diff --git a/paper/paper.tex b/paper/paper.tex index 11b5cc91..d47d293d 100644 --- a/paper/paper.tex +++ b/paper/paper.tex @@ -366,13 +366,13 @@ \subsection{Queuing methods for multivariate processes} \section{Implementation} \label{sec:implementation} -\texttt{JumpProcesses.jl} is a Julia library for simulating jump --- or point --- processes which is part of Julia's SciML organization. Jumps are handled via callbacks that are checked at the end of each time-step of some time evolution algorithm, \eg an ODE solver from \texttt{OrdinaryDiffEq.jl}, a stochastic differential equation solver from \texttt{StochasticDiffEq.jl}, or the pure-jump process \texttt{SSAStepper} provided by \texttt{JumpProcesses.jl}. In simple terms, callbacks are functions that can be arbitrarily called at each step of the main loop of a time-stepping method. +\texttt{JumpProcesses.jl} is a Julia library for simulating jump --- or point --- processes which is part of Julia's SciML organization. In the Julia ecosystem, there are other libraries that can sample certain TPPs including \path{Hawkes.jl} \footnote{\url{https://github.com/em1234321/Hawkes.jl}}, \path{HawkesProcesses.jl}~\footnote{\url{https://github.com/dm13450/HawkesProcesses.jl}}, \path{NetworkHawkesProcesses.jl} \footnote{\url{https://github.com/cswaney/NetworkHawkesProcesses.jl}}, \path{PointProcessInference.jl} \cite{schauer2020} \footnote{\url{https://github.com/mschauer/PointProcessInference.jl}}, \path{GeoStats.jl} \cite{hoffimann2020} \footnote{\url{https://github.com/JuliaEarth/GeoStats.jl}}, \path{PiecewiseDeterministicMarkovProcesses.jl}~\cite{veltz2015}~\footnote{\url{https://github.com/rveltz/PiecewiseDeterministicMarkovProcesses.jl}}, and \path{PointProcesses.jl}~\cite{dalle2024} \footnote{\url{https://github.com/gdalle/PointProcesses.jl}}. Apart from \texttt{PiecewiseDeterministicMarkovProcesses.jl}, these other libraries can only sample the Poisson and/or the Hawkes processes. \texttt{PointProcesses.jl} also offers a formalized interface that other packages can implement to leverage its TPP modelling functionality. While \texttt{JumpProcesses.jl} can be used to directly simulate TPPs, in its documentation we also show how it can be wrapped to conform to this interface \footnote{\url{https://docs.sciml.ai/JumpProcesses/stable/applications/advanced_point_process}}. Our discussion in Section~\ref{sec:act} identified three exact methods for simulating point processes. In all the cases, we identified two mathematical constructs required for simulation: the intensity rate and the mark distribution. In \texttt{JumpProcesses.jl}, these can be mapped to user defined functions \texttt{rate(u, p, t)} and \texttt{affect!(integrator)}. The former takes the current state of the system, \texttt{u}, user provided parameters, \texttt{p}, and the current time, \texttt{t}, and returns the value of the intensity function at this time. The latter takes the solver \texttt{integrator} object, which stores all solution information, and updates it, including the state \texttt{integrator.u}, for whatever changes should occur when the jump it encodes fires at the time \texttt{integrator.t}. The library provides APIs for defining processes based on the nature of the intensity rate and the intended simulation algorithm. Processes simulated using exact sampling methods can choose between \texttt{ConstantRateJump} and \texttt{VariableRateJump}. While the former expects the rate between jumps to be constant, the latter allows for time-dependent rates. The library also provides the \texttt{MassActionJump} API to define large systems of point processes that can be expressed as mass action type reaction equations. Finally, \texttt{RegularJump} is intended for tau-leaping methods. The \textit{inverse} method as described around