@article{Morrill2020,
  title={{A Generalised Signature Method for Time Series}},
  author={Morrill, James and Fermanian, Adeline and Kidger, Patrick and Lyons, Terry},
  journal={arXiv preprint},
  volume={},
  number={},
  pages={},
  year={2020},
  publisher={}
}

@article{Bonnier2019,
  title={{Deep Signature Transforms}},
  author={Bonnier \emph{et al.}, Patric},
  journal={Neural Information Processing Systems},
  volume={},
  number={},
  pages={},
  year={2019},
  publisher={}
}

@article{beaumont2019approximate,
  title={{Approximate Bayesian computation}},
  author={Beaumont, Mark A},
  journal={Annual review of statistics and its application},
  volume={6},
  pages={379--403},
  year={2019},
  publisher={Annual Reviews}
}

@article{Kiraly2019,
abstract = {We present a novel framework for learning with sequential data of any kind, such as multivariate time series, strings, or sequences of graphs. The main result is a "sequentialization" that transforms any kernel on a given domain into a kernel for sequences in that domain. This procedure preserves properties such as positive definiteness, the associated kernel feature map is an ordered variant of sample (cross-)moments, and this sequentialized kernel is consistent in the sense that it converges to a kernel for paths if sequences converge to paths (by discretization). Further, classical kernels for sequences arise as special cases of this method. We use dynamic programming and low-rank techniques for tensors to provide efficient algorithms to compute this sequentialized kernel.},
archivePrefix = {arXiv},
arxivId = {1601.08169},
author = {Kiŕaly, Franz J. and Oberhauser, Harald},
eprint = {1601.08169},
file = {:home/user/Downloads/Kernels-for-Sequentially-Ordered-Data.pdf:pdf},
issn = {15337928},
journal = {Journal of Machine Learning Research},
keywords = {Kernels,Ordered moments,Sequential data,Signature,Signature kernels},
mendeley-groups = {Rough paths/path signatures},
pages = {1--45},
title = {{Kernels for sequentially ordered data}},
volume = {20},
year = {2019}
}


@article{Cochrane2020,
abstract = {We introduce powerful but simple methodology for identifying anomalous observations against a corpus of ‘normal' observations. All data are observed through a vector-valued feature map. Our approach depends on the choice of corpus and that feature map but is invariant to affine transformations of the map and has no other external dependencies, such as choices of metric; we call it conformance. Applying this method to (signatures) of time series and other types of streamed data we provide an effective methodology of broad applicability for identifying anomalous complex multimodal sequential data. We demonstrate the applicability and effectiveness of our method by evaluating it against multiple data sets. Based on quantifying performance using the receiver operating characteristic (ROC) area under the curve (AUC), our method yields an AUC score of 98.9% for the PenDigits data set; in a subsequent experiment involving marine vessel traffic data our approach yields an AUC score of 89.1%. Based on comparison involving univariate time series from the UEA & UCR time series repository with performance quantified using balanced accuracy and assuming an optimal operating point, our approach outperforms a state-of-the-art shapelet method for 19 out of 28 data sets.},
archivePrefix = {arXiv},
arxivId = {2006.03487},
author = {Cochrane, Thomas and Foster, Peter and Lyons, Terry and Arribas, Imanol Perez},
eprint = {2006.03487},
file = {:home/user/Downloads/2006.03487.pdf:pdf},
issn = {23318422},
journal = {arXiv},
title = {{Anomaly detection on streamed data}},
year = {2020}
}

@article{Schmon2020,
  title={{Generalized Posteriors in Approximate Bayesian Computation}},
  author={Schmon, Sebastian M and Cannon, Patrick W and Knoblauch, Jeremias},
  journal={arXiv preprint arXiv:2011.08644},
  year={2020}
}

@Article{SchmonDeligiannidisDoucet2018a,
	author = {Schmon, S M and Deligiannidis, G and Doucet, A and Pitt, M K},
	title = "{Large-sample asymptotics of the pseudo-marginal method}",
	journal = {Biometrika},
	volume = {108},
	number = {1},
	pages = {37-51},
	year = {2021},
	month = {03},
	issn = {0006-3444},
	doi = {10.1093/biomet/asaa044},
	url = {https://doi.org/10.1093/biomet/asaa044},
	eprint = {https://academic.oup.com/biomet/article-pdf/108/1/37/36440998/asaa044.pdf},
}

@article{andrieu2009pseudo,
  title={{The pseudo-marginal approach for efficient Monte Carlo computations}},
  author={Andrieu, Christophe and Roberts, Gareth O and others},
  journal={The Annals of Statistics},
  volume={37},
  number={2},
  pages={697--725},
  year={2009},
  publisher={Institute of Mathematical Statistics}
}

@article{beaumont2003estimation,
  title={Estimation of population growth or decline in genetically monitored populations},
  author={Beaumont, Mark A},
  journal={Genetics},
  volume={164},
  number={3},
  pages={1139--1160},
  year={2003},
  publisher={Oxford University Press}
}

@article{Fearnhead2012,
abstract = {Many modern statistical applications involve inference for complex stochastic models, where it is easy to simulate from the models, but impossible to calculate likelihoods. Approximate Bayesian computation (ABC) is a method of inference for such models. It replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values, and comparing summary statistics of the simulated data with summary statistics of the observed data. Here we show how to construct appropriate summary statistics for ABC in a semi-automatic manner. We aim for summary statistics which will enable inference about certain parameters of interest to be as accurate as possible. Theoretical results show that optimal summary statistics are the posterior means of the parameters. Although these cannot be calculated analytically, we use an extra stage of simulation to estimate how the posterior means vary as a function of the data; and we then use these estimates of our summary statistics within ABC. Empirical results show that our approach is a robust method for choosing summary statistics that can result in substantially more accurate ABC analyses than the ad hoc choices of summary statistics that have been proposed in the literature. We also demonstrate advantages over two alternative methods of simulation-based inference. {\textcopyright} 2012 Royal Statistical Society.},
archivePrefix = {arXiv},
arxivId = {arXiv:1004.1112v2},
author = {Fearnhead, Paul and Prangle, Dennis},
doi = {10.1111/j.1467-9868.2011.01010.x},
eprint = {arXiv:1004.1112v2},
file = {:home/user/Downloads/1004.1112.pdf:pdf},
issn = {13697412},
journal = {Journal of the Royal Statistical Society. Series B: Statistical Methodology},
keywords = {Indirect inference,Likelihood-free inference,Markov chain Monte Carlo methods,Simulation,Stochastic kinetic networks},
mendeley-groups = {Calibration},
number = {3},
pages = {419--474},
title = {{Constructing summary statistics for approximate Bayesian computation: Semi-automatic approximate Bayesian computation}},
volume = {74},
year = {2012}
}

