Cyclic Tempograms: Local Spectral Analysis

• Cyclic tempograms are a narrow-band spectral tool that tracks the evolution of known periodicity intensity through sliding-window analysis.
• They compute localized periodograms and sum spectral energy around pre-specified frequencies, ensuring robustness even with uneven sampling or missing data.
• Applications in sensor data, virtual learning environments, and animal monitoring demonstrate the method’s ability to reveal temporal rhythmic variations and event-driven changes.

Searching arXiv for the cited paper and closely related work on tempograms/periodicity intensity. Cyclic tempograms are a sliding-window spectral methodology for tracking how the strength of a known periodicity evolves through time. In the formulation summarized for Smeaton and Hu’s "Periodicity Intensity Reveals Insights into Time Series Data: Three Use Cases" (Smeaton et al., 2023), the method computes a periodogram within successive windows and then extracts spectral energy at one or more pre-specified frequencies of interest, such as $1/24\,\mathrm{h}$ or $1/168\,\mathrm{h}$. The paper also uses the term "periodicity intensity" for this narrow-band measure, and describes the resulting representation either as a single temporal curve per target frequency or, when evaluated over a frequency grid, as a classic tempogram (Smeaton et al., 2023).

1. Definition and conceptual scope

For a uniformly sampled real signal $x[n]$, $n=0\ldots N-1$, the summary gives the standard sliding-window periodogram at frequency $f$ and window start $t$ as

$$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$

This is described as the "power spectral density" estimate within that window. The authors’ periodicity intensity then proceeds by computing $S(f)$ via FFT, or Lomb–Scargle if sampling is irregular, on the windowed data $X_w = x[t\ldots t+W-1]$, selecting a known frequency of interest $F_o$, and summing $1/168\,\mathrm{h}$0 over frequency bins $1/168\,\mathrm{h}$1 within a small neighborhood $1/168\,\mathrm{h}$2 of $1/168\,\mathrm{h}$3:

$1/168\,\mathrm{h}$4

In pseudocode, this is called IntensityFunc(Fo,S_f) (Smeaton et al., 2023).

The method is therefore a localized estimate of rhythmic strength rather than a global estimate of periodic structure. A periodogram over an entire series measures intensities of oscillations over the whole observation interval, whereas the cyclic tempogram measures how periodicity intensity at a specific frequency varies throughout the series. The supplied summary explicitly characterizes it as a narrow-band, single-frequency or few-frequency analog of the spectrogram, with computational effort concentrated on pre-specified frequencies of interest rather than on a full time-frequency decomposition (Smeaton et al., 2023).

2. Computational procedure

The algorithmic procedure is restated in the summary as "Compute cyclic tempogram." Its inputs are a time series $1/168\,\mathrm{h}$5, a window length $1/168\,\mathrm{h}$6, a window shift $1/168\,\mathrm{h}$7, a set of target frequencies $1/168\,\mathrm{h}$8, the frequency-bin width $1/168\,\mathrm{h}$9 determined by $x[n]$0, and a tolerance $x[n]$1 on frequency matching. The output is a set of sequences $x[n]$2, where each $x[n]$3 is the periodicity intensity at $x[n]$4 in window $x[n]$5, with

$x[n]$6

For each window, the procedure extracts $x[n]$7, computes $x[n]$8 using either $x[n]$9 for uniform sampling or Lomb–Scargle for uneven or missing data, and then assigns

$n=0\ldots N-1$0

This operationalization makes the method explicit as a sliding-window spectral-energy extraction scheme rather than a learned or model-based estimator. The summary also states that no de-meaning or window-tapering is explicitly mentioned, and that the authors rely on FFT or Lomb–Scargle’s built-in stability. For visualization, each per-subject, per-animal, or per-student time series $n=0\ldots N-1$1 is line-normalized to $n=0\ldots N-1$2 (Smeaton et al., 2023).

A plausible implication is that the cyclic tempogram is intentionally minimal in its signal-processing assumptions. Because it uses direct spectral estimates in local windows and a narrow-band readout around known periodicities, its interpretability is tied closely to the target rhythm itself: the output can be read as the strength of an $n=0\ldots N-1$3-hour rhythm as a function of time.

3. Parameterization and frequency selection

The supplied parameter guidelines identify the target frequency $n=0\ldots N-1$4 as usually known from the domain, with circadian $n=0\ldots N-1$5 and weekly $n=0\ldots N-1$6 given as examples. If the periodicity is unknown, the summary states that one can scan a coarse global periodogram to locate peaks. The window length $n=0\ldots N-1$7 must be long enough to resolve $n=0\ldots N-1$8; the example given is that $n=0\ldots N-1$9 is needed to see a stable 24 h peak. In practice, $f$0 days is used to "average out" weekday/weekend fluctuations in human-activity data. The hop size $f$1 is chosen to balance time resolution and computation, and the frequency resolution is $f$2, so larger $f$3 yields finer $f$4 but slower adaptation to change, while smaller $f$5 tracks faster but broadens peaks (Smeaton et al., 2023).

These settings define the basic trade-off of the method. A longer window improves spectral selectivity around the rhythm of interest, but it also smooths temporal transitions; a shorter window increases responsiveness but reduces frequency precision. Because the method sums power over a neighborhood of $f$6 to allow for leakage, it is explicitly tolerant to imperfect bin alignment, which is important when the frequency of interest does not fall exactly on a discrete Fourier bin.

