Pith. sign in

REVIEW 3 major objections 5 minor 30 references

In monitored one-dimensional quantum circuits, the largest entangled cluster grows as a power law with system size in both phases, with a measurement-tunable fractal dimension in its spatial support.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 22:47 UTC pith:FQBXUF2W

load-bearing objection Power-law entanglement depth is a credible new result; the fractal dimension is not yet established as a spatial property — the box-counting method may be measuring the clustering algorithm's own hierarchy. the 3 major comments →

arxiv 2511.08690 v2 pith:FQBXUF2W submitted 2025-11-11 quant-ph cond-mat.dis-nncond-mat.stat-mech

Fractal structure of multipartite entanglement in monitored quantum circuits

Vaibhav Sharma , Erich J Mueller This is my paper
classification quant-ph cond-mat.dis-nncond-mat.stat-mech
keywords measurement-induced phase transitionmultipartite entanglemententanglement depthmonitored quantum circuitsClifford circuitsfractal dimensionstabilizer statescoagulation-fragmentation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Using entanglement depth—the size of the largest cluster of qubits that cannot be factored into smaller entangled pieces—this paper shows that the steady states of a one-dimensional monitored Clifford circuit contain long-range multipartite entanglement on both sides of the measurement-induced phase transition. In the volume-law phase the depth grows linearly with system size; in the area-law phase it still grows as a power law, with an exponent that continuously falls toward zero as the measurement probability approaches 1. The spatial pattern of the largest cluster is approximately self-similar, with a fractal dimension between 0 and 1 that matches the depth exponent away from the critical point. This is a genuinely multipartite, basis-independent counterpart to the usual bipartite-entropy description, and it says that area-law phases can still carry substantial many-body entanglement.

Core claim

The paper's central claim is that the entanglement depth D of monitored-circuit steady states obeys D ~ L^gamma, with gamma = 1 in the volume-law phase and gamma decreasing continuously to 0 as p → 1, while the largest entangled cluster's spatial support is an approximate fractal whose box-counting dimension d equals gamma away from the critical point. This means multipartite entanglement persists even when bipartite entanglement saturates, and the geometric dimension of the entangled cluster is a continuously tunable, measurement-controlled observable. The authors attribute the structure to a coagulation-fragmentation competition: two-qubit unitaries grow or merge clusters, while single-qub

What carries the argument

The central object is the entanglement depth obtained from an entanglement-structure diagram: a recursive partition of qubits into w-clusters defined by total correlations, where clusters become indivisible elements in later iterations; the largest resulting cluster's size is D. The monitored circuit is a brickwork of random two-qubit Clifford gates with single-site Z measurements of probability p, simulated by stabilizer formalism up to L=240. To extract the fractal dimension, the authors coarse-grain the chain into boxes of b qubits, rerun the clustering at each scale, and count boxes N_b needed to tile the largest cluster; d = -ln N_b / ln b. The identity that carries the argument is d =

Load-bearing premise

The load-bearing premise is that the power-law and fractal behaviour seen for L ≤ 240 and box sizes b = 2–20 is the true steady-state structure rather than a finite-size crossover—a possibility the authors themselves flag—and that the recursive clustering method identifies the physically meaningful largest cluster rather than a shape manufactured by its own recursion.