@article{Park2016,
abstract = {Complicated generative models often result in a situation where computing the likelihood of observed data is intractable, while simulating from the conditional density given a parameter value is relatively easy. Approximate Bayesian Computation (ABC) is a paradigm that enables simulation-based posterior inference in such cases by measuring the similarity between simulated and observed data in terms of a chosen set of summary statistics. However, there is no general rule to construct sufficient summary statistics for complex models. Insufficient summary statistics will “leak” information, which leads to ABC algorithms yielding samples from an incorrect posterior. In this paper, we propose a fully nonparametric ABC paradigm which circumvents the need for manually selecting summary statistics. Our approach, K2-ABC, uses maximum mean discrepancy (MMD) to construct a dissimilarity measure between the observed and simulated data. The embedding of an empirical distribution of the data into a reproducing kernel Hilbert space plays a role of the summary statistic and is sufficient whenever the corresponding kernels are characteristic. Experiments on a simulated scenario and a real-world biological problem illustrate the effectiveness of the proposed algorithm.},
archivePrefix = {arXiv},
arxivId = {1502.02558},
author = {Park, Mijung and Jitkrittum, Wittawat and Sejdinovic, Dino},
eprint = {1502.02558},
file = {:home/user/Downloads/1502.02558.pdf:pdf},
journal = {Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, AISTATS 2016},
mendeley-groups = {Calibration},
pages = {398--407},
title = {{K2-ABC: Approximate Bayesian computation with kernel embeddings}},
volume = {41},
year = {2016}
}

@article{bissiri2016general,
  title={A general framework for updating belief distributions},
  author={Bissiri, Pier Giovanni and Holmes, Chris C and Walker, Stephen G},
  journal={Journal of the Royal Statistical Society. Series B, Statistical methodology},
  volume={78},
  number={5},
  pages={1103},
  year={2016},
  publisher={Wiley-Blackwell}
}

@article{knoblauch2019generalized,
  title={Generalized variational inference: Three arguments for deriving new posteriors},
  author={Knoblauch, Jeremias and Jewson, Jack and Damoulas, Theodoros},
  journal={arXiv preprint arXiv:1904.02063},
  year={2019}
}

@article{tavare1997inferring,
  title={Inferring coalescence times from DNA sequence data},
  author={Tavar{\'e}, Simon and Balding, David J and Griffiths, Robert C and Donnelly, Peter},
  journal={Genetics},
  volume={145},
  number={2},
  pages={505--518},
  year={1997},
  publisher={Genetics Soc America}
}

@article{beaumont2002approximate,
  title={{Approximate Bayesian computation in population genetics}},
  author={Beaumont, Mark A and Zhang, Wenyang and Balding, David J},
  journal={Genetics},
  volume={162},
  number={4},
  pages={2025--2035},
  year={2002},
  publisher={Oxford University Press}
}

@article{pritchard1999population,
  title={Population growth of human {Y} chromosomes: a study of {Y} chromosome microsatellites.},
  author={Pritchard, Jonathan K and Seielstad, Mark T and Perez-Lezaun, Anna and Feldman, Marcus W},
  journal={Molecular biology and evolution},
  volume={16},
  number={12},
  pages={1791--1798},
  year={1999},
  publisher={Oxford University Press}
}

@article{lyons2014rough,
  title={Rough paths, signatures and the modelling of functions on streams},
  author={Lyons, Terry},
  journal={arXiv preprint arXiv:1405.4537},
  year={2014}
}

@ARTICLE{2020SciPy-NMeth,
  author  = {Virtanen, Pauli and Gommers, Ralf and Oliphant, Travis E. and
            Haberland, Matt and Reddy, Tyler and Cournapeau, David and
            Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and
            Bright, Jonathan and {van der Walt}, St{\'e}fan J. and
            Brett, Matthew and Wilson, Joshua and Millman, K. Jarrod and
            Mayorov, Nikolay and Nelson, Andrew R. J. and Jones, Eric and
            Kern, Robert and Larson, Eric and Carey, C J and
            Polat, {\.I}lhan and Feng, Yu and Moore, Eric W. and
            {VanderPlas}, Jake and Laxalde, Denis and Perktold, Josef and
            Cimrman, Robert and Henriksen, Ian and Quintero, E. A. and
            Harris, Charles R. and Archibald, Anne M. and
            Ribeiro, Ant{\^o}nio H. and Pedregosa, Fabian and
            {van Mulbregt}, Paul and {SciPy 1.0 Contributors}},
  title   = {{{SciPy} 1.0: Fundamental Algorithms for Scientific
            Computing in Python}},
  journal = {Nature Methods},
  year    = {2020},
  volume  = {17},
  pages   = {261--272},
  adsurl  = {https://rdcu.be/b08Wh},
  doi     = {10.1038/s41592-019-0686-2},
}

@article{Morrill2019,
abstract = {Optimal feature selection leads to enhanced efficiency and accuracy when developing both supervised and unsupervised machine-learning models. In this work, a new signature-based regression model is proposed to automatically identify a patient's risk of sepsis based on physiological data streams and to make a positive or negative prediction ofsepsis for every time interval since admission to the intensive care unit. The gradient boosting machine algorithm that uses the features at the current time-points and the signature features extracted from the time-series to model the longitudinal effects ofsepsis yields the utility function score of 0.360 (officially ranked 1st, team name: 'Can I get your Signature?') on the full test set. The signature method shows a systematic and competitive approach to model sepsis by learning from health data streams.},
author = {Morrill, James and Kormilitzin, Andrey and Nevado-Holgado, Alejo and Swaminathan, Sumanth and Howison, Sam and Lyons, Terry},
doi = {10.23919/CinC49843.2019.9005805},
file = {:home/user/Downloads/14_CinCFinalPDF.pdf:pdf},
isbn = {9781728169361},
issn = {2325887X},
journal = {Computing in Cardiology},
mendeley-groups = {Rough paths/path signatures},
pages = {2--5},
title = {{The Signature-Based Model for Early Detection of Sepsis from Electronic Health Records in the Intensive Care Unit}},
volume = {2019-Septe},
year = {2019}
}


@book{BrentR.P1973Afmw,
series = {Prentice-Hall series in automatic computation},
publisher = {Prentice-Hall},
isbn = {9780130223357},
year = {1973},
title = {Algorithms for minimization without derivatives},
language = {eng},
address = {Englewood Cliffs ; [Hemel Hempstead},
author = {Brent, R. P},
keywords = {Maxima and minima; Algorithms; Approximation theory},
lccn = {gb 73021436},
}

@article{Wood2010,
abstract = {Chaotic ecological dynamic systems defy conventional statistical analysis. Systems with near-chaotic dynamics are little better. Such systems are almost invariably driven by endogenous dynamic processes plus demographic and environmental process noise, and are only observable with error. Their sensitivity to history means that minute changes in the driving noise realization, or the system parameters, will cause drastic changes in the system trajectory. This sensitivity is inherited and amplified by the joint probability density of the observable data and the process noise, rendering it useless as the basis for obtaining measures of statistical fit. Because the joint density is the basis for the fit measures used by all conventional statistical methods, this is a major theoretical shortcoming. The inability to make well-founded statistical inferences about biological dynamic models in the chaotic and near-chaotic regimes, other than on an ad hoc basis, leaves dynamic theory without the methods of quantitative validation that are essential tools in the rest of biological science. Here I show that this impasse can be resolved in a simple and general manner, using a method that requires only the ability to simulate the observed data on a system from the dynamic model about which inferences are required. The raw data series are reduced to phase-insensitive summary statistics, quantifying local dynamic structure and the distribution of observations. Simulation is used to obtain the mean and the covariance matrix of the statistics, given model parameters, allowing the construction of a synthetic likelihood that assesses model fit. This likelihood can be explored using a straightforward Markov chain Monte Carlo sampler, but one further post-processing step returns pure likelihood-based inference. I apply the method to establish the dynamic nature of the fluctuations in Nicholsons classic blowfly experiments. {\textcopyright} 2010 Macmillan Publishers Limited. All rights reserved.},
author = {Wood, Simon N.},
doi = {10.1038/nature09319},
file = {:home/user/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Wood - 2010 - Statistical inference for noisy nonlinear ecological dynamic systems.pdf:pdf},
issn = {00280836},
journal = {Nature},
mendeley-groups = {Calibration},
number = {7310},
pages = {1102--1104},
title = {{Statistical inference for noisy nonlinear ecological dynamic systems}},
volume = {466},
year = {2010}
}

