Was that change real? Quantifying uncertainty for change points

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Detecting changes in data is currently one of the most active areas of statistics. In many applications there is interest in segmenting the data into regions with the same statistical properties, either as a way to flexibly model data, to help with down-stream analysis or to ensure predictions are made based only on relevant data. Whilst in others the main interest lies in detecting when changes have occurred as they indicate features of interest, from potential failures of machinery to security breaches or the presence of genomic features such as copy number variations. To date most research in this area has been developing methods for detecting changes: algorithms that input data and output a best guess as to whether there have been relevant changes, and if so how many there have been and when they occurred. A comparatively ignored problem is assessing how confident we are that a specific change has occurred in a given part of the data. In many applications, quantifying the uncertainty around whether a change has occurred is of paramount importance. For example, if we are monitoring a large communication network, and changes indicate potential faults, it is helpful to know how confident we are that there is a fault at any given point in the network so that we can prioritise the use of limited resources available for investigating and repairing faults. When analysing calcium imaging data on neuronal activity, where changes correspond to times at which a neuron fires, it is helpful to know how certain we are that a neuron fired at each time point so as to improve down-stream analysis of the data.A naive approach to this problem is to first detect changes and then apply standard statistical tests for their presence. But this approach is flawed as it uses the data twice, first to decide where to test and then to perform the test. We can overcome this using sample splitting ideas - where we use half the data to detect a change, and the other half to perform the test. But such methods lose power, e.g. from using only part of the data to detect changes. This proposal will develop statistically valid approaches to quantifying uncertainty, that are more powerful than sample splitting approaches. These approaches are based on two complementary ideas (i) performing inference prior to detection; and (ii) develop tests for a change that account for earlier detection steps. The output will be a new general toolbox for change points encompassing both new general statistical methods and their implementation within software packages.