Infinite Sample Research Articles

A Bhattacharyya-type Conditional Error Bound for Quadratic Discriminant Analysis

We give an upper bound on the conditional error of Quadratic Discriminant Analysis (QDA), conditioned on parameter estimates. In the case of maximum likelihood estimation (MLE), our bound recovers the well-known Chernoff and Bhattacharyya bounds in the infinite sample limit. We perform an empirical assessment of the behaviour of our bound in a finite sample MLE setting, demonstrating good agreement with the out-of-sample error, in contrast with the simpler but uninformative estimated error, which exhibits unnatural behaviour with respect to the sample size. Furthermore, our conditional error bound is applicable whenever the QDA decision function employs parameter estimates that differ from the true parameters, including regularised QDA.

Relative permeability estimation using mercury injection capillary pressure measurements based on deep learning approaches

Relative permeability (RP) curves which provide fundamental insights into porous media flow behavior serve as critical parameters in reservoir engineering and numerical simulation studies. However, obtaining accurate RP curves remains a challenge due to expensive experimental costs, core contamination, measurement errors, and other factors. To address this issue, an innovative approach using deep learning strategy is proposed for the prediction of rock sample RP curves directly from mercury injection capillary pressure (MICP) measurements which include the mercury injection curve, mercury withdrawal curve, and pore size distribution. To capture the distinct characteristics of different rock samples' MICP curves effectively, the Gramian Angular Field (GAF) based graph transformation method is introduced for mapping the curves into richly informative image forms. Subsequently, these 2D images are combined into three-channel red, green, blue (RGB) images and fed into a Convolutional Long Short-Term Memory (ConvLSTM) model within our established self-supervised learning framework. Simultaneously the dependencies and evolutionary sequences among image samples are captured through the limited MICP-RP samples and self-supervised learning framework. After that, a highly generalized RP curve calculation proxy framework based on deep learning called RPCDL is constructed by the autonomously generated nearly infinite training samples. The remarkable performance of the proposed method is verified with the experimental data from rock samples in the X oilfield. When applied to 37 small-sample data spaces for the prediction of 10 test samples, the average relative error is 3.6%, which demonstrates the effectiveness of our approach in mapping MICP experimental results to corresponding RP curves. Moreover, the comparison study against traditional CNN and LSTM illustrated the great performance of the RPCDL method in the prediction of both So and Sw lines in oil–water RP curves. To this end, this method offers an intelligent and robust means for efficiently estimating RP curves in various reservoir engineering scenarios without costly experiments.

Uniform probability in cosmology

Problems with uniform probabilities on an infinite support show up in contemporary cosmology. This paper focuses on the context of inflation theory, where it complicates the assignment of a probability measure over pocket universes. The measure problem in cosmology, whereby it seems impossible to pick out a uniquely well-motivated measure, is associated with a paradox that occurs in standard probability theory and crucially involves uniformity on an infinite sample space. This problem has been discussed by physicists, albeit without reference to earlier work on this topic. The aim of this article is both to introduce philosophers of probability to these recent discussions in cosmology and to familiarize physicists and philosophers working on cosmology with relevant foundational work by Kolmogorov, de Finetti, Jaynes, and other probabilists. As such, the main goal is not to solve the measure problem, but to clarify the exact origin of some of the current obstacles. The analysis of the assumptions going into the paradox indicates that there exist multiple ways of dealing consistently with uniform probabilities on infinite sample spaces. Taking a pluralist stance towards the mathematical methods used in cosmology shows there is some room for progress with assigning probabilities in cosmological theories.

Geometry of sample spaces

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of Mn modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces.We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov–Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fréchet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Emergent Magnetic Field and Nonzero Gyrovector of the Toroidal Magnetic Hopfion

Magnetic hopfions are localized magnetic solitons with a nonzero 3D topological charge (Hopf index). Herein, an analytical calculation of the magnetic hopfion gyrovector is presented and it is shown that it does not vanish even in an infinite sample. The calculation method is based on the concept of the emergent magnetic field. The particular case of the simplest nontrivial toroidal hopfion with the Hopf index in the cylindrical magnetic dot is considered and dependencies of the gyrovector components on the dot sizes are calculated. Nonzero hopfion gyrovector is important in any description of the hopfion dynamics within the collective coordinate Thiele's approach.

Liquid-crystal composites of carbon nanotubes in a magnetic field: Bridging continuum theory and a molecular-statistical approach.