@article{Briol2019,
abstract = {While likelihood-based inference and its variants provide a statistically efficient and widely applicable approach to parametric inference, their application to models involving intractable likelihoods poses challenges. In this work, we study a class of minimum distance estimators for intractable generative models, that is, statistical models for which the likelihood is intractable, but simulation is cheap. The distance considered, maximum mean discrepancy (MMD), is defined through the embedding of probability measures into a reproducing kernel Hilbert space. We study the theoretical properties of these estimators, showing that they are consistent, asymptotically normal and robust to model misspecification. A main advantage of these estimators is the flexibility offered by the choice of kernel, which can be used to trade-off statistical efficiency and robustness. On the algorithmic side, we study the geometry induced by MMD on the parameter space and use this to introduce a novel natural gradient descent-like algorithm for efficient implementation of these estimators. We illustrate the relevance of our theoretical results on several classes of models including a discrete-time latent Markov process and two multivariate stochastic differential equation models.},
archivePrefix = {arXiv},
arxivId = {1906.05944},
author = {Briol, Fran{\c{c}}ois Xavier and Barp, Alessandro and Duncan, Andrew B. and Girolami, Mark},
eprint = {1906.05944},
file = {:home/user/Downloads/StatisticalInferenceMMD.pdf:pdf},
journal = {arXiv},
mendeley-groups = {Calibration},
pages = {1--57},
title = {{Statistical inference for generative models with maximum mean discrepancy}},
year = {2019}
}

@article{Nakagome2013,
author = {Shigeki Nakagome and Kenji Fukumizu and Shuhei Mano},
doi = {doi:10.1515/sagmb-2012-0050},
url = {https://doi.org/10.1515/sagmb-2012-0050},
title = {{Kernel approximate Bayesian computation in population genetic inferences}},
journal = {Statistical Applications in Genetics and Molecular Biology},
number = {6},
volume = {12},
year = {2013},
pages = {667--678}
}

@article{Marin2012,
abstract = {Approximate Bayesian Computation (ABC) methods, also known as likelihood-free techniques, have appeared in the past ten years as the most satisfactory approach to intractable likelihood problems, first in genetics then in a broader spectrum of applications. However, these methods suffer to some degree from calibration difficulties that make them rather volatile in their implementation and thus render them suspicious to the users of more traditional Monte Carlo methods. In this survey, we study the various improvements and extensions brought on the original ABC algorithm in recent years. {\textcopyright} 2011 Springer Science+Business Media, LLC.},
archivePrefix = {arXiv},
arxivId = {1101.0955},
author = {Marin, Jean Michel and Pudlo, Pierre and Robert, Christian P. and Ryder, Robin J.},
doi = {10.1007/s11222-011-9288-2},
eprint = {1101.0955},
file = {:home/user/Downloads/1101.0955.pdf:pdf},
issn = {09603174},
journal = {Statistics and Computing},
keywords = {ABC methodology,Bayesian model choice,Bayesian statistics,DIYABC,Likelihood-free methods},
number = {6},
pages = {1167--1180},
title = {{Approximate Bayesian computational methods}},
volume = {22},
year = {2012}
}

@article{Bernton2019,
abstract = {A growing number of generative statistical models do not permit the numerical evaluation of their likelihood functions. Approximate Bayesian computation has become a popular approach to overcome this issue, in which one simulates synthetic data sets given parameters and compares summaries of these data sets with the corresponding observed values. We propose to avoid the use of summaries and the ensuing loss of information by instead using the Wasserstein distance between the empirical distributions of the observed and synthetic data. This generalizes the well-known approach of using order statistics within approximate Bayesian computation to arbitrary dimensions. We describe how recently developed approximations of the Wasserstein distance allow the method to scale to realistic data sizes, and we propose a new distance based on the Hilbert space filling curve. We provide a theoretical study of the method proposed, describing consistency as the threshold goes to 0 while the observations are kept fixed, and concentration properties as the number of observations grows. Various extensions to time series data are discussed. The approach is illustrated on various examples, including univariate and multivariate g-and-k distributions, a toggle switch model from systems biology, a queuing model and a L{\'{e}}vy-driven stochastic volatility model.},
archivePrefix = {arXiv},
arxivId = {1905.03747},
author = {Bernton, Espen and Jacob, Pierre E. and Gerber, Mathieu and Robert, Christian P.},
doi = {10.1111/rssb.12312},
eprint = {1905.03747},
file = {:home/user/Downloads/1905.03747.pdf:pdf},
issn = {14679868},
journal = {Journal of the Royal Statistical Society. Series B: Statistical Methodology},
keywords = {Approximate Bayesian computation,Generative models,Likelihood-free inference,Optimal transport,Wasserstein distance},
mendeley-groups = {Calibration/ABC},
number = {2},
pages = {235--269},
title = {{Approximate Bayesian computation with the Wasserstein distance}},
volume = {81},
year = {2019}
}

@article{arribas2018derivatives,
      title={Derivatives pricing using signature payoffs}, 
      author={Imanol Perez Arribas},
      year={2018},
      eprint={1809.09466},
      archivePrefix={arXiv},
      primaryClass={q-fin.CP}
}

@article{KPCA,
  title = {Nonlinear Component Analysis as a Kernel Eigenvalue Problem},
  author = {Sch{\"o}lkopf, B. and Smola, AJ. and M{\"u}ller, K-R.},
  journal = {Neural Computation},
  volume = {10},
  number = {5},
  pages = {1299-1319},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  month = jul,
  year = {1998},
  month_numeric = {7}
}

@article{Christensen2015,
abstract = {Atrial fibrillation (AF) is the most common abnormal heart rhythm and the single biggest cause of stroke. Ablation, destroying regions of the atria, is applied largely empirically and can be curative but with a disappointing clinical success rate. We design a simple model of activation wave front propagation on an anisotropic structure mimicking the branching network of heart muscle cells. This integration of phenomenological dynamics and pertinent structure shows how AF emerges spontaneously when the transverse cell-to-cell coupling decreases, as occurs with age, beyond a threshold value. We identify critical regions responsible for the initiation and maintenance of AF, the ablation of which terminates AF. The simplicity of the model allows us to calculate analytically the risk of arrhythmia and express the threshold value of transversal cell-to-cell coupling as a function of the model parameters. This threshold value decreases with increasing refractory period by reducing the number of critical regions which can initiate and sustain microreentrant circuits. These biologically testable predictions might inform ablation therapies and arrhythmic risk assessment.},
author = {Christensen, Kim and Manani, Kishan A. and Peters, Nicholas S.},
doi = {10.1103/PhysRevLett.114.028104},
file = {:home/user/Downloads/PhysRevLett.114.028104.pdf:pdf},
issn = {10797114},
journal = {Physical Review Letters},
number = {2},
pages = {1--6},
pmid = {25635565},
title = {{Simple model for identifying critical regions in atrial fibrillation}},
volume = {114},
year = {2015}
}

