Sparse, rank-reduced and general smooth modelling

当一些感兴趣的变量与许多预测变量以复杂的方式相关时，平滑回归模型是有用的，我们想要了解这种关系。在许多情况下，变量之间的相关性很复杂，这意味着遵循传统的统计方法是不切实际的，即写下一个简单的统计模型来描述这种关系，在这种模型中，只需估计几个未知参数。取而代之的是，统计模型是用未知的预测因素的平滑函数来指定的，例如，对数血压是由年龄的平滑函数加上体重和身高的平滑函数加上每周锻炼小时的平滑函数得出的。然后，统计学上的挑战是估计光滑函数。考虑到几十年来在这些平滑模型的理论和计算方面的工作，它们的使用现在很广泛，几乎和传统的回归模型一样常规。然而，在它们最吸引人的复杂数据情况下，它们的使用仍然存在几个实际障碍1。目前的方法要么允许通过稀疏计算方法对短期空间、时间或时空相关性进行有效建模，要么允许通过降阶方法对涉及许多变量的复杂关系进行建模，但不能两者兼而有之。然而，具有短程残差相关的复杂模型恰恰是这种平滑模型最实用的地方。在降阶设置中，允许使用高度复杂的模型进行可行的计算，到目前为止，最可靠和最有效的计算方法仅限于感兴趣的变量来自指数分布族(正态、泊松、二项分布等)的情况。但鉴于这种方法已被证明具有广泛的实用性，同样可靠的方法在模型中也会有许多应用，其中感兴趣的变量服从与指数族中的变量非常不同的分布(例如，它可能是事件的等待时间，或者事件在空间位置上的发生)。越来越多的研究人员和公司正在寻求分析非常大的数据集，而这些数据集在目前的平滑建模技术下是根本不可行的。该项目旨在应对这些挑战，从而大幅增加这类模型的实用范围和实用性。特别是，该项目将寻求找到新的方法，将稀疏和降阶方法混合用于平滑建模，以解决问题1；根据指数族方法的经验，为来自更一般的分布类别的变量制定可靠和有效的方法；解决2；并为处理大型和复杂模型开发新的高效算法，这些算法可以在廉价的标准计算机硬件上容易地并行化，以解决3。所开发的方法将在R统计计算环境中的PIS成功mgcv包的基础上实施。还将通过教科书、短期课程和提供网络资源来传播这些方法。

Smooth regression models are useful when some variable of interest is related to a number of predictor variable in a complex manner, and we want to understand that relationship. In many cases the complexity of the dependence between the variables means that it is impractical to follow the traditional statistical approach of writing down a simple statistical model describing the relationship, in which only a few unknown parameters are to be estimated. Instead the statistical model is specified in terms of unknown smooth functions of predictors, for example `log blood pressure is given by a smooth function of age plus a smooth function of weight and height plus a smooth function of hours of exercise per week'. The statistical challenge is then to estimate the smooth functions. Given decades of work on the theory and computation of these smooth models, their use is now widespread and almost as routine as that of traditional regression models. However there remain several practical obstacles to their use, in exactly the complex data situations in which they should be most appealing.1. Current methods allow either the effective modelling of short range spatial, temporal or spatio-temporal correlation, via sparse computational methods, OR the modelling of complex relationships involving many variables, via reduced rank methods, but not both. However it is complex models with short range residual correlation are exactly where such smooth models are most practically appealing.2. In the reduced rank setting, that allows feasible computation with highly complex models, the most reliable and efficient computational methods are so far restricted to situations where variable of interest comes from the exponential family of distributions (normal, Poisson, binomial etc). But given the proven wide utility of such methods, there would also be many applications for similarly reliable methods for models where the variable of interest follows a very different distribution to those in the exponential family (for example it might be the waiting time to an event, or the occurrence of an event at a spatial location).3. Increasingly researchers and companies are seeking to analyze very large datasets, which are simply infeasible with current smooth modelling technology. This project aims to address these challenges, thereby massively increasing the practical scope and utility of this class of models. In particular the project will seek to find novel ways to hybridize the sparse and reduced rank approaches to smooth modelling to resolve issue 1; to build on experience with the exponential family methods to develop reliable and efficient methods for variables from a much more general class of distributions, to resolve 2; and to develop novel and efficient algorithms for handling large and complex models that can be readily parallelized on cheap standard computer hardware, to address 3. The methods developed will be implemented in free open source software, building on the PIs successfully mgcv package for generalized additive modelling, in the R statistical computing environment. The methods will also be disseminated via a textbook, short courses and the provision of web resources.