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arxiv: 2511.06999 · v2 · submitted 2025-11-10 · 📊 stat.AP

An Algebraic Approach to Evolutionary Accumulation Models

Jessica Renz , Frederik Witt , Iain G. Johnston This is my paper

Pith reviewed 2026-05-18 00:08 UTC · model grok-4.3

classification 📊 stat.AP
keywords evolutionary accumulation modelssemi-algebraic setspolynomial structurelikelihood maximizationEvAMalgebraic inferenceparameter consistency
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The pith

An algebraic method uses the polynomial structure of evolutionary processes to define a semi-algebraic set of consistent parameters before likelihood maximization.

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

This paper introduces an algebraic approach to evolutionary accumulation models that learn the sequence in which features appear over time. It begins by using the natural polynomial structure of the evolutionary process to construct a semi-algebraic set of parameters that match observed data. Likelihood maximization then proceeds inside that restricted set. A sympathetic reader would care because the method offers a structured alternative to direct numerical optimization and is shown to be compatible with existing statistical models while giving extra details on the feasible region.

Core claim

The central claim is that the evolutionary process possesses a natural underlying polynomial structure. This structure permits construction of a semi-algebraic set of candidate parameters consistent with a given data set. Likelihood maximization can then be carried out within the set, and the resulting solutions align with those obtained from various statistical evolutionary accumulation models while supplying additional information about the model.

What carries the argument

The semi-algebraic set of candidate parameters defined from the polynomial structure of the evolutionary process, which restricts the space before likelihood maximization.

If this is right

  • The algebraic construction yields parameters compatible with solutions from existing statistical evolutionary accumulation models.
  • The method supplies additional information about the feasible parameter region relative to purely optimization-based approaches.
  • Explicit examples confirm that the semi-algebraic sets align with known model outputs in concrete cases.

Where Pith is reading between the lines

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

  • The pre-filtering step could lower computational effort when searching high-dimensional parameter spaces for large evolutionary datasets.
  • Characterizing the full consistent set rather than a single optimum might improve uncertainty estimates in evolutionary inference.
  • The algebraic framing could be tested for extension to other sequential accumulation processes outside biology.

Load-bearing premise

The evolutionary process possesses a natural underlying polynomial structure that permits construction of a semi-algebraic set of parameters consistent with observed data.

What would settle it

A dataset for which the semi-algebraic set is empty or excludes the maximum-likelihood point found by standard optimization would falsify the compatibility claim.

Figures

Figures reproduced from arXiv: 2511.06999 by Frederik Witt, Iain G. Johnston, Jessica Renz.

Figure 1
Figure 1. Figure 1: HyperALG workflow. (1) The structure of the underlying evolutionary process can be represented by trajectories on a hypercube. Here we consider three features. An evolutionary trajectory always starts in 000 and moves along the edges towards 111. The aim is to determine the transition probabilities for the different edges. (2) A dataset that contains information about the presence or absence of the conside… view at source ↗
Figure 2
Figure 2. Figure 2: Solutions in the unit cube. Reported solutions of the different approaches, projected to three dimensions representing the parameters a000;100, a001;101 and a010;110. All three statistical models (HyperLAU, HyperTraPS and HyperHMM) find a solution (illustrated as points) on the same component reported by HyperALG (illustrated as lines). Feeding the input data D to HyperLAU [42], we obtain the solution set … view at source ↗
read the original abstract

We present an algebraic approach to evolutionary accumulation modelling (EvAM). EvAM is concerned with learning and predicting the order in which evolutionary features accumulate over time. Our approach is complementary to the more common optimisation-based inference methods used in this field. Namely, we first use the natural underlying polynomial structure of the evolutionary process to define a semi-algebraic set of candidate parameters consistent with a given data set before maximising the likelihood function. We consider explicit examples and show that this approach is compatible with the solutions given by various statistical evolutionary accumulation models. Furthermore, we discuss the additional information of our algebraic model relative to these models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents an algebraic approach to evolutionary accumulation models (EvAM) that first exploits a claimed natural underlying polynomial structure of the evolutionary process to define a semi-algebraic set of candidate parameters consistent with observed data, then performs likelihood maximization within that set. Compatibility with existing statistical EvAM models is illustrated through explicit examples, and the algebraic perspective is said to supply additional information relative to optimization-based methods.