We propose an approach combining the continuum theory and molecular-statistical approach for a suspension of carbon nanotubes based on a negative diamagnetic anisotropy liquid crystal. Using the continuum theory, we show that in the case of an infinite sample in suspension it is possible to observe peculiar magnetic Fréedericksz-like transitions between three nematic phases: planar, angular, and homeotropic with different mutual orientations of liquid-crystal and nanotube directors. The transition fields between these phases are found analytically as functions of material parameters of the continuum theory. To account for the effects associated with temperature changes, we propose a molecular-statistical approach that allows obtaining the equationsof orientational state for the orientation angles of the main axes of the nematic order, i.e., the liquid-crystal and carbon-nanotube directors in a similar form as was obtained within the continuum theory. Thus, it is possible to relate the parameters of the continuum theory, such as the surface-energy density of a coupling between molecules and nanotubes, to the parameters of the molecular-statistical model and the order parameters of the liquid crystal and carbon nanotubes. This approach allows determining the temperature dependencies of the threshold fields of transitions between different nematic phases, which is impossible in the framework of the continuum theory. In the framework of the molecular-statistical approach we predict the existence of an additional direct transition between the planar and homeotropic nematic phases of the suspension, which cannot be described based on the continuum theory. As the main results, the magneto-orientational response of the liquid-crystal composite is studied and a possible biaxial orientational ordering of the nanotubes in the magnetic field is shown.

Bayesian nonparametric mixtures of Exponential Random Graph Models for ensembles of networks

Ensembles of networks arise in various fields where multiple independent networks are observed, for example, a collection of student networks from different classes. However, there are few models that describe both the variations and characteristics of networks in an ensemble at the same time. In this manuscript, we propose to model ensembles of networks using a Dirichlet Process Mixture of Exponential Random Graph Models (DPM-ERGMs), which divides an ensemble into different clusters and models each cluster of networks using a separate Exponential Random Graph Model (ERGM). By employing a Dirichlet process mixture, the number of clusters can be determined automatically and changed adaptively with the data provided. Moreover, in order to perform full Bayesian inference for DPM-ERGMs, we develop a Metropolis-within-slice sampling algorithm to address the problem of sampling from the intractable ERGMs on an infinite sample space. We also demonstrate the performance of DPM-ERGMs with both simulated and real datasets.

Asymptotic bias of the hazard probability model under model mis-specification

We compare the sensitivity of the estimated effective strip half-width with respect to choice of hazard probability function (Q). This is done by fitting the model under an erroneous assumption about the parametric form of Q, and comparing the estimated and true effective strip half-width. An ‘infinite sample size’ setting is employed, where fitting the model by maximum likelihood amounts to minimising the Kullback Leibler distance between the assumed and true models. The experiment is carried out in a situation that is relevant to minke whale sighting surveys both in the Antarctic and in the northeastern Atlantic. It is found that the hazard probability model is fairly robust with respect to the choice of parametric class for Q. The largest observed bias in the resulting effective strip half-width is less than 10%, while for most situations there is almost no bias.

The risk-return relationship and volatility feedback in South Africa: a comparative analysis of the parametric and nonparametric Bayesian approach

<abstract> <p>This study aimed to investigate the risk-return relationship, provided volatility feedback was taken into account, in the South African market. Volatility feedback, a stronger measure of volatility, was treated as an important source of asymmetry in the investigation of the risk-return relationship. This study analyzed the JSE ALSI excess returns and realized variance for the sample period from 15 October 2009 to 15 October 2019. This study modelled the novel and robust Bayesian approach in a parametric and nonparametric framework. A parametric model has modelling assumptions, such as normality, and a finite sample space. A nonparametric approach relaxes modelling assumptions and allows for an infinite sample space; thus, taking into account every possible asymmetric risk-return relationship. Given that South Africa is an emerging market, which is subject to higher levels of volatility, the presence of volatility feedback was expected to be more pronounced. However, contrary to expectations, the test results from both the parametric and nonparametric Bayesian model showed that volatility feedback had an insignificant effect in the South African market. The risk-return relationship was then investigated free from empirical distortions that resulted from volatility feedback. The parametric Bayesian model found a positive risk-return relationship, in line with traditional theoretical expectations. However, the nonparametric Bayesian model found no relationship between risk and return, in line with early South African studies. Since the nonparametric Bayesian approach is more robust than the parametric Bayesian approach, this study concluded that there is no risk-return relationship. Therefore, investors can include South Africa in their investment portfolio with higher risk countries in order to spread their risk and derive diversification benefits. In addition, risk averse investors can find a safe environment within the South African market and earn a return in accordance to their risk tolerance.</p> </abstract>

Effect of Production Strategy Performance on Brand Royalty


 
 
 
 Background: Product strategy is a plan that identifies the goals and objectives of a product and then explains the vision to achieve those goals. This helps to relate the vision of the project with the actual methods used to implement it. Product strategy maps out key benchmarks for creating, marketing, and distributing any concept you plan to sell.
 Aim: The purpose of this study is to determine the effect of production strategy performance on brand royalty in companies and restaurants.
 Method: The sample used in this study can be determined using the infinite sample formula, with a 95% confidence level and a 10% error level, the sample used in this study was 100 people as respondents. The sampling technique in this study is non probability sampling with the Purposive Sampling.
 Findings: They also provide a central plan that people across the business can refer to to guide their efforts and fine-tune their overall business strategy. Business in the food sector is a potential business at this time. Factors that need to be considered in facing competition in the restaurant business include products and prices. If these two factors are getting better and more attractive, it can make customers feel satisfied, if customers feel satisfied, they will be loyal to the restaurant
 
