Many biological objects possess approximate bilateral symmetry about a midplane. This paper uses landmark-based methods to measure departures from bilateral symmetry. The landmarks for each object come in pairs either side of a midplane or as solos on or near the midplane. After registration, a vector of unsigned elementary features is constructed to describe the asymmetry of each object. Two approaches are proposed to compare the level of asymmetry between two groups of objects. In the first approach the elementary features are combined into a scalar composite asymmetry score for each object; then standard univariate tests are used on the composite asymmetry score to assess asymmetry. In the second approach a univariate test statistic is constructed for each elementary feature; then the maximum of these statistics is used for a union-intersection test. The methodology is illustrated on two data sets collected to assess the success of two types of surgery: (1) cleft lip surgery, where the level of asymmetry is compared between a group of cleft lip subjects and a group of normal subjects and (2) orthognathic surgery, where for each patient, the level of asymmetry after surgery is compared to that before surgery.

We address the problem of predicting high-detail RNA structure geometry from the information available in low-detail experimental maps. Here, low-detail refers to resolutions ≈ 2.5-3.5Å, where the location of the phosphate groups and the glycosidic bonds can be determined from experimental maps but all other backbone atom positions cannot. In contrast, higher-resolution maps allow high-detail determinations of all backbone atomic positions. To this end, we first create a gold standard dataset of highly curated, experimentally supported RNA suites. Second, we develop and employ a modified version of the previously devised algorithm MINT-AGE to learn clusters that are in high correspondence with the gold standard's conformational classes of suites based on 3D RNA structure. Since some of the gold standard classes are of very small size, a new modified version of MINT-AGE is able to also identify very small clusters. Third, we create a new conformer prediction algorithm, RNAprecis, which assigns low-detail structures to newly designed 3D shape coordinates. Our improvements include: (i) learned classes augmented to cover also very low sample sizes and (ii) replacing distances from clusters by Bayesian posterior probabilities. On test data containing suites modeled as conformational outliers, RNAprecis shows good results suggesting that our learning method generalizes well. In particular, we show that the modified MINT-AGE clustering can more finely delineate between previously unseen suite conformer separations. For example, the 0a conformer has been separated into two clusters seen in different structural contexts. Such new distinctions can have implications for biochemical interpretation of RNA structure.

It is not an exaggeration to say that R.A. Fisher is the Albert Einstein of Statistics. He pioneered almost all the main branches of statistics, but it is not as well known that he opened the area of Directional Statistics with his 1953 paper introducing a distribution on the sphere which is now known as the Fisher distribution. He stressed that for spherical data one should take into account that the data is on a manifold. We will describe this Fisher distribution and reanalyze his geological data. We also comment on the two goals he set himself in that paper, and on how he reinvented the von Mises distribution on the circle. Since then, many extensions of this distribution have appeared bearing Fisher's name such as the von Mises-Fisher distribution and the matrix Fisher distribution. In fact, the subject of Directional Statistics has grown tremendously in the last two decades with new applications emerging in life sciences, image analysis, machine learning and so on. We give a recent new method of constructing the Fisher type distributions on manifolds which has been motivated by some problems in machine learning. The number of directional distributions has increased since then, including the bivariate von Mises distribution and we describe its connection to work resulting in the 2024 Nobel-winning AlphaFold (in Chemistry). Further, the subject has evolved as Statistics on Manifolds which also includes the new field of Shape Analysis, and finally, we end with a historical note pointing out some correspondence between D'Arcy Thompson and R.A. Fisher related to Shape Analysis.

The seminal breakthrough of AlphaFold in protein structure prediction relied on a learned potential energy function parameterized by deep models, in contrast to its successors AlphaFold2 and AlphaFold3, which lack an explicit probabilistic interpretation. While AlphaFold's potential was originally justified by heuristic analogy to physical potentials of mean force, we show that it can instead be understood as a principled instance of probability kinematics (PK), also known as Jeffrey conditioning, a generalization of Bayesian updating. This reinterpretation reveals that AlphaFold is a generalized Bayesian model that explicitly defines a posterior distribution over structures, providing a deeper explanation of its success and a foundation for future model design. To demonstrate this framework with precision, we introduce a tractable synthetic model in which an angular random walk prior is updated with distance-based evidence via PK, directly mirroring AlphaFold's mechanism. This setting allows us to explore the probabilistic foundations of AlphaFold in a clear and interpretable way. Our work connects a landmark in protein structure prediction to a broader class of compositional deep generative models and points to new opportunities for principled probabilistic approaches.