Equation~\ref{eqn:inverse} uses root finding to calculate the next jump time. Jumps to be simulated via the \textit{inverse} method must be initialized as a \texttt{VariableRateJump}. \texttt{JumpProcesses.jl} builds a continuous callback following the algorithm in~\cite{salis2005} and passes the problem to an \texttt{OrdinaryDiffEq.jl} integrator, which easily interoperates with \texttt{JumpProcesses.jl} (both libraries are part of the \textit{SciML} organization, and by design built to easily compose). \texttt{JumpProcesses.jl} does not yet support the CHV ODE based approach. -Alternatively, \textit{thinning} methods can be simulated via discrete steps. In the context of the library, any method that uses thinning via a discrete callback is called an \textit{aggregator}. There are twelve different aggregators which we discuss below and are summarized in Table~\ref{tab:aggregators} in the \hyperref[sec:annex]{Annex}. Aggregator's handle sampling the next jump time and type, which is then read via the callback by the user-selected time-stepper. +Alternatively, \textit{thinning} methods can be simulated via discrete steps. In \texttt{JumpProcesses.jl}, simulation approaches that take discrete steps are handled via discrete callbacks that are checked at the end of each time-step of some time evolution algorithm, \eg an ODE solver from \texttt{OrdinaryDiffEq.jl}, a stochastic differential equation solver from \texttt{StochasticDiffEq.jl}, or the pure-jump process \texttt{SSAStepper} provided by \texttt{JumpProcesses.jl}. In simple terms, discrete callbacks involve two functions. Condition functions are checked at each step of the main loop of a time-stepping method to see if the callback should be executed, and if it should, an associated affect function is called. In the context of the library, any method that uses thinning via a discrete callback is called an \textit{aggregator}. There are twelve different aggregators which we discuss below and are summarized in Table~\ref{tab:aggregators} in the \hyperref[sec:annex]{Annex}. We start with constant rate \textit{thinning} aggregators for marked TPPs. Algorithm~\ref{algo:sim-thinning} assumes that there is a single process. In reality, all the implementations first assume a finite multivariate point process with \( M \) interdependent sub-processes. This can be easily conciliated, as we do now, using Definition 6.4.1~\cite{daley2003} which states the equivalence of such process with a point process with a finite space of marks. @@ -526,7 +526,7 @@ \subsection{Benchmarks} \label{subsec:benchmark} We fix the Hawkes parameters at \( \lambda = 0.5 , \alpha = 0.1 , \beta = 5.0 \) ensuring the process does not explode and simulate models in the range from \( 1 \) to \( 95 \) nodes for \( 25 \) units of time. We simulate \( 50 \) trajectories with a limit of ten seconds to complete execution. For this benchmark, we save the state of the system exactly after each jump. -We assess the benchmark in eight different settings. First, we run the \textit{inverse} method, \texttt{Coevolve} and \textit{CHV simple} using the brute force formula of the intensity rate which loops through the whole history of past events --- Equation~\ref{eqn:hawkes-brute}. Second, we simulate the same three methods with the recursive formula --- Equation~\ref{eqn:hawkes-recursive}. Next, we run the benchmark against \textit{CHV full}. All \textit{CHV} specifications are implemented with \texttt{PiecewiseDeterministicMarkovProcesses.jl}~\footnote{\url{https://github.com/rveltz/PiecewiseDeterministicMarkovProcesses.jl}} which is developed by Veltz, the author of the \textit{CHV} algorithm discussed in Subsection~\ref{subsec:sim-inverse}. Finally, we run the benchmark using the Python library Tick\footnote{\url{https://github.com/X-DataInitiative/tick}}. This