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A set of symbols is presented along with logical operators which represent the possible manipulations of the linear model. The use of these symbols and operators is to simplify the representation of analysis of variance models, correlation models and factor analysis models. (Author)

When latent variables are used as outcomes in regression analysis, a common approach that is used to solve the ignored measurement error issue is to take a multilevel perspective on item response modeling (IRT). Although recent computational advancement allow efficient and accurate estimation of multilevel IRT models, we argue that a two-stage divide-and-conquer strategy still has its unique advantages. Within the two-stage framework, three methods that take into account heteroscedastic measurement errors of the dependent variable in stage II analysis are introduced, they are the closed-form marginal MLE (MMLE), the Expectation Maximization (EM) algorithm, and the moment estimation method. They are compared to the naïve two-stage estimation and the one-stage MCMC estimation. A simulation study is conducted to compare the five methods in terms of model parameter recovery and their standard error estimation. The pros and cons of each method are also discussed to provide guidelines for practitioners. Finally, a real data example is given to illustrate the applications of various methods using the National Educational Longitudinal Survey data (NELS 88). [This paper was published in "Psychometrika" v84 p673-700 2019.]

When latent variables are used as outcomes in regression analysis, a common approach that is used to solve the ignored measurement error issue is to take a multilevel perspective on item response modeling (IRT). Although recent computational advancement allow efficient and accurate estimation of multilevel IRT models, we argue that a two-stage divide-and-conquer strategy still has its unique advantages. Within the two-stage framework, three methods that take into account heteroscedastic measurement errors of the dependent variable in stage II analysis are introduced, they are the closed-form marginal MLE (MMLE), the Expectation Maximization (EM) algorithm, and the moment estimation method. They are compared to the naïve two-stage estimation and the one-stage MCMC estimation. A simulation study is conducted to compare the five methods in terms of model parameter recovery and their standard error estimation. The pros and cons of each method are also discussed to provide guidelines for practitioners. Finally, a real data example is given to illustrate the applications of various methods using the National Educational Longitudinal Survey data (NELS 88). [This paper was published in "Psychometrika" v84 p673-700 2019.]

We present a new approach to study the properties of the sun. We consider small variations of the physical and chemical properties of the sun with respect to Standard Solar Model predictions and we linearize the structure equations to relate them to the properties of the solar plasma. By assuming that the (variation of) the present solar composition can be estimated from the (variation of) the nuclear reaction rates and elemental diffusion efficiency in the present sun, we obtain a linear system of ordinary differential equations which can be used to calculate the response of the sun to an arbitrary modification of the input parameters (opacity, cross sections, etc.). This new approach is intended to be a complement to the traditional methods for solar model calculation and allows to investigate in a more efficient and transparent way the role of parameters and assumptions in solar model construction. We verify that these Linear Solar Models recover the predictions of the traditional solar models with an high level of accuracy.

Mixtures of Zellner's g-priors have been studied extensively in linear models and have been shown to have numerous desirable properties for Bayesian variable selection and model averaging. Several extensions of g-priors to Generalized Linear Models (GLMs) have been proposed in the literature; however, the choice of prior distribution of g and resulting properties for inference have received considerably less attention. In this paper, we unify mixtures of g-priors in GLMs by assigning the truncated Compound Confluent Hypergeometric (tCCH) distribution to 1/(1 + g), which encompasses as special cases several mixtures of g-priors in the literature, such as the hyper-g, Beta-prime, truncated Gamma, incomplete inverse-Gamma, benchmark, robust, hyper-g/n, and intrinsic priors. Through an integrated Laplace approximation, the posterior distribution of 1/(1 + g) is in turn a tCCH distribution, and approximate marginal likelihoods are thus available analytically, leading to "Compound Hypergeometric Information Criteria" for model selection. We discuss the local geometric properties of the g-prior in GLMs and show how the desiderata for model selection proposed by Bayarri et al, such as asymptotic model selection consistency, intrinsic consistency, and measurement invariance may be used to justify the prior and specific choices of the hyper parameters. We illustrate inference using these priors and contrast them to other approaches via simulation and real data examples. The methodology is implemented in the R package BAS and freely available on CRAN.