In many applications of shape analysis, lengths between some landmarks are constrained. For instance, biomolecules often have some bond lengths and some bond angles constrained, and variation occurs only along unconstrained bonds and constrained bonds' torsions where the latter are conveniently modelled by dihedral angles. Our work has been motivated by low resolution biomolecular chain RNA where only some prominent atomic bonds can be well identified. Here, we propose a new modelling strategy for such constrained shape analysis starting with a product of polar coordinates (polypolars), where, due to constraints, for example, some radial coordinates should be omitted, leaving products of spheres (polyspheres). We give insight into these coordinates for particular cases such as five landmarks which are motivated by a practical RNA application. We also discuss distributions for polypolar coordinates and give a specific methodology with illustration when the constrained size-and-shape variables are concentrated. There are applications of this in clustering and we give some insight into a modified version of the MINT-AGE algorithm.

The method of Cartesian transformations introduced by D’Arcy Thompson a century ago in his celebrated book On Growth and Form precipitated an important development in 20th-century biometrics: a fusion of the geometrical and biological approaches to morphology. Some decades later this fusion, in turn, spun off another multidisciplinary focus, statistical shape analysis, that bridges between biostatistics and biomedical imaging. Our article is intended to seed a complementary focus: a bridge between biostatistics and medical anatomy, a field that has to this day remained mainly verbal rather than quantitative. Specifically, we are proposing a novel methodology for arriving at anatomical interpretations of statistical findings about large-scale contrasts of organismal morphology or its dynamics by combining two toolkits hitherto separate in their notation and their disciplinary housing: morphometrics and psychometrics. A contemporary morphometric analysis deals with patterns of shape coordinate covariation in terms of their geometric adjacency; psychometric factor analysis, the same patterns of covariation in terms of simplicity of interpretation. By combining these tools we account for the dynamics of a facial expression in terms of the actions of the underlying muscles, thereby realizing Thompson’s original metaphor, the “origins of form in force,” for systems that “vary in a more or less uniform manner.”This paper reviews the history of Thompson’s metaphor and then the current literature quantifying smiles, in order to set the stage for the combination of scaling analysis and factor analysis that we are putting forward. We demonstrate the new approach by reanalyzing a data set of ten landmarks around the vermilion borders of the human lip contrasting the dynamics of two socially stereotyped physiological cycles, the open-lip smile and the closed-lip smile, over a sample of 14 normal faces. Our analysis centers on just two dimensions of statistical shape space, those of largest geometrical scale, which can be identified with the action of two different muscles, orbicularis oris and zygomaticus major. The two smiles differ radically in their achieved deformations. Furthermore, while the closed-lip smiles arrived at their final forms along similar shape trajectories, the open-lip smiles did not. Our closing discussion explores some aspects of morphodynamics that are illuminated by the example here.### Competing Interest StatementThe authors have declared no competing interest.

Sir Ronald Aylmer Fisher opened many new areas in Multivariate Analysis, and the one which we will consider is discriminant analysis. Several papers by Fisher and others followed from his seminal paper in 1936 where he coined the name discrimination function. Historically, his four papers on discriminant analysis during 1936-1940 connect to the contemporaneous pioneering work of Hotelling and Mahalanobis. We revisit the famous iris data which Fisher used in his 1936 paper and in particular, test the hypothesis of multivariate normality for the data which he assumed. Fisher constructed his genetic discriminant motivated by this application and we provide a deeper insight into this construction; however, this construction has not been well understood as far as we know. We also indicate how the subject has developed along with the computer revolution, noting newer methods to carry out discriminant analysis, such as kernel classifiers, classification trees, support vector machines, neural networks, and deep learning. Overall, with computational power, the whole subject of Multivariate Analysis has changed its emphasis but the impact of this Fisher's pioneering work continues as an integral part of supervised learning in Artificial Intelligence (AI).

Fisher opened many new areas in Multivariate Analysis, and the one which we will consider is discriminant analysis. Several papers by Fisher and others followed from his seminal paper in 1936 where he coined the name discrimination function. Historically, his four papers on discriminant analysis during 1936-1940 connect to the contemporaneous pioneering work of Hotelling and Mahalanobis. We revisit the famous iris data which Fisher used in his 1936 paper and in particular, test the hypothesis of multivariate normality for the data which he assumed. Fisher constructed his genetic discriminant motivated by this application and we provide a deeper insight into this construction; however, this construction has not been well understood as far as we know. We also indicate how the subject has developed along with the computer revolution, noting newer methods to carry out discriminant analysis, such as kernel classifiers, classification trees, support vector machines, neural networks, and deep learning. Overall, with computational power, the whole subject of Multivariate Analysis has changed its emphasis but the impact of this Fisher's pioneering work continues as an integral part of supervised learning in Artificial Intelligence.

