100, pval}}]}]]

应用  (2)

Watson 检验的功效曲线：

Wolfram Language code: data = Table[RandomVariate[UniformDistribution[{-4, 4}], {500, i}], {i, n = {5, 7, 10, 15, 20, 25, 30}}];
Wolfram Language code: ℋ = Table[WatsonUSquareTest[data[[i, j]], NormalDistribution[]], {i, Length[data]}, {j, Length[data[[i]]]}];
Wolfram Language code: pC = Interpolation[Transpose[{n, Table[Probability[x ≤ 0.05, xi], {i, ℋ}]}], InterpolationOrder -> 1];

将近似功效曲线可视化：

Wolfram Language code: Plot[pC[x], {x, 5, 30}, PlotRange -> {0, 1}, Ticks -> {n, Automatic}, AxesOrigin -> {0, 0}]

当内在分布是一个 UniformDistribution[{-4,4}]，检验大小为0.05，样本数为 12 时，估计 Watson 检验的功效：

Wolfram Language code: pC[12.]

统计课中，决定测试一个旋转盘游戏来学习偏差. 班上的50名学生，每人旋转一次，并用一个设备来记录每次旋转的角度（以弧度为单位）：

Wolfram Language code: data = {25.1, 28.51, 15.58, 7.98, 7.87, 28.9, 28.72, 10.64, 27.33, 23.86, 6.49, 11.02, 8.66, 28.84, 31.14, 24.05, 15.92, 12.66, 20.31, 17.06, 17.45, 21.94, 9.8, 7.26, 11.44, 23.78, 18.83, 13.17, 23.12, 18.06, 21.79, 26.57, 22.11, 21.12, 13.24, 24.26, 18.48, 28.63, 20.51, 8.21, 21.82, 17.09, 30.67, 14.99, 28.42, 21.64, 16.75, 8.06, 9.5, 30.55};
Wolfram Language code: ListPlot[Table[{Cos[x], Sin[x]}, {x, data}], AspectRatio -> 1]

把每次测量转换到 上：

Wolfram Language code: conv[d_] := With[{r = d / 2π}, 2π(r - IntegerPart[r])]
Wolfram Language code: cData = conv /@ data;

圆上的均匀性测试显示旋转器是无偏的：

Wolfram Language code: WatsonUSquareTest[cData, UniformDistribution[{0, 2π}], "TestDataTable"]

属性和关系  (8)

默认情况下，单变量数据与一个 NormalDistribution 进行比较：

Wolfram Language code: data = RandomVariate[NormalDistribution[2, 3], 10^4];
Wolfram Language code: ℋ = WatsonUSquareTest[data, Automatic, "HypothesisTestData"];
Wolfram Language code: ℋ["TestDataTable"]

参数已经从数据中估计得到：

Wolfram Language code: ℋ["FittedDistribution"]

默认情况下，多变量数据与一个 MultinormalDistribution 进行比较：

Wolfram Language code: data = RandomVariate[MultinormalDistribution[{1, 2, 3}, IdentityMatrix[3]], 1000];
Wolfram Language code: ℋ = WatsonUSquareTest[data, Automatic, "HypothesisTestData"];
Wolfram Language code: ℋ["TestDataTable"]
Wolfram Language code: ℋ["FittedDistribution"]//TraditionalForm

若未指定，则检验分布的参数从数据中估计得到：

Wolfram Language code: data = RandomVariate[NormalDistribution[1, 2], 1000];
Wolfram Language code: WatsonUSquareTest[data, NormalDistribution[μ, σ], "FittedDistribution"]

不估计已指定的参数：

Wolfram Language code: WatsonUSquareTest[data, NormalDistribution[μ, 2], "FittedDistribution"]
Wolfram Language code: WatsonUSquareTest[data, NormalDistribution[1, 2], "FittedDistribution"]

对检验分布的未指明的参数，使用最大似然估计法：

Wolfram Language code: data = RandomVariate[ExponentialDistribution[3], 10^3];
Wolfram Language code: ℋ = WatsonUSquareTest[data, ExponentialDistribution[λ], "FittedDistribution"]
Wolfram Language code: WatsonUSquareTest[data, ExponentialDistribution[λ]]

如果参数未知，则当可能的情况下，WatsonUSquareTest 应用一次校正：

Wolfram Language code: data = RandomVariate[NormalDistribution[3, 4], 10^4];
Wolfram Language code: est = EstimatedDistribution[data, NormalDistribution[μ, σ]]

估计参数，但不应用校正：

Wolfram Language code: WatsonUSquareTest[data, est]
Wolfram Language code: ℋ = WatsonUSquareTest[data, NormalDistribution[μ, σ], "HypothesisTestData"];

拟合分布与之前的相同，而且 值被校正：

Wolfram Language code: ℋ["FittedDistribution"]
Wolfram Language code: ℋ["PValue"]

在多变量拟合优度的检验中，假设边缘密度是独立的：

Wolfram Language code: data = RandomVariate[MultinormalDistribution[{0, 0}, {{0.118, 0.252}, {0.252, 0.665}}], 100];
Wolfram Language code: WatsonUSquareTest[data, MultinormalDistribution[{0, 0}, {{0.118, 0.252}, {0.252, 0.665}}], "TestStatistic"]

假设具有独立性的情况下，检验统计量是相同的：

Wolfram Language code: WatsonUSquareTest[data, MultinormalDistribution[{0, 0}, {{0.118, 0}, {0, 0.665}}], "TestStatistic"]