Significance. If the polynomial-to-semi-algebraic construction is shown to be general and faithful to the underlying dynamics, the method could usefully constrain the feasible parameter region before numerical optimization, offering a complementary tool for EvAM inference with potential gains in interpretability. The examples suggest compatibility is achievable in selected cases, but the significance remains provisional pending a general derivation and quantitative validation.

major comments (2)
  1. [Abstract] Abstract: the central step of defining a semi-algebraic set of candidate parameters via the 'natural underlying polynomial structure' is asserted without a general derivation or explicit construction from the evolutionary dynamics; this mapping is load-bearing for the claim that the set encodes consistency constraints implied by the data before likelihood maximization.
  2. [Examples] Examples and results: compatibility with statistical EvAM models is demonstrated only on selected explicit examples, with no accompanying derivation details, error analysis, or quantitative validation metrics (e.g., parameter recovery error or comparison of maximized likelihood values); this leaves open whether the algebraic set excludes valid parameters or is redundant for arbitrary data.
minor comments (2)
  1. Notation for the polynomials and the resulting semi-algebraic sets should be introduced with explicit equations or definitions to improve readability.
  2. The manuscript would benefit from a brief discussion of computational considerations when constructing the semi-algebraic set for models with more features.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment below and outline the revisions we will make to strengthen the presentation of the algebraic approach.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central step of defining a semi-algebraic set of candidate parameters via the 'natural underlying polynomial structure' is asserted without a general derivation or explicit construction from the evolutionary dynamics; this mapping is load-bearing for the claim that the set encodes consistency constraints implied by the data before likelihood maximization.

    Authors: We acknowledge that the current manuscript introduces the semi-algebraic set via the polynomial structure without a fully general derivation. The approach is motivated by the structure of the evolutionary process, but we agree that an explicit general construction is needed to support the central claim. In the revised version, we will add a dedicated section deriving the semi-algebraic parameter set directly from the underlying polynomial relations in the evolutionary dynamics, showing how consistency constraints with observed data arise before likelihood maximization. revision: yes

  2. Referee: [Examples] Examples and results: compatibility with statistical EvAM models is demonstrated only on selected explicit examples, with no accompanying derivation details, error analysis, or quantitative validation metrics (e.g., parameter recovery error or comparison of maximized likelihood values); this leaves open whether the algebraic set excludes valid parameters or is redundant for arbitrary data.

    Authors: The examples were selected to illustrate compatibility in concrete cases, but we agree that this leaves the generality and fidelity open to question. We will expand the results section with additional examples, explicit derivation details for each case, and quantitative validation including parameter recovery errors and direct comparisons of maximized likelihood values between the algebraic and statistical approaches. This will clarify that the semi-algebraic set neither excludes valid parameters nor is redundant. revision: yes

Circularity Check

0 steps flagged

Algebraic semi-algebraic set construction presented as independent preprocessing step prior to likelihood maximization with no reduction to input data by construction.

full rationale

The paper asserts the existence of a 'natural underlying polynomial structure' of the evolutionary process to define a semi-algebraic set of candidate parameters consistent with observed data, then maximizes the likelihood within that set. It reports compatibility with solutions from statistical EvAM models on explicit examples but provides no equations or derivations showing that the semi-algebraic set is tautologically equivalent to the data constraints already implicit in the likelihood function or that the subsequent maximization is forced by the algebraic step. No self-citations, fitted inputs renamed as predictions, or definitional loops are indicated in the abstract or claims. The derivation chain therefore remains self-contained against external benchmarks, with the algebraic step serving as a complementary filter rather than a re-expression of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that evolutionary accumulation admits a natural polynomial representation whose semi-algebraic consistency sets can be constructed from data; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The evolutionary process possesses a natural underlying polynomial structure.
    Invoked to justify construction of the semi-algebraic set of candidate parameters.