Motivation Reconstructions of structure of biomolecules, for instance via X-ray crystallography or cryo-EM frequently contain clashes of atomic centers. Correction methods are usually based on simulations approximating biophysical chemistry, making them computationally expensive and often not correcting all clashes. Results We propose a computationally fast data-driven statistical method yielding suites free from within-suite clashes: From such a clash free training data set, devising mode hunting after torus PCA on adaptive cutting average linkage tree clustering (MINTAGE), we learn RNA suite shapes. With classification based on multiscale structure enhancement (CLEAN), for a given clash suite we determine its neighborhood on a mesoscopic scale involving several suites. As corrected suite we propose the Fréchet mean on a torus of the largest classes in this neighborhood. We validate CLEAN MINTAGE on a benchmark data set, compare it to a state of the art correction method and apply it, as proof of concept, to two exemplary suites adjacent to helical pieces of the frameshift stimulation element of SARS-CoV-2 which are difficult to reconstruct. In contrast to a recent reconstruction proposing several different structure models, CLEAN MINTAGE unanimously proposes structure corrections within the same clash free class for all suites. Code Availability https://gitlab.gwdg.de/henrik.wiechers1/clean-mintage-code

Mathematics in Ancient Jaina Literature, pp. 183-203 (2022) No AccessArticle 10: Jain Syllogism and Other Aspects of Jain Logic with Applications to Statistical ThinkingKanti V. Mardia and Anthony J. RudaKanti V. MardiaDepartment of Statistics, University of Leeds, Leeds, LS2 9JT, UKDepartment of Statistics, University of Oxford, Oxford, OX1 3LB, UK and Anthony J. RudaJain Noble Truths Association, 2021–2022 Bhagwan Mahavira Fellow, International School for Jain Studies, 844 BMCC Road Shivajinagar, Pune – 411 004, Maharashtra, Indiahttps://doi.org/10.1142/9789811255502_0010Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: It is well known that the core concept of Jain logic, the conditional holistic principle (known as Anekāntavāda), originated in the ancient Jain literature. From this core concept, the conditional predication principle (known as Syādvāda) was developed and has since become one of the important areas of ancient Jain logic in which some aspects have been well studied in modern terminology. However, some aspects of Jain logic have not yet been explored and are dealt with in this article. This article gives historical details on the five-fold Jain syllogism (known as Pañcāvayavavākya) as part of a comprehensive analysis, thereby unifying the literature and bringing the concept in line with modern mathematical terminology. Its five components — proposition (Pratijñā), reason (Hetu), example (Udāharaṇam), application (Upanaya), and conclusion (Nigamanam) — are discussed in depth to show how Pañcāvayavavākya matches the basis of the current statistical inference. In particular, Syādvāda in combination with Pañcāvayavavākya yields the Bayesian approach. Further, a formal and succinct notation is introduced to describe Syādvāda and to illustrate the modern mathematical connections, while Anekāntavāda is shown to have as its corollary a stratified sampling. Finally, an element of Syādvāda inherent to Turing's cryptographic work towards breaking the Enigma code — work that could fairly be called 'enigmatic statistics' — is revealed. Keywords: ancient Jain logicAristotleBayesian analysisconditional holistic principle (Anekāantavāda)conditional predication principle (Syādvāda)five-fold Jain syllogism (Pañcāvayavavākya)samplingTuring2020MSC: 01-06, 01A32, 01A99, 03A05, 62A01 FiguresReferencesRelatedDetails Recommended Mathematics in Ancient Jaina LiteratureMetrics History Keywordsancient Jain logicAristotleBayesian analysisconditional holistic principle (Anekāantavāda)conditional predication principle (Syādvāda)five-fold Jain syllogism (Pañcāvayavavākya)samplingTuringPDF download

Free Access References and Author Index Kanti Mardia, Kanti Mardia University of Leeds, UKSearch for more papers by this authorJohn Kent, John Kent University of Leeds, University of Oxford, UKSearch for more papers by this author Book Author(s):Kanti Mardia, Kanti Mardia University of Leeds, UKSearch for more papers by this authorJohn Kent, John Kent University of Leeds, University of Oxford, UKSearch for more papers by this author First published: 05 July 2022 https://doi.org/10.1002/9781118763551.refBook Series:Wiley Series in Probability and Statistics AboutPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShareShare a linkShare onFacebookTwitterLinked InRedditWechat References Abend , K. , Hartley , T.J. , and Kanal , L.N. ( 1965 ). Classification of binary random patterns . IEEE Transactions on Information Theory IT-11 : 538 – 544 . Abramowitz , M. and Stegun , I.A. ( 1964 ). Handbook of Mathematical Functions . New York : Dover . Adler , R.J. 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Finite mixture models are fitted to spherical data. Kent distributions are used for the components of the mixture because they allow considerable flexibility. Previous work on such mixtures has used an approximate maximum likelihood estimator for the parameters of a single component. However, the approximation causes problems when using the EM algorithm to estimate the parameters in a mixture model. Hence the exact maximum likelihood estimator is used here for the individual components. This paper is motivated by a challenging prize problem in structural bioinformatics of how proteins fold. It is known that hydrogen bonds play a key role in the folding of a protein. We explore this hydrogen bond geometry using a data set describing bonds between two amino acids in proteins. An appropriate coordinate system to represent the hydrogen bond geometry is proposed, with each bond represented as a point on a sphere. We fit mixtures of Kent distributions to different subsets of the hydrogen bond data to gain insight into how the secondary structure elements bond together, since the distribution of hydrogen bonds depends on which secondary structure elements are involved.

