Pith. sign in

REVIEW 3 major objections 5 minor 33 references

Information and Complexity Analysis of Spatial Data

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read The paper argues that generalized complexity $C_{\alpha,\beta}=e^{H_\alpha-H_\beta}$ is the common core of entropy-based spatial complexity measures, and that in multifractal settings its increments and derivatives read directly off the…

desk verdict A mostly review paper with a modest new relative-dimension formalism; the formal identities check out but the central new limit is asserted, not proved, and the empirical illustration lacks the details needed to support it. read the letter →

arxiv 2411.16871 v1 pith:KFHGOKNY submitted 2024-11-25 math.ST stat.TH

Jose M. Angulo , Francisco J. Esquivel , Ana E. Madrid , Francisco J. Alonso This is my paper · ORCID
classification math.STstat.TH MSC 62B1062M3028A80
keywords Rényientropydivergencegeneralizeddimensionsrelativemultifractalmeasurescomplexitymeasurespatialdataseismicseries
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 builds a single information-theoretic ladder for measuring the complexity of spatial data, connecting discrete entropy, divergence, and multifractal geometry. Its central claim is that the two-parameter generalized complexity measure $C_{\alpha,\beta}=e^{H_\alpha-H_\beta}$, understood as a ratio of Campbell diversity indices, contains the standard product-complexity measures and Batty's complexity difference as special or limiting cases, and that in the multifractal limit it scales as $\varepsilon^{D_\beta-D_\alpha}$, so the increments and derivatives of the generalized Rényi dimension curve are themselves complexity measures. The same construction is carried out for relative complexity using Rényi divergence, leading to generalized relative Rényi dimensions whose increments play the analogous role when comparing two spatial measures. The paper applies these tools to the 2011 El Hierro seismic sequence and shows that the phase containing the volcanic eruption is clearly separated from the surrounding phases by both the shape of the dimension curves and the dissociation between the event-frequency distribution and the magnitude-weighted energy distribution.

What carries the argument

The central object is the two-parameter exponential complexity family $C_{\alpha,\beta}=e^{H_\alpha-H_\beta}$, interpreted as a diversity ratio through Campbell's exponential entropy, together with its relative analogue $C_{\alpha,\beta}(\cdot\|\cdot)$ built on Rényi divergence. The argument moves from these finite-state objects to multifractal geometry through the box-counting scaling relation $e^{-H_q(P_\varepsilon)}\sim\varepsilon^{D_q}$, which defines the generalized Rényi dimension curve $q\mapsto D_q$ and the generalized relative dimension curve $q\mapsto D_q(\mu_1\|\mu_2)$. Substituting that scaling into the complexity family turns the two-parameter index into a power law whose exponent is a difference of dimensions, so the meaningful complexity diagnostics are the increment maps $(D_\alpha-D_\beta)/(\alpha-\beta)$ and their diagonal limit, the derivative curves $D'_\alpha$ and $D'_\alpha(\mu_1\|\mu_2)$.

What would settle it

Fix a box-width schedule and check log-log linearity of $C_{\alpha,\beta}(P_\varepsilon)$ against $\varepsilon$: the central relation predicts a line of slope $D_\beta-D_\alpha$, and the relative version predicts slope $D_\beta(\bar p\|\bar e)-D_\alpha(\bar p\|\bar e)$. Also permute event magnitudes across event times within each phase; if the phase differences in the derivative curves persist after permutation, the claimed structural dissociation is not a genuine feature of the joint event-magnitude process.

