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Detecting and dating structural breaks in functional data without dimension reduction
Alexander Aue, Gregory Rice, Ozan Sönmez
TL;DR
Functional break detection and dating often use fPCA, which can lose information about breaks orthogonal to the retained basis. This paper develops fully functional tests and break-date estimators, and reports advantages for breaks poorly aligned with leading principal components, including in simulations and an Australian temperature-curve application.
Problem
Dimension reduction can discard functional break information orthogonal to the projection basis, making consistent detection or dating impossible for such breaks.
Source: §1 · p. 2Method
Results
For breaks orthogonal to leading principal components, fully functional estimation asymptotically improves on fPCA-based estimation, and simulations report better performance especially for poorly aligned breaks.
Source: §1 · p. 2, §6 · p. 22Takeaways & Limitations
In Australian annual temperature curves, fPCA-based break dates can occur outside the confidence intervals from the fully functional procedure.
Source: §1 · p. 3The break-date confidence interval is derived under a shrinking-break assumption and tends to be conservative in practice.
Source: §2.2 · p. 7
Abstract
from arXiv · showhide
Methodology is proposed to uncover structural breaks in functional data that is "fully functional" in the sense that it does not rely on dimension reduction techniques. A thorough asymptotic theory is developed for a fully functional break detection procedure as well as for a break date estimator, assuming a fixed break size and a shrinking break size. The latter result is utilized to derive confidence intervals for the unknown break date. The main results highlight that the fully functional procedures perform best under conditions when analogous fPCA based estimators are at their worst, namely when the feature of interest is orthogonal to the leading principal components of the data. The theoretical findings are confirmed by means of a Monte Carlo simulation study in finite samples. An application to annual temperature curves illustrates the practical relevance of the proposed procedures.
1 Introduction
Functional break procedures often rely on fPCA, which can discard break information orthogonal to the retained basis. This paper develops fully functional detection and dating methods, with advantages for such breaks and an application to Australian temperature curves.
1 Introduction
- Earlier functional break research includes methods for independent curves, weakly dependent time series, and other settings, while many FDA procedures use fPCA for dimension reduction.Related work also includes functional break methods using self-normalization and approaches for functional linear models and spatially distributed data.Source: §1 · pp. 2–3
- When a mean break is orthogonal to the projection basis, dimension-reduced procedures cannot consistently test for or date it.Projection discards information orthogonal to the retained basis.Source: §1 · p. 2
- The paper develops fully functional procedures for detecting and dating breaks without dimension reduction, with asymptotic theory under weaker assumptions than fPCA-based approaches.The theory does not require accounting for convergence of empirical covariance eigenvalues and eigenfunctions; related assumptions are typically formulated as finite fourth moments.Source: §1 · pp. 2–3
- For breaks orthogonal to leading principal components, the fully functional estimator asymptotically improves mean-break estimation, while fPCA methods can make such breaks difficult or impossible to detect.The comparison is for mean breaks of the same size.Source: §1 · pp. 2–3
- In Australian annual temperature curves, fPCA-based break dates can fall outside confidence intervals from the fully functional procedure.The covariance eigenvalues decay slowly, and fPCA dating accuracy can decline even when the break loads almost exclusively on the first component.Source: §1 · p. 3
2 Main results
The paper develops fully functional procedures for detecting a mean-function break and estimating its date under weakly dependent errors. Its asymptotic results highlight settings where breaks orthogonal to leading variation can be difficult for fPCA methods but easier for the fully functional approach.
