Variable Bandpass Periodic Block Bootstrap

• VBPBB is a nonparametric resampling method designed for periodically correlated series that isolates target periodic components via bandpass filtering before phase-aware bootstrap resampling.
• It uses filters such as the KZFT to separate the desired periodic component from noise, then resamples seasonal subsequences to maintain phase-specific correlations.
• The method has been applied for inference on periodic means and for robust imputation in environmental, epidemiological, and energy time series, significantly enhancing accuracy.

Variable Bandpass Periodic Block Bootstrap (VBPBB) is a nonparametric resampling method for periodically correlated (PC) time series that combines narrowband frequency isolation with period-aware bootstrap resampling. Introduced by Valachovic to address the failure of ordinary bootstrap and conventional block-bootstrap schemes to preserve periodic dependence, VBPBB first separates a target periodic component from interference such as noise or other harmonics and then resamples the isolated component in a way that preserves its periodic correlation structure. Across the literature, the method is used for two closely related purposes: inference for periodic means and harmonics, and structure-preserving imputation for incomplete time series (Valachovic, 2024, Ahmad et al., 4 May 2025).

1. Emergence and problem formulation

VBPBB was developed for settings in which a time series contains one or more periodic or cyclostationary components, such as annual, weekly, daily, or harmonic structure. In this setting, the central statistical difficulty is not merely temporal dependence, but periodic dependence: observations separated by one full cycle or by harmonic multiples can remain strongly correlated, and that correlation depends on phase within the cycle. The literature therefore treats VBPBB as a method for PC series rather than for generic weakly dependent processes (Valachovic, 2024).

This development is motivated by limitations of earlier bootstrap procedures. Ordinary bootstrap destroys temporal dependence. The classical Moving-Block Bootstrap preserves short-range dependence up to the block length but destroys correlations at lags beyond that block length, so it cannot reproduce long-lag periodic correlations. Seasonal or Periodic Block Bootstrap methods improve on this by aligning resampling with the cycle, but when they operate on the unfiltered series they also resample other frequencies and noise, which inflates bootstrap variance and widens confidence bands (Chen et al., 3 Sep 2025).

The method quickly became a framework for empirical work on environmental, epidemiological, and energy time series. Applications in the supplied literature include US COVID-19 mortality, New York State COVID-19 hospitalizations, Manhattan PM$_{2.5}$, Turkey electricity consumption, ischemic heart disease hospitalizations in New York, and time-series imputation in daily temperature and simulation settings (Valachovic et al., 2024, Ahmad et al., 2024, Sun et al., 2024, Yao et al., 2024, Chen et al., 3 Sep 2025, Ahmad et al., 4 May 2025). A related extension, the Variable Multiple Bandpass Periodic Block Bootstrap (VMBPBB), was introduced for time series with multiple periodically correlated components, with separate filtering and resampling for each component before recombination (Valachovic, 11 Feb 2025).

2. Periodic correlation and bandpass separation

The mathematical setting is a PC process whose mean and covariance repeat with period $p$. One common formulation is

$$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$

while another writes

$$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$

Both formulations express the same idea: second-order structure depends on phase within the cycle rather than only on lag (Valachovic, 2024, Chen et al., 3 Sep 2025).

VBPBB uses frequency separation to isolate the PC component of interest before resampling. In the original exposition, the filtered series is obtained with the Kolmogorov–Zurbenko Fourier Transform (KZFT) filter,

$$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$

where $m$ is an odd window length, $k$ is the number of iterations, and $\nu=1/p$ is the target frequency. Its energy transfer function is

$$\bigl|B(\lambda-\nu)\bigr|^2 = \Bigl| \frac{1}{m}\frac{\sin(\pi m(\lambda-\nu))}{\sin(\pi(\lambda-\nu))} \Bigr|^{2k}.$$

The filter is designed so that energy near the target frequency passes while nearby interference is attenuated (Valachovic, 2024).

A closely related exposition writes the bandpass component as the difference of two low-pass Kolmogorov–Zurbenko filters,

$$B_p(\omega)=H_{m_1,k_1}(\omega)-H_{m_2,k_2}(\omega),$$

with

$p$0

In time domain form,

$p$1

and the residual is $p$2 (Ahmad et al., 4 May 2025).