Watch

Extended reading notes

Core claim

The paper's central formal claim is that the two-parameter generalized complexity measure $C_{\alpha,\beta}(\bar p)=e^{H_\alpha(\bar p)-H_\beta(\bar p)}$, equivalently the ratio of Campbell diversity indices $DI_\alpha(\bar p)/DI_\beta(\bar p)$, is the common core of the main entropy-based complexity notions for spatial data. In the discrete setting it contains Batty's complexity difference as the special case $ID(\bar p)=\ln C_{0,1}(\bar p)$, with the complexity ratio given by $\ln C_{0,1}(\bar p)/\ln n$. In the multifractal setting, using the scaling $e^{-H_q(P_\varepsilon)}\sim\varepsilon^{D_q}$, it becomes $C_{\alpha,\beta}(P_\varepsilon)\sim\varepsilon^{D_\beta-D_\alpha}$, so the increments of the generalized Rényi dimension curve, and their diagonal limit $D'_\alpha$, are complexity measures. The paper constructs the analogous two-parameter family for relative complexity from Rényi divergence and defines generalized relative Rényi dimensions $D_q(\mu_1\|\mu_2)$ such that $C_{\alpha,\beta}(P_{1\varepsilon}\|P_{2\varepsilon})\sim\varepsilon^{D_\beta(\mu_1\|\mu_2)-D_\alpha(\mu_1\|\mu_2)}$; the increments and derivatives of these relative dimension curves then quantify local structural coherence between two measures. On the El Hierro seismic data, these diagnostics separate the eruption-containing phase from its neighbors and reveal how strongly the temporal distribution of events is dissociated from the distribution of released energy.

Load-bearing premise

The argument rests on the El Hierro event series being genuinely multifractal at the box scales analyzed, so the estimated dimension curves are stable limits rather than finite-sample artifacts, and on the hand-chosen three-phase split not manufacturing the contrasts.

Editorial extensions

If this is right

  • If the scaling relation is correct, Batty's complexity difference and complexity ratio become special cases of the diversity-ratio family, so they inherit a direct information-theoretic interpretation rather than being ad hoc geographic indices.
  • For any multifractal measure, the whole generalized dimension curve becomes a complexity object: the increment maps $(D_\alpha-D_\beta)/(\alpha-\beta)$ and the derivative curve $D'_\alpha$ quantify structural complexity, complementing the usual multifractal spectrum.
  • The relative-dimension curves $D_q(\bar p\|\bar e)$ and $D_q(\bar e\|\bar p)$ give a directional, state-by-state reading of how event frequency and released energy separate, with asymmetry indicating which distribution is more concentrated.
  • Applied to El Hierro, the method distinguishes the eruption phase from pre- and post-eruption phases through a shortened multifractal step and an inversion in the relative-dimension derivative curves, supporting its use for monitoring structural change in space-time point processes.

Reading between the lines

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

  • The paper does not report convergence diagnostics; a natural check would be to randomize event magnitudes across event times within each phase and see whether the phase differences in $D'_q(\bar p\|\bar e)$ survive, which would confirm the dissociation is a property of the joint process rather than of the marginal magnitude distribution.
  • The scaling relation suggests a direct estimator, $\ln C_{\alpha,\beta}(P_\varepsilon)/\ln\varepsilon$ approximates $D_\beta-D_\alpha$; block-bootstrapping over the box grid could attach uncertainty bands to the derivative curves, something the paper leaves unexplored.
  • Because $C_{\alpha,\beta}$ is a ratio of effective numbers of states, the same machinery transfers to any diversity decomposition, such as species counts, land-use mixes, or communication networks, where the deformation parameter acts as a sensitivity weight.
  • A rolling-window version of the derivative-curve scan, rather than the three hand-chosen phases, could turn the static comparison into a prospective early-warning tool for volcanic or seismic unrest.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper reviews and connects information-theoretic measures of entropy, divergence, complexity, and multifractality, and proposes a divergence-based relative diversity index DI_q(p1||p2)=exp(H_q(p1||p2)). It shows that Batty's complexity difference is ln C0,1(p), interprets the two-parameter complexity measure as a ratio of Campbell diversity indices, recalls the scaling relation C_{α,β}(P_ε) ∼ ε^{D_β-D_α}, and introduces generalized relative Rényi dimensions D_q(µ1||µ2) with an analogous scaling relation. An application to El Hierro seismic data compares generalized and relative dimension curves across three phases around the 2011 eruption.

