Multivariate models and inference

在金融等许多领域的研究中，被测量的变量并不单独遵循经典的钟形分布，也不具有经过变换后可以成为多元高斯分布的依赖性。在过去的10年中，藤对-联结结构是应用最灵活的多元非高斯分布。藤对-联结结构可以处理连续、离散、二元或有序的测量变量。在需要从解释变量中预测一个（响应）变量的研究中，并且同时观察到这些变量，vine copula方法适合一个联合分布，然后从给定解释变量的响应的条件分布进行推断。与经典的多元回归、广义线性模型和生存回归方法相比，这种方法允许使用条件分位数和条件方差作为解释变量的函数的灵活形状模型进行预测。该方法对于无界预测空间特别有用，因为经典方法不能很好地处理这种情况。在涉及风险的应用领域中，尾推理是很重要的。为了评估不同模型的适用性，除了通常的中心依赖性度量之外，我们还开发了尾加权依赖性度量作为诊断。我们的推导是基于尾形式的copula，它允许对联合尾（风险）概率进行非参数估计，并评估参数模型的拟合。在一些应用中，我们使用类似于经典模型的潜在变量，但使用联结观察到的变量和潜在变量，以允许尾部依赖行为更灵活，更好地拟合多变量数据。一些近期和未来研究的主要目标是进一步开发多元非高斯响应的多元模型和推理/计算程序。当变量数量较大时，需要具有简洁的依赖结构和灵活的尾部行为的多变量模型。对于拟合柔性copula后的条件分布预测，正在研究将基于不同vine copula的模型与使用机器学习方法的模型进行比较；标准包括可解释性和样本外性能。变异选择是模型比较的一部分。

In research studies in many areas, such as finance, measured variables do not follow the classical bell-shaped distribution individually and do not have dependence that can be multivariate Gaussian after transforms. In the past 10 years, the vine pair-copula construction has been the most flexible multivariate non-Gaussian distributions for applications. The vine pair-copula construction can handle measured variables that are continuous, discrete, binary or ordinal. In studies that required prediction of one (response) variable from explanatory variables and where the variables are simultaneously observed, the vine copula approach fits a joint distribution followed by inference from the conditional distribution of the response given the explanatory variables. Compare with classical methods of multiple regression, generalized linear models and survival regression, this approach allows prediction with models for flexible shapes for conditional quantiles and conditional variance as a function of the explanatory variables. The approach is especially useful for unbounded predictor spaces, as classical methods do not handled this situation well. With use of copulas in areas of applications involving risk, tail inference is important. To evaluate suitability of different models, we have developed tail-weighted measures of dependence as diagnostics in addition to the usual measures of central dependence. Our derivations are based on a tail form of copulas that allow for non-parametric estimation of joint tail (risk) probabilities and assess fits of parametric models. In some applications, we use latent variables similar to classical models, but with the use of copulas linking observed variables to latent variables to allow for more flexibility in tail dependence behaviour and better fits to multivariate data. The main objectives of some recent and proposed future research is to further develop multivariate models and inference/computing procedures for multivariate non-Gaussian response. With a larger number of variables, it is desirable to have multivariate models with parsimonious dependence structures and flexible tail behaviour. For predictions from conditional distributions after fitting a flexible copula, research is underway to compare models based on different vine copulas with models using machine learning methods; criteria include interpretability and out-of-sample performance. Variation selection is part of the model comparison.