library implements a version of the thinning method for simulating the Hawkes process and implements a recursive algorithm for computing the intensity rate. +We assess the benchmark in eight different settings. First, we run the \textit{inverse} method, \texttt{Coevolve} and \textit{CHV simple} using the brute force formula of the intensity rate which loops through the whole history of past events --- Equation~\ref{eqn:hawkes-brute}. Second, we simulate the same three methods with the recursive formula --- Equation~\ref{eqn:hawkes-recursive}. Next, we run the benchmark against \textit{CHV full}. All \textit{CHV} specifications are implemented with \texttt{PiecewiseDeterministicMarkovProcesses.jl} which is developed by Veltz, the author of the \textit{CHV} algorithm discussed in Subsection~\ref{subsec:sim-inverse}. Finally, we run the benchmark using the Python library Tick\footnote{\url{https://github.com/X-DataInitiative/tick}}. This library implements a version of the thinning method for simulating the Hawkes process and implements a recursive algorithm for computing the intensity rate. Table~\ref{tab:benchmark-hawkes} shows that the \textit{Inverse} method which relies on root finding is the most inefficient of all methods for any system size. For large system size this method is unable to complete all \( 50 \) simulation runs because it needs to find an ever larger number of roots of an ever larger system of differential equations. diff --git a/paper/references.bib b/paper/references.bib index fbfd8519..ba93fe80 100644 --- a/paper/references.bib +++ b/paper/references.bib @@ -1,8 +1,9 @@ @article{bierkens2019, title = {The {{Zig-Zag}} Process and Super-Efficient Sampling for {{Bayesian}} Analysis of Big Data}, author = {Bierkens, Joris and Fearnhead, Paul and Roberts, Gareth}, - date = {2019-06}, - journaltitle = {The Annals of Statistics}, + year = {2019}, + month = jun, + journal = {The Annals of Statistics}, volume = {47}, number = {3}, publisher = {{Institute of Mathematical Statistics}}, @@ -13,10 +14,10 @@ @article{bierkens2019 @book{bjork2021, title = {Point {{Processes}} and {{Jump Diffusions}}: {{An Introduction}} with {{Finance Applications}}}, shorttitle = {Point {{Processes}} and {{Jump Diffusions}}}, - author = {Björk, Tomas}, - date = {2021}, + author = {Bj{\"o}rk, Tomas}, + year = {2021}, publisher = {{Cambridge University Press}}, - location = {{Cambridge}}, + address = {{Cambridge}}, doi = {10.1017/9781009002127}, isbn = {978-1-316-51867-0} } @@ -25,10 +26,10 @@ @incollection{bloznelis1994 title = {On the {{Central Limit Theorem}} for {{Multiparameter Stochastic Processes}}}, booktitle = {Probability in {{Banach Spaces}}, 9}, author = {Bloznelis, M. and Paulauskas, V.}, - editor = {Hoffmann-Jørgensen, Jørgen and Kuelbs, James and Marcus, Michael B.}, - date = {1994}, - publisher = {{Birkhäuser Boston}}, - location = {{Boston, MA}}, + editor = {{Hoffmann-J{\o}rgensen}, J{\o}rgen and Kuelbs, James and Marcus, Michael B.}, + year = {1994}, + publisher = {{Birkh{\"a}user Boston}}, + address = {{Boston, MA}}, doi = {10.1007/978-1-4612-0253-0_9}, isbn = {978-1-4612-6682-2} } @@ -36,12 +37,12 @@ @incollection{bloznelis1994 @book{daley2003, title = {An {{Introduction}} to the {{Theory}} of {{Point Processes}}: {{Volume I}}: {{Elementary Theory}} and {{Methods}}}, shorttitle = {An {{Introduction}} to the {{Theory}} of {{Point Processes}}}, - author = {Daley, Daryl J. and Vere-Jones, David}, - date = {2003}, + author = {Daley, Daryl J. and {Vere-Jones}, David}, + year = {2003}, series = {Probability and {{Its Applications}}, {{An Introduction}} to the {{Theory}} of {{Point Processes}}}, edition = {2}, publisher = {{Springer-Verlag}}, - location = {{New York}}, + address = {{New York}}, doi = {10.1007/b97277}, isbn = {978-0-387-95541-4} } @@ -49,20 +50,29 @@ @book{daley2003 @book{daley2007, title = {An {{Introduction}} to the {{Theory}} of {{Point Processes}}: {{Volume II}}: {{General Theory}} and {{Structure}}}, shorttitle = {An {{Introduction}} to the {{Theory}} of {{Point Processes}}}, - author = {Daley, Daryl J. and Vere-Jones, David}, - date = {2007-11-12}, + author = {Daley, Daryl J. and {Vere-Jones}, David}, + year = {2007}, + month = nov, edition = {2nd edition}, publisher = {{Springer}}, - location = {{New York}}, - isbn = {978-0-387-21337-8}, - pagetotal = {590} + address = {{New York}}, + isbn = {978-0-387-21337-8} +} + +@misc{dalle2024, + title = {{{PointProcesses}}.Jl}, + author = {Dalle, Guillaume}, + year = {2024}, + month = jan, + doi = {10.5281/zenodo.10477603}, + howpublished = {Zenodo} } @article{donsker1983, title = {Asymptotic Evaluation of Certain {{Markov}} Process Expectations for Large Time. {{IV}}}, author = {Donsker, M. D. and Varadhan, S. R. S.}, - date = {1983}, - journaltitle = {Communications on Pure and Applied Mathematics}, + year = {1983}, + journal = {Communications on Pure and Applied Mathematics}, volume = {36}, number = {2}, issn = {1097-0312}, @@ -71,9 +81,9 @@ @article{donsker1983 @article{erdos1959, title = {On Random Graphs {{I}}}, - author = {Erdős, Paul and Rényi, Alfréd}, - date = {1959}, - journaltitle = {Publicationes mathematicae (Debrecen)}, + author = {Erd{\H o}s, Paul and R{\'e}nyi, Alfr{\'e}d}, + year = {1959}, + journal = {Publicationes mathematicae (Debrecen)}, volume = {6}, issn = {0033-3883}, url = {https://www.renyi.hu/~p_erdos/1959-11.pdf} @@ -81,9 +91,9 @@ @article{erdos1959 @article{erdos1960, title = {On the Evolution of Random Graphs}, - author = {Erdős, P and Rényi, A}, - date = {1960}, - journaltitle = {Publications of the Mathematical Institute of the Hungarian Academy of Sciences}, + author = {Erd{\H o}s, P and R{\'e}nyi, A}, + year = {1960}, + journal = {Publications of the Mathematical Institute of the Hungarian Academy of Sciences}, volume = {5}, issn = {0541-9514} } @@ -91,9 +101,10 @@ @article{erdos1960 @article{farajtabar2017, title = {{{COEVOLVE}}: A Joint Point Process Model for Information Diffusion and Network Evolution}, shorttitle = {{{COEVOLVE}}}, - author = {Farajtabar, Mehrdad and Wang, Yichen and Gomez-Rodriguez, Manuel and Li, Shuang and Zha, Hongyuan and Song, Le}, - date = {2017-01-01}, - journaltitle = {The Journal of Machine Learning Research}, + author = {Farajtabar, Mehrdad and Wang, Yichen and {Gomez-Rodriguez}, Manuel and Li, Shuang and Zha, Hongyuan and Song, Le}, + year = {2017}, + month = jan, + journal = {The Journal of Machine Learning Research}, volume = {18}, number = {1}, issn = {1532-4435}, @@ -104,7 +115,8 @@ @incollection{feller1949 title = {On the {{Theory}} of {{Stochastic Processes}}, with {{Particular Reference}} to {{Applications}}}, booktitle = {Berkeley {{Symposium}} on {{Mathematical Statistics}} and {{Probability}}}, author = {Feller, W.}, - date = {1949-01-01}, + year = {1949}, + month = jan, volume = {1}, publisher = {{University of California Press}}, url = {https://projecteuclid.org/ebooks/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings-of-the-First-Berkeley-Symposium-on-Mathematical-Statistics-and/chapter/On-the-Theory-of-Stochastic-Processes-with-Particular-Reference-to/bsmsp/1166219215} @@ -113,8 +125,9 @@ @incollection{feller1949 @article{gibson2000, title = {Efficient {{Exact Stochastic Simulation}} of {{Chemical Systems}} with {{Many Species}} and {{Many Channels}}}, author = {Gibson, Michael A. and Bruck, Jehoshua}, - date = {2000-03-01}, - journaltitle = {The Journal of Physical Chemistry A}, + year = {2000}, + month = mar, + journal = {The Journal of Physical Chemistry A}, volume = {104}, number = {9}, issn = {1089-5639, 1520-5215}, @@ -124,8 +137,9 @@ @article{gibson2000 @article{gilbert1959, title = {Random Graphs}, author = {Gilbert, E. N.