Significance. If the central identities and scaling relations hold, the paper offers a useful unifying interpretation of existing complexity measures and a new diagnostic for relative multifractal structure. The paper's strengths include the clean formal embedding ID(p)=ln C0,1(p), the diversity-ratio interpretation of C_{α,β}, and the explicit proposal of generalized relative Rényi dimensions as a tool for comparing two spatial measures. However, the new relative-dimension bridge in Section 4.2 is asserted rather than derived, and the empirical illustration in Section 5 does not report the estimation details needed to rule out resolution artifacts; these gaps currently limit the strength of the contribution.

major comments (3)
  1. [Section 4.2, paragraph after the definition of D_q(µ1||µ2)] The limiting relation C_{α,β}(P_{1,ε}∥P_{2,ε}) ∼ ε^{D_β(µ1∥µ2)-D_α(µ1∥µ2)} is asserted with 'Similarly' and no proof. This is the central new mathematical claim of the paper. The authors should state the measure-theoretic hypotheses under which the limit defining D_q(µ1∥µ2) exists, justify the limit interchange or coarse-graining step that produces the scaling of the ratio of Rényi divergences, and show that any prefactor in C_{α,β}(P_{1,ε}∥P_{2,ε}) is ε-independent or subexponential at q=α,β. If a theorem is available in the cited literature, it should be quoted with precise conditions; otherwise a derivation is needed before the relative-dimension bridge can be considered established.
  2. [Section 4.2, displayed definition of D_1(µ1||µ2)] The definition of D_1(µ1∥µ2) is not the q→1 limit of the q≠1 formula. As q→1, the numerator in the q≠1 expression tends to −KL(P_ε∥Q_ε), so consistency requires D_1(µ1∥µ2)=lim [−KL(P_ε∥Q_ε)]/ln ε, whereas the displayed definition uses +KL(P_ε∥Q_ε)/ln ε. This sign inconsistency propagates to the claimed ε^{D_β−D_α} scaling at q=1 and to the derivative curves in Figure 16, and it should be corrected or explicitly justified as a different convention.
  3. [Section 5, Figures 11-16] The empirical demonstration assumes that the event-count process and the accumulated-energy process are multifractal and in the scaling regime at the available resolutions, but the paper never states the box-width schedule, the number of occupied boxes at each ε, the magnitude-to-energy conversion, or any convergence or plateau diagnostics. For a finite point process, once ε falls below the minimum inter-event gap every occupied box contains exactly one event and every estimated D_q collapses to zero, so the curves in Figures 11-16 depend on an unstated resolution cutoff. Please report these details, examine stability of the estimated exponents over a range of ε, and describe how the phase boundaries A/B/C were chosen; without this information the apparent phase separation could be a finite-sample or resolution artifact rather than a property of the underlying measures.
minor comments (5)
  1. [Section 4.2, definition of D_q(µ1||µ2)] There is a typographical error in the displayed formula: 'µ^q_1[Bε(k)]]µ^{1−q}_2[Bε(k)]' contains an extra closing bracket.
  2. [Section 2.2, 'A relative diversity index'] The phrase 'formulated here is terms of divergence' should read 'formulated here in terms of divergence'.
  3. [References] Several references contain typos or misspellings: 'North Nolland' should be 'North-Holland', 'Cecatto' should be 'Ceccato', 'Jaharb Regionalwissensc' should be corrected, and 'Annals of Mahematical Statistics' should be 'Annals of Mathematical Statistics'.
  4. [Section 5, first paragraph] The number 11.142 should be rendered as 11,142 with a thousands separator to avoid ambiguity.
  5. [Section 4.2, introductory paragraph] The phrase 'in contrast to related proposals in the literature' needs explicit citations to the related proposals so that the reader can assess the claimed difference.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the section 4 bridges are algebraic identities following from the definitions of generalized (relative) Rényi dimensions and of the complexity measures, and the seismic illustration is an interpretive application rather than a fitted prediction.

