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Hierarchical Bidirectional Transition Dispersion Entropy-based Lempel-Ziv Complexity and Its Application in Fault-Bearing Diagnosis

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper introduces BT-DELZC, a Lempel-Ziv complexity measure that counts new patterns in the in-link and out-link sequences of each dispersion pattern, and reports that it outperforms LZC, PLZC, and DELZC on two fault bearing datasets.

desk verdict Genuine LZC variant with a clean definition, but the top-line accuracy claim in the bearing experiments is not backed by significance tests or a reported K value. read the letter →

arxiv 2412.11123 v1 pith:QDJS3OUL submitted 2024-12-15 physics.data-an cs.LGeess.SPmath-phmath.MP

Runze Jiang , Pengjian Shang This is my paper · ORCID
classification physics.data-ancs.LGeess.SPmath-phmath.MP
keywords Lempel-ZivcomplexitytransitionnetworkdispersionentropyhierarchicaldecompositionfaultbearingdiagnosisnonlineartimeseriesMarkovchainmeasure
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a new Lempel-Ziv complexity measure, BT-DELZC, that captures not only which dispersion patterns appear in a time series but how those patterns transition into one another. Prior LZC variants such as PLZC and DELZC symbolise the series and count new patterns in the symbol sequence, ignoring the dynamics between consecutive embedding vectors. BT-DELZC builds a weighted bidirectional transition network over dispersion patterns, computes the DELZC of each pattern's in-link and out-link sequences, and weights these by pattern frequency. The authors report that the resulting measure is more robust across noise, more sensitive to gradual frequency changes, and achieves the highest fault-classification accuracy on both real-world bearing datasets they test.

What carries the argument

The load-bearing construction is the weighted bidirectional transition network over dispersion patterns. Each embedding vector of the time series is assigned a dispersion pattern; for every pattern, the algorithm collects the sequences of patterns that precede and follow it (excluding self-loops), computes a DELZC value on each of those two link sequences, averages them, and weights the result by the pattern's relative frequency. A hierarchical decomposition with averaging and differencing operators feeds multiple frequency components of the signal into this calculation, and PCA compresses the resulting feature vectors before classification.

What would settle it

Run the two bearing experiments under repeated stratified cross-validation over a grid of embedding dimension, class count, and hierarchical layer values, and compare the distributions of accuracy; if BT-DELZC's margin over DELZC falls within the cross-validation noise for most settings, the central claim fails.

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Extended reading notes

Core claim

The central claim is that incorporating transition dynamics into Lempel-Ziv complexity improves feature extraction from nonlinear time series. Concretely, the paper claims that BT-DELZC outperforms LZC, PLZC, and DELZC in distinguishing chaotic from noisy signals, in tracking the frequency sweep of a chirp signal and the stochastic-to-periodic drift of a mix process, and in classifying nine fault types from one bearing benchmark and twelve from another, under KNN, decision-tree, and feedforward-neural-network classifiers, with accuracy gains of several percentage points in most comparisons.

Load-bearing premise

The results assume that the parameter settings (embedding dimension, class count, time delay, hierarchical layer) and the single 70/30 train-test split are unbiased and representative, so that the measured accuracy differences reflect genuine method performance rather than sampling noise or favourable tuning.

Editorial extensions

If this is right

  • On both real-world bearing datasets, BT-DELZC yields the highest accuracy for every classifier tested (KNN, DT, FNN), with the largest single gain over DELZC reaching about 6.7 percentage points.
  • The method separates fully chaotic logistic maps from white noise at lower signal-to-noise ratios than LZC, PLZC, or DELZC in most tested hierarchies.
  • BT-DELZC tracks gradual dynamics better than the existing measures: it produces the smoothest decline in the mix process and the clearest frequency-change signal in the chirp experiment.
  • Because the measure is weighted by pattern frequency, common dispersion patterns dominate the complexity value, which the authors argue makes the measure more stable across noise realisations.
  • The hierarchical decomposition stage multiplies the number of features per sample, and the paper shows that classification accuracy stays high across different numbers of PCA components.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same bidirectional transition construction could be applied to permutation-based LZC or other symbolisation schemes, not just dispersion patterns; the paper does not test these alternatives.
  • Excluding self-loops may change the complexity estimate for highly repetitive or strongly chaotic signals; the paper flags this as future work but does not quantify the effect.
  • The accuracy results rest on a single 70/30 train-test split per dataset; repeated stratified cross-validation across parameter choices would show whether the reported margin is stable.
  • If the embedding dimension, class count, time delay, and hierarchical layer were tuned per dataset, part of the reported advantage could come from that tuning rather than from the transition network itself.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes BT-DELZC, a Lempel-Ziv complexity variant that combines dispersion-pattern-based DELZC with a weighted bidirectional transition network, and appends hierarchical decomposition to extract multi-frequency features. The method is evaluated on four simulated experiments (parameter analysis, robustness to noise, chirp signals, and a mix process) and on two public fault-bearing datasets (Paderborn University and Case Western Reserve University) using KNN, decision tree, and feedforward neural network classifiers. The authors report that BT-DELZC consistently yields the highest accuracy among LZC, PLZC, DELZC, and BT-DELZC, and claim that it 'significantly outperforms' the existing methods.