What would settle it

Extend the same calculation to L ≈ 1000 or more (or use a polynomial-scaling approximate scheme) and check whether gamma(p) remains 1 for p < p_c and whether d continues to track gamma away from the critical point; if the power-law bends toward saturation, gamma deviates from 1, or d departs from gamma at fixed p, the central claim fails. An independent check would compute a different multipartite witness (e.g., quantum Fisher information with an optimized local operator) on the same stabilizer states and see whether the hierarchical fractal survives.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Entanglement depth provides a multipartite, operator-independent order parameter for monitored circuits that remains nontrivial in the area-law phase.
  • States in the area-law phase of monitored circuits are not just weakly entangled; they contain clusters whose size grows as a power of the system size, so they may still be useful for distributed quantum information tasks.
  • The fractal dimension of the largest cluster offers a continuously tunable, measurement-controlled geometric descriptor of the steady state, with a plateau near p ~ 0.4–0.6 and a knee near p ~ 0.6.
  • The observed d = gamma relation away from the critical point ties a dynamical scaling exponent to a static geometric dimension, a concrete target for analytical coagulation-fragmentation models.
  • The small but statistically significant d versus gamma mismatch at the critical point points to logarithmic corrections or additional critical structure that future work can probe.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural extension: applying an independent multipartite witness—quantum Fisher information with optimized operators, or k-partite mutual information—to the same trajectories would test whether the fractal geometry is a property of the state or an artifact of the recursive clustering.
  • If the power-law is genuine, the pair (gamma, d) might sit in the universality class of classical coagulation-fragmentation; a mean-field rate-equation estimate for d(p) would be a testable prediction the paper does not make.
  • The knee at p ≈ 0.6 coincides with the coarse-graining floor D = 2; deep in the area-law phase the ensemble may be dominated by rare large clusters, so studying the full distribution of D rather than the mean would clarify whether the continuous decrease of gamma is a true exponent or a finite-size crossover.
  • The random-Cantor-set analogy suggests a multiplicative-cascade picture: treating each measurement as an independent deletion event with cluster survival probability depending on p would yield a simple prediction for d(p) that could be checked in the volume-law phase.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies one-dimensional monitored random Clifford circuits with local projective measurements at rate p. Using the authors' previously developed entanglement-structure method, they compute the entanglement depth D (the size of the largest cluster of mutually entangled qubits) and find D ~ L^γ with γ ≈ 1 for p < p_c ≈ 0.16 (volume-law phase) and γ continuously decreasing toward 0 as p → 1 (area-law phase). They further claim that the spatial support of the largest cluster is fractal, with a box-counting dimension d that matches γ away from the critical point. The authors propose a coagulation-fragmentation picture as the physical mechanism. The numerical work uses stabilizer simulation, 500 realizations, system sizes up to L=240, and robust Theil-Sen regression with MAD error bars, and it explicitly acknowledges the possibility of finite-size effects.

Significance. If correct, the paper would establish a new multipartite-entanglement structure in monitored circuits: long-range multipartite entanglement in the area-law phase that is invisible to bipartite entanglement entropy, organized as approximate self-similar fractals with a continuously tunable fractal dimension. This would be a genuinely novel and field-relevant contribution. The numerical methodology is careful: stabilizer simulation is exact for Clifford circuits, 500 realizations are used, the regression is robust, and the authors are transparent about limitations. However, the central fractal claim rests on a box-counting procedure that may not measure the spatial geometry of the base-level cluster, and the quantitative support is limited to roughly one decade of scales. The match d=γ is suggestive but requires an independent check before it can be accepted.