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Works this paper leans on

54 extracted references · 54 canonical work pages · 1 internal anchor

  1. [1]

    Olav N. L. Aga, Morten Brun, Kazeem A. Dauda, Ramon Diaz-Uriarte, Konstantinos Giannakis, and Iain G. Johnston. HyperTraPS-CT: Inference and prediction for accumulation pathways with flexible data and model structures.PLOS Computational Biology, 20(9):e1012393, 2024. doi: 10.1371/journal.pcbi.1012393

    work page doi:10.1371/journal.pcbi.1012393 2024

  2. [2]

    Johnston

    OlavN.L.Aga, SabrinaJ.Moyo, JoelManyahi, UpendoKibwana, IrenH.Löhr, NinaLangeland, BjørnBlomberg, and Iain G. Johnston. A natural history of AMR in Klebsiella pneumoniae: Global diversity, predictors, and predictions of evolutionary pathways, 2025. bioRxiv. doi: 10.1101/2025.09.20.677523

    work page doi:10.1101/2025.09.20.677523 2025

  3. [3]

    PMCE: efficient inference of expressive models of cancer evolution with high prognostic power.Bioinformatics, 38(3):754–762, 2022

    Fabrizio Angaroni, Kevin Chen, Chiara Damiani, Giulio Caravagna, Alex Graudenzi, and Daniele Ramazzotti. PMCE: efficient inference of expressive models of cancer evolution with high prognostic power.Bioinformatics, 38(3):754–762, 2022. doi: 10.1093/bioinformatics/btab717

    work page doi:10.1093/bioinformatics/btab717 2022

  4. [4]

    Bates, Jonathan D

    Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler. Bertini: Software for numerical algebraic geometry. Available at bertini.nd.edu with permanent doi: dx.doi.org/10.7274/R0H41PB5

    work page doi:10.7274/r0h41pb5

  5. [5]

    Beaulieu and Brian C

    Jeremy M. Beaulieu and Brian C. O’Meara. Detecting Hidden Diversification Shifts in Models of Trait-Dependent Speciation and Extinction.Systematic Biology, 65(4):583–601, 2016. doi: 10.1093/sysbio/syw022

    work page doi:10.1093/sysbio/syw022 2016

  6. [6]

    Beaulieu, Brian C

    Jeremy M. Beaulieu, Brian C. O’Meara, and Michael J. Donoghue. Identifying Hidden Rate Changes in the Evolution of a Binary Morphological Character: The Evolution of Plant Habit in Campanulid Angiosperms. Systematic Biology, 62(5):725–737, 2013. doi: 10.1093/sysbio/syt034

    work page doi:10.1093/sysbio/syt034 2013

  7. [7]

    Beerenwinkel and S

    N. Beerenwinkel and S. Sullivant. Markov models for accumulating mutations.Biometrika, 96(3):645–661, 2009. doi: 10.1093/biomet/asp023

    work page doi:10.1093/biomet/asp023 2009

  8. [8]

    Evolution on distributive lattices.Journal of Theoretical Biology, 242(2):409–420, 2006

    Niko Beerenwinkel, Nicholas Eriksson, and Bernd Sturmfels. Evolution on distributive lattices.Journal of Theoretical Biology, 242(2):409–420, 2006. doi: 10.1016/j.jtbi.2006.03.013

    work page doi:10.1016/j.jtbi.2006.03.013 2006

  9. [9]

    Conjunctive Bayesian networks.Bernoulli, 13(4): 893–909, 2007

    Niko Beerenwinkel, Nicholas Eriksson, and Bernd Sturmfels. Conjunctive Bayesian networks.Bernoulli, 13(4): 893–909, 2007. doi: 10.3150/07-BEJ6133

    work page doi:10.3150/07-bej6133 2007

  10. [10]

    Schwarz, Moritz Gerstung, and Florian Markowetz

    Niko Beerenwinkel, Roland F. Schwarz, Moritz Gerstung, and Florian Markowetz. Cancer Evolution: Mathemat- ical Models and Computational Inference.Systematic Biology, 64(1):e1–e25, 2015. doi: 10.1093/sysbio/syu081

    work page doi:10.1093/sysbio/syu081 2015

  11. [11]

    Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer

    Jeff Bezanson, Alan Edelman, Stefan Karpinski, and Viral B Shah. Julia: A fresh approach to numerical com- puting.SIAM Review, 59(1):65–98, 2017. doi: 10.1137/141000671

    work page doi:10.1137/141000671 2017

  12. [12]

    Boyko and Jeremy M

    James D. Boyko and Jeremy M. Beaulieu. Generalized hidden Markov models for phylogenetic comparative datasets.Methods in Ecology and Evolution, 12(3):468–478, 2021. doi: 10.1111/2041-210X.13534

    work page doi:10.1111/2041-210x.13534 2021

  13. [13]

    HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia

    Paul Breiding and Sascha Timme. HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia. In International Congress on Mathematical Software, pages 458–465. Springer, 2018

    work page 2018

  14. [14]

    Homotopy continuation method for solving systems of nonlinear and polynomial equations.Communications in Information and Systems, 15(2):119–307, 2015

    Tianran Chen and Tien-Yien Li. Homotopy continuation method for solving systems of nonlinear and polynomial equations.Communications in Information and Systems, 15(2):119–307, 2015. doi: 10.4310/CIS.2015.v15.n2.a1

    work page doi:10.4310/cis.2015.v15.n2.a1 2015

  15. [15]

    13 Graham Everest, Alf Van der Poorten, Igor Shparlinski, and Thomas Ward.Recurrence sequences

    David A. Cox, John Little, and Donal O’Shea.Ideals, Varieties, and Algorithms. Springer International Publish- ing, Cham, fourth edition, 2015. doi: 10.1007/978-3-319-16721-3

    work page doi:10.1007/978-3-319-16721-3 2015

  16. [16]

    Dalgıç, Haoran Wu, F

    Özden O. Dalgıç, Haoran Wu, F. Safa Erenay, Mustafa Y. Sir, Osman Y. Özaltın, Brian A. Crum, and Kalyan S. Pasupathy. Mapping of critical events in disease progression through binary classification: Application to amy- otrophic lateral sclerosis.Journal of Biomedical Informatics, 123:103895, 2021. doi: 10.1016/j.jbi.2021.103895. 14

    work page doi:10.1016/j.jbi.2021.103895 2021

  17. [17]

    Springer, 1 edition, 2025

    Wolfram Decker, Christian Eder, Claus Fieker, Max Horn, and Michael Joswig, editors.The Computer Algebra System OSCAR: Algorithms and Examples, volume 32 ofAlgorithms and Computation in Mathematics. Springer, 1 edition, 2025. doi: 10.1007/978-3-031-62127-7. URLhttps://link.springer.com/book/9783031621260

    work page doi:10.1007/978-3-031-62127-7 2025

  18. [18]

    Johnston

    Ramon Diaz-Uriarte and Iain G. Johnston. A Picture Guide to Cancer Progression and Evolutionary Accumula- tion Models: Systematic Critique, Plausible Interpretations, and Alternative Uses.IEEE Access, 13:62306–62340,

    work page

  19. [19]

    doi: 10.1109/ACCESS.2025.3558392

    work page doi:10.1109/access.2025.3558392 2025

  20. [20]

    Every which way? On predicting tumor evolution using cancer pro- gression models.PLOS Computational Biology, 15(8):e1007246, 2019

    Ramon Diaz-Uriarte and Claudia Vasallo. Every which way? On predicting tumor evolution using cancer pro- gression models.PLOS Computational Biology, 15(8):e1007246, 2019. doi: 10.1371/journal.pcbi.1007246

    work page doi:10.1371/journal.pcbi.1007246 2019

  21. [21]

    Oberwolfach Seminars

    Mathias Drton, Bernd Sturmfels, and Seth Sullivant.Lectures on Algebraic Statistics. Oberwolfach Seminars. Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8905-5

    work page doi:10.1007/978-3-7643-8905-5 2009

  22. [22]

    Notices of the American Mathematical Society , author =

    Timothy Duff and Margaret Regan. Polynomial systems, homotopy continuation and applications.Notices of the American Mathematical Society, 70(1):151 –155, 2023. doi: 10.1090/noti2592

    work page doi:10.1090/noti2592 2023

  23. [23]

    Quantifying cancer progression with conjunctive Bayesian networks.Bioinformatics, 25(21):2809–2815, 2009

    Moritz Gerstung, Michael Baudis, Holger Moch, and Niko Beerenwinkel. Quantifying cancer progression with conjunctive Bayesian networks.Bioinformatics, 25(21):2809–2815, 2009. doi: 10.1093/bioinformatics/btp505

    work page doi:10.1093/bioinformatics/btp505 2009

  24. [24]