Significance. If the empirical claims are substantiated, BT-DELZC would be a useful addition to the family of Lempel-Ziv complexity measures, particularly for bearing fault diagnosis, and the combination of transition networks with LZC is a reasonable novelty. The paper has several strengths: the algorithm is defined step-by-step in Section 2.3; the simulated experiments include quantitative robustness checks with repeated samples and Wilcoxon rank-sum tests; and the real-world evaluation uses two standard public benchmark datasets and multiple classifiers. These strengths make the paper a candidate for publication after the evaluation protocol is made statistically rigorous and the reporting of experimental settings is completed.

major comments (4)
  1. [Section 4.2, Tables 3–4; Section 4.3, Table 6] The central claim that BT-DELZC 'significantly outperforms' LZC, PLZC, and DELZC is not supported by any statistical test. All reported accuracy rates in Tables 3, 4, and 6 are point estimates from a single 70/30 split. With only 270 (PU) or 360 (CWRU) test samples, split-to-split variance can be substantial, and the margins over the best baseline in some cases are small (e.g., CWRU KNN: 99.72% vs. 98.61%). The authors should report repeated random splits or k-fold cross-validation with means, standard deviations or confidence intervals, and apply a paired test such as McNemar's test between BT-DELZC and each baseline. Without this, the word 'significantly' in the abstract and conclusion is not justified.
  2. [Section 4.1, Step 2; Sections 4.2 and 4.3] The hierarchical layer K is never specified for the real-world experiments. Section 4.1, Step 2 says 'according to the selected layer K', but Sections 4.2 and 4.3 do not state the K value used; the number of features fed into PCA depends on K. This makes the experimental results irreproducible and leaves open the possibility that K was chosen favorably. The authors must report K for both datasets and, ideally, provide a sensitivity analysis over K (e.g., K = 2, 3, 4) to demonstrate that the reported accuracies are not an artifact of this choice.
  3. [Eq. (17) and the surrounding text] The hierarchical decomposition matrix in Eq. (17) is dimensionally inconsistent with the stated scale. The text claims that 'the scale of Q_K^ε becomes N - 2^K + 1 × N - 2^{K-1} + 1', but composing the operators in Eq. (16) K times should yield an (N-K)×N matrix (or similar, depending on the alignment convention), not the dimensions given. For K=2, the claimed scale would be (N-3)×(N-1), which does not match the result of applying two operators to X. Please clarify the exact construction of Q_K^ε, correct the matrix and dimension statement, and provide the precise matrix form used in the experiments.
  4. [Sections 4.2 and 4.3] The parameters (m,c) differ between the two datasets (PU: m=3, c=5; CWRU: m=4, c=4) without a systematic justification or sensitivity analysis. While the same parameters are used for all methods within each dataset, the per-dataset choice must be shown not to favor BT-DELZC. The authors should provide a sensitivity study (e.g., a grid over m in {3,4,5} and c in {3,4,5}) on at least one dataset to demonstrate that the reported performance advantage is robust to parameter settings rather than a result of per-dataset tuning.
minor comments (6)
  1. [Table 6 caption] The table header reads 'DETLZC' instead of 'BT-DELZC'; please correct the typo.
  2. [After Eq. (18)] The statement 'Consequently, ∑_{j=1}^K 2^j components including the original time series X, can be obtained' is mathematically inconsistent: the sum equals 2^{K+1}-2, and including X gives 2^{K+1}-1 components. Please rephrase to accurately count the components.
  3. [Section 2.3, Step 2] The index range for dispersion patterns is stated as 'i = 2, 3, ..., N − 1', but the actual range should account for the embedding dimension and delay, i.e., i = 2, ..., N − (m−1)τ − 1, as used in Eq. (14). Please correct for consistency.
  4. [Table 1] The entries '**' are not defined in the table or caption, and the caption contains a typo ('statictical'). Please define the meaning of '**' and correct the typo.
  5. [Section 2.2.1] The reference to Bandt and Pompe appears as 'Bandt and Pompe[]' with an empty bracket; please insert the proper citation.
  6. [Section 5] The paper claims 'the first attempt to integrate transition network methods with LZC approaches.' The authors should verify that no prior work combines transition networks (even without the bidirectional weighting) with LZC-based metrics, and temper the claim if any exists.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: BT-DELZC is a definitional composition of DELZC and transition links, evaluated on external benchmarks; observed weaknesses are evaluation-validity issues, not circular reductions.