major comments (3)
  1. [End Matter B; Fig. 4] The box-counting procedure does not measure the spatial support of the largest cluster found at the base scale. For each b, the entanglement structure is recomputed on coarse-grained superqubits, and N_b is the number of boxes in the largest cluster of that new structure. This cluster need not be related to the b=1 cluster by inclusion or geometric cover. A block can be dropped or added based on the recursive total-correlation criterion rather than on whether it intersects the original cluster. The holes visualized in Fig. 4 are therefore not necessarily holes in the b=1 cluster. To support the fractal claim, the authors must compare this algorithm-derived N_b with the geometric box count of the fixed b=1 (or b=2) largest cluster: count intervals of length b that contain at least one qubit of that cluster. Without such a cross-check, d cannot be interpreted as the fractal dimension of th
  2. [Fractal Structures; End Matter B; Fig. 5] The fractal dimension is extracted from b=2..20 at L=240, roughly one decade of box sizes, with the largest box being 1/12 of the system. This is a narrow scaling window, and the authors themselves concede a possible finite-size effect. In the area-law phase, the averaged entanglement depth approaches the coarse-graining floor D≈2, and the knee at p≈0.6 is stated to be sensitive to the degree of coarse graining. I request a robustness analysis: fit d over subranges of b (e.g., 2–10 and 4–20) and show the exponents are stable; report the L values and number of realizations used for each p; show representative fits with residuals for both γ and d. Without this, the continuous variation of d and γ may be a crossover artifact rather than asymptotic scaling.
  3. [Figs. 2–3; End Matter A] The power-law exponent γ is obtained from log-log fits of D vs L, but the paper does not list the L values used for each p. For deep volume law, only L=100,160 are mentioned, for p=0.04 and 0.08. The claims that γ=1 for all p<p_c and that d=γ quantitatively require that the fits are stable and that the same scaling window is used across p. Please provide the full set of L values and the fitted data (or a plot showing all fits), and state whether γ is extracted from the same L values for every p. Since D is defined using a two-qubit coarse graining, a check with a different coarse-graining size (e.g., four qubits) would show whether the exponent is robust to this choice.
minor comments (5)
  1. [Fig. 1 caption] Typo: 'abreviated' should be 'abbreviated'. Also 'heirarchical' appears in the Fractal Structures section and 'Sierpienski' should be 'Sierpiński'.
  2. [Throughout] The estimator is named after Theil and Sen; 'Thiel-Sen' should be 'Theil-Sen' (also in End Matter A).
  3. [End Matter A] The error bar is the median absolute deviation of all pairwise slopes. This is a dispersion measure, not a rigorous confidence interval. The authors should clarify that the quoted error bars are not standard errors or confidence intervals, and that they are used only as a robustness indicator.
  4. [Entanglement depth and long-range entanglement] The statement 'At p=0, we expect all qubits to be entangled and the largest cluster exactly spans the entire system' is plausible but not derived. A short argument (e.g., from the scrambling nature of Clifford circuits) would strengthen the p→0 baseline.
  5. [Fig. 3] The text refers to 'blue data points' and 'yellow data points'. For accessibility, consider using different marker shapes in addition to colors, and define them clearly in the caption.

Circularity Check

2 steps flagged

The fractal dimension d and the depth exponent γ are both extracted from the same recursive clustering algorithm, so their match is at least partly a consistency relation of the method rather than an independent spatial measurement.

specific steps
  1. self definitional [Fractal Structures (main text) and End Matter B: box-counting method]
    "For a given state in our ensemble, we first coarse-grain our system by grouping b neighboring qubits together. ... These boxes are considered as indivisible entities, coarse-graining any entanglement structure that we obtain. We then compute the coarse-grained entanglement structure using the method in [23]. ... We count the number of boxes involved in the largest cluster within this entanglement structure and term it as Nb. ... The fractal dimension, d is then defined as, d = -ln Nb/ln b."

    Nb is not the geometric box-count of the b=1 largest cluster; it is the size, in boxes, of the largest cluster obtained by re-running the same recursive clustering algorithm at scale b. Thus Nb is literally the coarse-grained entanglement depth, the same observable whose L-dependence defines gamma. The reported equality d = gamma therefore compares the scaling of this algorithmically defined depth under two rescalings (system size and box size), both generated by the identical hierarchical-clustering rule. If the clustering rule is scale-invariant, this equality follows by construction; the paper provides no independent check in which the b=1 cluster is spatially covered by b-boxes, so the holes and self-similarity may be artifacts of the coarse-graining/clustering procedure rather than pr

  2. self citation load bearing [Entanglement Structure Calculation; Summary and Outlook]
    "We use the method developed in [23] to extract the entanglement depth and construct the largest cluster of entangled qubits. ... By using the formalism developed in [23], we construct a number of metrics of multipartite entanglement. This procedure allows us to extract the largest non-separable cluster of qubits."

