@article{amaral2007pi,
  ids = {amaral2007pi-1},
  title = {Pivotal {{Bootstrap Methods}} for k-{{Sample Problems}} in {{Directional Statistics}} and {{Shape Analysis}}},
  author = {Amaral, G. J. A. and Dryden, I. L. and Wood, Andrew T. A.},
  year = {2007},
  journal = {Journal of the American Statistical Association},
  volume = {102},
  number = {478},
  eprint = {27639898},
  eprinttype = {jstor},
  pages = {695--707},
  publisher = {[American Statistical Association, Taylor \& Francis, Ltd.]},
  url = {http://www.jstor.org/stable/27639898},
  abstract = {We propose a novel bootstrap hypothesis testing approach for the problem of testing a null hypothesis of a common mean direction, mean polar axis, or mean shape across several populations of real unit vectors (the directional case) or complex unit vectors (the two-dimensional shape case). Multisample testing problems of this type arise frequently in directional statistics and shape analysis (as in other areas of statistics), but to date there has been relatively little discussion of nonparametric bootstrap approaches to this problem. The bootstrap approach described here is based on a statistic that can be expressed as the smallest eigenvalue of a certain positive definite matrix. We prove that this statistic has a limiting chi-squared distribution under the null hypothesis of equality of means across populations. Although we focus mainly on the version of the statistic in which neither isotropy within populations nor constant dispersion structure across populations is assumed, we explain how to modify the statistic so that either or both of these assumptions can be incorporated. Our numerical results indicate that the bootstrap approach proposed here may be expected to perform well in practice.},
}

@misc{bell2023cp,
  title = {{{CppAD}}: {{A Package}} for {{Differentiation}} of {{C}}++ {{Algorithms}}},
  shorttitle = {{{CppAD}}},
  author = {Bell, Bradley},
  year = {2023},
  month = feb,
  url = {https://github.com/coin-or/CppAD},
  abstract = {A C++ Algorithmic Differentiation Package: Home Page},
}

@article{hyvarinen2005es,
  title = {Estimation of {{Non-Normalized Statistical Models}} by {{Score Matching}}},
  author = {Hyv{\"a}rinen, Aapo},
  year = {2005},
  journal = {Journal of Machine Learning Research},
  volume = {6},
  number = {24},
  pages = {695--709},
  url = {https://jmlr.org/papers/v6/hyvarinen05a.html},
}

@article{hyvarinen2007ex,
  title = {Some extensions of score matching},
  author = {Hyv{\"a}rinen, Aapo},
  year = {2007},
  month = feb,
  journal = {Computational Statistics \& Data Analysis},
  volume = {51},
  number = {5},
  pages = {2499--2512},
  doi = {10.1016/j.csda.2006.09.003},
  abstract = {Many probabilistic models are only defined up to a normalization constant. This makes maximum likelihood estimation of the model parameters very difficult. Typically, one then has to resort to Markov Chain Monte Carlo methods, or approximations of the normalization constant. Previously, a method called score matching was proposed for computationally efficient yet (locally) consistent estimation of such models. The basic form of score matching is valid, however, only for models which define a differentiable probability density function over Rn. Therefore, some extensions of the framework are proposed. First, a related method for binary variables is proposed. Second, it is shown how to estimate non-normalized models defined in the non-negative real domain, i.e. R+n. As a further result, it is shown that the score matching estimator can be obtained in closed form for some exponential families.},
}

@misc{liu2019es,
  title = {Estimating {{Density Models}} with {{Truncation Boundaries}} using {{Score Matching}}},
  author = {Liu, Song and Kanamori, Takafumi and Williams, Daniel J.},
  year = {2019},
  doi = {10.48550/arXiv.1910.03834},
  abstract = {Truncated densities are probability density functions defined on truncated domains. They share the same parametric form with their non-truncated counterparts up to a normalizing constant. Since the computation of their normalizing constants is usually infeasible, Maximum Likelihood Estimation cannot be easily applied to estimate truncated density models. Score Matching (SM) is a powerful tool for fitting parameters using only unnormalized models. However, it cannot be directly applied here as boundary conditions used to derive a tractable SM objective are not satisfied by truncated densities. In this paper, we study parameter estimation for truncated probability densities using SM. The estimator minimizes a weighted Fisher divergence. The weight function is simply the shortest distance from a data point to the boundary of the domain. We show this choice of weight function naturally arises from minimizing the Stein discrepancy as well as upperbounding the finite-sample estimation error. The usefulness of our method is demonstrated by numerical experiments and a study on the Chicago crime data set. We also show that the proposed density estimation can correct the outlier-trimming bias caused by aggressive outlier detection methods.},
}

