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One of the important problems in the statistical shape analysis context is to predict the shape change. Rather than taking the change in terms of time, we confine ourselves to the deformation of the objects represented by a regression-type model. We properly combine the shape definition in terms of the Kendall and Bookstein shape coordinate systems to break down the problem on the sphere. This is achieved through the triangulation of objects; a very popular technique in geometrical mathematics. A novel idea on tracing the residuals of the spherical regression is then proposed, enabling us to invoke the well-known spherical distributions, including von Mises–Fisher density, to make the statistical inference. New directional residuals not only lie on the sphere but also help to easily simulate the spherical responses. The performance of our proposed method is highlighted via running a simulation study as well as analysing a real data set.

Some of the spherical distributions can be constructed through proper transformation of the densities on plane. Since the logistic density on the Euclidean space has similar behavior to the normal distribution, it is of interest to extend it for spherical data. In this paper, we introduce spherical logistic distribution on the unit sphere and then study relevant statistical inferences including parameters estimation through method of moments and maximum likelihood techniques. It is shown that the spherical logistic distribution is a multimodal distribution with the marginal logistic density function. Proposed density has rotational symmetry property and this plays a key role to drive some important results related to first two moments. To investigate the proposed density in more details, some simulation studies along with analyzing real-life data are also considered.

Multivariate circular observations, i.e. points on a torus arise frequently in fields where instruments such as compass, protractor, weather vane, sextant or theodolite are used. Multivariate wrapped models are often appropriate to describe data points scattered on p-dimensional torus. However, the statistical inference based on such models is quite complicated since each contribution in the log-likelihood function involves an infinite sum of indices in $${\mathbb {Z}}^p$$ , where p is the dimension of the data. To overcome this problem, for moderate dimension p, we propose two estimation procedures based on Expectation-Maximisation and Classification Expectation-Maximisation algorithms. We study the performance of the proposed techniques on a Monte Carlo simulation and further illustrate the advantages of the new procedures on three real-world data sets.

Spherical distributions, including the von Mises-Fisher density, have received a great attention in the literature because of their usefulness to model circular data lying on the unit sphere. However, there is a paucity of research on proposing spherical densities possing multimode tuned with a single parameter. To fill in this gap, we extend von Mises-Fisher distribution to construct a new density. Moreover, some of the important statistical properties of the proposed distribution including the estimation of parameters are highlighted. To evaluate the performance of the proposed distribution, some simulation studies and analyzing three real-life examples are presented.

The Pearson type family densities are among the most important classes of distributions that play a key role in the directional statistics. Their particular structures make them suitable candidates to analysis data on non-Euclidean space, such as sphere. To model data scattered asymmetrically on such spaces, the researchers confined themselves to extend particular distributions from the class of the Pearson type family densities. Those specific distributions are symmetric in nature but their extended versions are usually heavy tailed. In this paper, we introduce some alternative probability density functions in the class of Pearson type distributions on the sphere having the spherical Student’s $t$, Fisher and Chi-square densities as the subfamilies. Via investigating various theoretical properties of this new subclass, we show that it is inheritably asymmetric. To further evaluating this subclass, some simulation studies are conducted. Also, modeling of two real-life data using the proposed densities and then comparing their fitting consequences with those resulted from invoking other common spherical distributions are considered.

It is known that the shapes of planar triangles can be represented by a set of points on the surface of the unit sphere. On the other hand, most of the objects can easily be triangulated and so each triangle can accordingly be treated in the context of shape analysis. There is a growing interest to fit a smooth path going through the cloud of shape data available in some time instances. To tackle this problem, we propose a longitudinal model through a triangulation procedure for the shape data. In fact, our strategy initially relies on a spherical regression model for triangles, but is extended to shape data via triangulation. Regarding modeling of directional data, we use the bivariate von Mises–Fisher distribution for density of the errors. Various forms of the composite likelihood functions, constructed by altering the assumptions considered for the angles defined for each triangle, are invoked. The proposed regression model is applied to rat skull data. Also, some simulations results are presented along with the real data results.