    Every quantitative result in the paper - the entanglement depth D, the power-law exponent gamma, the box count Nb, and hence the fractal dimension d - is defined through the authors' own prior formalism [23]. The manuscript contains no independent verification, code-level reproduction, or alternate witness (e.g., quantum Fisher information or k-party mutual information) confirming that the identified clusters and their holes are properties of the quantum state rather than of the hierarchical clustering construction. The main conclusions therefore inherit their content from an unverified self-citation; if the clustering rule in [23] is not the correct operational definition, all reported exponents change.

full rationale

The paper's non-circular core is the numerical observation that the algorithmically defined entanglement depth D grows as a power law in L with an exponent gamma(p), and that long-range multipartite entanglement survives in the area-law phase. This could in principle be wrong and is not itself circular. The circularity enters in the fractal claim: in End Matter B, the box count Nb is obtained by recomputing the entanglement structure after grouping qubits into boxes, so Nb is the coarse-grained entanglement depth, not an independent spatial cover of the b=1 cluster. The headline match d = gamma is therefore at least partly a consistency relation of the same recursive clustering rule applied at different scales. The authors themselves concede the finite-size possibility: 'it is possible that we are observing a finite size effect which would not be present in numerical experiments on larger systems.' Because the central 'tunable fractal dimension' result reduces, in part, to the self-consistency of the self-cited clustering construction, a score of 6 is appropriate. The D(L) power law and the qualitative robustness of area-law multipartite entanglement retain some independent numerical content, preventing a higher score.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The paper's central claims rest on (i) a published but author-owned clustering algorithm that defines the observable, (ii) a conjectured self-similarity that is only probed over about one decade, and (iii) prior determinations of p_c and the validity of stabilizer simulation. Beyond the measurement probability p (a physical control parameter), the only numbers chosen by hand that materially affect the results are the 2-qubit coarse-graining block size and the box-size window 2..20; both are documented in the paper, which is to its credit.

free parameters (3)
  • Coarse-graining block size for entanglement depth (2 qubits) = 2
    Chosen by hand to floor D at 2 and speed computation; the paper states the location of the knee in gamma(p) at p approx 0.6 is sensitive to this choice, so features of the central gamma(p) curve depend on it.
  • Box-size range for fractal dimension fit = b = 2..20
    The fractal dimension d is estimated from ln N_b vs ln b over this chosen range — only about one decade — at L = 240; a different window would change d.
  • System sizes sampled per p = up to L=240 (p >= 0.12); L=100,160 for p=0.04,0.08
    Exponent gamma is fit over these L-ranges; deep in the volume-law phase the range is short (L <= 160) because of computational cost of the hierarchy method.
axioms (5)
  • standard math Gottesman-Knill theorem: Clifford circuits + Pauli measurements are classically simulable in polynomial time
    Basis for simulating the monitored circuits at L = 240; standard formal result.
  • domain assumption The entanglement-structure hierarchy of ref. 23 faithfully identifies the largest genuinely multipartite entangled cluster (entanglement depth)
    The central observable D is defined by the authors' own prior clustering algorithm (w-clusters via total correlation, iterative coarse-graining). All conclusions about 'the largest cluster' inherit the properties of this particular construction; no independent check (e.g., operator-based witness) is given in this paper.
  • domain assumption MIPT critical point p_c approx 0.16 for this brickwork Clifford circuit, taken from refs. 3-4
    The division into volume-law and area-law regimes and the interpretation of gamma(p) behavior at p_c rely on the prior determination of the transition point.
  • domain assumption Approximate self-similarity of the largest cluster over the accessible length scales (fractal hypothesis)
    The paper assumes the intermediate-scale power-law behavior (box sizes 2..20, L <= 240) persists to the thermodynamic limit; the text explicitly concedes this might be a finite-size effect.
  • standard math Theil-Sen (with MAD) supplies unbiased slope estimates for correlated log-log data
    Used for all exponent extractions; a robust-regression assumption appropriate for median-based estimation, applied to ensemble-averaged data with few points per curve.

pith-pipeline@v1.3.0-alltime-deepseek · 8209 in / 23219 out tokens · 231331 ms · 2026-08-03T22:47:38.424640+00:00 · methodology