full rationale

The paper's central formal claims reduce to definitions without any fitted parameter or imported self-citation bearing the load. Section 3 embeds Batty's complexity difference as ID(p)=ln C_{0,1}(p); this is immediate from H_0(p)=ln n and C_{0,1}=e^{H_0-H_1}, and no data are involved. In Section 4.1, D_q is defined as the scaling exponent of e^{-H_q(P_epsilon)} (equivalently D_q = lim H_q(P_epsilon)/ln epsilon), so C_{alpha,beta}(P_epsilon)=e^{H_alpha-H_beta}=e^{-H_beta}/e^{-H_alpha} ~ epsilon^{D_beta-D_alpha} follows by construction. The same holds in Section 4.2 for the relative quantities: D_q(mu1||mu2) is defined via the scaling of e^{-H_q(P_{1,epsilon}||P_{2,epsilon})}, making C_{alpha,beta}(P_{1,epsilon}||P_{2,epsilon}) ~ epsilon^{D_beta-D_alpha} a direct consequence. The attributions to Angulo and Esquivel (2014) and Esquivel, Alonso and Angulo (2017) are not load-bearing because the derivation is reproduced in the text; removing those citations would not change the equations. The El Hierro section is an exploratory illustration, not a statistical prediction: the phase partition is chosen by the eruption date, the box-counting estimates are not fitted to the complexity curves, and no parameter is calibrated to a subset and then used to 'predict' the same subset. Therefore no fitted-input-called-prediction or self-definitional circularity is present. Limitations of the empirical analysis (unreported box-width schedule, missing convergence diagnostics, hand-selected phases) are validity concerns rather than circularity, as is a possible sign inconsistency in the displayed q=1 relative-dimension formula; these do not affect the circularity score.

Assumptions & free parameters 4 free parameters · 4 assumptions · 2 invented entities

The core formalism relies only on standard entropy and divergence results, but the new relative-dimension quantities are introduced without external validation, and the empirical illustration depends on several hand-chosen settings (phase split, q range, box widths, energy weighting) that are not specified in detail.

free parameters (4)
  • Phase boundaries A/B/C = 19 Jul 2011 to 7 Jan 2012, split around 10 Oct 2011 eruption
    The series is divided into three subperiods with the eruption in phase B; this choice is made by the authors and affects all observed contrasts.
  • Rényi order range q = -25 to 25
    Dimension curves and derivative plots are computed over this fixed q range; results depend on the chosen range.
  • Box-counting resolution range ε = not specified
    Generalized dimensions are estimated from lattice partitions at decreasing box width; the paper does not state the ε values or convergence criteria.
  • Magnitude-to-energy weighting = not specified
    The accumulated energy distribution weights events according to magnitude; the conversion formula is not given and is deferred to prior references.
assumptions (4)
  • standard math Standard definitions and properties of Shannon/Rényi entropy and Kullback-Leibler/Rényi divergence are taken as given.
    Section 2 collects these known results; the framework builds on them.
  • domain assumption The multifractal formalism: generalized Rényi dimensions D_q exist and characterize the scaling of partition entropies.
    Section 4.1 assumes the limit defining D_q and the asymptotic e^{-H_q(Pε)} ~ ε^{D_q}; no regularity conditions are stated.
  • domain assumption The empirical seismic event distribution and the accumulated energy distribution are realizations of multifractal measures with well-defined relative Rényi dimensions.
    Section 5 applies relative dimensions to the El Hierro series without testing for multifractality or convergence.
  • ad hoc to paper The proposed definition of generalized relative Rényi dimensions D_q(µ1∥µ2) is a meaningful measure of local structural dissimilarity.
    Section 4.2 introduces the definition in contrast to related proposals in the literature but does not justify it with external benchmarks or prove existence.
invented entities (2)
  • Generalized relative Rényi dimensions D_q(µ1∥µ2)
    purpose: Quantify the rate of divergence scaling between two multifractal measures, enabling local (state-by-state) complexity comparisons.
    Defined in Section 4.2 and used in Section 5; no independent validation or comparison with existing relative-dimension proposals.
  • Relative diversity index DI_q(p1∥p2)
    purpose: Provide a diversity-ratio interpretation for relative complexity measures.
    Introduced in Section 2.2 as the exponential of Rényi divergence; it is a reformulation of divergence rather than an independently measured quantity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Information and Complexity Analysis of Spatial Data." pith.science (2026). https://pith.science/paper/KFHGOKNY

@misc{pith2026241116871,
  author       = {Pith},
  title        = {Pith review of: Information and Complexity Analysis of Spatial Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KFHGOKNY}},
  note         = {Machine review of arXiv:2411.16871}
}
read the original abstract

Information Theory provides a fundamental basis for analysis, and for a variety of subsequent methodological approaches, in relation to uncertainty quantification. The transversal character of concepts and derived results justifies its omnipresence in scientific research, in almost every area of knowledge, particularly in Physics, Communications, Geosciences, Life Sciences, etc. Information-theoretic aspects underlie modern developments on complexity and risk. A proper use and exploitation of structural characteristics inherent to spatial data motivates, according to the purpose, special considerations in this context. In this paper, some of the most relevant approaches introduced, in particular recent contributions and directions, regarding the informational analysis of spatial data and related aspects concerning complexity analysis, are reviewed under a conceptually connective evolutionary perspective. The discussion involves the cases of spatial data from magnitude measurements and spatial point patterns, with the latter possibly being of a multifractal nature.

