arXiv subjects

Piotr Fryzlewicz

Publications and source records attributed to Piotr Fryzlewicz.

32 records · Page 2Linked to original sources

arXiv2020

Detecting possibly frequent change-points: Wild Binary Segmentation 2 and steepest-drop model selection

Many existing procedures for detecting multiple change-points in data sequences fail in frequent-change-point scenarios. This article proposes a new change-point detection methodology designed to work well in both infrequent and frequent change-point settings. It is made up of two ingredients: one is "Wild Binary Segmentation 2" (WBS2), a recursive algorithm for producing what we call a `complete' solution path to the change-point detection problem, i.e. a sequence of estimated nested models containing $0, \ldots, T-1$ change-points, where $T$ is the data length. The other ingredient is a new model selection procedure, referred to as "Steepest Drop to Low Levels" (SDLL). The SDLL criterion acts on the WBS2 solution path, and, unlike many existing model selection procedures for change-point problems, it is not penalty-based, and only uses thresholding as a certain discrete secondary check. The resulting WBS2.SDLL procedure, combining both ingredients, is shown to be consistent, and to significantly outperform the competition in the frequent change-point scenarios tested. WBS2.SDLL is fast, easy to code and does not require the choice of a window or span parameter.

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arXiv2020

Detection of gamma-ray transients with wild binary segmentation

In the context of time domain astronomy, we present an offline detection search of gamma-ray transients using a wild binary segmentation analysis called FWBSB targeting both short and long gamma-ray bursts (GRBs) and covering the soft and hard gamma-ray bands. We use NASA Fermi/GBM archival data as a training and testing data set. This paper describes the analysis applied to the 12 NaI detectors of the Fermi/GBM instrument. This includes background removal, change-point detection that brackets the peaks of gamma-ray flares, the evaluation of significance for each individual GBM detector and the combination of the results among the detectors. We also explain the calibration of the 10 parameters present in the method using one week of archival data. Finally, we present our detection performance result for 60 days of a blind search analysis with FWBSB by comparing to both the on-board and offline GBM search as well as external events found by others surveys such as Swift-BAT. We detect 42/44 on-board GBM{events but also other gamma-ray flares at a rate of 1 per hour in the 4-50 keV band. Our results show that FWBSB is capable of recovering gamma-ray flares, including the detection of soft X-ray long transients. FWBSB offers an independent identification of GRBs in combination with methods for determining spectral and temporal properties of the transient as well as localization. This is particularly useful for increasing the GRB rate and that will help the joint detection with gravitational-wave events.

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arXiv2019

Predictive, finite-sample model choice for time series under stationarity and non-stationarity

In statistical research there usually exists a choice between structurally simpler or more complex models. We argue that, even if a more complex, locally stationary time series model were true, then a simple, stationary time series model may be advantageous to work with under parameter uncertainty. We present a new model choice methodology, where one of two competing approaches is chosen based on its empirical, finite-sample performance with respect to prediction, in a manner that ensures interpretability. A rigorous, theoretical analysis of the procedure is provided. As an important side result we prove, for possibly diverging model order, that the localised Yule-Walker estimator is strongly, uniformly consistent under local stationarity. An R package, forecastSNSTS, is provided and used to apply the methodology to financial and meteorological data in empirical examples. We further provide an extensive simulation study and discuss when it is preferable to base forecasts on the more volatile time-varying estimates and when it is advantageous to forecast as if the data were from a stationary process, even though they might not be.

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arXiv2018

Simultaneous multiple change-point and factor analysis for high-dimensional time series

We propose the first comprehensive treatment of high-dimensional time series factor models with multiple change-points in their second-order structure. We operate under the most flexible definition of piecewise stationarity, and estimate the number and locations of change-points consistently as well as identifying whether they originate in the common or idiosyncratic components. Through the use of wavelets, we transform the problem of change-point detection in the second-order structure of a high-dimensional time series, into the (relatively easier) problem of change-point detection in the means of high-dimensional panel data. Also, our methodology circumvents the difficult issue of the accurate estimation of the true number of factors in the presence of multiple change-points by adopting a screening procedure. We further show that consistent factor analysis is achieved over each segment defined by the change-points estimated by the proposed methodology. In extensive simulation studies, we observe that factor analysis prior to change-point detection improves the detectability of change-points, and identify and describe an interesting `spillover' effect in which substantial breaks in the idiosyncratic components get, naturally enough, identified as change-points in the common components, which prompts us to regard the corresponding change-points as also acting as a form of `factors'. Our methodology is implemented in the R package {\tt factorcpt}, available from CRAN.