One of the important topics in morphometry that received high attention recently is the longitudinal analysis of shape variation. According to Kendall's definition of shape, the shape of object appertains on non-Euclidean space, making the longitudinal study of configuration somehow difficult. However, to simplify this task, triangulation of the objects and then constructing a non-parametric regression-type model on the unit sphere is pursued in this paper. The prediction of the configurations in some time instances is done using both properties of triangulation and the size of great baselines. Moreover, minimizing a Euclidean risk function is proposed to select feasible weights in constructing smoother functions in a non-parametric smoothing manner. These will provide some proper shape growth models to analysis objects varying in time. The proposed models are applied to analysis of two real-life data sets.

Among the statistical methods to model stochastic behaviours of objects, clustering is a preliminary technique to recognize similar patterns within a group of observations in a data set. Various distances to measure differences among objects could be invoked to cluster data through numerous clustering methods. When variables in hand contain geometrical information of objects, such metrics should be adequately adapted. In fact, statistical methods for these typical data are endowed with a geometrical paradigm in a multivariate sense. In this paper, a procedure for clustering shape data is suggested employing appropriate metrics. Then, the best shape distance candidate as well as a suitable agglomerative method for clustering the simulated shape data are provided by considering cluster validation measures. The results are implemented in a real life application.

Statistics, as one of the applied sciences, has great impacts in vast area of other sciences. Prediction of protein structures with great emphasize on their geometrical features using dihedral angles has invoked the new branch of statistics, known as directional statistics. One of the available biological techniques to predict is molecular dynamics simulations producing high-dimensional molecular structure data. Hence, it is expected that the principal component analysis (PCA) can response some related statistical problems particulary to reduce dimensions of the involved variables. Since the dihedral angles are variables on non-Euclidean space (their locus is the torus), it is expected that direct implementation of PCA does not provide great information in this case. The principal geodesic analysis is one of the recent methods to reduce the dimensions in the non-Euclidean case. A procedure to utilize this technique for reducing the dimension of a set of dihedral angles is highlighted in this paper. We fur...

Multilevel models are popular models for analysing data inheriting a hierarchical structure. They are used in diverse fields including social, medical, economical and biological sciences. These models encounter some problems in estimating the parameters, if there are measurement errors in either explanatory or response variables. A common approach to tackle this obstacle is to consider the pseudo variables and follow some simulation methods to estimate the parameters. We propose a new algorithm constituting the iterative and simulation extrapolation steps in turn. To evaluate the proposed algorithm, various simulation studies are also conducted. Moreover, we investigate the implementation of our method on a real data set concerning the cost and expenditure of the households in Tehran city in the year 2007.

An active area of research in bioinformatics is finding structural similarity of proteins by alignment. Among many methods, the popular one is to find the similarity based on statistical features. This method involves gathering information from the complex biomolecule structure and obtaining the best alignment by maximizing the number of matched features. In this paper, after reviewing statistical models for matching the structural biomolecule, it is shown that local alignment based on the Delaunay tetrahedralization (DT) can be used for Bayesian alignment of proteins. In this method, we use DT to add a priori structural information of protein in the Bayesian methodology. We demonstrate that this method shows advantages over competing methods in achieving a global alignment of proteins, accelerating the convergence rate and improving the parameter estimates.

Most of the linear statistics deal with data lying in a Euclidean space. However, there are many examples, such as DNA molecule topological structures, in which the initial or the transformed data lie in a non-Euclidean space. To get a measure of variability in these situations, the principal component analysis (PCA) is usually performed on a Euclidean tangent space as it cannot be directly implemented on a non-Euclidean space. Instead, principal geodesic analysis (PGA) is a new tool that provides a measure of variability for nonlinear statistics. In this paper, the performance of this new tool is compared with that of the PCA using a real data set representing a DNA molecular structure. It is shown that due to the nonlinearity of space, the PGA explains more variability of the data than the PCA.

One of the new area of research emerging in the field of statistics is the shape analysis. Shape is defined as all the geometrical information of an object whose location, scale and orientation is not of interest. Diffusion in shape analysis can be studied via either perturbation of the key coordi- nates identifying the initial object or random evolution of the shape itself. Reviewing the first case, we mainly consider the second case and particularly define a new family of diffusion processes. It can be used to model diffusion phenomena represented by shape evolution such as cell motion.