}, - date = {1959-12}, - journaltitle = {The Annals of Mathematical Statistics}, + year = {1959}, + month = dec, + journal = {The Annals of Mathematical Statistics}, volume = {30}, number = {4}, issn = {0003-4851, 2168-8990}, @@ -135,9 +149,9 @@ @article{gilbert1959 @article{gillespie1976, title = {A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions}, author = {Gillespie, Daniel T}, - date = {1976-12-01}, - journaltitle = {Journal of Computational Physics}, - shortjournal = {Journal of Computational Physics}, + year = {1976}, + month = dec, + journal = {Journal of Computational Physics}, volume = {22}, number = {4}, issn = {0021-9991}, @@ -147,8 +161,9 @@ @article{gillespie1976 @article{gillespie1977, title = {Exact Stochastic Simulation of Coupled Chemical Reactions}, author = {Gillespie, Daniel T.}, - date = {1977-12-01}, - journaltitle = {The Journal of Physical Chemistry}, + year = {1977}, + month = dec, + journal = {The Journal of Physical Chemistry}, volume = {81}, number = {25}, issn = {0022-3654}, @@ -158,8 +173,9 @@ @article{gillespie1977 @article{gillespie2001, title = {Approximate Accelerated Stochastic Simulation of Chemically Reacting Systems}, author = {Gillespie, Daniel T.}, - date = {2001-07-22}, - journaltitle = {The Journal of Chemical Physics}, + year = {2001}, + month = jul, + journal = {The Journal of Chemical Physics}, volume = {115}, number = {4}, issn = {0021-9606}, @@ -169,8 +185,9 @@ @article{gillespie2001 @article{gillespie2007, title = {Stochastic {{Simulation}} of {{Chemical Kinetics}}}, author = {Gillespie, Daniel T.}, - date = {2007-05-01}, - journaltitle = {Annual Review of Physical Chemistry}, + year = {2007}, + month = may, + journal = {Annual Review of Physical Chemistry}, volume = {58}, number = {1}, issn = {0066-426X, 1545-1593}, @@ -180,8 +197,9 @@ @article{gillespie2007 @article{granovetter1973, title = {The Strength of Weak Ties}, author = {Granovetter, Mark S.}, - date = {1973-05-01}, - journaltitle = {American Journal of Sociology}, + year = {1973}, + month = may, + journal = {American Journal of Sociology}, volume = {78}, number = {6}, issn = {0002-9602}, @@ -191,20 +209,31 @@ @article{granovetter1973 @article{gupta2018, title = {An {{Overview}} of {{Network-Based}} and -{{Free Approaches}} for {{Stochastic Simulation}} of {{Biochemical Systems}}}, author = {Gupta, Abhishekh and Mendes, Pedro}, - date = {2018-03}, - journaltitle = {Computation}, + year = {2018}, + month = mar, + journal = {Computation}, volume = {6}, number = {1}, issn = {2079-3197}, doi = {10.3390/computation6010009} } +@misc{hoffimann2020, + title = {{{JuliaEarth}}/{{GeoStats}}.Jl: V0.11.7}, + shorttitle = {{{JuliaEarth}}/{{GeoStats}}.Jl}, + author = {Hoffimann, J{\'u}lio and Ekre, Fredrik and Visser, Martijn and Singhvi, Anshul and D'souza, Durand and Siddique, M. A. and Piibeleht, Morten and Kelman, Tony and Vasovi{\'c}, Zlatan}, + year = {2020}, + month = jun, + doi = {10.5281/zenodo.3875233}, + howpublished = {Zenodo} +} + @article{holme2021, title = {Fast and Principled Simulations of the {{SIR}} Model on Temporal Networks}, author = {Holme, Petter}, - date = {2021-02-12}, - journaltitle = {PLOS ONE}, - shortjournal = {PLOS ONE}, + year = {2021}, + month = feb, + journal = {PLOS ONE}, volume = {16}, number = {2}, issn = {1932-6203}, @@ -213,17 +242,17 @@ @article{holme2021 @book{last2017, title = {Lectures on the {{Poisson Process}}}, - author = {Last, Günter and Penrose, Mathew}, - date = {2017-10-26}, + author = {Last, G{\"u}nter and Penrose, Mathew}, + year = {2017}, + month = oct, edition = {1st edition}, - publisher = {{Cambridge University Press}}, - pagetotal = {310} + publisher = {{Cambridge University Press}} } @book{laub2021, title = {The {{Elements}} of {{Hawkes Processes}}}, author = {Laub, Patrick J. and Lee, Young and Taimre, Thomas}, - date = {2021}, + year = {2021}, publisher = {{Springer International Publishing}}, doi = {10.1007/978-3-030-84639-8}, isbn = {978-3-030-84638-1 978-3-030-84639-8} @@ -233,8 +262,9 @@ @article{lee2021 title = {Gibbs Sampler and Coordinate Ascent Variational Inference: {{A}} Set-Theoretical Review}, shorttitle = {Gibbs Sampler and Coordinate Ascent Variational Inference}, author = {Lee, Se Yoon}, - date = {2021-05-10}, - journaltitle = {Communications in Statistics - Theory and Methods}, + year = {2021}, + month = may, + journal = {Communications in Statistics - Theory and Methods}, volume = {51}, number = {2022}, issn = {0361-0926}, @@ -243,10 +273,10 @@ @article{lee2021 @article{lemaire2018, title = {Exact {{Simulation}} of the {{Jump Times}} of a {{Class}} of {{Piecewise Deterministic Markov Processes}}}, - author = {Lemaire, Vincent and Thieullen, Michèle and Thomas, Nicolas}, - date = {2018-06-01}, - journaltitle = {Journal of Scientific Computing}, - shortjournal = {J Sci Comput}, + author = {Lemaire, Vincent and Thieullen, Mich{\`e}le and Thomas, Nicolas}, + year = {2018}, + month = jun, + journal = {Journal of Scientific Computing}, volume = {75}, number = {3}, issn = {1573-7691}, @@ -256,8 +286,8 @@ @article{lemaire2018 @article{lewis1976, title = {Simulation of {{Nonhomogeneous Poisson Processes}} with {{Log Linear Rate Function}}}, author = {Lewis, P. A. W. and Shedler, G. S.}, - date = {1976}, - journaltitle = {Biometrika}, + year = {1976}, + journal = {Biometrika}, volume = {63}, number = {3}, eprint = {2335727}, @@ -269,8 +299,8 @@ @article{lewis1976 @article{lewis1979, title = {Simulation of Nonhomogeneous Poisson Processes by Thinning}, author = {Lewis, P. a. W and Shedler, G. S.}, - date = {1979}, - journaltitle = {Naval Research Logistics Quarterly}, + year = {1979}, + journal = {Naval Research Logistics Quarterly}, volume = {26}, number = {3}, issn = {1931-9193}, @@ -280,10 +310,10 @@ @article{lewis1979 @book{marchetti2017, title = {Simulation {{Algorithms}} for {{Computational Systems Biology}}}, author = {Marchetti, Luca and Priami, Corrado and Thanh, Vo Hong}, - date = {2017}, + year = {2017}, series = {Texts in {{Theoretical Computer Science}}. {{An EATCS Series}}}, publisher = {{Springer International Publishing}}, - location = {{Cham}}, + address = {{Cham}}, doi = {10.1007/978-3-319-63113-4}, isbn = {978-3-319-63111-0 978-3-319-63113-4} } @@ -292,10 +322,10 @@ @incollection{masuda2013a title = {Self-Exciting Point Process Modeling of Conversation Event Sequences}, booktitle = {Temporal {{Networks}}}, author = {Masuda, Naoki and Takaguchi, Taro and Sato, Nobuo and Yano, Kazuo}, - editor = {Holme, Petter and Saramäki, Jari}, - date = {2013}, + editor = {Holme, Petter and Saram{\"a}ki, Jari}, + year = {2013}, publisher = {{Springer Berlin Heidelberg}}, - location = {{Berlin, Heidelberg}}, + address = {{Berlin, Heidelberg}}, doi = {10.1007/978-3-642-36461-7_12}, isbn = {978-3-642-36461-7} } @@ -304,7 +334,8 @@ @book{masuda2016 ids = {masuda2016a}, title = {A Guide to Temporal Networks}, author = {Masuda, Naoki and Lambiotte, Renaud}, - date = {2016-09}, + year = {2016}, + month = sep, publisher = {{World Scientific}}, doi = {10.1142/q0033}, isbn = {978-1-78634-114-3} @@ -313,9 +344,9 @@ @book{masuda2016 @article{mccollum2006, title = {The Sorting Direct Method for Stochastic Simulation of Biochemical Systems with Varying Reaction Execution Behavior}, author = {McCollum, James M. and Peterson, Gregory D. and Cox, Chris D. and Simpson, Michael L. and Samatova, Nagiza F.