    Greenbury, Mauricio Barahona, and Iain G

    Sam F. Greenbury, Mauricio Barahona, and Iain G. Johnston. HyperTraPS: Inferring Probabilistic Patterns of Trait Acquisition in Evolutionary and Disease Progression Pathways.Cell Systems, 10(1):39–51.e10, 2020. doi: 10.1016/j.cels.2019.10.009

    work page doi:10.1016/j.cels.2019.10.009 2020

  25. [25]

    Hauenstein and Frank Sottile

    Jonathan D. Hauenstein and Frank Sottile. Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems.ACM Transactions on Mathematical Software, 38(4):28:1–28:20, 2012. doi: 10.1145/2331130.2331136

    work page doi:10.1145/2331130.2331136 2012

  26. [26]

    Tree inference for single-cell data.Genome Biology, 17 (1):86, 2016

    Katharina Jahn, Jack Kuipers, and Niko Beerenwinkel. Tree inference for single-cell data.Genome Biology, 17 (1):86, 2016. doi: 10.1186/s13059-016-0936-x

    work page doi:10.1186/s13059-016-0936-x 2016

  27. [27]

    A hypercubic Mk model framework for capturing reversibility in disease, cancer, and evolutionary accumulation modelling.Bioinformatics, 41(1):btae737, 2025

    Iain G Johnston and Ramon Diaz-Uriarte. A hypercubic Mk model framework for capturing reversibility in disease, cancer, and evolutionary accumulation modelling.Bioinformatics, 41(1):btae737, 2025. doi: 10.1093/ bioinformatics/btae737

    work page 2025

  28. [28]

    Johnston and Ellen C

    Iain G. Johnston and Ellen C. Røyrvik. Data-Driven Inference Reveals Distinct and Conserved Dynamic Pathways of Tool Use Emergence across Animal Taxa.iScience, 23(6):101245, 2020. doi: 10.1016/j.isci.2020.101245

    work page doi:10.1016/j.isci.2020.101245 2020

  29. [29]

    Johnston and Ben P

    Iain G. Johnston and Ben P. Williams. Evolutionary Inference across Eukaryotes Identifies Specific Pressures Favoring Mitochondrial Gene Retention.Cell Systems, 2(2):101–111, 2016. doi: 10.1016/j.cels.2016.01.013

    work page doi:10.1016/j.cels.2016.01.013 2016

  30. [30]

    Johnston, Till Hoffmann, Sam F

    Iain G. Johnston, Till Hoffmann, Sam F. Greenbury, Ornella Cominetti, Muminatou Jallow, Dominic Kwiatkowski, Mauricio Barahona, Nick S. Jones, and Climent Casals-Pascual. Precision identification of high-risk phenotypes and progression pathways in severe malaria without requiring longitudinal data.npj Digital Medicine, 2(1):1–9, 2019. doi: 10.1038/s41746-...

    work page doi:10.1038/s41746-019-0140-y 2019

  31. [31]

    Strausberg, Ilan R

    Turid Knutsen, Vasuki Gobu, Rodger Knaus, Hesed Padilla-Nash, Meena Augustus, Robert L. Strausberg, Ilan R. Kirsch, Karl Sirotkin, and Thomas Ried. The interactive online sky/m-fish & cgh database and the entrez cancer chromosomes search database: Linkage of chromosomal aberrations with the genome sequence.Genes, Chromosomes and Cancer, 44(1):52–64, 2005....

    work page doi:10.1002/gcc.20224 2005

  32. [32]

    Paul O. Lewis. A Likelihood Approach to Estimating Phylogeny from Discrete Morphological Character Data. Systematic Biology, 50(6):913–925, 2001. doi: 10.1080/106351501753462876

    work page doi:10.1080/106351501753462876 2001

  33. [33]

    Joint inference of exclusivity patterns and recurrent trajec- tories from tumor mutation trees.Nature Communications, 14(1):3676, 2023

    Xiang Ge Luo, Jack Kuipers, and Niko Beerenwinkel. Joint inference of exclusivity patterns and recurrent trajec- tories from tumor mutation trees.Nature Communications, 14(1):3676, 2023. doi: 10.1038/s41467-023-39400-w

    work page doi:10.1038/s41467-023-39400-w 2023

  34. [34]