@book{mardia2000di,
  title = {Directional {{Statistics}}},
  author = {Mardia, Kanti V. and Jupp, Peter E.},
  year = {2000},
  series = {Probability and {{Statistics}}},
  publisher = {Wiley},
  address = {Great Britain},
  isbn = {0-471-95333-4},
  keywords = {bib data checked,October2022report}
}

@misc{mardia2016sc,
  type = {{{arXiv}}},
  title = {Score matching estimators for directional distributions},
  author = {Mardia, Kanti V. and Kent, John T. and Laha, Arnab K.},
  year = {2016},
  doi = {10.48550/arXiv.1604.08470},
  abstract = {One of the major problems for maximum likelihood estimation in the well-established directional models is that the normalising constants can be difficult to evaluate. A new general method of "score matching estimation" is presented here on a compact oriented Riemannian manifold. Important applications include von Mises-Fisher, Bingham and joint models on the sphere and related spaces. The estimator is consistent and asymptotically normally distributed under mild regularity conditions. Further, it is easy to compute as a solution of a linear set of equations and requires no knowledge of the normalizing constant. Several examples are given, both analytic and numerical, to demonstrate its good performance.},
}

@inproceedings{mardia2018ne,
  title = {A {{New Estimation Methodology}} for {{Standard Directional Distributions}}},
  booktitle = {2018 21st {{International Conference}} on {{Information Fusion}} ({{FUSION}})},
  author = {Mardia, Kanti},
  year = {2018},
  month = jul,
  pages = {724--729},
  publisher = {IEEE},
  address = {Cambridge, UK},
  doi = {10.23919/ICIF.2018.8455640},
  abstract = {One of the major stumbling blocks in the use of standard directional distributions is difficulty in implementing the well-established maximum likelihood estimation method. The root of the problem is that their normalizing constants are complicated. To circumvent this problem, sometimes a member from a non-exponential family is used and then a relevant member from the exponential family is matched. These alternative distributions have simple moment estimators though the matching can be somewhat arbitrary. Firstly, we resolve the estimation problem by introducing the score matching estimators for directional distributions. Secondly, we resolve the matching problem optimally by introducing the score matching approximation. We give some insight into these proposals through the basic directional distributions, though the results are optimal and universally applicable for distributions on compact manifolds.},
}

@article{martin2019mi,
  title = {The mixed model for the analysis of a repeated-measurement multivariate count data},
  author = {Martin, Ivonne and Uh, Hae-Won and Supali, Taniawati and Mitreva, Makedonka and Houwing-Duistermaat, Jeanine J.},
  year = {2019},
  journal = {Statistics in Medicine},
  volume = {38},
  number = {12},
  pages = {2248--2268},
  doi = {10.1002/sim.8101},
  abstract = {Clustered overdispersed multivariate count data are challenging to model due to the presence of correlation within and between samples. Typically, the first source of correlation needs to be addressed but its quantification is of less interest. Here, we focus on the correlation between time points. In addition, the effects of covariates on the multivariate counts distribution need to be assessed. To fulfill these requirements, a regression model based on the Dirichlet-multinomial distribution for association between covariates and the categorical counts is extended by using random effects to deal with the additional clustering. This model is the Dirichlet-multinomial mixed regression model. Alternatively, a negative binomial regression mixed model can be deployed where the corresponding likelihood is conditioned on the total count. It appears that these two approaches are equivalent when the total count is fixed and independent of the random effects. We consider both subject-specific and categorical-specific random effects. However, the latter has a larger computational burden when the number of categories increases. Our work is motivated by microbiome data sets obtained by sequencing of the amplicon of the bacterial 16S rRNA gene. These data have a compositional structure and are typically overdispersed. The microbiome data set is from an epidemiological study carried out in a helminth-endemic area in Indonesia. The conclusions are as follows: time has no statistically significant effect on microbiome composition, the correlation between subjects is statistically significant, and treatment has a significant effect on the microbiome composition only in infected subjects who remained infected.},
}