full rationale

The derivation of BT-DELZC is self-contained and definitional. Equation (15) defines BT-DELZC as a weighted sum of DELZC values computed on in-link and out-link sequences of each dispersion pattern, where the DELZC values are obtained by the standard LZC counting procedure applied to those link sequences. No parameter of this construction is fitted to the classification labels in the real-data experiments. The simulated experiments in Sections 3.2-3.4 compare the new measure with LZC, PLZC, and DELZC on externally generated white noise, blue noise, chaotic logistic maps, chirp signals, and mix processes, which are independent benchmarks. The bearing experiments use public Paderborn University and Case Western Reserve University datasets with standard classifiers, and while the paper reports the highest accuracies for BT-DELZC, the claim of 'significantly outperforming' is not supported by repeated resampling or significance testing. Likewise, the dataset-specific (m,c) choices and the unreported hierarchical layer K are reproducibility and robustness concerns, not circularity. No step in the paper reduces a predicted or derived result to a fitted input or to a load-bearing self-citation chain, so per the hard rules no circular step is identified.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the validity of LZC on short per-pattern link sequences, the hierarchical decomposition, and the representativeness of the sampling. These are accepted or assumed, and the paper introduces no new physical or mathematical entities.

free parameters (6)
  • embedding dimension m = 3 (PU), 4 (CWRU/simulation)
    User-selected; the paper recommends m=3 or 4 but gives no formal selection criterion for the real-data experiments.
  • number of dispersion classes c = 5 (PU), 4 (CWRU/simulation)
    User-selected; parameter analysis suggests c>=3 but the exact values vary across datasets without justification.
  • time delay tau = 2
    Chosen to prevent overlap of adjacent embedding vectors; the paper recommends tau>=2.
  • hierarchical layer K = 3 in simulation; unspecified in real experiments
    The layer count is set to 3 in the simulated chirp experiment but is not stated for the fault diagnosis experiments, affecting feature extraction.
  • PCA feature count = 3 (also 2 to 6 in one ablation)
    The number of principal components used for classification is set to 3 in the main results; a small ablation varies 2-6.
  • train/test split ratio = 70/30
    A single split is used; no cross-validation or repeated splits.
assumptions (5)
  • standard math The LZC normalization C* ~ N/log_M(N) is a valid finite-sample approximation for the in-link and out-link sequences.
    Used in Eqs. (9)-(10) to normalize per-pattern complexity; the sequences can be very short for rare patterns, and finite-size effects are not quantified.
  • domain assumption Adjacent dispersion patterns carry the relevant dynamic information, modeled as Markov transitions.
    Section 2.3 states the transition framework reflects Markov chain theory, but no Markov property is verified for the signals.
  • standard math The hierarchical decomposition operators Q0 and Q1 extract meaningful low- and high-frequency components.
    Adopted from Jiang et al. (2011) and Zhao et al. (2023); the matrix form in Eq. (17) is garbled but the recurrence is standard.
  • domain assumption The NCDF-based dispersion symbolization preserves signal information relevant to complexity.
    Inherited from DELZC (Li et al., 2022) and Rostaghi & Azami (2016); assumed valid for bearing vibration signals.
  • domain assumption The 100 randomly sampled windows per fault class are representative, and the classifiers are treated as reliable black boxes.
    The sampling details are not given, and classifier implementations come from MATLAB's toolbox without specification of hyperparameters.