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arXiv2018

Narrowest-Over-Threshold Detection of Multiple Change-points and Change-point-like Features

We propose a new, generic and flexible methodology for nonparametric function estimation, in which we first estimate the number and locations of any features that may be present in the function, and then estimate the function parametrically between each pair of neighbouring detected features. Examples of features handled by our methodology include change-points in the piecewise-constant signal model, kinks in the piecewise-linear signal model, and other similar irregularities, which we also refer to as generalised change-points. Our methodology works with only minor modifications across a range of generalised change-point scenarios, and we achieve such a high degree of generality by proposing and using a new multiple generalised change-point detection device, termed Narrowest-Over-Threshold (NOT). The key ingredient of NOT is its focus on the smallest local sections of the data on which the existence of a feature is suspected. Crucially, this adaptive localisation technique prevents NOT from considering subsamples containing two or more features, a key factor that ensures the general applicability of NOT. For selected scenarios, we show the consistency and near-optimality of NOT in detecting the number and locations of generalised change-points. Furthermore, we propose to select NOT's threshold (automatically) via the strengthened Schwarz Information Criterion (sSIC) and give theoretical justifications. The NOT estimators are easy to implement and rapid to compute: the entire threshold-indexed solution path can be computed in close-to-linear time. Importantly, the NOT approach is easy to extend by the user to tailor to their own needs. There is no single competitor, but we show that the performance of NOT matches or surpasses the state of the art in the scenarios tested. Our methodology is implemented in the R package \textbf{not}.

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arXiv2017

Likelihood ratio Haar variance stabilization and normalization for Poisson and other non-Gaussian noise removal

We propose a new methodology for denoising, variance-stabilizing and normalizing signals whose both mean and variance are parameterized by a single unknown varying parameter, such as Poisson or scaled chi-squared. Key to our methodology is the observation that the signed and square-rooted generalized log-likelihood ratio test for the equality of the local means is approximately and asymptotically distributed as standard normal under the null. We use these test statistics within the Haar wavelet transform at each scale and location, referring to them as the likelihood ratio Haar (LRH) coefficients of the data. In the denoising algorithm, the LRH coefficients are used as thresholding decision statistics, which enables the use of thresholds suitable for i.i.d. Gaussian noise, despite the standard Haar coefficients of the signal being heteroscedastic. In the variance-stabilizing and normalizing algorithm, the LRH coefficients replace the standard Haar coefficients in the Haar basis expansion. To the best of our knowledge, the variance-stabilizing and normalizing properties of the generalized likelihood ratio test have not been interpreted or exploited in this manner before. We prove the consistency of our LRH smoother for Poisson counts with a near-parametric rate, and various numerical experiments demonstrate the good practical performance of our methodology.

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arXiv2016

Multiscale and multilevel technique for consistent segmentation of nonstationary time series

In this paper, we propose a fast, well-performing, and consistent method for segmenting a piecewise-stationary, linear time series with an unknown number of breakpoints. The time series model we use is the nonparametric Locally Stationary Wavelet model, in which a complete description of the piecewise-stationary second-order structure is provided by wavelet periodograms computed at multiple scales and locations. The initial stage of our method is a new binary segmentation procedure, with a theoretically justified and rapidly computable test criterion that detects breakpoints in wavelet periodograms separately at each scale. This is followed by within-scale and across-scales post-processing steps, leading to consistent estimation of the number and locations of breakpoints in the second-order structure of the original process. An extensive simulation study demonstrates good performance of our method.

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arXiv2016

High-dimensional variable selection via tilting

The paper considers variable selection in linear regression models where the number of covariates is possibly much larger than the number of observations. High dimensionality of the data brings in many complications, such as (possibly spurious) high correlations between the variables, which result in marginal correlation being unreliable as a measure of association between the variables and the response. We propose a new way of measuring the contribution of each variable to the response which takes into account high correlations between the variables in a data-driven way. The proposed tilting procedure provides an adaptive choice between the use of marginal correlation and tilted correlation for each variable, where the choice is made depending on the values of the hard thresholded sample correlation of the design matrix. We study the conditions under which this measure can successfully discriminate between the relevant and the irrelevant variables and thus be used as a tool for variable selection. Finally, an iterative variable screening algorithm is constructed to exploit the theoretical properties of tilted correlation, and its good practical performance is demonstrated in a comparative simulation study.

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arXiv2016

Multiscale interpretation of taut string estimation and its connection to Unbalanced Haar wavelets

We compare two state-of-the-art non-linear techniques for nonparametric function estimation via piecewise constant approximation: the taut string and the Unbalanced Haar methods. While it is well-known that the latter is multiscale, it is not obvious that the former can also be interpreted as multiscale. We provide a unified multiscale representation for both methods, which offers an insight into the relationship between them as well as suggesting lessons both methods can learn from each other.

