Efficient bayesian inference and recycling of features in data analysis problems

该项目与英国电信公司合作，收集了大量的时间序列数据，以及大量的潜在预测指标，例如具有位置和天气的每月电信数据。随着许多领域产生和存储的数据越来越多，对这些数据集进行统计分析可以深入了解预测者和感兴趣的反应之间的关系。然而，当模型拟合和预测在高维环境中变得具有挑战性时，存在太多预测值的情况下会出现困难，其中可能有许多预测值是强相关的。子集选择方法提供了一种有吸引力的解决方案，通过减少预测器的数量来确保所产生的统计模型是可解释的，然而，困难在于识别最能解释响应的相关预测器组合。此外，随着预测因子数量的增加，评估所有潜在模型的计算成本很高，因此需要能够大幅降低这一成本的方法。此外，时间离散化数据显示季节性，具有复杂的非线性关系，可以随时间变化。因此，跟踪预测因素的变化是至关重要的，这些变化可能会导致关系的变化。这可以使用参数估计的贝叶斯框架来实现，该框架将关于预测者的不确定性和先验信念建模为概率分布。然后结合关于来自观测数据的预测器的附加信息来更新分布，使得这反映了参数信息的变化。传统的马尔可夫链蒙特卡罗方法在贝叶斯推理中需要知道一定比例的目标分布，但由于似然函数的难解性，这种方法对于时变数据是不可行的。因此，粒子滤波方法为具有复杂结构和难以处理的似然性的动态时间序列模型提供了数值近似。这个项目的目的是结合子集选择方法和适当的粒子滤波贝叶斯方法来建立动态时间序列模型。传统的统计模型都是手工拟合，耗时长，成本高，因此需要考虑模型拟合和预报器选择的自动化过程。所开发的方法将应用于BT，目的是与目前工业上使用的方法相比，提高洞察力和预测能力。这个项目与EPSRC相关，因为我们正在开发和应用统计方法，以提高工业环境中的预测能力和决策过程。此外，开发的方法还旨在降低目前使用的相关方法的计算成本，因此属于EPSRC的数学和数据科学重点。

The project is in collaboration with BT where large amounts of time-series data are collected along with vast amounts of potential predictors e.g. monthly telecommunications data with location and weather. With an increasing number of data generated and stored in many fields, applying statistical analysis to these data sets gives insight on the relationships between predictors and the response of interest. However, difficulties arise in situations where too many predictors exist as model fitting and prediction become challenging in the high dimensional setting with potentially many predictors being strongly correlated. Subset selection methods provide an attractive solution to ensure the statistical model produced is interpretable by reducing the number of predictors, however the difficulty is in identifying the relevant combination of predictors which best explain the response. Furthermore, evaluating all potential models as the number of predictors increases is computationally expensive and so requires methods which can greatly reduce this cost.In addition, time-discretised data display seasonalities with complex non-linear relationships which can change over time. Therefore, it is crucial to keep track of the changes in predictors which potentially causes the changes in the relationship. This could be achieved using the Bayesian framework for parameter estimation which models uncertainty and prior beliefs about predictors as probability distributions. Additional information about predictors from observed data are then incorporated to update the distribution such that this reflects changes in parameter information. Traditional Markov Chain Monte Carlo methods used in Bayesian inference require target distributions to be known up to some proportionality however, such methods are infeasible for time dependent data due to the intractability of the likelihood function. Therefore, particle filtering methods offer numerical approximations for dynamic time series models with complex structures and intractable likelihoods. The aim of this project is to incorporate subset selection methods and appropriate particle filter Bayesian methodology for dynamic time series models. Traditional statistical models are fit by hand which is time consuming and costly so automating the process of model fitting and predictor selection should be taken into consideration. The developed method would be applied in BT with the aims to improve insight and predictive ability in comparison to the current methods used in industry. This project is relevant to EPSRC as we are developing and applying statistical methods which can improve the predictive ability and decision-making process in industrial settings. In addition, the developed method also aims to reduce the computational cost of the relevant methods that are currently used, and so falls within the mathematics and data science focus of EPSRC.