@article{scealy2023sc,
  title = {Score matching for compositional distributions},
  author = {Scealy, Janice L. and Wood, Andrew T. A.},
  year = {2023},
  journal = {Journal of the American Statistical Association},
  volume = {118},
  number = {543},
  pages = {1811--1823},
  publisher = {Taylor \& Francis},
  doi = {10.1080/01621459.2021.2016422},
  abstract = {Compositional data are challenging to analyse due to the non-negativity and sum-to-one constraints on the sample space. With real data it is often the case that many of the compositional components are highly right-skewed, with large numbers of zeros. Major limitations of currently available models for compositional data include one or more of the following: insufficient flexibility in terms of distributional shape; difficulty in accommodating zeros in the data in estimation; and lack of computational viability in moderate to high dimensions. In this article we propose a new model, the polynomially-tilted pairwise interaction (PPI) model, for analysing compositional data. Maximum likelihood estimation is difficult for the PPI model. Instead, we propose novel score matching estimators, which entails extending the score matching approach to Riemannian manifolds with boundary. These new estimators are available in closed form and simulation studies show that they perform well in practice. As our main application we analyse real microbiome count data with fixed totals using a multinomial latent variable model with a PPI model for the latent variable distribution. We prove that, under certain conditions, the new score matching estimators are consistent for the parameters in the new multinomial latent variable model.},
}

@article{scealy2024ro,
  title = {Robust score matching for compositional data},
  author = {Scealy, Janice L. and Hingee, Kassel L. and Kent, John T. and Wood, Andrew T. A.},
  year = {2024},
  journal = {Statistics and Computing},
  volume = {34},
  pages = {93},
  doi = {10.1007/s11222-024-10412-w},
  abstract = {The restricted polynomially-tilted pairwise interaction (RPPI) distribution gives a flexible model for compositional data. It is particularly well-suited to situations where some of the marginal distributions of the components of a composition are concentrated near zero, possibly with right skewness. This article develops a method of tractable robust estimation for the model by combining two ideas. The first idea is to use score matching estimation after an additive log-ratio transformation. The resulting estimator is automatically insensitive to zeros in the data compositions. The second idea is to incorporate suitable weights in the estimating equations. The resulting estimator is additionally resistant to outliers. These properties are confirmed in simulation studies where we further also demonstrate that our new outlier-robust estimator is efficient in high concentration settings, even in the case when there is no model contamination. An example is given using microbiome data. A user-friendly R package accompanies the article.},
  langid = {english}
}

@article{windham1995ro,
  title = {Robustifying {{Model Fitting}}},
  author = {Windham, Michael P.},
  year = {1995},
  journal = {Journal of the Royal Statistical Society. Series B (Methodological)},
  volume = {57},
  number = {3},
  eprint = {2346159},
  eprinttype = {jstor},
  pages = {599--609},
  publisher = {[Royal Statistical Society, Wiley]},
  url = {http://www.jstor.org/stable/2346159},
  abstract = {A robust method for fitting a model from a parametric family is fundamental to effective statistical analysis. A procedure is given for robustifying any model fitting process by using weights from the family from which the model is to be chosen. The weighting reduces the influence of information that is not compatible with the model family, while maintaining the basic structure of a familiar model fitting process. The procedure produces a parametric family of model fitting functions. The value of the parameter determines the degree to which the weighting influences the robustified model fit. A mechanism for determining an appropriate value for the parameter is described.},
}