@article{Geanakoplos2012,
Author = {Geanakoplos, John and Axtell, Robert and Farmer, J. Doyne and Howitt, Peter and Conlee, Benjamin and Goldstein, Jonathan and Hendrey, Matthew and Palmer, Nathan M. and Yang, Chun-Yi},
Title = {Getting at Systemic Risk via an Agent-Based Model of the Housing Market},
Journal = {American Economic Review},
Volume = {102},
Number = {3},
Year = {2012},
Month = {May},
Pages = {53-58},
DOI = {10.1257/aer.102.3.53},
URL = {https://www.aeaweb.org/articles?id=10.1257/aer.102.3.53}}

@article{Alsing_2018,
   title={Massive optimal data compression and density estimation for scalable, likelihood-free inference in cosmology},
   volume={477},
   ISSN={1365-2966},
   url={http://dx.doi.org/10.1093/mnras/sty819},
   DOI={10.1093/mnras/sty819},
   number={3},
   journal={Monthly Notices of the Royal Astronomical Society},
   publisher={Oxford University Press (OUP)},
   author={Alsing, Justin and Wandelt, Benjamin and Feeney, Stephen},
   year={2018},
   month={Mar},
   pages={2874–2885}
}

@misc{kidger2021signatory,
      title={Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU}, 
      author={Patrick Kidger and Terry Lyons},
      year={2021},
      eprint={2001.00706},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@article{Xie2018,
abstract = {Online handwritten Chinese text recognition (OHCTR) is a challenging problem as it involves a large-scale character set, ambiguous segmentation, and variable-length input sequences. In this paper, we exploit the outstanding capability of path signature to translate online pen-tip trajectories into informative signature feature maps, successfully capturing the analytic and geometric properties of pen strokes with strong local invariance and robustness. A multi-spatial-context fully convolutional recurrent network (MC-FCRN) is proposed to exploit the multiple spatial contexts from the signature feature maps and generate a prediction sequence while completely avoiding the difficult segmentation problem. Furthermore, an implicit language model is developed to make predictions based on semantic context within a predicting feature sequence, providing a new perspective for incorporating lexicon constraints and prior knowledge about a certain language in the recognition procedure. Experiments on two standard benchmarks, Dataset-CASIA and Dataset-ICDAR, yielded outstanding results, with correct rates of 97.50 and 96.58 percent, respectively, which are significantly better than the best result reported thus far in the literature.},
archivePrefix = {arXiv},
arxivId = {1610.02616},
author = {Xie, Zecheng and Sun, Zenghui and Jin, Lianwen and Ni, Hao and Lyons, Terry},
doi = {10.1109/TPAMI.2017.2732978},
eprint = {1610.02616},
file = {:home/user/Downloads/TPAMI-2016-08-0656-R2.pdf:pdf},
issn = {01628828},
journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
keywords = {Handwritten Chinese text recognition,implicit language model,multiple spatial contexts,path signature,residual recurrent network},
number = {8},
pages = {1903--1917},
pmid = {28767364},
title = {{Learning Spatial-Semantic Context with Fully Convolutional Recurrent Network for Online Handwritten Chinese Text Recognition}},
volume = {40},
year = {2018}
}

@article{Li2017,
abstract = {Hand gesture recognition is gaining more attentions because it's a natural and intuitive mode of human computer interaction. Hand gesture recognition still faces great challenges for the real-world applications due to the gesture variance and individual difference. In this paper, we propose the LPSNet, an end-to-end deep neural network based hand gesture recognition framework with novel log path signature features. We pioneer a robust feature, path signature (PS) and its compressed version, log path signature (LPS) to extract effective feature of hand gestures. Also, we present a new method based on PS and LPS to effectively combine RGB and depth videos. Further, we propose a statistical method, DropFrame, to enlarge the data set and increase its diversity. By testing on a well-known public dataset, Sheffield Kinect Gesture (SKIG), our method achieves classification rate as 96.7{\%} (only use RGB videos) and 98.7{\%} (combining RGB and Depth videos), which is the best result comparing with state-of-the-art methods.},
author = {Li, Chenyang and Zhang, Xin and Jin, Lianwen},
doi = {10.1109/ICCVW.2017.80},
file = {:home/user/Downloads/Li{\_}LPSNet{\_}A{\_}Novel{\_}ICCV{\_}2017{\_}paper.pdf:pdf},
isbn = {9781538610343},
journal = {Proceedings - 2017 IEEE International Conference on Computer Vision Workshops, ICCVW 2017},
pages = {631--639},
title = {{LPSNet: A Novel Log Path Signature Feature Based Hand Gesture Recognition Framework}},
volume = {2018-January},
year = {2017}
}


@article{Moore2019,
abstract = {The path signature is a means of feature generation that can encode nonlinear interactions in data in addition to the usual linear terms. It provides interpretable features and its output is a fixed length vector irrespective of the number of input points or their sample times. In this paper we use the path signature to provide features for identifying people whose diagnosis subsequently converts to Alzheimer's disease. In two separate classification tasks we distinguish converters from 1) healthy individuals, and 2) individuals with mild cognitive impairment. The data used are time-ordered measurements of the whole brain, ventricles and hippocampus from the Alzheimer's Disease Neuroimaging Initiative (ADNI). We find two nonlinear interactions which are predictive in both cases. The first interaction is change of hippocampal volume with time, and the second is a change of hippocampal volume relative to the volume of the whole brain. While hippocampal and brain volume changes are well known in Alzheimer's disease, we demonstrate the power of the path signature in their identification and analysis without manual feature selection. Sequential data is becoming increasingly available as monitoring technology is applied, and the path signature method is shown to be a useful tool in the processing of this data.},
archivePrefix = {arXiv},
arxivId = {1808.05865},
author = {Moore, P. J. and Lyons, T. J. and Gallacher, J.},
doi = {10.1371/journal.pone.0222212},
eprint = {1808.05865},
file = {:home/user/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Moore, Lyons, Gallacher - 2019 - Using path signatures to predict a diagnosis of Alzheimer's disease.pdf:pdf},
isbn = {1111111111},
issn = {19326203},
journal = {PLoS ONE},
mendeley-groups = {Rough paths/path signatures},
number = {9},
pages = {1--16},
title = {{Using path signatures to predict a diagnosis of Alzheimer's disease}},
url = {http://dx.doi.org/10.1371/journal.pone.0222212},
volume = {14},
year = {2019}
}

@article{Hambly_2010,
   title={Uniqueness for the signature of a path of bounded variation and the reduced path group},
   volume={171},
   ISSN={0003-486X},
   url={http://dx.doi.org/10.4007/annals.2010.171.109},
   DOI={10.4007/annals.2010.171.109},
   number={1},
   journal={Annals of Mathematics},
   publisher={Annals of Mathematics, Princeton U},
   author={Hambly, Ben and Lyons, Terry},
   year={2010},
   month={Mar},
   pages={109–167}
}

@inproceedings{Lyons1998DiEE,
  title={Di erential equations driven by rough signals},
  author={Terry Lyons},
  year={1998}
}

@article{PhysRevA.33.1134,
  title = {Independent coordinates for strange attractors from mutual information},
  author = {Fraser, Andrew M. and Swinney, Harry L.},
  journal = {Phys. Rev. A},
  volume = {33},
  issue = {2},
  pages = {1134--1140},
  numpages = {0},
  year = {1986},
  month = {Feb},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevA.33.1134},
  url = {https://link.aps.org/doi/10.1103/PhysRevA.33.1134}
}