    Allen, and William F

    Uwe-G Maier, Stefan Zauner, Christian Woehle, Kathrin Bolte, Franziska Hempel, John F. Allen, and William F. Martin. Massively Convergent Evolution for Ribosomal Protein Gene Content in Plastid and Mitochondrial Genomes.Genome Biology and Evolution, 5(12):2318–2329, 2013. doi: 10.1093/gbe/evt181

    work page doi:10.1093/gbe/evt181 2013

  35. [35]

    HyperHMM: efficient inference of evolutionary and progressive dynamics on hypercubic transition graphs.Bioinformatics, 39(1):btac803, 2023

    Marcus T Moen and Iain G Johnston. HyperHMM: efficient inference of evolutionary and progressive dynamics on hypercubic transition graphs.Bioinformatics, 39(1):btac803, 2023. doi: 10.1093/bioinformatics/btac803

    work page doi:10.1093/bioinformatics/btac803 2023

  36. [36]

    Optim: A mathematical optimization package for Julia

    Patrick Kofod Mogensen and Asbjørn Nilsen Riseth. Optim: A mathematical optimization package for Julia. Journal of Open Source Software, 3(24):615, 2018. doi: 10.21105/joss.00615

    work page doi:10.21105/joss.00615 2018

  37. [37]

    Fletcher, Robert A

    Daniel Nichol, Peter Jeavons, Alexander G. Fletcher, Robert A. Bonomo, Philip K. Maini, Jerome L. Paul, Robert A. Gatenby, Alexander R. A. Anderson, and Jacob G. Scott. Steering Evolution with Sequential Therapy to Prevent the Emergence of Bacterial Antibiotic Resistance.PLOS Computational Biology, 11(9):e1004493, 2015. doi: 10.1371/journal.pcbi.1004493. 15

    work page doi:10.1371/journal.pcbi.1004493 2015

  38. [38]

    Nicol, Kevin R

    Phillip B. Nicol, Kevin R. Coombes, Courtney Deaver, Oksana Chkrebtii, Subhadeep Paul, Amanda E. Toland, and Amir Asiaee. Oncogenetic network estimation with disjunctive Bayesian networks.Computational and Systems Oncology, 1(2):e1027, 2021. doi: 10.1002/cso2.1027

    work page doi:10.1002/cso2.1027 2021

  39. [39]

    Inferrig tree causal models of cancer progression with probability raising.PLoS ONE, 9(10):e108358, 2014

    L Olde Loohuis, G Caravagna, A Graudenzi, D Ramazzotti, and G Mauri. Inferrig tree causal models of cancer progression with probability raising.PLoS ONE, 9(10):e108358, 2014. doi: 10.1371/journal.pone.0108358

    work page doi:10.1371/journal.pone.0108358 2014

  40. [40]

    OSCAR – Open Source Computer Algebra Research system, Version 1.5.0, 2025

    OSCAR. OSCAR – Open Source Computer Algebra Research system, Version 1.5.0, 2025. URLhttps://www. oscar-system.org

    work page 2025

  41. [41]

    Mark Pagel. Detecting correlated evolution on phylogenies: a general method for the comparative analysis of discrete characters.Proceedings of the Royal Society B: Biological Sciences, 255(1342):37–45, 1994. doi: 10.1098/rspb.1994.0006

    work page doi:10.1098/rspb.1994.0006 1994

  42. [42]

    Peach, Sam F

    Robert L. Peach, Sam F. Greenbury, Iain G. Johnston, Sophia N. Yaliraki, David J. Lefevre, and Mauricio Bara- hona. Understanding learner behaviour in online courses with Bayesian modelling and time series characterisation. Scientific Reports, 11(1):2823, 2021. doi: 10.1038/s41598-021-81709-3

    work page doi:10.1038/s41598-021-81709-3 2021

  43. [43]

    Flexible inference of evolutionary accumulation dynamics using uncertain observational data

    Jessica Renz, Morten Brun, and Iain G. Johnston. Flexible inference of evolutionary accumulation dynamics using uncertain observational data, 2025. arXiv:2502.05872 [q-bio.PE]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  44. [44]

    Dauda, Olav N

    Jessica Renz, Kazeem A. Dauda, Olav N. L. Aga, Ramon Diaz-Uriarte, Iren H. Löhr, Bjørn Blomberg, and Iain G. Johnston. Evolutionary accumulation modeling in AMR: machine learning to infer and predict evolutionary dynamics of multi-drug resistance.mBio, 0(0):e00488–25, 2025. doi: 10.1128/mbio.00488-25

    work page doi:10.1128/mbio.00488-25 2025

  45. [45]