Big data, high dimensional data, sparse data, large scale data, and imaging data are all becoming new frontiers of statistics. Changing technologies have created this flood and have led to a real hunger for new modeling strategies and data analysis by scientists. In many cases data are not Euclidean; for example, in molecular biology, the data sit on manifolds. Even in a simple non-Euclidean manifold (circle), to summarize angles by the arithmetic average cannot make sense and so more care is needed. Thus non-Euclidean settings throw up many major challenges, both mathematical and statistical. This paper will focus on the PCA and clustering methods for some manifolds. Of course, the PCA and clustering methods in multivariate analysis are one of the core topics. We basically deal with two key manifolds from a practical point of view, namely spheres and tori. It is well known that dimension reduction on non-Euclidean manifolds with PCA-like methods has been a challenging task for quite some time but recently there has been some breakthrough. One of them is the idea of nested spheres and another is transforming a torus into a sphere effectively and subsequently use the technology of nested spheres PCA. We also provide a new method of clustering for multivariate analysis which has a fundamental property required for molecular biology that penalizes wrong assignments to avoid chemically no go areas. We give various examples to illustrate these methods. One of the important examples includes dealing with COVID-19 data.

We give a unified treatment of constructing families of circular discrete distributions. Some of these families are deduced from established distributions such as von Mises and wrapped Cauchy. Some others are derived directly such as a flexible family based on trigonometric sums and the circular location family. Results interrelating these families are discussed. These distributions have been motivated by two examples of discrete circular data: casino roulette spins and smart health acrophase monitoring, and these data are analyzed using our proposed models. We discuss how using continuous circular models for circular discrete data can be misleading.

Molecular structures of RNA molecules reconstructed from X-ray crystallography frequently contain errors. Motivated by this problem we examine clustering on a torus since RNA shapes can be described by dihedral angles. A previously developed clustering method for torus data involves two tuning parameters and we assess clustering results for different parameter values in relation to the problem of so-called RNA clashes. This clustering problem is part of the dynamically evolving field of statistics on manifolds. Statistical problems on the torus highlight general challenges for statistics on manifolds. Therefore, the torus PCA and clustering methods we propose make an important contribution to directional statistics and statistics on manifolds in general.

A new two-parameter "full exponential cardioid" radial growth model for two-dimensional geometric objects is proposed and analyzed. The model depends additionally on two rotation parameters and on two seeds about which the growth is centered, plus a choice of three possible assumptions about statistical errors. If the seeds are assumed known, the remaining parameters can be estimated in closed form. Comparisons are given to earlier approaches. Two examples are given, one for a set of simulated data and one for a set of rat calvarial data.

We reexamine the the classical multidimensional scaling (MDS). We study some special cases, in particular, the exact solution for the sub-space formed by the 3 dimensional principal coordinates is derived. Also we give the extreme case when the points are collinear. Some insight into the effect on the MDS solution of the excluded eigenvalues (could be both positive as well as negative) of the doubly centered matrix is provided. As an illustration, we work through an example to understand the distortion in the MDS construction with positive and negative eigenvalues.

Probabilistic programming languages (PPLs) are at the interface between statistics and the theory of programming languages. PPLs formulate statistical models as stochastic programs that enable automatic inference algorithms and optimization. Pyro [1] and its sibling NumPyro [2] are PPLs built on top of the deep learning frameworks PyTorch [3] and Jax [4], respectively. Both PPLs provide simple, highly similar interfaces for inference using efficient implementations of Hamiltonian Monte Carlo (HMC), the No-U-Turn Sampler (NUTS), and Stochastic Variational Inference (SVI). They automatically generate variational distributions from a model, automatically enumerate discrete variables, and support formulating deep probabilistic models such as variational autoencoders and deep Markov models. The Sine von Mises distribution and its skewed variant are toroidal distributions relevant to protein bioinformatics. They provide a natural way to model the dihedral angles of protein structures, which is important in protein structure prediction, simulation and analysis. We present efficient implementations of the Sine von Mises distribution and its skewing in Pyro and NumPyro, and devise a simulation method that increases efficiency with several orders of magnitude when using parallel hardware (i.e., modern CPUs, GPUs, and TPUs). We demonstrate the use of the skewed Sine von Mises distribution by modeling dihedral angles of proteins using a Bayesian mixture model inferred using NUTS, exploiting NumPyro's facilities for automatic enumeration [5].