@book{HastieTrevor2001Teos,
series = {Springer series in statistics},
publisher = {Springer},
isbn = {9780387952840},
year = {2001},
title = {The elements of statistical learning : data mining, inference, and prediction},
language = {eng},
address = {New York},
author = {Hastie, Trevor and Tibshirani, Robert and Friedman, J. H},
keywords = {Supervised learning (Machine learning)},
lccn = {2001031433},
}

@article {Mengersen1321,
	author = {Mengersen, Kerrie L. and Pudlo, Pierre and Robert, Christian P.},
	title = {Bayesian computation via empirical likelihood},
	volume = {110},
	number = {4},
	pages = {1321--1326},
	year = {2013},
	doi = {10.1073/pnas.1208827110},
	publisher = {National Academy of Sciences},
	abstract = {Approximate Bayesian computation has become an essential tool for the analysis of complex stochastic models when the likelihood function is numerically unavailable. However, the well-established statistical method of empirical likelihood provides another route to such settings that bypasses simulations from the model and the choices of the approximate Bayesian computation parameters (summary statistics, distance, tolerance), while being convergent in the number of observations. Furthermore, bypassing model simulations may lead to significant time savings in complex models, for instance those found in population genetics. The Bayesian computation with empirical likelihood algorithm we develop in this paper also provides an evaluation of its own performance through an associated effective sample size. The method is illustrated using several examples, including estimation of standard distributions, time series, and population genetics models.},
	issn = {0027-8424},
	URL = {https://www.pnas.org/content/110/4/1321},
	eprint = {https://www.pnas.org/content/110/4/1321.full.pdf},
	journal = {Proceedings of the National Academy of Sciences}
}

@article{Bradley_2015,
   title={Nonlinear time-series analysis revisited},
   volume={25},
   ISSN={1089-7682},
   url={http://dx.doi.org/10.1063/1.4917289},
   DOI={10.1063/1.4917289},
   number={9},
   journal={Chaos: An Interdisciplinary Journal of Nonlinear Science},
   publisher={AIP Publishing},
   author={Bradley, Elizabeth and Kantz, Holger},
   year={2015},
   month={Sep},
   pages={097610}
}

@article{Gillespie,
author = {Gillespie, Daniel T.},
title = {Exact stochastic simulation of coupled chemical reactions},
journal = {The Journal of Physical Chemistry},
volume = {81},
number = {25},
pages = {2340-2361},
year = {1977},
doi = {10.1021/j100540a008},

URL = { 
        https://doi.org/10.1021/j100540a008
    
},
eprint = { 
        https://doi.org/10.1021/j100540a008
    
}

}

@article{rahimi2007random,
  title={Random features for large-scale kernel machines},
  author={Rahimi, Ali and Recht, Benjamin},
  journal={Advances in neural information processing systems},
  volume={20},
  year={2007}
}

@article{yang2012nystrom,
  title={Nystr{\"o}m method vs random fourier features: A theoretical and empirical comparison},
  author={Yang, Tianbao and Li, Yu-Feng and Mahdavi, Mehrdad and Jin, Rong and Zhou, Zhi-Hua},
  journal={Advances in neural information processing systems},
  volume={25},
  year={2012}
}

@article{williams2000using,
  title={Using the Nystr{\"o}m method to speed up kernel machines},
  author={Williams, Christopher and Seeger, Matthias},
  journal={Advances in neural information processing systems},
  volume={13},
  year={2000}
}

@phdthesis{Kypraios2007EfficientBI,
    title        = {{Efficient Bayesian inference for partially observed stochastic epidemics and a new class of semi-parametric time series models}},
    author       = {Theo Kypraios},
    month        = {June},
    year         = {2007},
    school       = {Lancaster University},
    type         = {{PhD thesis}}
}

@article{Bonneel2015,
	abstract = {
This article details two approaches to compute barycenters of measures using 1-D Wasserstein distances along radial projections of the input measures. The first method makes use of the Radon transform of the measures, and the second is the solution of a convex optimization problem over the space of measures. We show several properties of these barycenters and explain their relationship. We show numerical approximation schemes based on a discrete Radon transform and on the resolution of a non-convex optimization problem. We explore the respective merits and drawbacks of each approach on applications to two image processing problems: color transfer and texture mixing.
},
	affiliation = {Harvard University; CNRS-LIRIS; GREYC, Université de Caen; CNRS-CEREMADE, Université Paris-Dauphine},
	author = {Bonneel, Nicolas and Rabin, Julien and Peyré, Gabriel and Pfister, Hanspeter},
	copyright = {Springer Science+Business Media New York},
	doi = {10.1007/s10851-014-0506-3},
	journal = {Journal of Mathematical Imaging and Vision},
	keywords = {Optimal transport; Radon transform; Wasserstein distance; Barycenter of measures},
	language = {English},
	number = {1},
	pages = {22-45},
	title = {Sliced and Radon Wasserstein Barycenters of Measures},
	volume = {51},
	year = {2015},
}

@INPROCEEDINGS{9054735,

  author={Nadjahi, Kimia and Bortoli, Valentin De and Durmus, Alain and Badeau, Roland and Şimşekli, Umut},

  booktitle={ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)}, 

  title={Approximate Bayesian Computation with the Sliced-Wasserstein Distance}, 

  year={2020},

  volume={},

  number={},

  pages={5470-5474},

  doi={10.1109/ICASSP40776.2020.9054735}}


@book{schervish1995theory,
  title={Theory of Statistics},
  author={Schervish, M.J.},
  isbn={9781461242512},
  series={Springer series in statistics},
  url={https://books.google.co.uk/books?id=fvOCoAEACAAJ},
  year={1995},
  publisher={3Island Press}
}

@misc{levin2016learning,
      title={Learning from the past, predicting the statistics for the future, learning an evolving system}, 
      author={Daniel Levin and Terry Lyons and Hao Ni},
      year={2016},
      eprint={1309.0260},
      archivePrefix={arXiv},
      primaryClass={q-fin.ST}
}

@misc{liao2019learning,
      title={Learning stochastic differential equations using RNN with log signature features}, 
      author={Shujian Liao and Terry Lyons and Weixin Yang and Hao Ni},
      year={2019},
      eprint={1908.08286},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@article{FLINT20162593,
title = {{Discretely sampled signals and the rough Hoff process}},
journal = {Stochastic Processes and their Applications},
volume = {126},
number = {9},
pages = {2593-2614},
year = {2016},
issn = {0304-4149},
doi = {https://doi.org/10.1016/j.spa.2016.02.011},
url = {https://www.sciencedirect.com/science/article/pii/S0304414916000429},
author = {Guy Flint and Ben Hambly and Terry Lyons},
keywords = {Rough path theory, Lead–lag path, Hoff process, Wong–Zakai approximations, Itô–Stratonovich correction},
abstract = {We introduce a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead–lag increments. In particular, by sampling a d-dimensional continuous semimartingale X:[0,1]→Rd at a set of times D={ti}, we construct a piecewise linear, axis-directed process XD:[0,1]→R2d comprised of a past and a future component. We call such an object the Hoff process associated with the discrete data {Xt}ti∈D. The Hoff process can be lifted to its natural rough path enhancement and we consider the question of convergence as the sampling frequency increases. We prove that the Itô integral can be recovered from a sequence of random ODEs driven by the components of XD. This is in contrast to the usual Stratonovich integral limit suggested by the classical Wong–Zakai Theorem (Wong and Zakai, 1965). Such random ODEs have a natural interpretation in the context of mathematical finance.}
}

