In this paper, we assume that observations are independent from the regression model
where is a one-dimensional exponential family distribution with densities . The parametric function is a right-continuous piecewise constant function. The model (1) includes the Gaussian mean regression as a special case, that is,
where and the standard Gaussian.
The (multiple) change-point problem amounts to estimating the number and locations of change-points and the value of functionon each segment. The study of change-point detection problems has a long and rich history in the statistical literatures (Carlstein et al., 1994; Csörgö and Horvàth, 2011; DavidSiegmund, 2013), and has experienced a revival in recent years, mainly due to modern large scale applications, for example in bioinformatics, predicting transmembrane helix locations (Lio and Vannucci, 2000), detecting changes in the DNA copy number (Olshen et al., 2004; Venkatraman and Olshen, 2007); in climate, analyzing undocumented change-points in climate data (Reeves et al., 2007); and in economics and finance, identifying change-points in financial volatility (Spokoiny, 2009).
Among the vast literature of change-point problems, we consider the so-called multiscale change-point segmentation methods (see e.g. Frick et al., 2014; Li et al., 2016), which are statistically well-understood and meanwhile practically well-performed, see also (Davies et al., 2012; Hotz et al., 2013; Li et al., 2017). These multiscale segmentation methods minimize the number of the change-points subjected to a side-constraint that multiscale statistics does not exceed a specified threshold (see Section 2.1 for a formal definition). The threshold , as a balancing parameter between the data-fit and model complexity, is often chosen as the quantile of under null distribution (e.g., ). Unfortunately, the computation of such a quantile involves the evaluation of , which has quadratic computational time in terms of sample size, and has to be repeated sufficiently many times to guarantee a proper estimation accuracy. This makes the multiscale segmentation methods impractical for large scale applications (e.g, for ). To overcome this computation bottleneck, we proposed, in this paper, a fast algorithm with linear computational complexity for the evaluation of , see Figure 1 for an illustration.
The rest of the paper is organized as follows. In Section 2, we introduce the multiscale change-point segmentation methods, and some basic concepts from algorithmic geometry. In Section 3 we propose a linear algorithm for the evaluation of and give its complexity analysis. The performance of the proposed algorithm is examined by simulations in Section 4. Section 5 concludes this paper.
2.1 Multiscale change-point segmentation
We start with a brief introduction of the multiscale change-point segmentation methods for the change-point problem in (1). Recall that the underlying truth is right-continuous and piecewise constant, i.e.,
where denote the locations of change-points, and the function value on the th segment with . Let denote the space of all right continuous step functions, and, for every in , let denote the set of change-points and the number of change-points.
where is a user-specified threshold, and a multiscale statistic. By we denote the collection of all subintervals of . The multiscale statistic is defined as the maximum of penalized local likelihood ratio statistic on every interval where is constant, that is,
Here the penalty terms play a role as scale (i.e., length of ) calibration, which aim to put different scales on equal baseline especially for small intervals (see Dümbgen and Spokoiny, 2001; Frick et al., 2014). The local likelihood ratio statistic
is a testing statistic on the hypothesisversus the alternative with on interval , more precisely,
The user-specific threshold in (3, see Li et al. (2017). In practice, it is recommended to select the quantile of null asymptotic distribution of with certain significance level , which allows for an immediate statistical interpretation
see Frick et al. (2014). Given such choices of , the solution to problem (3) exists but may be non-unique, in which case one is free to choose the solution, such as the constrained maximum likelihood estimator (Frick et al., 2014). Note that the value of can be estimated via Monte Carlo simulations, because the distribution of or its asymptotic distribution (Frick et al., 2014, Theorem 2.1) is independent of the unknown truth .
2.2 Constrained Minkowski sum
We now restrict ourselves to the Euclidean space . The Minkowski sum, a fundamental concept in algorithmic geometry, is defined as for . As in Bernholt et al. (2009), we define the constrained Minkowski sum as
By we denote the convex hull of , and by the set of vertices of . Bernholt and Hofmeister (2006), Bernholt et al. (2007, 2009) have shown that can be computed in time, if and are sorted with respect to the first coordinate. More precisely, a set can be computed such that and .
We recall some basic results of quasiconvexity, a useful generalization of convexity, see e.g., (Boyd and Vandenberghe, 2004, Section 4 in Chapter 3) for further details and the proofs.
Let be a nonempty convex set. A function is called quasiconvex if its sublevel set is convex for every .
Note that convex functions are clearly quasiconvex, and that many properties of convex functions carry over to quasiconvex functions.
let be a nonempty convex set. A function is quasiconvex if and only if for any and any it holds that
3 An method for quantile simulation
In this section, we first consider the computation of quantiles for multiscale change-point segmentation methods. We will show that the evaluation of is equivalent to finding the maximal value of a quasiconvex function over a constrained Minkowski sum.
3.1 Fast quantile simulation
We start with the Gaussian mean regression model (2), and the penalty term given in Frick et al. (2014). Note that in model (2) the distribution of is independent of . Thus, it is sufficient to consider
) can be speeded up by utilizing fast Fourier transforms, resulting inruntime (which is implemented in CRAN R-package “stepR”), see e.g. Hotz et al. (2013). A further speedup is possible by means of cumulative sum transformation , which reduces a summation over to a single subtraction. This leads to an algorithm of complexity (which is implemented in CRAN R-package “FDRSeg”), see also Allison (2003). In what follows, we will present a fast algorithm for evaluating (6) in a linear runtime, i.e., .