    Modelling cancer progression using Mutual Hazard Networks.Bioinformatics, 36(1):241–249, 2020

    Rudolf Schill, Stefan Solbrig, Tilo Wettig, and Rainer Spang. Modelling cancer progression using Mutual Hazard Networks.Bioinformatics, 36(1):241–249, 2020. doi: 10.1093/bioinformatics/btz513

    work page doi:10.1093/bioinformatics/btz513 2020

  46. [46]

    Linda Hu, Andreas Lösch, Peter Georg, Simon Pfahler, Stefan Vocht, Stefan Hansch, Tilo Wettig, Lars Grasedyck, and Rainer Spang

    Rudolf Schill, Maren Klever, Kevin Rupp, Y. Linda Hu, Andreas Lösch, Peter Georg, Simon Pfahler, Stefan Vocht, Stefan Hansch, Tilo Wettig, Lars Grasedyck, and Rainer Spang. Reconstructing Disease Histories in Huge Discrete State Spaces.KI - Künstliche Intelligenz, 39:33–43, 2025. doi: 10.1007/s13218-023-00822-9

    work page doi:10.1007/s13218-023-00822-9 2025

  47. [47]

    Schäffer

    Russell Schwartz and Alejandro A. Schäffer. The evolution of tumour phylogenetics: principles and practice. Nature Reviews Genetics, 18(4):213–229, 2017. doi: 10.1038/nrg.2016.170

    work page doi:10.1038/nrg.2016.170 2017

  48. [48]

    Sommese and Charles W

    Andrew J. Sommese and Charles W. Wampler.The Numerical Solution Of Systems Of Polynomials Arising In Engineering And Science. World Scientific Publishing Company, Singapore, 2005. doi: 10.1142/5763

    work page doi:10.1142/5763 2005

  49. [49]

    Richard P. Stanley. Two poset polytopes.Discrete & Computational Geometry, 1(1):9–23, 1986. doi: 10.1007/ BF02187680

    work page 1986

  50. [50]

    Estimating an oncogenetic tree when false negatives and positives are present

    Aniko Szabo and Kenneth Boucher. Estimating an oncogenetic tree when false negatives and positives are present. Mathematical Biosciences, 176(2):219–236, 2002. doi: 10.1016/S0025-5564(02)00086-X

    work page doi:10.1016/s0025-5564(02)00086-x 2002

  51. [51]

    Accumulation of genetic and epigenetic alterations in normal cells and cancer risk.npj Precision Oncology, 3(1):1–8, 2019

    Hideyuki Takeshima and Toshikazu Ushijima. Accumulation of genetic and epigenetic alterations in normal cells and cancer risk.npj Precision Oncology, 3(1):1–8, 2019. doi: 10.1038/s41698-019-0079-0

    work page doi:10.1038/s41698-019-0079-0 2019

  52. [52]

    Hidden Randomness between Fitness Landscapes Limits Reverse Evolution.Physical Review Letters, 106(19):198102, 2011

    Longzhi Tan, Stephen Serene, Hui Xiao Chao, and Jeff Gore. Hidden Randomness between Fitness Landscapes Limits Reverse Evolution.Physical Review Letters, 106(19):198102, 2011. doi: 10.1103/PhysRevLett.106.198102

    work page doi:10.1103/physrevlett.106.198102 2011

  53. [53]

    Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy con- tinuation.ACM Transactions on Mathematical Software, 25(2):251–276, 1999

    Jan Verschelde. Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy con- tinuation.ACM Transactions on Mathematical Software, 25(2):251–276, 1999. doi: 10.1145/317275.317286

    work page doi:10.1145/317275.317286 1999

  54. [54]

    Polynomial homotopy continuation with PHCpack.ACM Communications in Compututer Alge- bra, 44(3/4):217–220, 2011

    Jan Verschelde. Polynomial homotopy continuation with PHCpack.ACM Communications in Compututer Alge- bra, 44(3/4):217–220, 2011. doi: 10.1145/1940475.1940524. 16 A Necessity of considering earlier and later states The following example illustrates what problems can occur when working with closed equations and not taking into account earlier and later stat...

    work page doi:10.1145/1940475.1940524 2011