@article{thorpe2017transportation,
  title={A {T}ransportation ${L}^p$ {D}istance for {S}ignal {A}nalysis},
  author={Thorpe, Matthew and Park, Serim and Kolouri, Soheil and Rohde, Gustavo K and Slep{\v{c}}ev, Dejan},
  journal={Journal of mathematical imaging and vision},
  volume={59},
  number={2},
  pages={187--210},
  year={2017},
  publisher={Springer}
}

@article{flamary2021pot,
  author  = {R{\'e}mi Flamary and Nicolas Courty and Alexandre Gramfort and Mokhtar Z. Alaya and Aur{\'e}lie Boisbunon and Stanislas Chambon and Laetitia Chapel and Adrien Corenflos and Kilian Fatras and Nemo Fournier and L{\'e}o Gautheron and Nathalie T.H. Gayraud and Hicham Janati and Alain Rakotomamonjy and Ievgen Redko and Antoine Rolet and Antony Schutz and Vivien Seguy and Danica J. Sutherland and Romain Tavenard and Alexander Tong and Titouan Vayer},
  title   = {POT: Python Optimal Transport},
  journal = {Journal of Machine Learning Research},
  year    = {2021},
  volume  = {22},
  number  = {78},
  pages   = {1-8},
  url     = {http://jmlr.org/papers/v22/20-451.html}
}

@article{Lyons2014,
abstract = {Rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems. It draws on the analysis of LC Young and the geometric algebra of KT Chen. The concepts and the uniform estimates, have widespread application and have simplified proofs of basic questions from the large deviation theory and extended Ito's theory of SDEs; the recent applications contribute to (Graham) automated recognition of Chinese handwriting and (Hairer) formulation of appropriate SPDEs to model randomly evolving interfaces. At the heart of the mathematics is the challenge of describing a smooth but potentially highly oscillatory and vector valued path $x_{t}$ parsimoniously so as to effectively predict the response of a nonlinear system such as $dy_{t}=f(y_{t})dx_{t}$, $y_{0}=a$. The Signature is a homomorphism from the monoid of paths into the grouplike elements of a closed tensor algebra. It provides a graduated summary of the path $x$. Hambly and Lyons have shown that this non-commutative transform is faithful for paths of bounded variation up to appropriate null modifications. Among paths of bounded variation with given Signature there is always a unique shortest representative. These graduated summaries or features of a path are at the heart of the definition of a rough path; locally they remove the need to look at the fine structure of the path. Taylor's theorem explains how any smooth function can, locally, be expressed as a linear combination of certain special functions (monomials based at that point). Coordinate iterated integrals form a more subtle algebra of features that can describe a stream or path in an analogous way; they allow a definition of rough path and a natural linear "basis" for functions on streams that can be used for machine learning.},
archivePrefix = {arXiv},
arxivId = {1405.4537},
author = {Lyons, Terry},
eprint = {1405.4537},
file = {:home/user/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Lyons - 2014 - Rough paths, Signatures and the modelling of functions on streams.pdf:pdf},
keywords = {acknowledges the support of,erc ad-,functional regres-,machine learning,numerical approximation of parabolic,pde,regularity structures,rough paths,shuffle product,sion,tensor algebra,the oxford-man institute,the support provided by},
mendeley-groups = {Rough paths/path signatures},
number = {291244},
title = {{Rough paths, Signatures and the modelling of functions on streams}},
url = {http://arxiv.org/abs/1405.4537},
year = {2014}
}


@article{Chevyrev2016,
abstract = {In these notes, we wish to provide an introduction to the signature method, focusing on its basic theoretical properties and recent numerical applications. The notes are split into two parts. The first part focuses on the definition and fundamental properties of the signature of a path, or the path signature. We have aimed for a minimalistic approach, assuming only familiarity with classical real analysis and integration theory, and supplementing theory with straightforward examples. We have chosen to focus in detail on the principle properties of the signature which we believe are fundamental to understanding its role in applications. We also present an informal discussion on some of its deeper properties and briefly mention the role of the signature in rough paths theory, which we hope could serve as a light introduction to rough paths for the interested reader. The second part of these notes discusses practical applications of the path signature to the area of machine learning. The signature approach represents a non-parametric way for extraction of characteristic features from data. The data are converted into a multi-dimensional path by means of various embedding algorithms and then processed for computation of individual terms of the signature which summarise certain information contained in the data. The signature thus transforms raw data into a set of features which are used in machine learning tasks. We will review current progress in applications of signatures to machine learning problems.},
archivePrefix = {arXiv},
arxivId = {1603.03788},
author = {Chevyrev, Ilya and Kormilitzin, Andrey},
eprint = {1603.03788},
file = {:home/user/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Chevyrev, Kormilitzin - 2016 - A Primer on the Signature Method in Machine Learning.pdf:pdf},
mendeley-groups = {Rough paths/path signatures},
title = {{A Primer on the Signature Method in Machine Learning}},
url = {http://arxiv.org/abs/1603.03788},
year = {2016}
}

@article{Gyurk2014,
archivePrefix = {arXiv},
arxivId = {arXiv:1307.7244v2},
author = {Gyurk, Lajos Gergely},
eprint = {arXiv:1307.7244v2},
file = {:home/user/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Gyurk - 2014 - Extracting information from the signature of a financial data stream.pdf:pdf},
mendeley-groups = {Rough paths/path signatures},
pages = {1--22},
title = {{Extracting information from the signature of a financial data stream}},
year = {2014}
}

@article{10.1534/genetics.109.102509,
    author = {Wegmann, Daniel and Leuenberger, Christoph and Excoffier, Laurent},
    title = "{Efficient Approximate Bayesian Computation Coupled With Markov Chain Monte Carlo Without Likelihood}",
    journal = {Genetics},
    volume = {182},
    number = {4},
    pages = {1207-1218},
    year = {2009},
    month = {08},
    abstract = "{Approximate Bayesian computation (ABC) techniques permit inferences in complex demographic models, but are computationally inefficient. A Markov chain Monte Carlo (MCMC) approach has been proposed (Marjoram  et al. 2003), but it suffers from computational problems and poor mixing. We propose several methodological developments to overcome the shortcomings of this MCMC approach and hence realize substantial computational advances over standard ABC. The principal idea is to relax the tolerance within MCMC to permit good mixing, but retain a good approximation to the posterior by a combination of subsampling the output and regression adjustment. We also propose to use a partial least-squares (PLS) transformation to choose informative statistics. The accuracy of our approach is examined in the case of the divergence of two populations with and without migration. In that case, our ABC–MCMC approach needs considerably lower computation time to reach the same accuracy than conventional ABC. We then apply our method to a more complex case with the estimation of divergence times and migration rates between three African populations.}",
    issn = {1943-2631},
    doi = {10.1534/genetics.109.102509},
    url = {https://doi.org/10.1534/genetics.109.102509},
    eprint = {https://academic.oup.com/genetics/article-pdf/182/4/1207/37542192/genetics1207.pdf},
}

@article{drovandi2021comparison,
  title={A Comparison of Likelihood-Free Methods With and Without Summary Statistics},
  author={Drovandi, Christopher and Frazier, David T},
  journal={arXiv preprint arXiv:2103.02407},
  year={2021}
}