For , we define , and . The evaluation of in (6) can be written as an optimization of a bivariate function over finite collection of points, more precisely,
The bivariate function in (7) is quasiconvex over .
By Definition 1, it is sufficient to show that sublevel set
is convex for all . Define for . Notice that it is trivial when because sublevel set is empty. If is not empty, it follows that . Noting that implies , we have
Thus, is concave, and it follows that is convex for all . ∎
Moreover, it is known that there is a linear algorithm for finding (see Section 2.2). Based on it, we can derive a linear algorithm for the evaluation of , the details of which is given in Algorithm 1.
In Algorithm 1, the incremental Graham scan algorithm (Graham, 1972) is employed in first step to compute the convex hull of in runtime on line 2. For each point , we consider in order to satisfy the constraint (line 9). Among such points, we compute a set that contains the vertices involving in (line 11-16). After recording to (line 18), we delete and from (line 21), because there is no point in of the form for and , see Bernholt et al. (2009) for a proof. Then the algorithm proceeds recursively; each time we update and . In the end, the set , being a subset of , contains . The maximal value can be obtained on .
As in (6) is independent of , we can always assume . Given realization , we compute via Algorithm 1. The quantile of is computed via repetitions of such a procedure. Thus, the quantile of can be computed in runtime. This is significantly faster than the best existing algorithm, which is of runtime. In general, larger leads to more precise estimation of the quantile. In practice, we find that the estimate is quite stable for (see Figure 2), and thus suggest as the default choice.
3.2 Quasiconvexity of exponential family regression
By Proposition 1, a larger class of quasiconvex objective functions attain the maximum over on . In exponential family regression model, the multiscale statistic (4) is independent of . Similar to Gaussian mean model (7), it can be mapped into a bivariate function . Next we will discuss the computation of multiscale statistics for the exponential family regression model (1) and extend Algorithm 1 to a general form.
We assume in model (1), the independent observations ( come from a exponential family with standard density
where the cumulant function is strictly convex on . Then the maximum scanning statistics in (5) through exponential family regression can be simplified as:
Let and as defined before, then the evaluation of in (8) can be written as a supremum over of a bivariate function:
The bivariate function in (9) is convex over .
Since is linear about , it follows that ,
By the definition of convex function, is convex on . ∎
The multiscale statistic in (4) is made up of the scanning statistic and a penalty term . The penalty function working as a scale calibration only depends on interval length . So multiscale statistic can be written as in (9) added by a penalty function:
By Lemma 1, the bivariate function is convex and it keeps convex if it is substracted by a concave penalty . According to Proposition 1 the maximum of (10) over can be attained on the vertices of the convex hull of . Thus, the optimization of multiscale statistic can also be solved by Algorithm 1. We state this result in Theorem 1.
The multiscale statistic for exponential family regression model with concave penalty terms can be evaluated in a linear runtime.
In summary, the proposed algorithm is a general method for simulating multiscale statistic from exponential family regression model with convex penalization. It speeds up the existing algorithms to a linear runtime. Meanwhile, the memory space mainly used for storing points is bounded by the number of vertices in , i.e., .
4 Simulation study
This section examines the empirical performance of the proposed Algorithm 1. We provide the implementation of the proposed method in R package “linearQ”, available from CRAN.
We start with the Gaussian mean regression in (2), and compare the proposed method with other existing methods. To this end, we consider the Fourier transform based algorithm, implemented in CRAN R package “stepR” (Frick et al., 2014), and the cumulative sum based algorithm, implemented in CRAN R package “FDRSeg” (Li et al., 2016), see Section 3.1
. The simulation data is generated as i.i.d. realizations of standard normal random variables, for different sample sizes ranging fromto . For a given sample size, we repeat times, which is set to . The average computation time for the evaluation of for different methods is reported in Figure 3. It shows that the proposed method is significantly faster than the other two, achieving one order speed-up, with its computation complexity .
In addition, the proposed method applies to every distribution in exponential family provided that the penalty term is convex, see Section 3.2. As a demonstration, we consider the Poisson case, i.e., in (1
) is the Poisson distribution with mean. Figure 4 illustrates the computation time of evaluating without scale penalty, with the data sizes from to and repetition , via the proposed method. Again the empirical performance supports our theoretical complexity analysis that the computation time is linear in terms of sample size .
The multiscale change-point segmentation methods are recognized as the-state-of-the-art in change-point inference, and have been playing an important role in various applications. In this paper, we propose a fast algorithm for the computation of the only tuning parameter of such multiscale change-point segmentation methods. The proposed method has a linear computation complexity and a linear memory complexity, in terms of the sample size, in sharp contrast to the existing methods with at least quadratic computation complexity. The crucial idea behind is to transform the original problem into the maximization of a quasiconvex function over a constrained Minkowski sum. The theoretical complexity is well supported by the empirical performance. Extension to general models beyond exponential family is a possible line of future research.
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