Figures

Figures reproduced from arXiv: 2411.16871 by the authors.

Figure 1
Figure 1. Shannon entropy for ¯p = (p1, p2, p3), with values varying from 0, the minimum un￾certainty at the degenerate distributions corresponding to the vertices, to ln(3), the maximum uncertainty associated with the equiprobability central point. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Paths followed by ¯p q,∗ = (p q,∗ 1 , p q,∗ 2 , p q,∗ 3 ) as q tends from 1 to ∞ (left plot) or to 0 (right plot), from different starting distributions. In the first case, power distortion makes any non-equiprobable distribution to move towards the edges and/or vertices; in particular, middle edge points attract those distributions having two dominant equiprobable states. In the second case, conversely, any distrib… view at source ↗
Figure 3
Figure 3. R´enyi entropy of orders q = 0.5 (top left), 2 (top right), 5 (bottom left) and 100 (bottom right) for ¯p = (p1, p2, p3). In all cases, values vary from 0, the minimum uncertainty at the degenerate distributions corresponding to the vertices, to ln(3), the maximum uncer￾tainty associated with the equiprobability central point, with dissimilar transition patterns reflecting different distortion effects derived from t… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Campbell diversity indices DI1(¯p) (left) and DI2(¯p) (right) for ¯p = (p1, p2, p3). The values continuously vary from 1, the minimum number of intrinsic states corresponding to the degenerate distributions at the vertices, to 3, the maximum associated with the equipro…
Figure 5
Figure 5. Figure 5: Information difference (or ‘complexity difference’) [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Generalized complexity measure Cα,β(¯p) for ¯p = (p1, p2, p3), with (α, β) = (1, 2) (top left), (2,10) (top right), (0.5,1) ((bottom left) and (0.5,10) (bottom right). Noting that in all the cases α has been taken lower than β (complexity values would be their correspo…
Figure 7
Figure 7. Figure 7: Derivative of R´enyi entropy, H′ α, at α = 2, for ¯p = (p1, p2, p3). The structure of this plot shows the intrinsic value of −H′ α as a one-parameter generalized complexity measure, under the notion of ‘complexity’ as departure from both equilibrium and degeneracy. and…
Figure 8
Figure 8. Figure 8: Generalized relative complexity measure Cα,β(¯p∥q¯) for ¯p = (p1, p2, p3), ¯q = (0.2, 0.3, 0.5), with (α, β) = (1, 2) (top left), (2,10) (top right), (0.5,1) (bottom left) and (0.5,10) (bottom righ). In all the cases, the maximum value 1 is reached for ¯p equal to the …
Figure 9
Figure 9. Figure 9: Generalized relative complexity measure Cα,β(¯p∥q¯) for varying ¯p = (p1, p2, p3) and fixed ¯q = (0.2, 0.3, 0.5) (left), and for varying ¯q = (q1, q2, q3) and fixed ¯p = (0.2, 0.3, 0.5) (right), in both cases with (α, β) = (0.5, 1). The different patterns in these plot…
Figure 10
Figure 10. Figure 10: El Hierro data: epicenters on contoured island (top), and temporal sequence [PITH_FULL_IMAGE:figures/full_fig_p032_10.png]
Figure 11
Figure 11. Figure 11: Generalized R´enyi dimension curves, showing different multifractal patterns for [PITH_FULL_IMAGE:figures/full_fig_p033_11.png]
Figure 12
Figure 12. Figure 12: Maps of relative increments Dα−Dβ α−β for the three subperiods (from left to right: phases A, B and C). The multifractal complexity structural pattern for phase B notably differs from the corresponding patterns for phases A and C, which are relatively similar. 33 [PI…
Figure 13
Figure 13. Figure 13: Derivatives of the generalized R´enyi dimension curves for the three subperiods [PITH_FULL_IMAGE:figures/full_fig_p034_13.png]
Figure 14
Figure 14. Figure 14: Generalized relative R´enyi dimension curves for the three subperiods (from left to [PITH_FULL_IMAGE:figures/full_fig_p035_14.png]
Figure 15
Figure 15. Figure 15: Values of Dα(p||e)−Dβ(p||e) α−β (top) and Dα(e||p)−Dβ(e||p) α−β (bottom), for the three subperiods (from left to right: phases A, B and C). Here, clear dissimilarities between the three patterns reveal the different multifractal complexity structures for phases A, B a…
Figure 16
Figure 16. Figure 16: Derivatives of the generalized relative R´enyi dimension curves for the three subpe [PITH_FULL_IMAGE:figures/full_fig_p036_16.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Dependence assessment based on generalized complexity: Application to sampling network design