We provide a novel approach to model space-time random fields where the temporal argument is decomposed into two parts. The former captures the linear argument, which is related, for instance, to the annual evolution of the field. The latter is instead a circular variable describing, for instance, monthly observations. The basic intuition behind this construction is to consider a random field defined over space (a compact set of the $d$-dimensional Euclidean space) across time, which is considered as the product space $\mathbb{R} \times \mathbb{S}^1$, with $\mathbb{S}^1$ being the unit circle. Under such framework, we derive new parametric families of covariance functions. In particular, we focus on two classes of parametric families. The former being parenthetical to the Gneiting class of covariance functions. The latter is instead obtained by proposing a new Lagrangian framework for the space-time domain considered in the manuscript. Our findings are illustrated through a real dataset of surface air temperatures. We show that the incorporation of both temporal variables can produce significant improvements in the predictive performances of the model. We also discuss the extension of this approach for fields defined spatially on a sphere, which allows to model space-time phenomena over large portions of planet Earth.

The scalar perturbations in inflationary models, based on a two-component diagonal non-linear sigma model, are considered. For inhomogeneities generated at an inflationary stage, the law of motion of the comoving curvature ${\cal R}$ is obtained (without using the slow roll approximations). A formal expressions for the power spectrum and its spectral index are obtained, which are valid and after the slow-roll stage. As an example, an inflationary model with a massive scalar field and quadratic curvature corrections is studied numerically.

Introduction The permissible exposure time to vibration for the operator is one of the key factors in maintaining the operator's health while optimizing machinery and equipment. The tractor studied was the ITM475, manufactured in Iran. The purpose of this study was to calculate the operator's permissible vibration exposure time while using the tractor to ensure the driver can maintain good bodily health. Materials and Methods In this study, experiments were conducted using a 3-axis vibration meter based on the ISO 2631 standard. The obtained data were analyzed through a factorial experiment using 18 treatments and 3 replications. The factors studied were engine rotation speed (at three levels of 1000, 1500, and 2000 rpm), road type (dirt and asphalt), and gear position (at three levels of 1, 2, and 3). Results and Discussion Various total vibration models were obtained for the tractor, and their determination coefficient varied from 90.11% for gear No. 3 on an asphalt road to 100% for gear No. 1 on an asphalt road and gear No. 2 on a dirt road. The maximum whole-body vibration, and consequently the minimum permissible exposure time, was observed for gear No. 3 at an engine rotation speed of 2000 rpm on a dirt road, which was 1.49 and 1.16 hours, respectively. Conclusion The maximum whole-body vibration experienced during an 8-hour tractor-driving session was measured at 0.85 m s -2 . It is important to note that the permissible exposure time decreases as vibration levels increase, and it reaches a limit of 1.16 hours. To ensure drivers adhere to these permissible exposure times across various driving conditions, measures must be implemented to reduce tractor vibration and minimize its transmission to the driver. By reducing overall tractor vibration and minimizing its impact on the driver, it becomes possible to increase the permissible exposure time for drivers.