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Pith. "Pith review of Hierarchical Bidirectional Transition Dispersion Entropy-based Lempel-Ziv Complexity and Its Application in Fault-Bearing Diagnosis." pith.science (2026). https://pith.science/paper/QDJS3OUL

@misc{pith2026241211123,
  author       = {Pith},
  title        = {Pith review of: Hierarchical Bidirectional Transition Dispersion Entropy-based Lempel-Ziv Complexity and Its Application in Fault-Bearing Diagnosis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QDJS3OUL}},
  note         = {Machine review of arXiv:2412.11123}
}
read the original abstract

Lempel-Ziv complexity (LZC) is a key measure for detecting the irregularity and complexity of nonlinear time series and has seen various improvements in recent decades. However, existing LZC-based metrics, such as Permutation Lempel-Ziv complexity (PLZC) and Dispersion-Entropy based Lempel-Ziv complexity (DELZC), focus mainly on patterns of independent embedding vectors, often overlooking the transition patterns within the time series. To address this gap, this paper introduces a novel LZC-based method called Bidirectional Transition Dispersion Entropy-based Lempel-Ziv complexity (BT-DELZC). Leveraging Markov chain theory, this method integrates a bidirectional transition network framework with DELZC to better capture dynamic signal information. Additionally, an improved hierarchical decomposition algorithm is used to extract features from various frequency components of the time series. The proposed BT-DELZC method is first evaluated through four simulated experiments, demonstrating its robustness and effectiveness in characterizing nonlinear time series. Additionally, two fault-bearing diagnosis experiments are conducted by combining the hierarchical BT-DELZC method with various classifiers from the machine learning domain. The results indicate that BT-DELZC achieves the highest accuracy across both datasets, significantly outperforming existing methods such as LZC, PLZC, and DELZC in extracting features related to fault bearings.

Figures

Figures reproduced from arXiv: 2412.11123 by the authors.