    Alonso FJ, Bueso MC and Angulo JM (2016). Dependence assessment based on generalized complexity: Application to sampling network design. Method- ology and Computing in Applied Probability 18:921–933 38

    work page 2016

  2. [2]

    A new approach to spatial entropy measures

    Altieri L, Cocchi D and Roli G (2017). A new approach to spatial entropy measures. Environmental and Ecological Statistics 25: 95–110

    work page 2017

  3. [3]

    Advances in spatial entropy measures

    Altieri L, Cocchi D and Roli G (2019). Advances in spatial entropy measures. Stochastic Environmental Research and Risk Assessment 33: 1223–1240

    work page 2019

  4. [4]

    Structural complexity in space-time seismic event data

    Angulo JM and Esquivel FJ (2014). Structural complexity in space-time seismic event data. Stochastic Environmental Research and Risk Assessment 28:1187–1206

    work page 2014

  5. [5]

    Multifractal dimensional dependence assessment based on Tsallis mutual information

    Angulo JM and Esquivel FJ (2015). Multifractal dimensional dependence assessment based on Tsallis mutual information. Entropy 17:5382–5401

    work page 2015

  6. [6]

    Spatial entropy

    Batty M (1974). Spatial entropy. Geographical Analysis 6:1–31

    work page 1974

  7. [7]

    Entropy, complexity, and spatial information

    Batty M, Morphet R, Masucci P and Stanilov K (2014). Entropy, complexity, and spatial information. Journal of Geographical Systems 16:363–385

    work page 2014

  8. [8]

    Exponential entropy as a measure of extent of a dis- tribution

    Campbell LL (1966). Exponential entropy as a measure of extent of a dis- tribution. Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und Verwandte Gebiete 5:217–225

    work page 1966

Show all 33 references
  1. [9]

    Features of the extension of a statistical measure of complexity to continuous systems

    Catal´ an RG, Garay J and L´ opez-Ruiz R (2002). Features of the extension of a statistical measure of complexity to continuous systems. Physical Review E 66:011102

    2002

  2. [10]

    Multifractal complexity analysis in space–time based on the generalized dimensions derivatives.Spatial Statistics 22:469–480

    Esquivel FJ, Alonso FJ and Angulo JM (2017). Multifractal complexity analysis in space–time based on the generalized dimensions derivatives.Spatial Statistics 22:469–480

    2017

  3. [11]

    Non-extensive analysis of the seismic activity involving the 2011 volcanic eruption in El Hierro

    Esquivel FJ and Angulo JM (2015). Non-extensive analysis of the seismic activity involving the 2011 volcanic eruption in El Hierro. Spatial Statistics 14(B):208–221

    2015

  4. [12]

    Generalized dimensions of strange attractors

    Grassberger P (1983). Generalized dimensions of strange attractors. Physics Letters A 97:227–230 39

    1983

  5. [13]

    Multifractals: Theory and Applications

    Harte D (2001). Multifractals: Theory and Applications . CRC Press: Boca Raton

    2001

  6. [14]

    Transmission of information.The Bell System Technical Journal 7:535–563

    Hartley R V (1928). Transmission of information.The Bell System Technical Journal 7:535–563

    1928

  7. [15]

    The infinite number of generalized dimensions of fractals and strange attractors

    Hentschel HGE and Procaccia I (1983). The infinite number of generalized dimensions of fractals and strange attractors. Physica D 1983:435–444

    1983

  8. [16]

