This paper addresses the problem of detecting and characterizing local variability in time series and other forms of sequential data. The goal is to identify and characterize statistically significant variations, at the same time suppressing the inevitable corrupting observational errors. We present a simple nonparametric modeling technique and an algorithm implementing it-an improved and generalized version of Bayesian Blocks [Scargle 1998]-that finds the optimal segmentation of the data in the observation interval. The structure of the algorithm allows it to be used in either a real-time trigger mode, or a retrospective mode. Maximum likelihood or marginal posterior functions to measure model fitness are presented for events, binned counts, and measurements at arbitrary times with known error distributions. Problems addressed include those connected with data gaps, variable exposure, extension to piecewise linear and piecewise exponential representations, multi-variate time series data, analysis of variance, data on the circle, other data modes, and dispersed data. Simulations provide evidence that the detection efficiency for weak signals is close to a theoretical asymptotic limit derived by [Arias-Castro, Donoho and Huo 2003]. In the spirit of Reproducible Research [Donoho et al. (2008)] all of the code and data necessary to reproduce all of the figures in this paper are included as auxiliary material.

Abstract-Many signal processing problems can be solved by maximizing the fitness of a segmented model over all possible partitions of the data interval. This letter describes a simple but powerful algorithm that searches the exponentially large space of partitions of N data points in time O(N 2 ). The algorithm is guaranteed to find the exact global optimum, automatically determines the model order (the number of segments), has a convenient real-time mode, can be extended to higher dimensional data spaces, and solves a surprising variety of problems in signal detection and characterization, density estimation, cluster analysis and classification.

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A nondecreasing sequence of positive integers is (α, β)-Conolly, or Conollylike for short, if for every positive integer m the number of times that m occurs in the sequence is α + βr m , where r m is 1 plus the 2-adic valuation of m. A recurrence relation is (α, β)-Conolly if it has an (α, β)-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the formA(n − a ij )) with appropriate initial conditions. For any fixed integers k and p 1 , p 2 , . . . , p k we prove that there are only finitely many pairs (α, β) for which A(n) can be (α, β)-Conolly. For the case where α = 0 and β = 1, we provide a bijective proof using labelled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence H(n) = H(n − H(n − 2)) + H(n − 3 − H(n − 5)) also has the Conolly sequence as a solution. When k = 2 and p 1 = p 2 , we construct an example of an (α, β)-Conolly recursion for every possible (α, β) pair, thereby providing the first examples of nested recursions with p i > 1 whose solutions are completely understood. Finally, in the case where k = 2 and p 1 = p 2 , we provide an if and only if condition for a given nested recurrence A(n) to be (α, 0)-Conolly by proving a very general ceiling function identity.