@article{Toth2019,
abstract = {We introduce a Bayesian approach to learn from stream-valued data by using Gaussian processes with the recently introduced signature kernel as covariance function. To cope with the computational complexity in time and memory that arises with long streams that evolve in large state spaces, we develop a variational Bayes approach with sparse inducing tensors. We provide an implementation based on GPFlow and benchmark this variational Gaussian process model on supervised classification tasks for time series and text (a stream of words).},
archivePrefix = {arXiv},
arxivId = {1906.08215},
author = {Toth, Csaba and Oberhauser, Harald},
eprint = {1906.08215},
file = {:home/user/Downloads/1906.08215.pdf:pdf},
mendeley-groups = {Rough paths/path signatures},
title = {{Variational Gaussian Processes with Signature Covariances}},
url = {http://arxiv.org/abs/1906.08215},
year = {2019}
}

@article{andrieu2010particle,
  title={{Particle Markov chain Monte Carlo methods}},
  author={Andrieu, Christophe and Doucet, Arnaud and Holenstein, Roman},
  journal={Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
  volume={72},
  number={3},
  pages={269--342},
  year={2010},
  publisher={Wiley Online Library}
}

@incollection{prangle2018handbook,
  author      = "Dennis Prangle",
  title       = "{Summary statistics in approximate Bayesian computation}",
  editor      = "Sisson, Scott A and Fan, Yanan and Beaumont, Mark",
  booktitle   = "Handbook of approximate Bayesian computation",
  publisher   = "FL: CRC",
  year        = 2018,
  pages       = "125-152"
}

@article{schmon2021optimal,
  title={Optimal scaling of random walk Metropolis algorithms using Bayesian large-sample asymptotics},
  author={Schmon, Sebastian M and Gagnon, Philippe},
  journal={Statistics and Computing},
  volume={32},
  number={2},
  pages={1--16},
  year={2022},
  publisher={Springer}
}

@article{10.1093/biomet/asaa044,
    author = {Schmon, S M and Deligiannidis, G and Doucet, A and Pitt, M K},
    title = "{Large-sample asymptotics of the pseudo-marginal method}",
    journal = {Biometrika},
    volume = {108},
    number = {1},
    pages = {37-51},
    year = {2020},
    month = {07},
    abstract = "{The pseudo-marginal algorithm is a variant of the Metropolis–Hastings algorithm which samples asymptotically from a probability distribution when it is only possible to estimate unbiasedly an unnormalized version of its density. Practically, one has to trade off the computational resources used to obtain this estimator against the asymptotic variances of the ergodic averages obtained by the pseudo-marginal algorithm. Recent works on optimizing this trade-off rely on some strong assumptions, which can cast doubts over their practical relevance. In particular, they all assume that the distribution of the difference between the log-density, and its estimate is independent of the parameter value at which it is evaluated. Under regularity conditions we show that as the number of data points tends to infinity, a space-rescaled version of the pseudo-marginal chain converges weakly to another pseudo-marginal chain for which this assumption indeed holds. A study of this limiting chain allows us to provide parameter dimension-dependent guidelines on how to optimally scale a normal random walk proposal, and the number of Monte Carlo samples for the pseudo-marginal method in the large-sample regime. These findings complement and validate currently available results.}",
    issn = {0006-3444},
    doi = {10.1093/biomet/asaa044},
    url = {https://doi.org/10.1093/biomet/asaa044},
    eprint = {https://academic.oup.com/biomet/article-pdf/108/1/37/36645753/asaa044.pdf},
}

@article{
SPABM,
author = {Brian Skyrms  and Robin Pemantle },
title = {A dynamic model of social network formation},
journal = {Proceedings of the National Academy of Sciences},
volume = {97},
number = {16},
pages = {9340-9346},
year = {2000},
doi = {10.1073/pnas.97.16.9340},
URL = {https://www.pnas.org/doi/abs/10.1073/pnas.97.16.9340},
eprint = {https://www.pnas.org/doi/pdf/10.1073/pnas.97.16.9340},
abstract = {We consider a dynamic social network model in which agents play
 repeated games in pairings determined by a stochastically evolving
 social network. Individual agents begin to interact at random, with the
 interactions modeled as games. The game payoffs determine which
 interactions are reinforced, and the network structure emerges as a
 consequence of the dynamics of the agents' learning behavior. We study
 this in a variety of game-theoretic conditions and show that the
 behavior is complex and sometimes dissimilar to behavior in the absence
 of structural dynamics. We argue that modeling network structure as
 dynamic increases realism without rendering the problem of analysis
 intractable.}}

 @book{LyonsT.J2007Dedb,
series = {Lecture notes in mathematics (Springer-Verlag) ; 1908},
abstract = {Each year young mathematicians congregate in Saint Flour, France, and listen to extended lecture courses on new topics in Probability Theory. The goal of these notes, representing a course given by Terry Lyons in 2004, is to provide a straightforward and self supporting but minimalist account of the key results forming the foundation of the theory of rough paths. The proofs are similar to those in the existing literature, but have been refined with the benefit of hindsight. The theory of rough paths aims to create the appropriate mathematical framework for expressing the relationships between evolving systems, by extending classical calculus to the natural models for noisy evolving systems, which are often far from differentiable.},
publisher = {Springer},
booktitle = {Differential equations driven by rough paths : École d'été de probabilités de Saint-Flour XXXIV-2004},
isbn = {9783540712855},
year = {2007},
title = {Differential equations driven by rough paths : École d'été de probabilités de Saint-Flour XXXIV-2004 [electronic resource]},
language = {eng},
address = {Berlin ; New York},
author = {Lyons, T. J and Caruana, Michael and Lévy, Thierry},
keywords = {Differential equations; Stochastic processes; Probabilities; Équations différentielles; Processus stochastiques; Probabilités; Differential equations; Probabilities; Stochastic processes; Differentiaalvergelijkingen; Equações diferenciais; Processos estocásticos; Probabilidade},
}

@article{Chevyrev_2020,
	doi = {10.1109/tpami.2018.2885516},
  
	url = {https://doi.org/10.1109\%2Ftpami.2018.2885516},
  
	year = 2020,
	month = {jan},
  
	publisher = {Institute of Electrical and Electronics Engineers ({IEEE})},
  
	volume = {42},
  
	number = {1},
  
	pages = {192--202},
  
	author = {Ilya Chevyrev and Vidit Nanda and Harald Oberhauser},
  
	title = {{Persistence Paths and Signature Features in Topological Data Analysis}},
  
	journal = {{IEEE} Transactions on Pattern Analysis and Machine Intelligence}
}

@misc{chevyrev2018signature,
      title={Signature moments to characterize laws of stochastic processes}, 
      author={Ilya Chevyrev and Harald Oberhauser},
      year={2018},
      eprint={1810.10971},
      archivePrefix={arXiv},
      primaryClass={math.ST}
}

@book{steinwart2008support,
  title={Support Vector Machines},
  author={Steinwart, I. and Christmann, A.},
  isbn={9780387772424},
  lccn={2008932159},
  series={Information Science and Statistics},
  url={https://books.google.co.uk/books?id=HUnqnrpYt4IC},
  year={2008},
  publisher={Springer New York}
}

@article{JMLR:v20:16-314,
  author  = {Franz J. Kiraly and Harald Oberhauser},
  title   = {Kernels for Sequentially Ordered Data},
  journal = {Journal of Machine Learning Research},
  year    = {2019},
  volume  = {20},
  number  = {31},
  pages   = {1--45},
  url     = {http://jmlr.org/papers/v20/16-314.html}
}