}, - date = {2006-02-01}, - journaltitle = {Computational Biology and Chemistry}, - shortjournal = {Computational Biology and Chemistry}, + year = {2006}, + month = feb, + journal = {Computational Biology and Chemistry}, volume = {30}, number = {1}, issn = {1476-9271}, @@ -325,20 +356,20 @@ @article{mccollum2006 @book{meiss2017, title = {Differential {{Dynamical Systems}}, {{Revised Edition}}}, author = {Meiss, James}, - date = {2017-01-24}, + year = {2017}, + month = jan, series = {Mathematical {{Modeling}} and {{Computation}}}, publisher = {{Society for Industrial and Applied Mathematics}}, doi = {10.1137/1.9781611974645}, - isbn = {978-1-61197-463-8}, - pagetotal = {400} + isbn = {978-1-61197-463-8} } @article{newman2018, title = {Network Structure from Rich but Noisy Data}, author = {Newman, M. E. J.}, - date = {2018-06}, - journaltitle = {Nature Physics}, - shortjournal = {Nature Phys}, + year = {2018}, + month = jun, + journal = {Nature Physics}, volume = {14}, number = {6}, issn = {1745-2481}, @@ -348,8 +379,9 @@ @article{newman2018 @article{newman2018a, title = {Estimating Network Structure from Unreliable Measurements}, author = {Newman, M. E. J.}, - date = {2018-12-26}, - journaltitle = {Physical Review E}, + year = {2018}, + month = dec, + journal = {Physical Review E}, volume = {98}, number = {6}, doi = {10.1103/PhysRevE.98.062321} @@ -358,8 +390,9 @@ @article{newman2018a @article{ogata1981, title = {On {{Lewis}}' Simulation Method for Point Processes}, author = {Ogata, Y.}, - date = {1981-01}, - journaltitle = {IEEE Transactions on Information Theory}, + year = {1981}, + month = jan, + journal = {IEEE Transactions on Information Theory}, volume = {27}, number = {1}, issn = {1557-9654}, @@ -370,37 +403,40 @@ @book{pervin2014 title = {Foundations of {{General Topology}}}, author = {Pervin, William J.}, editor = {Boas, Ralph P.}, - date = {2014-12-24}, + year = {2014}, + month = dec, publisher = {{Academic Press}}, - isbn = {978-1-4832-1172-5}, - pagetotal = {222} + isbn = {978-1-4832-1172-5} } @article{rackauckas2017, - title = {{{DifferentialEquations}}.Jl – {{A Performant}} and {{Feature-Rich Ecosystem}} for {{Solving Differential Equations}} in {{Julia}}}, + title = {{{DifferentialEquations}}.Jl {\textendash} {{A Performant}} and {{Feature-Rich Ecosystem}} for {{Solving Differential Equations}} in {{Julia}}}, author = {Rackauckas, Christopher and Nie, Qing}, - date = {2017-05-25}, + year = {2017}, + month = may, volume = {5}, number = {1}, issn = {2049-9647}, doi = {10.5334/jors.151} } -@online{rodrigues2021, +@misc{rodrigues2021, title = {A Stochastic Model of Hippocampal Synaptic Plasticity with Geometrical Readout of Enzyme Dynamics}, - author = {Rodrigues, Yuri E. and Tigaret, Cezar M. and Marie, Hélène and O’Donnell, Cian and Veltz, Romain}, - date = {2021-03-30}, - eprinttype = {bioRxiv}, + author = {Rodrigues, Yuri E. and Tigaret, Cezar M. and Marie, H{\'e}l{\`e}ne and O'Donnell, Cian and Veltz, Romain}, + year = {2021}, + month = mar, + publisher = {{bioRxiv}}, doi = {10.1101/2021.03.30.437703}, - pubstate = {preprint} + archiveprefix = {bioRxiv} } @article{saichev2007, title = {Theory of Earthquake Recurrence Times}, shorttitle = {Theory of Earthquake Recurrence Times}, author = {Saichev, Alexander and Sornette, Didier}, - date = {2007-04}, - journaltitle = {Journal of Geophysical Research: Solid Earth}, + year = {2007}, + month = apr, + journal = {Journal of Geophysical Research: Solid Earth}, volume = {112}, number = {B4}, issn = {01480227}, @@ -410,20 +446,31 @@ @article{saichev2007 @article{salis2005, title = {Accurate Hybrid Stochastic Simulation of a System of Coupled Chemical or Biochemical Reactions}, author = {Salis, Howard and Kaznessis, Yiannis}, - date = {2005-02}, - journaltitle = {The Journal of Chemical Physics}, - shortjournal = {J. Chem. 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