@article{wiria2010do,
  title = {Does treatment of intestinal helminth infections influence malaria?},
  shorttitle = {Does treatment of intestinal helminth infections influence malaria?},
  author = {Wiria, Aprilianto E and Prasetyani, Margaretta A and Hamid, Firdaus and Wammes, Linda J and Lell, Bertrand and Ariawan, Iwan and Uh, Hae Won and Wibowo, Heri and Djuardi, Yenny and Wahyuni, Sitti and Sutanto, Inge and May, Linda and Luty, Adrian JF and Verweij, Jaco J and Sartono, Erliyani and Yazdanbakhsh, Maria and Supali, Taniawati},
  year = {2010},
  journal = {BMC Infectious Diseases},
  volume = {10},
  pages = {77},
  doi = {10.1186/1471-2334-10-77},
  abstract = {Background Given that helminth infections are thought to have strong immunomodulatory activity, the question whether helminth infections might affect responses to malaria antigens needs to be addressed. Different cross-sectional studies using diverse methodologies have reported that helminth infections might either exacerbate or reduce the severity of malaria attacks. The same discrepancies have been reported for parasitemia. Methods/Design To determine the effect of geohelminth infections and their treatment on malaria infection and disease outcome, as well as on immunological parameters, the area of Nangapanda on Flores Island, Indonesia, where malaria and helminth parasites are co-endemic was selected for a longitudinal study. Here a Double-blind randomized trial will be performed, incorporating repeated treatment with albendazole (400 mg) or placebo at three monthly intervals. Household characteristic data, anthropometry, the presence of intestinal helminth and Plasmodium spp infections, and the incidence of malaria episodes are recorded. In vitro cultures of whole blood, stimulated with a number of antigens, mitogens and toll like receptor ligands provide relevant immunological parameters at baseline and following 1 and 2 years of treatment rounds. The primary outcome of the study is the prevalence of Plasmodium falciparum and P. vivax infection. The secondary outcome will be incidence and severity of malaria episodes detected via both passive and active follow-up. The tertiary outcome is the inflammatory cytokine profile in response to parasite antigens. The project also facilitates the transfer of state of the art methodologies and technologies, molecular diagnosis of parasitic diseases, immunology and epidemiology from Europe to Indonesia. Discussion The study will provide data on the effect of helminth infections on malaria. It will also give information on anthelminthic treatment efficacy and effectiveness and could help develop evidence-based policymaking. Trial registration This study was approved by The Ethical Committee of Faculty of Medicine, University of Indonesia, ref:194/PT02.FK/Etik/2006 and has been filed by ethics committee of the Leiden University Medical Center. Clinical trial number:ISRCTN83830814. The study is reported in accordance with the CONSORT guidelines for cluster-randomized studies.},
}

@article{yu2019ge,
  title = {Generalized {{Score Matching}} for {{Non-Negative Data}}},
  author = {Yu, Shiqing and Drton, Mathias and Shojaie, Ali},
  year = {2019},
  journal = {Journal of Machine Learning Research},
  volume = {20},
  number = {76},
  pages = {1--70},
  url = {https://jmlr.org/papers/v20/18-278.html},
  abstract = {A common challenge in estimating parameters of probability density functions is the intractability of the normalizing constant. While in such cases maximum likelihood estimation may be implemented using numerical integration, the approach becomes computationally intensive. The score matching method of Hyv{\"a}rinen (2005) avoids direct calculation of the normalizing constant and yields closed-form estimates for exponential families of continuous distributions over \${\textbackslash}mathbb\{R\}\^{}m\$. Hyv{\"a}rinen (2007) extended the approach to distributions supported on the non-negative orthant, \${\textbackslash}mathbb\{R\}\_+\^{}m\$. In this paper, we give a generalized form of score matching for non-negative data that improves estimation efficiency. As an example, we consider a general class of pairwise interaction models. Addressing an overlooked inexistence problem, we generalize the regularized score matching method of Lin et al. (2016) and improve its theoretical guarantees for non-negative Gaussian graphical models.},
}

@misc{yu2020ge,
  title = {Generalized {{Score Matching}} for {{General Domains}}},
  author = {Yu, Shiqing and Drton, Mathias and Shojaie, Ali},
  year = {2020},
  month = sep,
  doi = {10.48550/arXiv.2009.11428},
  abstract = {Estimation of density functions supported on general domains arises when the data is naturally restricted to a proper subset of the real space. This problem is complicated by typically intractable normalizing constants. Score matching provides a powerful tool for estimating densities with such intractable normalizing constants, but as originally proposed is limited to densities on \${\textbackslash}mathbb\{R\}\^{}m\$ and \${\textbackslash}mathbb\{R\}\_+\^{}m\$. In this paper, we offer a natural generalization of score matching that accommodates densities supported on a very general class of domains. We apply the framework to truncated graphical and pairwise interaction models, and provide theoretical guarantees for the resulting estimators. We also generalize a recently proposed method from bounded to unbounded domains, and empirically demonstrate the advantages of our method.},
}