Introduction The color of soil depends on its composition and this feature is easily available and rather stable. Fast and accurate determination of soil organic matter distribution in the agricultural fields is required, especially in precision farming. General laboratory methods for determining the soil organic carbon are expensive, time-consuming with many repetitions, and high consumption of chemicals. Soil scientists use the Munsell soil color diagrams to define the soil color. Due to the nature of Munsell color diagrams; this system is less suitable for recognizing exact color of the soil because of weak relationship and limited number of chips. Fast methods like image processing, colorimetric and spectroscopy provide a description of most physical characteristics of the soil color. Some of the advantages of using digital cameras was used in this study, are simple sampling of screened soil, being free from chemicals and toxic materials and being fast, inexpensive and precise. Materials and Methods In this research, 80 A-horizon (0-10 cm) soil samples were collected from various agricultural soils in west Azerbaijan, in the North West of Iran. Soil texture of these fields was loam clay and clay. The amount of organic carbon in samples was determined. The camera was installed at the distance of 0.5 m from the Petri dish on the lighting compartment. Captured images with the digital camera were saved in RGB color space. Processing operations were done by MATLAB 2012 software. The features extracted from the color images are used to model the soil organic carbon including the color features in different spaces. Four-color spaces including RGB, HSI, LAB and LUV were studied to find the relation between the color and the soil organic carbon. Results and Discussion The correlation of R component in the RGB model shows a strong single-parameter relation with the organic carbon as R 2 =0.83. This good relationship can be due to the compound information of the red color component on both brightness and chromaticity dimension. The results also show that the organic carbon has a relatively strong correlation with the light parameters, intensity and lightness in the HSI, Lab and LUV color spaces respectively. It also has a weak correlation with other parameters, since they cannot have a proper linear correlation with organic carbon due to their structural nature. Results show that the highest correlation is obtained when the R and G components participate in modeling and the component B is omitted. One explanation of this high correlation could be the high sensitivity of component B to the intensity and the angle of light. Even a small change in light changes this component. Thus, in order to reduce the effect of this component, it is better to omit it from the models and make models independent of it. In next section, 51 data were used to train neural network, 14 data were used to test the network and 12 data for network validating. The amount of soil organic carbon was output of the neural networks that was estimated after training using the color component values of each segment. To assess the accuracy of the network, estimated values and actual values of each sample were plotted in a graph. The minimum MSE values were 7.28e-6 with 16 neurons, 3.77e-6 with 14 neurons, 4.8e-3 with 10 neurons and 3.77e-6 with 12 neurons for RGB, HSI, Lab and LUV color spaces respectively. Conclusion The availability of digital cameras, possibility to use it in different situations, being inexpensive and providing many samples are the advantages of this method to estimate the soil organic carbon amount. Therefore, digital photography can be used as an analytical method to evaluate the soil fertility. It also requires a low cost of sample testing, and can provide a good possibility of time and place classification for studying a vast area. However to develop more reliable models, more effort is needed, such as collecting more soil samples of different areas that include a wide range of soil features.

Thesis (M.S. in Operations Research and M.S. in Comp. Sci.)--Naval Postgraduate School, 1979

We study N=2 nonlinear two dimensional sigma models with boundaries and their massive generalizations (the Landau-Ginzburg models). These models are defined over either Kahler or bihermitian target space manifolds. We determine the most general local N=2 superconformal boundary conditions (D-branes) for these sigma models. In the Kahler case we reproduce the known results in a systematic fashion including interesting results concerning the coisotropic A-type branes. We further analyse the N=2 superconformal boundary conditions for sigma models defined over a bihermitian manifold with torsion. We interpret the boundary conditions in terms of different types of submanifolds of the target space. We point out how the open sigma models correspond to new types of target space geometry. For the massive Landau-Ginzburg models (both Kahler and bihermitian) we discuss an important class of supersymmetric boundary conditions which admits a nice geometrical interpretation.

This paper explores the feasibility of neural computing methods such as artificial neural networks (ANNs) and abductory induction mechanisms (AIM) for use in educational measurement. ANNs and AIMS methods are contrasted with more traditional statistical techniques, such as multiple regression and discriminant function analyses, for making classification or placement decisions in schools and colleges. Classification rates obtained with multiple regression and discriminant analysis were compared with ANN (back propagation) and AIM methods across a number of plausible models of algebra proficiency that included measures of arithmetic ability, high school achievement, test anxiety, and gender. Analyses were conducted on a sample of 290 male and 310 female college freshmen for the entire sample and for each gender. At each stage 10 randomly selected subsets were used to train and test the neural computing methods. In general, ANN and AIM methods outperformed the more traditional methods. Results suggest that neural computing methods may lead to higher rates of classification accuracy, particularly when underlying models are nonlinear. Included are four tables, and one figure. (Contains 17 references.) (Author/SLD)

We propose a way how to generate a max-stable process in $C[0,1]$ from a max-stable random vector in $\R^d$ by generalizing the \emph{max-linear model} established by Wang and Stoev (2011). It turns out that if the random vector follows some finite dimensional distribution of some initial max-stable process, the approximating processes converge uniformly to the original process and the pointwise mean squared error can be represented in a closed form. The obtained results carry over to the case of generalized Pareto processes. The introduced method enables the reconstruction of the initial process only from a finite set of observation points and, thus, reasonable prediction of max-stable processes gets possible.

We propose a new Bayesian tracking and parameter learning algorithm for non-linear non-Gaussian multiple target tracking (MTT) models. We design a Markov chain Monte Carlo (MCMC) algorithm to sample from the posterior distribution of the target states, birth and death times, and association of observations to targets, which constitutes the solution to the tracking problem, as well as the model parameters. In the numerical section, we present performance comparisons with several competing techniques and demonstrate significant performance improvements in all cases.