    Complexity and adaptation

    Huberman BA and Hogg T (1986). Complexity and adaptation. Physica D 22:376–384

    1986

  9. [17]

    A new information theoretical measure of global and local spatial association

    Karlstr¨ om A and Cecatto V (2002). A new information theoretical measure of global and local spatial association. Jaharb Regionalwissensc 22:13–40

    2002

  10. [18]

    On information and sufficiency

    Kullback S and Leibler RA (1951). On information and sufficiency. Annals of Mahematical Statistics 22:79–86

    1951

  11. [19]

    Defining spatial entropy from multivariate distribu- tions of co-occurrences

    Leibovici DG (2009). Defining spatial entropy from multivariate distribu- tions of co-occurrences. In Spatial information theory. COSIT 2009. Lecture Notes in Computer Science , Hornsby KS, Claramunt C, Denis M, Ligozat G (eds), vol. 5756:392–404. Springer: Berlin, Heidelberg

    2009

  12. [20]

    Higher-order co-occurrences for exploratory point pattern analysis and decision tree clustering on spatial data

    Leibovici DG, Bastin L and Jackson M (2011). Higher-order co-occurrences for exploratory point pattern analysis and decision tree clustering on spatial data. Computers & Geosciences 37:382–389

    2011

  13. [21]

    Local and global spatio-temporal entropy indices based on distance ratios and co- occurrences distributions

    Leibovici DG, Claramunt C, Le Guyader C and Brosset D (2014). Local and global spatio-temporal entropy indices based on distance ratios and co- occurrences distributions. International Journal of Geographical Information Science 28:1061–1084

    2014

  14. [22]

    A new contagion index to quantify spatial patterns of landscapes

    Li H and Reynolds JF (1993). A new contagion index to quantify spatial patterns of landscapes. Landscape Ecology 8:155–162

    1993

  15. [23]

    A statistical measure of complexity

    L´ opez-Ruiz R, Mancini HL and Calbet X (1995). A statistical measure of complexity. Physics Letters A 209:321–326 40

    1995

  16. [24]

    A generalized statistical complexity measure: Applications to quantum systems

    L´ opez-Ruiz R, Nagy A, Romera E and Sa˜ nudo J (2009). A generalized statistical complexity measure: Applications to quantum systems. Journal of Mathematical Physics 50:123528

    2009

  17. [25]

    Generalized statistical com- plexity measures: geometrical and analytical properties

    Martin MT, Plastino O and Rosso A (2006). Generalized statistical com- plexity measures: geometrical and analytical properties. Physica A 369:439– 462

    2006

  18. [26]

    Indices of landscape pattern

    O’Neill R V, Krummel JR, Gardner RH, Sugihara G, Jackson B, DeAngelis DL, Milne BT, Turner MG, Zygmunt B, Christensen SW, Dale VH and Gra- ham RL (1988). Indices of landscape pattern. Landscape Ecology 1:153–162

    1988

  19. [27]

    On measures of entropy and information

    R´ enyi A (1961). On measures of entropy and information. InProceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability , Berkeley, CA, USA, June 20–July 30 1960; Neyman, J (ed) vol. 1:547–561. University of California Press: Berkeley

    1961

  20. [28]

    A note on contagion indices for landscape analysis

    Riitters KH, O’Neill R V, Wickham JD and Jones KB (1996). A note on contagion indices for landscape analysis. Landscape Ecology 11:197–202

    1996

  21. [29]

    A generalized relative complexity measure

    Romera E, Sen KD and Nagy ´A (2011). A generalized relative complexity measure. Journal of Statistical Mechanics: Theory and Experiment 09:P09016

    2011

  22. [30]

    A mathematical theory of communication

    Shannon CE (1948). A mathematical theory of communication. The Bell System Technical Journal 27:379–423

    1948

  23. [31]

    Economics and Information Theory

    Theil H (1967). Economics and Information Theory . North Nolland: Am- sterdam

    1967

  24. [32]

    Statistical Decomposition Analysis

    Theil H (1972). Statistical Decomposition Analysis. North Holland: Ams- terdam

    1972

  25. [33]

    Possible generalization of Boltzmann-Gibbs statistics

    Tsallis C (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics 52:479–487 41

    1988

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.