0 comments
read the original abstract

We study the structure of multipartite entanglement in monitored quantum circuits exhibiting measurement-induced phase transitions (MIPTs). Using a one-dimensional Clifford circuit subject to local measurements with a probability $p$, we show numerically that the entanglement depth, corresponding to the size of the largest cluster of entangled qubits scales as a power law with system size on both sides of the transition. The power law exponent is 1 in the entangling phase and continuously decreases to 0 as $p \to 1$ in the disentangling phase. In addition, we find that the spatial support of the largest cluster exhibits an approximate fractal geometry with a tunable fractal dimension controlled by the measurement rate. We argue that this structure arises from a competition between unitary-driven coagulation of entangled clusters and measurement-induced fragmentation, giving rise to a fractal steady state reminiscent of classical coagulation-fragmentation models. Away from the MIPT critical point, the fractal dimension matches the entanglement depth power law exponent. These results show that multipartite entanglement structure provides a fresh perspective on the emergent quantum correlations in monitored quantum circuits and noisy quantum dynamics.

Figures

Figures reproduced from arXiv: 2511.08690 by Erich J Mueller, Vaibhav Sharma.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Entanglement structure schematic of a twelve [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Log-log plot of average entanglement depth as a [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. A typical 120 qubit state generated at measurement [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Power law exponent of entanglement depth growth [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references · 3 linked inside Pith

  1. [1]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annual Review of Condensed Matter Physics14, 335 (2023)

    2023

  2. [2]

    S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum er- ror correction in scrambling dynamics and measurement- induced phase transition, Phys. Rev. Lett.125, 030505 (2020)

    2020

  3. [3]

    Y. Li, X. Chen, and M. P. A. Fisher, Quantum zeno effect and the many-body entanglement transition, Phys. Rev. B98, 205136 (2018)

    2018

  4. [4]

    Y. Li, X. Chen, and M. P. A. Fisher, Measurement- driven entanglement transition in hybrid quantum cir- cuits, Phys. Rev. B100, 134306 (2019)

    2019

  5. [5]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- induced phase transitions in the dynamics of entangle- ment, Phys. Rev. X9, 031009 (2019)

    2019

  6. [6]

    Ippoliti, M

    M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, Entanglement phase transitions in measurement-only dynamics, Phys. Rev. X11, 011030 (2021)

    2021

  7. [7]

    Lavasani, Y

    A. Lavasani, Y. Alavirad, and M. Barkeshli, Measurement-induced topological entanglement transi- tions in symmetric random quantum circuits, Nature Physics17, 342 (2021)

    2021

  8. [8]

    Jian, Y.-Z

    C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Measurement-induced criticality in random quantum cir- 6 cuits, Phys. Rev. B101, 104302 (2020), arXiv:1908.08051 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2020

  9. [9]

    Sang and T

    S. Sang and T. H. Hsieh, Measurement-protected quan- tum phases, Phys. Rev. Research3, 023200 (2021)

    2021

  10. [10]

    Turkeshi, R

    X. Turkeshi, R. Fazio, and M. Dalmonte, Measurement- induced criticality in (2+1)-dimensional hybrid quantum circuits, Phys. Rev. B102, 014315 (2020)

    2020

  11. [11]

    Sierant, M

    P. Sierant, M. Schir` o, M. Lewenstein, and X. Turkeshi, Measurement-induced phase transitions in (d +1 ) - dimensional stabilizer circuits, Phys. Rev. B106, 214316 (2022), arXiv:2210.11957 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2022

  12. [12]

    Lavasani, Z.-X

    A. Lavasani, Z.-X. Luo, and S. Vijay, Monitored quantum dynamics and the kitaev spin liquid, Phys. Rev. B108, 115135 (2023)

    2023

  13. [13]

    Sharma, C.-M

    V. Sharma, C.-M. Jian, and E. J. Mueller, Subsystem symmetry, spin-glass order, and criticality from random measurements in a two-dimensional bacon-shor circuit, Phys. Rev. B108, 024205 (2023)

