RUI: A Bayesian Approach to Sequential Change Point Detection

该项目将开发一种有效的变更点算法，它不仅可以指示变更点发生的时间，还可以提供关于这些变更的数量和准确时间的不确定性估计。该模型的应用非常广泛，包括收集长序列数据的任何领域，例如医学（例如EEG读数），经济学（例如股票市场数据，煤矿灾害）和气候（例如温度读数，冰川记录）。更具体地说，500万年的全球冰量记录至少显示了两个明显的变化。第一次发生在大约270万年前，代表了地球上冰量的增加，因为北半球开始形成永久冰川，而更近的一次发生在大约80万年前，代表了主要冰川融化事件的频率从每4万年逐渐变化到每10万年。一个更突出的例子是NCDC的全球温度异常数据集，许多人将其作为全球变暖的证据。这一记录表明，在过去133年里，地球上的温度上升速度发生了三次变化——1906年、1945年以及1963年或1976年。该算法将能够处理连续数据，使其能够在记录每个新观察结果时快速更新自己，并能够准确预测数据集中发生变化的位置。众所周知，长时间序列在本质上往往是异构的，任何对这些数据集建模的尝试都必须考虑随时间变化的参数。这种差异可以简单到底层信号的平均值、斜率或频率的变化。然而，鉴定？改变点吗?并不总是一项微不足道的任务，因为潜在解决方案的数量随着数据集的长度呈指数级增长，使得蛮力尝试解决问题变得不可行的。在贝叶斯变点算法的前期工作中，通过使用类动态规划递归将计算复杂度从指数型降低到二次型，得到了多变点问题的高效精确概率解。从变化点位置的联合后验分布中提取的样本量化了数据集中变化的数量和时间的不确定性。在这个项目中，现有的变更点模型将被修改以处理顺序数据。一旦这个初始目标完成，研究将转向进一步的修改，包括处理相关误差项的能力和具有线性复杂性的近似算法，将计算复杂性降低到可以分析任何长度的时间序列的程度。该项目自然适合本科教育，并将作为夏季研究项目、高级论文和新统计项目的潜在研讨会课程的基础。通过这个项目开发的软件将向公众提供，以便使各种领域的研究人员能够使用这种尖端的统计方法。

This project will develop an efficient change point algorithm that not only will indicate when change points occur, but also provide uncertainty estimates as to the number and exact timing of these changes. Applications of this model are widespread and include any field where long sequences of data are collected such as medicine (e.g. EEG readings), economics (e.g. stock market data, coal mining disasters), and climate (e.g. temperature readings, glacial records). More specifically, a 5 million year record of global ice volume shows at least two distinct changes. The first, around 2.7 million years ago, represents an increase in the amount of ice volume on the Earth as permanent glaciers began to form in the northern hemisphere, whereas a more recent change around 0.8 million years ago represents a gradual change in the frequency of major glacial melting events from every 40,000 to every 100,000 years. A more prominent example concerns NCDC's global temperature anomalies data set that many have cited as evidence of global warming. This record indicates three changes in the rate of temperature increases on the Earth over the last 133 years - in 1906, 1945, and either 1963 or 1976. The algorithm will be able to handle sequential data, giving it the ability to quickly update itself as each new observation is recorded, and will be able to accurately predict where in the data set a change point has occurred. It is well known that long time series are often heterogeneous in nature, any attempt to model these data sets may have to account for parameters that change through time. The difference can be as simple as a change in the mean, slope, or frequency of the underlying signal. However, the identification of ?change points? is not always a trivial task as the number of potential solutions grows exponentially with the length of the data set, rendering brute force attempts to solve the problem infeasible. Previous work on a Bayesian change point algorithm has produced an efficient and exact probabilistic solution to the multiple change point problem by using dynamic programming-like recursions to reduce the computational complexity from exponential to quadratic. Samples drawn from the joint posterior distribution of the change point locations quantify the uncertainty in both the number and timing of changes in the data set. In this project, the existing change point model will be modified to handle sequential data. Once this initial objective is complete, research will turn towards further modifications that include the ability to handle correlated error terms and an approximate algorithm that has linear complexity, bringing the computational complexity down to a point where a time series of any length can be analyzed. The project fits naturally with undergraduate education and will serve as the basis of summer research projects, senior theses, and a potential seminar course for a new statistics program. The software developed through this project will be made publicly available so as to make this cutting-edge statistical methodology accessible to researchers in a wide variety of fields.