The common decomposition in later work is

$p$3

where each $p$4 is a narrowband PC component and $p$5 is remainder noise. Reported practical guidance ties the filter width to frequency separation. Examples in the literature include choosing $p$6 as the smallest odd integer exceeding a rule based on the gap between adjacent target frequencies, or choosing $p$7 so that the passband extends no farther than halfway to the nearest unwanted frequency. Reported iteration counts vary by study, including $p$8, $p$9, and settings in which $$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$0 is often $$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$1 (Chen et al., 3 Sep 2025, Yao et al., 2024, Ahmad et al., 4 May 2025).

3. Bootstrap construction

Once a periodic component has been isolated, VBPBB resamples it in a way that respects periodic phase. The core idea is that resampling must occur on the filtered component, not on the raw series. The original formulation partitions the filtered series into seasonal subsequences

$$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$2

then cycles through these subsequences while sampling with replacement to form a bootstrap replicate. Other expositions describe nonoverlapping blocks of length exactly equal to the period, or phase strata

$$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$3

within which observations are resampled and then written back to their phase-aligned positions. These formulations differ operationally, but they share the same structural principle: preserve the phase-specific correlation pattern of the periodic component (Valachovic, 2024, Chen et al., 3 Sep 2025).

In the imputation literature, the algorithm is often presented componentwise. For each significant period $$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$4, the method filters the series to obtain $$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$5, partitions the filtered component into blocks of length $$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$6, samples those blocks with replacement, concatenates the sampled blocks, and reconstructs

$$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$7

for bootstrap replicate $$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$8. Amplitude summaries such as $$\mu_{t+p}=\mu_t,\qquad \gamma(t+p,\tau)=\gamma(t,\tau),$$9 may then be computed from the resampled blocks (Ahmad et al., 4 May 2025).

A related bias study allows block length $$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$0 to be an integer multiple of the period, such as $$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$1 or $$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$2, and resamples overlapping blocks from each filtered component. This formulation emphasizes the bias-variance trade-off in block choice: larger $$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$3 reduces truncation bias but can increase bootstrap variance (Sun et al., 10 Sep 2025).

Implementation guidance in the supplied papers is consistent on several points. The block length is ordinarily set equal to the period of the component under study. Reported numbers of bootstrap replications include $$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$4–$$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$5, and one practical recommendation is $$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$6 for stable median estimates. The original computational summary gives total cost

$$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$7

combining one KZFT pass and $$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$8 linear-time bootstrap replicates (Valachovic, 2024, Ahmad et al., 4 May 2025, Ahmad et al., 27 Aug 2025).

4. Inference for periodic means and significant components

The main inferential target in early VBPBB work is the periodic mean of a filtered component or of a sum of significant components. For a component with period $$\mathrm{Cov}\bigl(X(t),X(t+h)\bigr)=\gamma_h\bigl(t \bmod p\bigr).$$9, one computes phase-wise or time-wise bootstrap summaries and obtains confidence bands from empirical quantiles. One representative expression is

$$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$0

where

$$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$1

aggregates the bootstrap replicates of the significant components (Chen et al., 3 Sep 2025).

The literature reports several closely related significance rules. In one formulation, a component is declared significant if a bootstrap confidence interval for its amplitude excludes zero,

$$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$2

An equivalent test statistic is

$$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$3

compared with $$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$4 (Ahmad et al., 4 May 2025).

In another formulation, significance is based on the periodic-mean confidence band not admitting a horizontal line, equivalently

$$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$5

Application papers often state the rule more simply: a component is significant when the 95% pointwise confidence band for the bootstrapped periodic mean excludes zero (Chen et al., 3 Sep 2025, Sun et al., 2024).

This inferential architecture generalizes naturally to multiple components. VMBPBB formalizes the separate filtering, separate componentwise resampling, and final recombination needed to preserve several periodic correlations simultaneously in a multiple-periodic setting (Valachovic, 11 Feb 2025).

5. Use in missing-data imputation

A major later use of VBPBB is missing-data imputation for univariate time series with strong seasonal or cyclic structure. In this setting, significant periodic components extracted by VBPBB are supplied to the Amelia II multiple-imputation framework as auxiliary variables. Ahmad et al. describe using median vectors of periodic blocks and/or the bootstrap replicates of the significant components, after which Amelia II’s bootstrap-EM algorithm imputes missing values while preserving the overall sample mean and the sample variance-covariance structure. Their formulation states that, in practice, the resulting imputations match the original periodic means and variances up to Monte Carlo error (Ahmad et al., 4 May 2025).