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arXiv2016

Multiple-change-point detection for high dimensional time series via sparsified binary segmentation

Time series segmentation, a.k.a. multiple change-point detection, is a well-established problem. However, few solutions are designed specifically for high-dimensional situations. In this paper, our interest is in segmenting the second-order structure of a high-dimensional time series. In a generic step of a binary segmentation algorithm for multivariate time series, one natural solution is to combine CUSUM statistics obtained from local periodograms and cross-periodograms of the components of the input time series. However, the standard "maximum" and "average" methods for doing so often fail in high dimensions when, for example, the change-points are sparse across the panel or the CUSUM statistics are spuriously large. In this paper, we propose the Sparsified Binary Segmentation (SBS) algorithm which aggregates the CUSUM statistics by adding only those that pass a certain threshold. This "sparsifying" step reduces the impact of irrelevant, noisy contributions, which is particularly beneficial in high dimensions. In order to show the consistency of SBS, we introduce the multivariate Locally Stationary Wavelet model for time series, which is a separate contribution of this work.

stat.ME↗

arXiv2014

Wild binary segmentation for multiple change-point detection

We propose a new technique, called wild binary segmentation (WBS), for consistent estimation of the number and locations of multiple change-points in data. We assume that the number of change-points can increase to infinity with the sample size. Due to a certain random localisation mechanism, WBS works even for very short spacings between the change-points and/or very small jump magnitudes, unlike standard binary segmentation. On the other hand, despite its use of localisation, WBS does not require the choice of a window or span parameter, and does not lead to a significant increase in computational complexity. WBS is also easy to code. We propose two stopping criteria for WBS: one based on thresholding and the other based on what we term the `strengthened Schwarz information criterion'. We provide default recommended values of the parameters of the procedure and show that it offers very good practical performance in comparison with the state of the art. The WBS methodology is implemented in the R package wbs, available on CRAN. In addition, we provide a new proof of consistency of binary segmentation with improved rates of convergence, as well as a corresponding result for WBS.

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arXiv2011

Mixing properties of ARCH and time-varying ARCH processes

There exist very few results on mixing for non-stationary processes. However, mixing is often required in statistical inference for non-stationary processes such as time-varying ARCH (tvARCH) models. In this paper, bounds for the mixing rates of a stochastic process are derived in terms of the conditional densities of the process. These bounds are used to obtain the $α$, 2-mixing and $β$-mixing rates of the non-stationary time-varying $\operatorname {ARCH}(p)$ process and $\operatorname {ARCH}(\infty)$ process. It is shown that the mixing rate of the time-varying $\operatorname {ARCH}(p)$ process is geometric, whereas the bound on the mixing rate of the $\operatorname {ARCH}(\infty)$ process depends on the rate of decay of the $\operatorname {ARCH}(\infty)$ parameters. We note that the methodology given in this paper is applicable to other processes.

math.ST↗

arXiv2008

Data-driven wavelet-Fisz methodology for nonparametric function estimation

We propose a wavelet-based technique for the nonparametric estimation of functions contaminated with noise whose mean and variance are linked via a possibly unknown variance function. Our method, termed the data-driven wavelet-Fisz technique, consists of estimating the variance function via a Nadaraya-Watson estimator, and then performing a wavelet thresholding procedure which uses the estimated variance function and local means of the data to set the thresholds at a suitable level. We demonstrate the mean-square near-optimality of our wavelet estimator over the usual range of Besov classes. To achieve this, we establish an exponential inequality for the Nadaraya-Watson variance function estimator. We discuss various implementation issues concerning our wavelet estimator, and demonstrate its good practical performance. We also show how it leads to a new wavelet-domain data-driven variance-stabilising transform. Our estimator can be applied to a variety of problems, including the estimation of volatilities, spectral densities and Poisson intensities, as well as to a range of problems in which the distribution of the noise is unknown.

math.ST↗

arXiv2008

Normalized least-squares estimation in time-varying ARCH models

We investigate the time-varying ARCH (tvARCH) process. It is shown that it can be used to describe the slow decay of the sample autocorrelations of the squared returns often observed in financial time series, which warrants the further study of parameter estimation methods for the model. Since the parameters are changing over time, a successful estimator needs to perform well for small samples. We propose a kernel normalized-least-squares (kernel-NLS) estimator which has a closed form, and thus outperforms the previously proposed kernel quasi-maximum likelihood (kernel-QML) estimator for small samples. The kernel-NLS estimator is simple, works under mild moment assumptions and avoids some of the parameter space restrictions imposed by the kernel-QML estimator. Theoretical evidence shows that the kernel-NLS estimator has the same rate of convergence as the kernel-QML estimator. Due to the kernel-NLS estimator's ease of computation, computationally intensive procedures can be used. A prediction-based cross-validation method is proposed for selecting the bandwidth of the kernel-NLS estimator. Also, we use a residual-based bootstrap scheme to bootstrap the tvARCH process. The bootstrap sample is used to obtain pointwise confidence intervals for the kernel-NLS estimator. It is shown that distributions of the estimator using the bootstrap and the ``true'' tvARCH estimator asymptotically coincide. We illustrate our estimation method on a variety of currency exchange and stock index data for which we obtain both good fits to the data and accurate forecasts.

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