We consider the problem of estimating the error variance in a general linear model when the error distribution is assumed to be spherically symmetric, but not necessary Gaussian. In particular we study the case of a scale mixture of Gaussians including the particularly important case of the multivariate-t distribution. Under Stein's loss, we construct a class of estimators that improve on the usual best unbiased (and best equivariant) estimator. Our class has the interesting double robustness property of being simultaneously generalized Bayes (for the same generalized prior) and minimax over the entire class of scale mixture of Gaussian distributions.

Modelling outliers and structural breaks in dynamic linear models with a novel use of a heavy tailed prior for the variances: An alternative to the Inverted Gamma

This paper introduces a novel class of models for binary data, which we call log-mean linear models. The characterizing feature of these models is that they are specified by linear constraints on the log-mean linear parameter, defined as a log-linear expansion of the mean parameter of the multivariate Bernoulli distribution. We show that marginal independence relationships between variables can be specified by setting certain log-mean linear interactions to zero and, more specifically, that graphical models of marginal independence are log-mean linear models. Our approach overcomes some drawbacks of the existing parameterizations of graphical models of marginal independence.

This paper introduces a novel class of models for binary data, which we call log-mean linear models. The characterizing feature of these models is that they are specified by linear constraints on the log-mean linear parameter, defined as a log-linear expansion of the mean parameter of the multivariate Bernoulli distribution. We show that marginal independence relationships between variables can be specified by setting certain log-mean linear interactions to zero and, more specifically, that graphical models of marginal independence are log-mean linear models. Our approach overcomes some drawbacks of the existing parameterizations of graphical models of marginal independence.

Functional linear regression has recently attracted considerable interest. Many works focus on asymptotic inference. In this paper we consider in a non asymptotic framework a simple estimation procedure based on functional Principal Regression. It revolves in the minimization of a least square contrast coupled with a classical projection on the space spanned by the m first empirical eigenvectors of the covariance operator of the functional sample. The novelty of our approach is to select automatically the crucial dimension m by minimization of a penalized least square contrast. Our method is based on model selection tools. Yet, since this kind of methods consists usually in projecting onto known non-random spaces, we need to adapt it to empirical eigenbasis made of data-dependent - hence random - vectors. The resulting estimator is fully adaptive and is shown to verify an oracle inequality for the risk associated to the prediction error and to attain optimal minimax rates of convergence over a certain class of ellipsoids. Our strategy of model selection is finally compared numerically with cross-validation.

Generalized linear models (GLMs) have been used widely for modelling the mean response both for discrete and continuous random variables with an emphasis on categorical response. Recently Yang, Mandal and Majumdar (2013) considered full factorial and fractional factorial locally D-optimal designs for binary response and two-level experimental factors. In this paper, we extend their results to a general setup with response belonging to a single-parameter exponential family and for multi-level predictors.

Exact inference in the linear regression model with spike and slab priors is often intractable. Expectation propagation (EP) can be used for approximate inference. However, the regular sequential form of EP (R-EP) may fail to converge in this model when the size of the training set is very small. As an alternative, we propose a provably convergent EP algorithm (PC-EP). PC-EP is proved to minimize an energy function which, under some constraints, is bounded from below and whose stationary points coincide with the solution of R-EP. Experiments with synthetic data indicate that when R-EP does not converge, the approximation generated by PC-EP is often better. By contrast, when R-EP converges, both methods perform similarly.

We introduce a new approach to a linear-circular regression problem that relates multiple linear predictors to a circular response, bringing a new modeling perspective on a circular variable. Some previous works model a circular variable as projection of a bivariate Gaussian random vector on the unit square, and the statistical inference of the resulting model involves complicated sampling steps. The proposed model treats circular responses as the result of the modulo operation on unobserved linear responses. The resulting model is a mixture of multiple linear-linear regression models. We present two EM algorithms for maximum likelihood estimation of the mixture model, one for parametric estimation and another for non-parametric estimation. Numerical examples are presented to demonstrate the performance of the proposed approach.