    2023

  14. [14]

    Lira-Solanilla, X

    A. Lira-Solanilla, X. Turkeshi, and S. Pappalardi, Mul- tipartite entanglement structure of monitored quantum circuits, Phys. Rev. Lett.135, 080401 (2025)

    2025

  15. [15]

    Paviglianiti and A

    A. Paviglianiti and A. Silva, Multipartite entanglement in the measurement-induced phase transition of the quan- tum ising chain, Phys. Rev. B108, 184302 (2023)

    2023

  16. [16]

    Di Fresco, B

    G. Di Fresco, B. Spagnolo, D. Valenti, and A. Car- ollo, Metrology and multipartite entanglement in measurement-induced phase transition, Quantum8, 1326 (2024)

    2024

  17. [17]

    P. M. Poggi and M. H. Mu˜ noz-Arias, Measurement- induced multipartite-entanglement regimes in collective spin systems, Quantum8, 1229 (2024)

    2024

  18. [18]

    S. J. Avakian, T. Pereg-Barnea, and W. Witczak- Krempa, Long-range multipartite entanglement near measurement-induced transitions, Phys. Rev. Res.7, 023135 (2025)

    2025

  19. [19]

    Xu and Y.-X

    G. Xu and Y.-X. Zhang, Multipartite greenberger-horne- zeilinger entanglement in monitored random clifford cir- cuits (2024)

    2024

  20. [20]

    Allen and W

    J. Allen and W. Witczak-Krempa, Spatial structure of multipartite entanglement at measurement induced phase transitions (2025), arXiv:2509.12109 [quant-ph]

    arXiv 2025

  21. [21]

    Hyllus, W

    P. Hyllus, W. Laskowski, R. Krischek, C. Schwem- mer, W. Wieczorek, H. Weinfurter, L. Pezz´ e, and A. Smerzi, Fisher information and multiparticle entan- glement, Phys. Rev. A85, 022321 (2012)

    2012

  22. [22]

    Y. Wang, Y. Fang, F. Xie, and Q. Si, Local and non-local entanglement witnesses of fermi liquid (2025), arXiv:2502.13958 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  23. [23]

    Sharma and E

    V. Sharma and E. J. Mueller, Multipartite entanglement structures in quantum stabilizer states, Phys. Rev. A 112, 012411 (2025)

    2025

  24. [24]

    Aaronson and D

    S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A70, 052328 (2004)

    2004

  25. [25]

    Theil, A rank-invariant method of linear and polyno- mial regression analysis, inHenri Theil’s Contributions to Economics and Econometrics(Springer Netherlands,

    H. Theil, A rank-invariant method of linear and polyno- mial regression analysis, inHenri Theil’s Contributions to Economics and Econometrics(Springer Netherlands,

  26. [26]

    P. K. Sen, Estimates of the regression coefficient based on kendall’s tau, Journal of the American Statistical As- sociation63, 1379–1389 (1968)

    1968

  27. [27]

    R. R. Tremblay and A. P. Siebesma, Void distribution of random cantor sets, Phys. Rev. A40, 5377 (1989)

    1989

  28. [28]

    J. T. Chayes, L. Chayes, and R. Durrett, Connectivity properties of mandelbrot’s percolation process, Probabil- ity Theory and Related Fields77, 307–324 (1988)

    1988

  29. [29]

    C. Leys, C. Ley, O. Klein, P. Bernard, and L. Licata, De- tecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median, Journal of Experimental Social Psychology49, 764–766 (2013)

    2013

  30. [30]

    M. G. Akritas, S. A. Murphy, and M. P. Lavalley, The theil-sen estimator with doubly censored data and appli- cations to astronomy, Journal of the American Statistical Association90, 170–177 (1995). I. END MA TTER A. Thiel-Sen Linear Regression Here we briefly describe the Thiel-Sen regression method we used to extract the slope and the error bars corresp...

    1995