Background: Although the matrix model for kinship networks includes many demographic processes, it is deterministic. It provides values of age-stage distributions of kin, but no information on (co)variances. Because kin populations are small, demographic stochasticity is expected to create appreciable inter-individual variation. Objective: To develop a stochastic kinship model that includes demographic stochasticity and projects (co)variances of kin age distributions, and functions thereof. Methods: Kin populations are described by multitype branching processes. Means and covariances are projected using matrices that are generalizations of the deterministic model. The analysis requires only an age-specific mortality and fertility schedule. Both linear and nonlinear transformations of the kin age distribution are treated as outputs accompanying the state equations. Results: The stochastic model follows the same mathematical framework as the deterministic model, modified to treat initial conditions as mixture distributions. Variances in numbers of most kin are compatible with Poisson distributions. Variances for parents and ancestors are compatible with binomial distributions. Prediction intervals are provided, as are probabilities of having at least one or two kin of each type. Prevalences of conditions are treated either as fixed or random proportions. Dependency ratios and their variances are calculated for any desired group of kin types. An example compares Japan under 1947 rates (high mortality, high fertility) and 2019 rates (low mortality, low fertility). Contribution: Previous presentations of the kinship model have acknowledged the limitation to expected values. That limitation is now removed; both means and variances are easily calculated with minimal modification of code.

中文翻译：

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亲属关系的正式人口学 VI：亲属关系网络中的人口随机性和方差（作者 Hal Caswell）


背景：尽管亲属关系网络的矩阵模型包括许多人口过程，但它是确定性的。它提供了亲属的年龄阶段分布的值，但没有提供有关（协）方差的信息。由于亲属种群规模小，预计人口随机性会产生明显的个体间差异。目的：开发一个随机亲属关系模型，其中包括人口统计随机性和亲属年龄分布的项目（协）方差及其功能。方法： 亲缘种群由多类型分支过程描述。均值和协方差是使用确定性模型的泛化矩阵进行投影的。该分析只需要一个特定年龄的死亡率和生育率时间表。亲属年龄分布的线性和非线性变换都被视为状态方程随附的输出。结果：随机模型遵循与确定性模型相同的数学框架，经过修改以将初始条件视为混合分布。大多数亲属的数量方差与 Poisson 分布兼容。父级和上级的方差与二项分布兼容。提供了预测区间，以及每种类型至少有一个或两个亲属的概率。病症的患病率被视为固定比例或随机比例。为任何所需的亲属类型组计算抚养比及其方差。一个例子比较了 1947 年和2019 年比率（低死亡率、低生育率）下的日本。贡献： 亲缘关系模型的先前演示已经承认了期望值的局限性。 该限制现已删除;均值和方差都可以通过对代码进行最少的修改来轻松计算。