Traditionally, the Dirichlet-multinomial distribution has been recognized as a key model for contingency tables generated by cluster sampling schemes. There are, however, other possible distributions appropriate for these contingency tables. This paper introduces new test-statistics capable to test log-linear modeling hypotheses with no distributional specification, when the individuals of the clusters are possibly homogeneously correlated. The estimator for the intracluster correlation coefficient proposed in Alonso-Revenga et al. (2016), valid for different cluster sizes, plays a crucial role in the construction of the goodness-of-fit test-statistic.

We consider the problem of estimating the slope function in a functional regression with a scalar response and a functional covariate. This central problem of functional data analysis is well known to be ill-posed, thus requiring a regularised estimation procedure. The two most commonly used approaches are based on spectral truncation or Tikhonov regularisation of the empirical covariance operator. In principle, Tikhonov regularisation is the more canonical choice. Compared to spectral truncation, it is robust to eigenvalue ties, while it attains the optimal minimax rate of convergence in the mean squared sense, and not just in a concentration probability sense. In this paper, we show that, surprisingly, one can strictly improve upon the performance of the Tikhonov estimator in finite samples by means of a linear estimator, while retaining its stability and asymptotic properties by combining it with a form of spectral truncation. Specifically, we construct an estimator that additively decomposes the functional covariate by projecting it onto two orthogonal subspaces defined via functional PCA; it then applies Tikhonov regularisation to the one component, while leaving the other component unregularised. We prove that when the covariate is Gaussian, this hybrid estimator uniformly improves upon the MSE of the Tikhonov estimator in a non-asymptotic sense, effectively rendering it inadmissible. This domination is shown to also persist under discrete observation of the covariate function. The hybrid estimator is linear, straightforward to construct in practice, and with no computational overhead relative to the standard regularisation methods. By means of simulation, it is shown to furnish sizeable gains even for modest sample sizes.

Constrained approaches to maximum likelihood estimation in the context of finite mixtures of normals have been presented in the literature. A fully data-dependent constrained method for maximum likelihood estimation of clusterwise linear regression is proposed, which extends previous work in equivariant data-driven estimation of finite mixtures of Gaussians for classification. The method imposes plausible bounds on the component variances, based on a target value estimated from the data, which we take to be the homoscedastic variance. Nevertheless, the present work does not only focus on classification recovery, but also on how well model parameters are estimated. In particular, the paper sheds light on the shrinkage-like interpretation of the procedure, where the target is the homoscedastic model: this is not only related to how close to the target the estimated scales are, but extends to the estimated clusterwise linear regressions and classification. We show, based on simulation and real-data based results, that our approach yields a final model being the most appropriate-to-the-data compromise between the heteroscedastic model and the homoscedastic model.

A recently proposed single progenitor cell model for skin cell proliferation [Clayton et al., Nature v446, 185 (2007)] is extended to incorporate homeostasis as a fixed point of the dynamics. Unlimited cell proliferation in such a model can be viewed as a paradigm for the onset of cancer. A novel way in which this can arise is if the homeostatic fixed point becomes metastable, so that the cell populations can escape from the homeostatic basin of attraction by a large but rare stochastic fluctuation. Such an event can be viewed as the final step in a multi-stage model of carcinogenesis. This offers a possible explanation for the peculiar epidemiology of lung cancer in ex-smokers.

Dark energy as a quintessence component causes a typical modification of the background cosmic expansion, which in addition to its clustering properties, can leave a potentially distinctive signature on large scale structures. Many previous studies have investigated this topic, particularly in relation to the non-linear regime of structure formation. However, no careful pre-selection of viable quintessence models with high precision cosmological data was performed. Here we show that this has led to a misinterpretation (and underestimation) of the imprint of quintessence on the distribution of large scale structures. To this purpose we perform a likelihood analysis of the combined Supernova Ia UNION dataset and WMAP5-years data to identify realistic quintessence models. Differences from the vanilla LambdaCDM are especially manifest in the predicted amplitude and shape of the linear matter power spectrum, though these remain within the uncertainties of the SDSS data. We use these models as benchmark for studying the clustering properties of dark matter halos by performing a series of high resolution N-body simulations. We find that realistic quintessence models allow for relevant differences of the dark matter distribution with the respect to the LambdaCDM scenario well into the non-linear regime, with deviations up to 40% in the non-linear power spectrum. Such differences are shown to depend on the nature of DE, as well as the scale and epoch considered. At small scales (k~1-5 h Mpc^{-1}, depending on the redshift) the structure formation process is about 20% more efficient than in LambdaCDM. We show that these imprints are a specific record of the cosmic structure formation history in DE cosmologies and therefore cannot be accounted in standard fitting functions of the non-linear matter power spectrum.