A later formalization introduces frequency-specific covariates $$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$6 and writes the augmented imputation model as

$$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$7

The stated objective is to ensure that the imputed values inherit the true underlying periodic structure of the data, especially annual, semiannual, and monthly cycles (Ahmad et al., 27 Aug 2025).

In the daily-temperature comparison reported by Ahmad et al., missing at random was imposed at 13% overall and 30% on weekends. Against standard Amelia II, Amelia II with VBPBB achieved $$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$8 and $$Y_t=\mathrm{KZFT}_{m,k,\nu}(X)_t =\sum_{u=-k(m-1)/2}^{k(m-1)/2} a_u^{(m,k)} e^{-2\pi i \nu u} X_{t+u},$$9, compared with $m$0 and $m$1 without VBPBB. The same study reports a reduction of MAE by 57%, a reduction of RMSE by 56%, and an increase in correlation with observed data from 0.862 to 0.972 (Ahmad et al., 4 May 2025).

Method	MAE	RMSE
Amelia II with VBPBB	1.326	4.630
Amelia II without VBPBB	3.015	10.353

A broader simulation assessment uses $m$2 daily observations, annual, harmonic, and monthly signals, noise variances $m$3, and MCAR missingness from 5% to 70%. Across all three signal stages, the VBPBB-enhanced approach consistently outperforms standard imputation. Reported gains are largest when multiple periodic components are retained, and under very low noise with 5% missingness the reductions in MAE and RMSE are about 93%; even at $m$4 and 70% missingness, improvements remain on the order of about 9%–16% depending on the stage (Ahmad et al., 27 Aug 2025).

6. Empirical record, extensions, and limitations

The empirical record supplied for VBPBB is dominated by comparisons of confidence-band width and component detectability against unfiltered periodic bootstrap procedures. Across these studies, the recurring result is that pre-filtering removes extraneous variance before resampling, producing much tighter confidence bands while preserving periodic correlation structure.

Application	Reported significant components	Reported comparative result
US COVID-19 mortality (Valachovic et al., 2024)	Seasonal, 2nd–5th annual harmonics, weekly	PBB/VBPBB median CI-width ratios 3.49, 5.98, 6.41, 13.87, 32.93, and 5.47
Manhattan PM$m$5 (Sun et al., 2024)	Semi-annual and weekly	GSBB CIs 11.00× wider for semi-annual and 15.87× wider for weekly
Turkey electricity (Yao et al., 2024)	Daily, weekly, annual-related components	Median CI-width ratios 1.64×, 4.61×, 3.65×, combined 1.58×
NY COVID-19 hospitalizations (Ahmad et al., 2024)	Annual, 3rd–5th harmonics	VBPBB CIs 1.64×, 5.26×, 6.55×, and 12.36× narrower
IHD hospitalizations in New York (Chen et al., 3 Sep 2025)	Weekly, annual, 2nd, 3rd, 5th, and 6th harmonics	GSBB-to-VBPBB band-width ratio 35.6 for annual and up to 179.1 for the fifth annual harmonic

The method’s theoretical and practical limits are also explicit in the literature. VBPBB is designed for time series that contain narrowband cyclical components, or that can be represented as a sum of a small number of PC components plus residual noise. It depends on filter specification, can suffer from end effects, and requires sufficiently long series so that filtering does not remove too much data at the edges. Several papers note that poor choices of $m$6, $m$7, or bandwidth may under-filter, over-filter, or damp true peaks. One application summary states that VBPBB reduces to ordinary PBB in the extreme case $m$8 or, more generally, when the passband is chosen wide enough to pass all frequencies (Valachovic, 2024, Valachovic et al., 2024).

The bias analysis in later work makes these issues more precise. Under conditions including second-order periodic stationarity of each component, $m$9, $k$0, and weak dependence of the residual noise, the overall-mean bias and the pointwise-mean bias of the bootstrap distribution are both

$k$1

That study recommends centering each bandpass filter at $k$2, choosing half-widths to exclude nearby harmonics, and balancing the bias-variance trade-off in block length. It also states that VBPBB will not reproduce nonperiodic trends or localized events unless they repeat each cycle (Sun et al., 10 Sep 2025).

Current future-direction statements in the supplied papers focus on automated selection of frequencies and bandwidths, extension beyond univariate MCAR settings, structural breaks, multivariate VBPBB exploiting cross-series periodic correlations, and parallelized filtering and blocked bootstrap schemes (Ahmad et al., 27 Aug 2025). Together, these directions place VBPBB within a broader program of frequency-aware resampling for cyclostationary inference and temporally structured imputation.