SummaryWe consider the application of Markov chain Monte Carlo (MCMC) estimation methods to random-effects models and in particular the family of discrete time survival models. Survival models can be used in many situations in the medical and social sciences and we illustrate their use through two examples that differ in terms of both substantive area and data structure. A multilevel discrete time survival analysis involves expanding the data set so that the model can be cast as a standard multilevel binary response model. For such models it has been shown that MCMC methods have advantages in terms of reducing estimate bias. However, the data expansion results in very large data sets for which MCMC estimation is often slow and can produce chains that exhibit poor mixing. Any way of improving the mixing will result in both speeding up the methods and more confidence in the estimates that are produced. The MCMC methodological literature is full of alternative algorithms designed to improve mixing of chains and we describe three reparameterization techniques that are easy to implement in available software. We consider two examples of multilevel survival analysis: incidence of mastitis in dairy cattle and contraceptive use dynamics in Indonesia. For each application we show where the reparameterization techniques can be used and assess their performance.

We discuss Brownian motion and Ornstein–Uhlenbeck processes specified directly in planar shape space. In particular, we obtain the drift and diffusion coefficients of Brownian motion in terms of Kendall shape variables and Goodall–Mardia polar shape variables. Stochastic differential equations are given and the stationary distributions are obtained. By adding in extra drift to a reference figure, Ornstein–Uhlenbeck processes can be studied, for example with stationary distribution given by the complex Watson distribution. The triangle case is studied in particular detail, and some simulations given. Connections with existing work are made, in particular with the diffusion of Euclidean shape. We explore statistical inference for the parameters in the model with an application to cell shape modelling.

Background: There is growing interest in the identification of novel therapeutic agents for the treatment of affective disorders, with modafinil being one promising substance. The purpose of the present investigation was to compare the efficacy of a combination of fluoxetine plus modafinil with that of fluoxetine plus placebo in the treatment of major depression in a 6-week double blind and placebo-controlled trial. Methods: Forty-six adult outpatients who met the DSM-IV-TR criteria for major depression participated in the trial. Patients had a baseline Hamilton Rating Scale for depression score of at least 18. Patients were allocated in a random fashion, 23 to fluoxetine 40 mg/day plus modafinil 400 mg/day (200 mg bid) (morning and evening) and 23 to fluoxetine 40 mg/day plus placebo. Patients were assessed at baseline and after 1, 2, 4, and 6 weeks after start of medication. Results: Forty-four patients completed the trial. Fluoxetine+modafinil and fluoxetine+placebo significantly decreased the Hamilton Rating Scale score for Depression over the trial period. However, the combination of fluoxetine and modafinil was significantly superior over fluoxetine alone in the treatment of symptoms of major depression. The difference between the two treatments was significant as indicated by the effect of group, the between-subjects factor (df = 1, F = 4.42, P = 0.046). There were no significant differences in the two groups in terms of observed side-effects. Conclusion: These findings suggest modafinil as a well-tolerated and potentially effective agent in combination with fluoxetine in the management of patients with major depression. Depression and Anxiety, 2011. © 2011 Wiley-Liss, Inc.

Among many algorithms, gradient descent algorithm (GDA) is a simple tool to derive an optimal quantity in dealing with an optimization problem in the linear space. Apart from the initial value, the step size has a great impact on the convergence rate of this algorithm. Its affect on the geometric structure of the consecutive configurations is more crucial if one works with an optimization problem in the statistical shape analysis. In other words, if the step size of the GDA is not properly tuned, the geometry might not be preserved while the algorithm is moving forward to reach an optimal mean shape. In order to improve the performance of the GDA, we introduce a dynamic step size and a new criterion both to check the geometry in each step of the algorithm and to accelerate the convergence rate. These lead to a new robust algorithm on deriving the intrinsic mean on the shape space. We compare the performance of our proposed procedure to the usual GDA using a real shape data accompanied with simulation studies.