In this paper we investigate the non-linear sigma model arising in the reduction of D = 5 supergravity to D = 3, and present the application of this sigma model to supersymmetric black ring solutions in five-dimensional minimal supergravity. With the ansatz of stationary solutions with $R \times U(1)\times U(1)$ isometry, we obtain a two-dimensional Lagrangian corresponding to geodesic motion of a string-like object on the coset $G_{2(+2)}/SO(4)$, and study the algebra of conserved charges and supersymmetry constraints of supersymmetric black rings. We also obtain the semi-classical wave function of supersymmetric black rings.

We study semiparametric varying-coefficient partially linear models when some linear covariates are not observed, but ancillary variables are available. Semiparametric profile least-square based estimation procedures are developed for parametric and nonparametric components after we calibrate the error-prone covariates. Asymptotic properties of the proposed estimators are established. We also propose the profile least-square based ratio test and Wald test to identify significant parametric and nonparametric components. To improve accuracy of the proposed tests for small or moderate sample sizes, a wild bootstrap version is also proposed to calculate the critical values. Intensive simulation experiments are conducted to illustrate the proposed approaches.

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The purpose of this paper is to provide a discussion, with illustrating examples, on Bayesian forecasting for dynamic generalized linear models (DGLMs). Adopting approximate Bayesian analysis, based on conjugate forms and on Bayes linear estimation, we describe the theoretical framework and then we provide detailed examples of response distributions, including binomial, Poisson, negative binomial, geometric, normal, log-normal, gamma, exponential, Weibull, Pareto, beta, and inverse Gaussian. We give numerical illustrations for all distributions (except for the normal). Putting together all the above distributions, we give a unified Bayesian approach to non-Gaussian time series analysis, with applications from finance and medicine to biology and the behavioural sciences. Throughout the models we discuss Bayesian forecasting and, for each model, we derive the multi-step forecast mean. Finally, we describe model assessment using the likelihood function, and Bayesian model monitoring.

This paper compares a family of methods for characterizing neural feature selectivity with natural stimuli in the framework of the linear-nonlinear model. In this model, the neural firing rate is a nonlinear function of a small number of relevant stimulus components. The relevant stimulus dimensions can be found by maximizing one of the family of objective functions, Renyi divergences of different orders. We show that maximizing one of them, Renyi divergence of order 2, is equivalent to least-square fitting of the linear-nonlinear model to neural data. Next, we derive reconstruction errors in relevant dimensions found by maximizing Renyi divergences of arbitrary order in the asymptotic limit of large spike numbers. We find that the smallest rrors are obtained with Renyi divergence of order 1, also known as Kullback-Leibler divergence. This corresponds to finding relevant dimensions by maximizing mutual information. We numerically test how these optimization schemes perform in the regime of low signal-to-noise ratio (small number of spikes and increasing neural noise) for model visual neurons. We find that optimization schemes based on either least square fitting or information maximization perform well even when number of spikes is small. Information maximization provides slightly, but significantly, better reconstructions than least square fitting. This makes the problem of finding relevant dimensions, together with the problem of lossy compression, one of examples where information-theoretic measures are no more data limited than those derived from least squares.

This paper presents a construction which transforms categorical models of additive-free propositional linear logic, closely based on de Paiva's dialectica categories and Oliva's functional interpretations of classical linear logic. The construction is defined using dependent type theory, which proves to be a useful tool for reasoning about dialectica categories. Abstractly, we have a closure operator on the class of models: it preserves soundness and completeness and has a monad-like structure. When applied to categories of games we obtain `games with bidding', which are hybrids of dialectica and game models, and we prove completeness theorems for two specific such models.

