| Citation: |
Zabidi Abu Hasan, R. U. Gobithaasan, Roslan Hasni. ASSESSING PERIODICITY IN TIME SERIES DATA USING PERSISTENT HOMOLOGY[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1413-1453. doi: 10.11948/20250083
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ASSESSING PERIODICITY IN TIME SERIES DATA USING PERSISTENT HOMOLOGY
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Abstract
Topological Data Analysis (TDA), a novel method rooted in algebraic topology, offers a powerful framework for quantifying the structure of high-dimensional datasets. This study focuses on Persistent Homology (PH), a key TDA method, to elucidate the underlying shape and structure of time series data, thereby facilitating the understanding of patterns and recurrence. By employing Takens’ embedding to represent the time series as a high-dimensional point cloud, this approach effectively encodes its topological characteristics, particularly periodicity. We utilize various PH summaries, in the form of persistence vectors (PV), to assess periodicity by quantifying connected components $(H_0)$ and one-dimensional holes $(H_1)$. The methodology is validated on ten synthetic datasets, each comprising 200 data points and exhibiting varying levels of periodicity, amplitude, and frequency. The optimal window size for the sliding window embedding is determined by maximizing the $PD_{\text{max}}$ score. Subsequently, the $L^1$-norm of the PVs is computed to quantify periodicity. Under optimal window selection $(D=14)$, the cosine signal achieves the highest periodicity, followed by damped cosine and linear trend. Formal quantitative stability analysis identifies one of the two Sliding Window 1-Persistence Scoring (SW1PerS) baselines, SW2, as the most stable representation under both Gaussian noise (Rank Instability Index, RII$=0.243$) and missing data (Mean Absolute Rank Deviation, MARD=0.117), with the Persistence Landscape Score (PLS) and SW1 ranking closely behind; differences under noise are not statistically significant (Friedman $p=0.42$), while those under missing data are significant $(p=0.001)$. A comparison against non-topological baselines (FFT-based spectral entropy, autocorrelation, and recurrence quantification analysis) demonstrates that PLS achieves Kendall $\tau=0.867$, substantially outperforming spectral entropy (SE: $\tau=0.733$) which is the only non-topological baseline that achieves statistically significant concordance $(p=0.002)$, while the autocorrelation peak ratio (APR: $\tau=-0.289$) and recurrence rate (RR: $\tau=-0.200$) yield non-significant negative concordance on these synthetic signals.
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References
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Zabidi Abu Hasan, R. U. Gobithaasan, Roslan Hasni. ASSESSING PERIODICITY IN TIME SERIES DATA USING PERSISTENT HOMOLOGY[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1413-1453. doi: 10.11948/20250083
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Figure 1.
Example of PD construction using 200 points with
$ H_0 $ $ H_1 $ -
Figure 2.
PL representation from PD in equation (2.5).
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Figure 3.
Persistent Image (PI) representation from PD in equation (2.5). Left is PI for
$ H_0 $ $ H_1 $ -
Figure 4.
Betti curve representation from PD in equation (2.5).
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Figure 5.
Lifetime curve (LC) representation from PD in equation (2.5). Left is the LC for
$ H_0 $ $ H_1 $ $ H_0 $ $ H_1 $ -
Figure 6.
Workflow to measure the periodicity score.
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Figure 7.
The rank plot of PV with Gaussian noise (SD(%)=0, 10, 25, 50) with embedding dimension
$ D=14 $ $ w=93 $ -
Figure 8.
The fixed different optimal window size used for missing values data with embedding dimension
$ D=14 $ -
Figure 9.
The rank plot PV with missing value (mv(%)=0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50) with embedding dimension
$ D=14 $ -
Figure 10.
The rank plot PV using DTM
$ (m=0.1 , \; p=1) $ $ D=14 $ -
Figure 11.
The rank plot PV using DTM
$ (m=0.1 , \; p=2) $ $ D=14 $ -
Figure 12.
The rank plot PV using DTM
$ (m=0.2 , \; p=1) $ $ D=14 $ -
Figure 13.
The rank plot PV using DTM
$ (m=0.2 , \;p=2) $ $ D=14 $ -
Figure 14.
The rank plot PV using DTM
$ (m=0.1 , p=1) $ $ D=14 $ -
Figure 15.
The PLS score plot for normalized and non-normalized point cloud with
$ w=93 $ -
Figure 16.
Power Spectral Density (PSD) for all ten synthetic signals, computed via Hann-windowed periodogram. The annotated SE_norm value (lower = more periodic) and rank are shown for each signal. Method: Spectral Entropy [28, 45].
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Figure 17.
Autocorrelation function (ACF) for all ten synthetic signals. The red dashed vertical line and dot mark the dominant ACF peak (APR value). Gray dotted lines indicate the 95% confidence interval
$ (\pm1.96/\sqrt{N}) $ -
Figure 18.
Recurrence plots for all ten synthetic signals
$ (\varepsilon=0.10\,\sigma_x $ $ m=1 $ $ \tau=1) $ -
Figure 19.
Left: Periodicity rank comparison among SW1-ref, SW1, SE, APR, and RR (rank 1 = most periodic). Right: Kendall’s τ against SW1-ref for each method. The blue dashed line marks PLS
$ \tau=0.867 $