2 Main results
- The section models a single mean-function break and develops fully functional detection and dating procedures based on a scaled functional CUSUM under weakly dependent errors.The observations are functions with a baseline mean shifted by δ after the unknown break date k*; the errors are assumed stationary and weakly dependent.Source: §2 · pp. 4–5
2.1 Asymptotic properties of structural break detector
- Under no break, the detector has a limit determined by long-run covariance eigenvalues and Brownian bridges; unlike fPCA procedures, it requires no nonorthogonality condition on the break.The null limit supports a test calibrated using its corresponding quantiles, approximated by Monte Carlo simulation with estimated eigenvalues.Source: §2.1 · pp. 5–6
2.2 Asymptotic properties of the break date estimator
- For fixed breaks, the date estimator's limit depends on the error distribution, whereas shrinking breaks yield a Brownian-motion limit with triangular drift and nuisance parameters.The shrinking-break result assumes ∥δn∥→0 while n∥δn∥²→∞.Source: §2.2 · pp. 6–7
- Breaks orthogonal to primary variation can be easier to detect and estimate asymptotically; when δ is orthogonal to error functions, the date limit's stochastic term vanishes.Alignment of shrinking break functions with eigenfunctions for smaller long-run covariance eigenvalues also drives σ² toward zero.Source: §2.2 · p. 7
- The shrinking-break date limit yields an asymptotic confidence interval for k*, but the intervals tend to be conservative in practice.The construction depends on unknown break fraction and variance parameters, which the paper discusses estimating; conservatism is attributed to the shrinking-break assumption.Source: §2.2 · p. 7
2.3 Two fPCA based approaches
- An fPCA date estimator using one projection can be inconsistent when the leading empirical component is orthogonal to the break, while the fully functional approach avoids that projection-based constraint.The paper identifies sizable breaks whose influence is not expressed in leading principal components as a setting where fully functional dating is especially advantageous.Source: §2.3 · p. 9
3 Implementation details
Implementation estimates the long-run covariance operator, uses its estimated eigenvalues to simulate test critical values, and constructs break-date confidence intervals by plugging in estimated parameters. The covariance estimator is consistent under an additional dependence condition, while the interval is asymptotically valid under the stated conditions.
3.1 Estimation of long-run covariance operator
- The lag-window estimator of the long-run covariance operator is consistent, and each fixed number of its leading eigenvalues is consistently estimated.This result requires Assumption 3.1 in addition to the conditions of Theorem 2.4.Source: §3.1 · pp. 9–10
- For moderately correlated errors, the tested weight and bandwidth choices performed similarly; test-size differences across bandwidths were at most 2% at a 5% level.Larger discrepancies and advantages for the empirical bandwidth were expected as dependence approached non-stationarity; reported results use h = n^1/4 and the Bartlett weight.Source: §3.1 · p. 11
3.2 Computation of critical values
- Critical values for the detector are approximated by simulating Brownian bridges using estimated eigenvalues, then taking quantiles of repeated Monte Carlo realizations.The reported Sections 4 and 5 use 1,000 repetitions.Source: §3.2 · pp. 11–12
3.3 Construction of confidence intervals
- The confidence interval for the break date uses plug-in estimates of the break fraction, variance parameter, and long-run covariance operator.The construction estimates the distributional quantities used for the interval from their population counterparts.Source: §3.3 · p. 12
- The resulting interval is asymptotically 1 − α confidence for k*, while replacing the variance estimate with the largest covariance eigenvalue yields a conservative interval.The theorem’s conditions include Assumption 3.1.Source: §3.3 · p. 12
4 Simulation Study
The simulations compare fully functional procedures with fPCA-based and change-aligned methods across eigenvalue settings, break shapes, and error distributions. Fully functional methods are especially competitive when fPCA performs poorly, while confidence intervals can be conservative and heavy tails challenge performance at two degrees of freedom.