This paper investigates correct variable selection in finite samples via $\ell_1$ and $\ell_1+\ell_2$ type penalization schemes. The asymptotic consistency of variable selection immediately follows from this analysis. We focus on logistic and linear regression models. The following questions are central to our paper: given a level of confidence $1-\delta$, under which assumptions on the design matrix, for which strength of the signal and for what values of the tuning parameters can we identify the true model at the given level of confidence? Formally, if $\widehat{I}$ is an estimate of the true variable set $I^*$, we study conditions under which $\mathbb{P}(\widehat{I}=I^*)\geq 1-\delta$, for a given sample size $n$, number of parameters $M$ and confidence $1-\delta$. We show that in identifiable models, both methods can recover coefficients of size $\frac{1}{\sqrt{n}}$, up to small multiplicative constants and logarithmic factors in $M$ and $\frac{1}{\delta}$. The advantage of the $\ell_1+\ell_2$ penalization over the $\ell_1$ is minor for the variable selection problem, for the models we consider here. Whereas the former estimates are unique, and become more stable for highly correlated data matrices as one increases the tuning parameter of the $\ell_2$ part, too large an increase in this parameter value may preclude variable selection.

We propose a roughness regularization approach in making nonparametric inference for generalized functional linear models. In a reproducing kernel Hilbert space framework, we construct asymptotically valid confidence intervals for regression mean, prediction intervals for future response and various statistical procedures for hypothesis testing. In particular, one procedure for testing global behaviors of the slope function is adaptive to the smoothness of the slope function and to the structure of the predictors. As a by-product, a new type of Wilks phenomenon [Ann. Math. Stat. 9 (1938) 60-62; Ann. Statist. 29 (2001) 153-193] is discovered when testing the functional linear models. Despite the generality, our inference procedures are easy to implement. Numerical examples are provided to demonstrate the empirical advantages over the competing methods. A collection of technical tools such as integro-differential equation techniques [Trans. Amer. Math. Soc. (1927) 29 755-800; Trans. Amer. Math. Soc. (1928) 30 453-471; Trans. Amer. Math. Soc. (1930) 32 860-868], Stein's method [Ann. Statist. 41 (2013) 2786-2819] [Stein, Approximate Computation of Expectations (1986) IMS] and functional Bahadur representation [Ann. Statist. 41 (2013) 2608-2638] are employed in this paper.

Subsampling methods have been recently proposed to speed up least squares estimation in large scale settings. However, these algorithms are typically not robust to outliers or corruptions in the observed covariates. The concept of influence that was developed for regression diagnostics can be used to detect such corrupted observations as shown in this paper. This property of influence -- for which we also develop a randomized approximation -- motivates our proposed subsampling algorithm for large scale corrupted linear regression which limits the influence of data points since highly influential points contribute most to the residual error. Under a general model of corrupted observations, we show theoretically and empirically on a variety of simulated and real datasets that our algorithm improves over the current state-of-the-art approximation schemes for ordinary least squares.

In the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated. A necessary condition for the optimality of a given design is provided, which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data. If the regression functions are eigenfunctions of an integral operator defined by the covariance kernel, it is shown that the corresponding measure defines a universally optimal design. For several models universally optimal designs can be identified explicitly. In particular, it is proved that the uniform distribution is universally optimal for a class of trigonometric regression models with a broad class of covariance kernels and that the arcsine distribution is universally optimal for the polynomial regression model with correlation structure defined by the logarithmic potential. To the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations, which are not restricted to the location scale model.

In this book, we present a study of linear models and algorithms to find the optimal solution for them using the concepts of neuroscientific science. We know that the linear programming method is one of the important methods of operations research, the science that was the product of the great scientific development that our contemporary world is witnessing. The name operations research is given to the group of scientific methods used. In analyzing problems and searching for optimal solutions, it is a science whose applications have achieved widespread success in various fields of life.   What is meant by neutrosophic models are models in which the data are neutrosophic values, that is, variables such as in the objective function, which expresses profit if the model is a maximization model, and expresses a cost if the model is a minimization model, which in turn is affected by environmental conditions.

Least squares fitting is in general not useful for high-dimensional linear models, in which the number of predictors is of the same or even larger order of magnitude than the number of samples. Theory developed in recent years has coined a paradigm according to which sparsity-promoting regularization is regarded as a necessity in such setting. Deviating from this paradigm, we show that non-negativity constraints on the regression coefficients may be similarly effective as explicit regularization. For a broad range of designs with Gram matrix having non-negative entries, we establish bounds on the $\ell_2$-prediction error of non-negative least squares (NNLS) whose form qualitatively matches corresponding results for $\ell_1$-regularization. Under slightly stronger conditions, it is established that NNLS followed by hard thresholding performs excellently in terms of support recovery of an (approximately) sparse target, in some cases improving over $\ell_1$-regularization. A substantial advantage of NNLS over regularization-based approaches is the absence of tuning parameters, which is convenient from a computational as well as from a practitioner's point of view. Deconvolution of positive spike trains is presented as application.