4. Representations and visualization

The summary distinguishes two output forms. If periodicity intensity is computed over a grid of frequencies $f$7, the result is a matrix $f$8 with time on the x-axis, frequency on the y-axis, and color representing intensity; this is identified as the classic tempogram. In the three use cases described, however, the authors compute only $f$9 or $t$0 plus its harmonic, and therefore plot a single curve $t$1 over time. Curves are min–max normalized per individual to $t$2. In the student case, the plots overlay vertical colored bands marking the lowest (red), highest (green), and steepest-change (yellow) intervals. For cohort-level behavior, normalized curves are stacked as ribbons in a stacked line chart, with the top-most envelope showing the aggregate pattern (Smeaton et al., 2023).

This visualization scheme emphasizes relative temporal modulation rather than absolute spectral power. Because normalization is performed per individual, cross-subject comparisons in the visual domain reflect shape and timing more directly than raw magnitude. This suggests that the method is especially suited to identifying synchrony, rhythm disruption, and temporal phase-specific changes in regularity across a cohort.

5. Empirical use cases

The summary presents three use cases drawn from the paper: in-home sensor data from older adults, student access logs from a virtual learning environment, and newborn calf accelerometry. The applications differ in signal source and hop size, but each uses the same sliding-window periodicity-intensity framework (Smeaton et al., 2023).

Use case	Data source	Reported settings or observations
In-home sensors	Doors, appliances, environment; fused into one activity measure	Compute 24 h periodicity intensity with $t$3 days, $t$4 h; dips or flatlines reveal health events
Student VLE access logs	Timestamped URL hits from 169 students	Compute 24 h periodicity intensity with $t$5 days, $t$6 h; stacked trace shows semester peaks
Newborn calf accelerometry	Axivity AX3 collars; processed to per-minute SVM	Compute 24 h periodicity intensity with $t$7 days, $t$8 min; herd trace shows synchronous drop after disbudding

In the in-home sensor case, the paper describes sensors on doors, appliances, and environmental devices, which are fused into a single activity measure. A 24 h periodicity-intensity curve is then computed using $t$9 days and $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$0 h. The summary states that individual traces reveal health events, with COVID isolation or hospital move-out appearing as dips or flatlines (Smeaton et al., 2023).

In the student VLE case, the input consists of timestamped URL hits to the online VLE from 169 students. The summary states that 24 h periodicity intensity is computed with $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$1 days and $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$2 h, and that per-student curves support personal reflection. The aggregated stacked trace shows semester peaks at the start, during mid-semester reading week, and in pre-exam revision (Smeaton et al., 2023).

In the calf study, data are collected using Axivity AX3 collars and processed to per-minute SVM after reduction from raw 12.5 Hz accelerometer data and band-pass filtering from $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$3 to $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$4 Hz to remove noise. The resulting 24 h periodicity intensity is computed with $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$5 days and $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$6 min. Individual curves are described as noisy, whereas the stacked herd trace shows a synchronous drop in rhythm following disbudding and recovery about a week later (Smeaton et al., 2023).

Collectively, these examples demonstrate the paper’s central claim that periodicity intensity reveals insights into time series data not visible using other forms of analysis. The common structure across cases is that rhythmic stability itself becomes the observable of interest.

6. Relation to spectrograms, wavelets, and missing-data settings

The discussion section in the summary compares the cyclic tempogram with several adjacent methodologies. It is described as a narrow-band, single-frequency or few-frequency analog of the spectrogram, focusing computational effort on pre-specified $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$7, $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$8, rather than computing a full spectral representation. It is also stated to be faster and more robust to missing data because Lomb–Scargle can be substituted when data are uneven or missing. The method is further distinguished from wavelets and EMD by the absence of data-driven decomposition into basis functions; instead, it performs a direct spectral-energy pull at known periodicities (Smeaton et al., 2023).

This comparison situates the cyclic tempogram within local spectral analysis rather than adaptive decomposition. Its principal advantages are listed as unsupervised operation, no training or model fitting, tunability to domain frequencies of interest, support for uneven sampling and moderate missing data through Lomb–Scargle, and easy interpretability as the "strength of $$I(f; t) = \left| \sum_{\tau=0}^{W-1} x[t+\tau] e^{-2\pi i f \tau} \right|^2.$$9-hour rhythm over time" (Smeaton et al., 2023).

A plausible implication is that the method is most appropriate when the periodicity of interest is semantically known in advance, such as circadian or weekly rhythms. It is less a discovery tool for unknown multiscale structure than a targeted monitor for temporal variation in known oscillatory regimes.

7. Complexity, interpretive boundaries, and an internal discrepancy in the use-case summary

The computational complexity reported in the summary is $S(f)$0 per window for FFT and $S(f)$1 per window for Lomb–Scargle, giving a total cost of $S(f)$2 with $S(f)$3. The summary adds that the method is in practice fast enough for multi-week data when $S(f)$4 is chosen coarsely (Smeaton et al., 2023).

These complexity estimates clarify why the method can be deployed on long observational records while retaining local temporal sensitivity. They also explain the design emphasis on a narrow-band readout rather than an exhaustive time-frequency analysis: computational savings arise from restricting attention to specific periodicities.

The same summary also exposes an internal discrepancy concerning the student VLE application. The abstract states that the third application concerns "the intensity of 7-day periodicity" in VLE access, whereas the detailed use-case summary states "Compute 24 h periodicity intensity with $S(f)$5 days, $S(f)$6 h" for the student data (Smeaton et al., 2023). This inconsistency should be read as a discrepancy within the supplied account rather than as a settled methodological distinction. A cautious interpretation is that the paper’s broader framework supports either daily or weekly target frequencies, and that the student case, as summarized, contains conflicting descriptions of which periodicity was emphasized.

More generally, the cyclic tempogram should not be conflated with a full spectrogram, a wavelet scalogram, or an adaptive mode decomposition. In the supplied formulation, it is a sliding-window estimate of local spectral power concentrated around a pre-specified rhythm. Its interpretive force derives from that narrow focus: it measures fluctuation in rhythmic intensity, not arbitrary time-local structure.