Figure 1
Figure 1. An example of finding the In-link and Out-link sequences corresponding to the dispersion pattern 𝜋231 𝐶 ∗ 𝑜𝑢𝑡(𝜋𝑖 ) = lim 𝑁𝑜𝑢𝑡→∞ 𝐶(𝑁𝑜𝑢𝑡)𝜋𝑖 ≈ 𝑁𝑜𝑢𝑡 log𝑐𝑚−1 𝑁𝑜𝑢𝑡 (10) 𝐷𝐸𝐿𝑍𝐶𝑖𝑛(𝜋𝑖 ) = 𝐶(𝑁𝑖𝑛)𝜋𝑖 𝐶 ∗ 𝑖𝑛(𝜋𝑖 ) (11) 𝐷𝐸𝐿𝑍𝐶𝑜𝑢𝑡(𝜋𝑖 ) = 𝐶(𝑁𝑜𝑢𝑡)𝜋𝑖 𝐶 ∗ 𝑜𝑢𝑡(𝜋𝑖 ) (12) where 𝑁𝑖𝑛 and 𝑁𝑜𝑢𝑡 represents the lengths of 𝑆 𝑖𝑛 𝜋𝑖 and 𝑆 𝑜𝑢𝑡 𝜋𝑖 , respectively. Note that 𝑁𝑖𝑛 and 𝑁𝑜𝑢𝑡 may differ due to varying numbers of self-loops in the series. The… view at source ↗
Figure 2
Figure 2. The flow diagram of the procedure for calculating BT-DELZC of a given time series 𝑋 𝐵𝑇 − 𝐷𝐸𝐿𝑍𝐶(𝑋) = 𝑐∑𝑚 𝑖=1 𝑝(𝜋𝑖 )𝐷𝐸𝐿𝑍𝐶(𝜋𝑖 ). (15) For a clearer understanding, the flow diagram for calculating BT-DELZC is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. The hierarchy diagram of 𝐾 = 3. ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 0 ⋯ 0 ⏟⏟⏟ 2𝐾−1−1 (−1)𝜀 2 0 ⋯ 0 0 0 0 1 2 0 ⋯ 0 ⏟⏟⏟ 2𝐾−1−1 (−1)𝜀 2 ⋯ 0 0 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 0 0 0 0 ⋯ 1 2 0 ⋯ 0 ⏟⏟⏟ 2𝐾−1−1 (−1)𝜀 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (17) where 𝐾 represents the hierarchical layer and 𝜀 ∈ {0, 1} denotes the frequency indicator. As the number of decomposition layers 𝐾 increases, the scale of 𝑄𝐾 𝜀 becomes 𝑁 −2𝐾 +1×𝑁 −2𝐾−1+1. Depending on the r… view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: The BT-DELZC results for 𝑚 = 3 and 𝑚 = 4 with varying time series length 𝑁 and different classification number 𝑐 of the models. The embedding dimension 𝑚 is typically set to 𝑚 ≥ 3. If 𝑚 is too small, the phase space trajectory may not be fully opened, making it difficu…
Figure 5
Figure 5. Figure 5: The standard deviations of BT-DELZC and three previous methods (LZC, PLZC, DELZC) with different time series length 𝑁) the generated white noise. The results indicate that LZC cannot distinguish between white noise and the fully chaotic logistic map when using the orig…
Figure 6
Figure 6. Figure 6: The errorbar plots of the whitenoise and the fully chaotic logistic map (a=4) under the varying SNR, including four methods (LZC, PLZC, DELZC, BT-DELZC) and three hierarchies (𝑋, 𝑋1,0 , 𝑋2,0 ). The minimum SNR for 𝑝 < 0.05 is marked by red line in each cases. 0 5 10 15…
Figure 7
Figure 7. Figure 7: The generated chirp signal with the frequency ranges from 10 Hz to 25 Hz. The sampling frequency is 1000 Hz. 𝑀𝐼𝑋 = (1 − 𝑈)𝑋 + 𝑈 𝑌 (21) In the formulation, 𝑋 represents a sinusoidal periodic series defined by 𝑋𝑘 = √ 2 sin 2𝜋𝑘 12 . 𝑌 consists of uniformly distributed ran…
Figure 8
Figure 8. Figure 8: The values of LZC, PLZC, DELZC, and BT-DELZC with the frequency changes in the chirp signals under 15 hierarchies. 0 5000 10000 15000 sample point -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Mix Process [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: The generated mix process with the sampling frequency selected as 1000 Hz. which supports four test bearings of type 6203. The bearings are subjected to radial force generated by a spring-screw mechanism. The vibration signals are collected from the test rig depicted i…
Figure 10
Figure 10. Figure 10: The values of LZC, PLZC, DELZC, and BT-DELZC with the characteristic varying in the mix process. Original time series Hierarchical Decomposition Calculate LZC-based value Dimension Reduction by PCA Obtain Accuracy Rate by Classifier [PITH_FULL_IMAGE:figures/full_fig_…
Figure 11
Figure 11. Figure 11: The procedures for the fault bearing diagnosis in the following experiments [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: The experimental machinery of KAt Research Data Center at Padrrborn University. (a) The setup for the accelerated lifetime test. (b) The test rig to collect the vibration signals [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: The needle charts of first three features extracted by the PCA algorithm from vibration signals of different fault bearings from KAt Data Centers at Paderborn University for four methods (LZC, PLZC, DELZC, BT-DELZC) and KA16, making these types hard to distinguish. In…
Figure 14
Figure 14. Figure 14: The accuracy rate results of four methods (LZC, PLZC, DELZC, BT-DELZC) under three classifiers (KNN, DT, FNN) [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]
Figure 15
Figure 15. Figure 15: The accuracy rate results of PU bearing dataset for four methods (LZC, PLZC, DELZC, BT-DELZC) by FNN classifier under the different 𝐹𝑛𝑢𝑚 results for these different 𝐹𝑛𝑢𝑚 selections are illustrated in [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]
Figure 16
Figure 16. Figure 16: The bearing test stand of Case Western Reserve University. dynamometer, and control electronics (not depicted). Vibration signals are recorded from accelerometers mounted on the fan end, with a sampling frequency of 12 kHz and a motor speed of 1797 revolutions per min…
Figure 17
Figure 17. Figure 17: The needle chart of first three features extracted by the PCA algorithm from fan end vibration signals of different fault bearings from Case Western Reserve University for four methods (LZC, PLZC, DELZC, BT-DELZC). Recent developments have seen time series network met…
Figure 18
Figure 18. Figure 18: The accuracy rate results of CWRU bearing datasets for four methods (LZC, PLZC, DELZC, BT-DELZC) under three classifiers (KNN, DT, FNN) 1 2 3 4 5 6 7 8 9 10 11 12 Predicted Class 1 2 3 4 5 6 7 8 9 10 11 12 True Class LZC 1 1 1 2 6 30 30 29 30 30 29 24 30 30 29 30 28 1…
Figure 19
Figure 19. Figure 19: The confusion matrices for four methods (LZC, PLZC, DELZC, BT-DELZC) of CWRU bearing dataset under FNN classifier. In our design, the dispersion pattern of embedding vectors is chosen for its ability to account for amplitude values in time series. This method has demo…

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