@article{BOEDIHARDJO2016720,
title = {The signature of a rough path: Uniqueness},
journal = {Advances in Mathematics},
volume = {293},
pages = {720-737},
year = {2016},
issn = {0001-8708},
doi = {https://doi.org/10.1016/j.aim.2016.02.011},
url = {https://www.sciencedirect.com/science/article/pii/S0001870816301104},
author = {Horatio Boedihardjo and Xi Geng and Terry Lyons and Danyu Yang},
keywords = {Rough paths, Chen series, Magnus series, Controlled differential equation, Reduced path group, Signature},
abstract = {In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly–Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.}
}

@article{bonnier2021proper,
  title={Proper Scoring Rules, Gradients, Divergences, and Entropies for Paths and Time Series},
  author={Bonnier, Patric and Oberhauser, Harald},
  journal={arXiv preprint arXiv:2111.06314},
  year={2021}
}

@article{cornuet2008inferring,
  title={{Inferring population history with DIY ABC: a user-friendly approach to approximate Bayesian computation}},
  author={Cornuet, Jean-Marie and Santos, Filipe and Beaumont, Mark A and Robert, Christian P and Marin, Jean-Michel and Balding, David J and Guillemaud, Thomas and Estoup, Arnaud},
  journal={Bioinformatics},
  volume={24},
  number={23},
  pages={2713--2719},
  year={2008},
  publisher={Oxford University Press}
}

@article{salvi2020computing,
  title={The Signature Kernel is the solution of a Goursat PDE},
  author={Salvi, Cristopher and Cass, Thomas and Foster, James and Lyons, Terry and Yang, Weixin},
  journal={SIAM Journal on Mathematics of Data Science},
  volume={3},
  number={3},
  pages={873--899},
  year={2021},
  publisher={SIAM}
}

@phdthesis{geng2015a,
  edition = {},
  number = {},
  journal = {},
  pages = {},
  publisher = {Oxford University, UK},
  school = {Oxford University, UK},
  title = {The signature of a rough path: uniqueness},
  volume = {},
  author = {Geng, X},
  editor = {},
  year = {2015},
  series = {}
}


@article{zhang2017random,
  title={Random graph models for dynamic networks},
  author={Zhang, Xiao and Moore, Cristopher and Newman, Mark EJ},
  journal={The European Physical Journal B},
  volume={90},
  number={10},
  pages={1--14},
  year={2017},
  publisher={Springer}
}

@article{chen1958integration,
  title={Integration of paths--A faithful representation of paths by noncommutative formal power series},
  author={Chen, Kuo-Tsai},
  journal={Transactions of the American Mathematical Society},
  volume={89},
  number={2},
  pages={395--407},
  year={1958},
  publisher={JSTOR}
}

@book{lyons2002system,
  title={System control and rough paths},
  author={Lyons, Terry and Qian, Zhongmin and others},
  year={2002},
  publisher={Oxford University Press}
}

@article{BROCK19981235,
title = {Heterogeneous beliefs and routes to chaos in a simple asset pricing model},
journal = {Journal of Economic Dynamics and Control},
volume = {22},
number = {8},
pages = {1235-1274},
year = {1998},
issn = {0165-1889},
doi = {https://doi.org/10.1016/S0165-1889(98)00011-6},
url = {https://www.sciencedirect.com/science/article/pii/S0165188998000116},
author = {William A. Brock and Cars H. Hommes},
keywords = {Asset pricing, Heterogeneous beliefs, Bifurcation},
abstract = {This paper investigates the dynamics in a simple present discounted value asset pricing model with heterogeneous beliefs. Agents choose from a finite set of predictors of future prices of a risky asset and revise their ‘beliefs’ in each period in a boundedly rational way, according to a ‘fitness measure’ such as past realized profits. Price fluctuations are thus driven by an evolutionary dynamics between different expectation schemes (‘rational animal spirits’). Using a mixture of local bifurcation theory and numerical methods, we investigate possible bifurcation routes to complicated asset price dynamics. In particular, we present numerical evidence of strange, chaotic attractors when the intensity of choice to switch prediction strategies is high.}
}

@article{PLATT2020103859,
title = {A comparison of economic agent-based model calibration methods},
journal = {Journal of Economic Dynamics and Control},
volume = {113},
pages = {103859},
year = {2020},
issn = {0165-1889},
doi = {https://doi.org/10.1016/j.jedc.2020.103859},
url = {https://www.sciencedirect.com/science/article/pii/S0165188920300294},
author = {Donovan Platt},
keywords = {Agent-based modelling, Calibration, Simulated minimum distance, Bayesian estimation},
abstract = {Despite significant expansion in recent years, the literature on quantitative and data-driven approaches to economic agent-based model validation and calibration consists primarily of studies that have focused on the introduction of new calibration methods that are neither benchmarked against existing alternatives nor rigorously tested in terms of the quality of the estimates they produce. In response, we compare a number of prominent agent-based model calibration methods, both established and novel, through a series of computational experiments in an attempt to determine the respective strengths and weaknesses of each approach. Overall, we find that a simple, likelihood-based approach to Bayesian estimation consistently outperforms several members of the more popular class of simulated minimum distance methods and results in reasonable parameter estimates in many contexts, with a degradation in performance observed only when considering a large-scale model and attempting to fit a substantial number of its parameters.}
}

@article{erdos1959,
  title={{On random graphs}},
  author={Erd{\H{o}}s, Paul and R{\'e}nyi, Alfr{\'e}d},
  journal={Publicationes Mathematicae},
  volume={6},
  pages={290--297},
  year={1959}
}

@article{erdos1960,
  title={{On the evolution of random graphs}},
  author={Erd{\H{o}}s, Paul and R{\'e}nyi, Alfr{\'e}d},
  journal={Publications of the Mathematical Institute of the Hungarian Academy of Sciences},
  volume={5},
  pages={17--61},
  year={1960}
}

@article{shervashidze2011weisfeiler,
  title={Weisfeiler-lehman graph kernels.},
  author={Shervashidze, Nino and Schweitzer, Pascal and Van Leeuwen, Erik Jan and Mehlhorn, Kurt and Borgwardt, Karsten M},
  journal={Journal of Machine Learning Research},
  volume={12},
  number={9},
  year={2011}
}

@article{miller2018robust,
  title={{Robust Bayesian inference via coarsening}},
  author={Miller, Jeffrey W and Dunson, David B},
  journal={Journal of the American Statistical Association},
  year={2018},
  publisher={Taylor \& Francis}
}

@article{dyer2022black,
  title={Black-box {B}ayesian inference for agent-based models},
  author={Dyer, Joel and Cannon, Patrick and Farmer, J Doyne and Schmon, Sebastian M},
  journal={Journal of Economic Dynamics and Control},
  pages={104827},
  year={2024},
  publisher={Elsevier}
}


@article{nguyen2020approximate,
  title={{Approximate Bayesian computation via the energy statistic}},
  author={Nguyen, Hien Duy and Arbel, Julyan and L{\"u}, Hongliang and Forbes, Florence},
  journal={IEEE Access},
  volume={8},
  pages={131683--131698},
  year={2020},
  publisher={IEEE}
}

@inproceedings{jiang2018approximate,
  title={{Approximate Bayesian computation with Kullback-Leibler divergence as data discrepancy}},
  author={Jiang, Bai},
  booktitle={International conference on artificial intelligence and statistics},
  pages={1711--1721},
  year={2018},
  organization={PMLR}
}