4.1 Setting
- The simulations used 21 Fourier basis functions and three eigenvalue-decay settings, and examined temporal dependence by comparing independent innovations with first-order functional autoregressive curves (|κ| = 0.5).The settings represented finite-dimensional errors, fast eigenvalue decay, and slow eigenvalue decay.Source: §4.1 · pp. 12–13
4.2 Level and power of the detection procedures
- Across the tested settings, the proposed detector kept empirical levels reasonably well, whereas the change-aligned procedure produced the most variable results.The simulations considered iid and FAR(1) processes, with sample sizes n = 50 and n = 100.Source: §4.2 · p. 14, Table 4.1 · p. 15
- When breaks were largely orthogonal to the errors, the fully functional detector showed its expected power advantage over dimension-reduction methods.The comparison used Setting 1 with m = 20; power also improved as m increased.Source: §4.2 · p. 15, Figure 4.1 · p. 16
- Additional simulations found the expected power improvement as sample size increased, without more power loss than typical under serial correlation.Source: §4.2 · p. 15
4.3 Performance of the break dating procedures
- For break dating, the fully functional method was competitive, with boxplots no wider than fPCA counterparts and an advantage at m = 1 under slow eigenvalue decay.fPCA performance was sensitive to TVE, and often only its best TVE choice was competitive.Source: §4.3 · pp. 16–17, Figure 4.2 · p. 17
- The break-date confidence intervals were conservative and narrowed when the break was spread across more directions.The widest intervals occurred at m = 1; larger samples and higher SNR were also expected to improve width.Source: §4.3 · p. 17, Figure 4.3 · p. 18
4.4 Heavy tails
- Under heavy-tailed innovations, the proposed method was superior in the reported cases, while its performance deteriorated at two degrees of freedom.The fPCA procedure failed to produce reasonable results across the cases considered; the fully functional method was not theoretically justified at two degrees of freedom.Source: §4.4 · pp. 18–19, Figure 4.4 · p. 19
5 Application to annual temperature curves
Across eight Australian stations, the fully functional procedure rejected the no-break null. At Gayndah, its estimated break date and confidence interval differed from fPCA dates for reasonable choices of projection dimension, despite the break function loading mainly on the first eigenfunction.
5 Application to annual temperature curves
- For all eight Australian stations, the fully functional detection procedure rejected the null hypothesis of no structural break in the mean function.Both fully functional and fPCA dating procedures were then applied.Source: §5 · p. 19, Table 5.1 · p. 20
- At Gayndah, fully functional detection estimated a 1962 break date (95% CI: 1952–1966); fPCA selected 1953 at d = 1, 1962 at d = 3, and 1968 for other reported dimensions.Source: §5 · p. 20
- At Gayndah, almost 90% of estimated break-function variation lay in the first sample eigenfunction, yet fPCA SNRs for d > 1 were below the fully functional estimate.Figure 5.1 shows the break function and eigenfunction contributions alongside the estimated SNRs across fPCA dimensions.Source: §5 · p. 20, Figure 5.1 · p. 21
- The application and simulations indicate that methods can differ substantially in practice, and the authors judge the fully functional method perhaps more trustworthy in this case.The authors conclude that the proposed method can outperform alternatives, with no obvious evidence that another procedure does better.Source: §5 · pp. 20, 22
6 Conclusions
The paper develops fully functional procedures for detecting and dating mean-curve breaks under assumptions that include weak dependence and heavy-tailed observations. Simulations and the temperature-curve application show advantages over fPCA in several settings, especially when breaks do not align with leading covariance eigenfunctions.
6 Conclusions
- The paper introduces fully functional methodology to detect and date mean-curve breaks without dimension reduction.Its assumptions accommodate time-series curves and include heavy-tailed functional observations.Source: §6 · p. 22
- The assumptions are weaker than those usually made in fPCA settings, with rates formulated through approximations by ℓ-dependent sequences.The conclusion states that the resulting asymptotic theory is widely applicable.Source: §6 · p. 22
- Simulations show the fully functional method tends to outperform fPCA, especially for breaks poorly aligned with the leading covariance eigenfunctions and in some slower-decay settings.The annual-temperature application illustrates the practical relevance of the slower-eigenvalue-decay setting.Source: §6 · p. 22
- When a signal is sparse because it is not contained in leading principal components, the conclusion advises considering alternatives to dimension-reduction methods.The paper says such alternatives are likely to be more effective in that setting.Source: §6 · p. 22