Watson 统计量可以使用 NExpectation 定义：

Wolfram Language code: n = 10; h0 = NormalDistribution[1, 2]; data = RandomVariate[h0, n];
Wolfram Language code: f[x_] := CDF[h0, x] Overscript[f, ^ ][x_] := CDF[EmpiricalDistribution[data], x]
Wolfram Language code: d[x_] := Overscript[f, ^ ][x] - f[x] Overscript[d, _] = NExpectation[d[x], xh0];
Wolfram Language code: n NExpectation[(d[x] - Overscript[d, _]) ^ 2, xh0]
Wolfram Language code: WatsonUSquareTest[data, h0, "TestStatistic"]

仅当输入为 TimeSeries 时，Watson 检验适用于值：

Wolfram Language code: ts = TemporalData[TimeSeries, {{{1.224578634529677, 0.47929635789978015, 0.6572781300178168, 0.21496048742669355, 0.7299608014554928, -0.2495111111278263, -1.3286551762002712, 0.552725018274874, 0.19272112205837066, 1.1809144012420882, -1.1671 ... 40938613662046, 1.052394590214582, 0.9345044123980388, 0.38537803109557855, -0.48660931166089394, -0.71203560340161}}, {{0, 100, 1}}, 1, {"Continuous", 1}, {"Discrete", 1}, 1, {ValueDimensions -> 1, ResamplingMethod -> None}}, False, 10.1];
Wolfram Language code: WatsonUSquareTest[ts]
Wolfram Language code: WatsonUSquareTest[ts["Values"]]

可能存在的问题  (3)

Watson 检验不用于离散分布：

Wolfram Language code: data = RandomVariate[DiscreteUniformDistribution[{-10, 10}], 35];
Wolfram Language code: WatsonUSquareTest[data, DiscreteUniformDistribution[{-10, 10}]]

连续性校正通常可以很好地保留检验的容量：

Wolfram Language code: sim = RandomVariate[DiscreteUniformDistribution[{-10, 10}], {500, 35}];
Wolfram Language code: p = Quiet[WatsonUSquareTest[#, DiscreteUniformDistribution[{-10, 10}]]]& /@ sim;
Wolfram Language code: Show[ListLinePlot[Table[{α, Probability[pv ≤ α, pvp]}, {α, .01, 1, .01}]], Plot[x, {x, 0, 1}, PlotStyle -> Dashed]]

某些情况下可能并不如此：

Wolfram Language code: sim = RandomVariate[DiscreteUniformDistribution[{1, 3}], {500, 35}];
Wolfram Language code: p = Quiet[WatsonUSquareTest[#, DiscreteUniformDistribution[{1, 3}]]]& /@ sim;
Wolfram Language code: Show[ListLinePlot[Table[{α, Probability[pv ≤ α, pvp]}, {α, .01, 1, .01}]], Plot[x, {x, 0, 1}, PlotStyle -> Dashed]]

在这些情况下，使用蒙特卡洛方法或 PearsonChiSquareTest：

Wolfram Language code: WatsonUSquareTest[sim[[1]], DiscreteUniformDistribution[{1, 3}], Method -> "MonteCarlo"]
Wolfram Language code: PearsonChiSquareTest[sim[[1]], DiscreteUniformDistribution[{1, 3}]]

当参数已经从数据中估计得到时，Watson 检验对某些分布是无效的：

Wolfram Language code: data = RandomVariate[BetaDistribution[1, 2], 100];
Wolfram Language code: WatsonUSquareTest[data, BetaDistribution[1, b]]

如果参数已知，则提供参数值：

Wolfram Language code: WatsonUSquareTest[data, BetaDistribution[1, 2]]

或者，使用蒙特卡罗方法来求近似 值：

Wolfram Language code: WatsonUSquareTest[data, BetaDistribution[1, b], Method -> "MonteCarlo"]

相同的数据被忽略：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 1000];
Wolfram Language code: WatsonUSquareTest[Join[data, {First[data]}]]
Wolfram Language code: PearsonChiSquareTest[Join[data, {First[data]}]]

如果具有很多相同的数据，那么差异将很明显：

Wolfram Language code: WatsonUSquareTest[Join[data, data]]
Wolfram Language code: PearsonChiSquareTest[Join[data, data]]

巧妙范例  (1)

当零假设 为真，计算统计量：

Wolfram Language code: data = RandomVariate[NormalDistribution[], {500, 100}];
Wolfram Language code: T1 = WatsonUSquareTest[#, NormalDistribution[], "TestStatistic"]& /@ data;

给定特定的备择假设的检验统计量：

Wolfram Language code: T2 = WatsonUSquareTest[#, LaplaceDistribution[1, 2], "TestStatistic"]& /@ data;

比较检验统计量的分布：

Wolfram Language code: SmoothHistogram[{T1, T2}, Filling -> Axis, PlotLegends -> {"SubscriptBox[H, 0] is True", "SubscriptBox[H, 0] is False"}, PlotRange -> All]
Wolfram Research (2010)，WatsonUSquareTest，Wolfram 语言函数，https://reference.wolfram.com/language/ref/WatsonUSquareTest.html.

文本

Wolfram Research (2010)，WatsonUSquareTest，Wolfram 语言函数，https://reference.wolfram.com/language/ref/WatsonUSquareTest.html.

CMS

Wolfram 语言. 2010. "WatsonUSquareTest." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/WatsonUSquareTest.html.

APA

Wolfram 语言. (2010). WatsonUSquareTest. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/WatsonUSquareTest.html 年

BibTeX

@misc{reference.wolfram_2026_watsonusquaretest, author="Wolfram Research", title="{WatsonUSquareTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/WatsonUSquareTest.html}", note=[Accessed: 03-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_watsonusquaretest, organization={Wolfram Research}, title={WatsonUSquareTest}, year={2010}, url={https://reference.wolfram.com/language/ref/WatsonUSquareTest.html}, note=[Accessed: 03-November-2026]}