We determine all the terms that are gauge-invariant up to a total spacetime derivative ("semi-invariant terms") for gauged non-linear sigma models. Assuming that the isotropy subgroup $H$ of the gauge group is compact or semi-simple, we show that (non-trivial) such terms exist only in odd dimensions and are equivalent to the familiar Chern-Simons terms for the subgroup $H$. Various applications are mentioned, including one to the gauging of the Wess-Zumino-Witten terms in even spacetime dimensions. Our approach is based on the analysis of the descent equation associated with semi-invariant terms.

A master equation description of the inherent dynamics is used to calculate the frequency-dependent linear thermo-visco-elastic response functions of a glass-forming liquid. From the imaginary parts of the isobaric specific heat, isothermal bulk modulus, and isobaric thermal expansion coefficient, we define a quantity $\Lambda_{Tp}(\omega)$ with the property that $\Lambda_{Tp}(\omega)=1$ is equivalent to having a one-parameter description of the linear thermo-visco-elastic response. This provides an alternative to the well-known criterion based on the Prigogine-Defay ratio.

We study d=2, N=(2,2) non-linear sigma-models in (2,2) superspace. By analyzing the most general constraints on a superfield, we show that through an appropriate choice of coordinates, there are no other superfields than chiral, twisted chiral and semi-chiral ones. We study the resulting sigma-models and we speculate on the possibility that all (2,2) non-linear sigma-models can be described using these fields. We apply the results to two examples: the SU(2) x U(1) and the SU(2) x SU(2) WZW model. Pending upon the choice of complex structures, the former can be described in terms of either one semi-chiral multiplet or a chiral and a twisted chiral multiplet. The latter is formulated in terms of one semi-chiral and one twisted chiral multiplet. For both cases we obtain the potential explicitely.

The qualitative analysis of the initial value problem P related to a non linear third order parabolic equation typical of diffusive models is discussed. Some basic properties of the the fundamental solution of a related linear operator are determined and are applied to an equivalent integro differential formulation of the problem. By the fixed point theorem, existence and uniqueness results are obtained

The models of the non-linear optics in which solitons were appeared are considered. These models are of paramount importance in studies of non-linear wave phenomena. The classical examples of phenomena of this kind are the self-focusing, self-induced transparency, and parametric interaction of three waves. At the present time there are a number of the theories based on completely integrable systems of equations, which are both generations of the original known models and new ones. The modified Korteweg-de Vries equation, the non- linear Schrodinger equation, the derivative non-linear Schrodinger equation, Sine-Gordon equation, the reduced Maxwell-Bloch equation, Hirota equation, the principal chiral field equations, and the equations of massive Thirring model are gradually putting together a list of soliton equations, which are usually to be found in non-linear optics theory.

In this paper, we obtain the central limit theorems for LS estimator in simple linear errors-in-variables (EV) regression models under some mild conditions. And we also show that those conditions are necessary in some sense.

A restricted dynamics, previously introduced in a kinetic model for relaxation phenomena in linear polymer chains, is used to study the dynamic critical exponent of one-dimensional Ising models. Both the alternating isotopic chain and the alternating-bond chain are considered. In contrast with what occurs for the Glauber dynamics, in these two models the dynamic critical exponent turns out to be the same. The alternating isotopic chain with the restricted dynamics is shown to lead to Nagel scaling for temperatures above some critical value. Further support is given relating the Nagel scaling to the existence of multiple (simultaneous) relaxation processes, the dynamics apparently not playing the most important role in determining such scaling.

Structural properties of two well-known families of keystream generators, Shrinking Generators and Cellular Automata, have been analyzed. Emphasis is on the equivalence of the binary sequences obtained from both kinds of generators. In fact, Shrinking Generators (SG) can be identified with a subset of linear Cellular Automata (mainly rule 90, rule 150 or a hybrid combination of both rules). The linearity of these cellular models can be advantageously used in the cryptanalysis of those keystream generators.