206 research outputs found
Optimal change point detection and localization in sparse dynamic networks
We study the problem of change point localization in dynamic networks models. We assume that we observe a sequence of independent adjacency matrices of the same size, each corresponding to a realization of an unknown inhomogeneous Bernoulli model. The underlying distribution of the adjacency matrices are piecewise constant, and may change over a subset of the time points, called change points. We are concerned with recovering the unknown number and positions of the change points. In our model setting, we allow for all the model parameters to change with the total number of time points, including the network size, the minimal spacing between consecutive change points, the magnitude of the smallest change and the degree of sparsity of the networks. We first identify a region of impossibility in the space of the model parameters such that no change point estimator is provably consistent if the data are generated according to parameters falling in that region. We propose a computationally-simple algorithm for network change point localization, called network binary segmentation, that relies on weighted averages of the adjacency matrices. We show that network binary segmentation is consistent over a range of the model parameters that nearly cover the complement of the impossibility region, thus demonstrating the existence of a phase transition for the problem at hand. Next, we devise a more sophisticated algorithm based on singular value thresholding, called local refinement, that delivers more accurate estimates of the change point locations. Under appropriate conditions, local refinement guarantees a minimax optimal rate for network change point localization while remaining computationally feasible
Univariate Mean Change Point Detection: Penalization, CUSUM and Optimality
The problem of univariate mean change point detection and localization based
on a sequence of independent observations with piecewise constant means has
been intensively studied for more than half century, and serves as a blueprint
for change point problems in more complex settings. We provide a complete
characterization of this classical problem in a general framework in which the
upper bound on the noise variance, the minimal spacing
between two consecutive change points and the minimal magnitude of the
changes, are allowed to vary with . We first show that consistent
localization of the change points, when the signal-to-noise ratio , is impossible. In contrast, when
diverges with at the rate of at least
, we demonstrate that two computationally-efficient change
point estimators, one based on the solution to an -penalized least
squares problem and the other on the popular wild binary segmentation
algorithm, are both consistent and achieve a localization rate of the order
. We further show that such rate is minimax
optimal, up to a term
Divide and Conquer Dynamic Programming: An Almost Linear Time Change Point Detection Methodology in High Dimensions
We develop a novel, general and computationally efficient framework, called
Divide and Conquer Dynamic Programming (DCDP), for localizing change points in
time series data with high-dimensional features. DCDP deploys a class of greedy
algorithms that are applicable to a broad variety of high-dimensional
statistical models and can enjoy almost linear computational complexity. We
investigate the performance of DCDP in three commonly studied change point
settings in high dimensions: the mean model, the Gaussian graphical model, and
the linear regression model. In all three cases, we derive non-asymptotic
bounds for the accuracy of the DCDP change point estimators. We demonstrate
that the DCDP procedures consistently estimate the change points with sharp,
and in some cases, optimal rates while incurring significantly smaller
computational costs than the best available algorithms. Our findings are
supported by extensive numerical experiments on both synthetic and real data.Comment: 84 pages, 4 figures, 6 table
- …