 
 

Effect of Production Strategy Performance on Brand Royalty


 
 
 
 Background: Product strategy is a plan that identifies the goals and objectives of a product and then explains the vision to achieve those goals. This helps to relate the vision of the project with the actual methods used to implement it. Product strategy maps out key benchmarks for creating, marketing, and distributing any concept you plan to sell.
 Aim: The purpose of this study is to determine the effect of production strategy performance on brand royalty in companies and restaurants.
 Method: The sample used in this study can be determined using the infinite sample formula, with a 95% confidence level and a 10% error level, the sample used in this study was 100 people as respondents. The sampling technique in this study is non probability sampling with the Purposive Sampling.
 Findings: They also provide a central plan that people across the business can refer to to guide their efforts and fine-tune their overall business strategy. Business in the food sector is a potential business at this time. Factors that need to be considered in facing competition in the restaurant business include products and prices. If these two factors are getting better and more attractive, it can make customers feel satisfied, if customers feel satisfied, they will be loyal to the restaurant
 
 
 

Distributional regression and its evaluation with the CRPS: Bounds and convergence of the minimax risk

The theoretical advances in the properties of scoring rules over the past decades have broadened the use of scoring rules in probabilistic forecasting. In meteorological forecasting, statistical postprocessing techniques are essential to improve the forecasts made by deterministic physical models. Numerous state-of-the-art statistical postprocessing techniques are based on distributional regression evaluated with the continuous ranked probability score (CRPS). However, the theoretical properties of such evaluations with the CRPS have solely considered the unconditional framework (i.e. without covariates) and infinite sample sizes. We extend these results and study the rate of convergence in terms of the CRPS of distributional regression methods. We find the optimal minimax rate of convergence for a given class of distributions and show that the k-nearest neighbor method and the kernel method reach this optimal minimax rate.

Nonconvex regularization for sparse neural networks

Convex ℓ 1 regularization using an infinite dictionary of neurons has been suggested for constructing neural networks with desired approximation guarantees, but can be affected by an arbitrary amount of over-parametrization. This can lead to a loss of sparsity and result in networks with too many active neurons for the given data, in particular if the number of data samples is large. As a remedy, in this paper, a nonconvex regularization method is investigated in the context of shallow ReLU networks: We prove that in contrast to the convex approach, any resulting (locally optimal) network is finite even in the presence of infinite data (i.e., if the data distribution is known and the limiting case of infinite samples is considered). Moreover, we show that approximation guarantees and existing bounds on the network size for finite data are maintained.

Accuracy, probabilism and Bayesian update in infinite domains

Scoring rules measure the accuracy or epistemic utility of a credence assignment. A significant literature uses plausible conditions on scoring rules on finite sample spaces to argue for both probabilism—the doctrine that credences ought to satisfy the axioms of probabilism—and for the optimality of Bayesian update as a response to evidence. I prove a number of formal results regarding scoring rules on infinite sample spaces that impact the extension of these arguments to infinite sample spaces. A common condition in the arguments for probabilism and Bayesian update is strict propriety: that according to each probabilistic credence, the expected accuracy of any other credence is worse. Much of the discussion needs to divide depending on whether we require finite or countable additivity of our probabilities. I show that in a number of natural infinite finitely additive cases, there simply do not exist strictly proper scoring rules, and the prospects for arguments for probabilism and Bayesian update are limited. In many natural infinite countably additive cases, on the other hand, there do exist strictly proper scoring rules that are continuous on the probabilities, and which support arguments for Bayesian update, but which do not support arguments for probabilism. There may be more hope for accuracy-based arguments if we drop the assumption that scores are extended-real-valued. I sketch a framework for scoring rules whose values are nets of extended reals, and show the existence of a strictly proper net-valued scoring rules in all infinite cases, both for f.a. and c.a. probabilities. These can be used in an argument for Bayesian update, but it is not at present known what is to be said about probabilism in this case.

Effect on essential health services during COVID-19 at the Primary level in India.

Coronavirus diesease (COVID-19) led to increased demand on the Indian health system due to the pandemic as well as other communicable and non-communicable diseases. Guidance was thus issued by the Ministry of Health and Family Welfare (MoHFW), India, in April 2020 to maintain the delivery of essential health services. To determine the extent of disruptions of essential healthcare services, identify associated factors, and establish pertinent correlations to address specific needs. The Mother and child tracking facilitation centre (MCTFC) conducted a telephonic survey with the front-line workers (FLWs) and beneficiaries in 21 Indian states. The sample size was determined using the infinite population sample size formula, and respondents were selected through a computer-generated random sequence technique. Data were quantitatively analysed using STATA-16. Descriptive univariate analysis was conducted using the Chi-square test. The majority of the essential health services were being satisfactorily delivered by FLWs (N = 1596; accredited social health activist (ASHA) = 798, auxiliary nurse midwife (ANM) = 798), where most of the beneficiaries (N = 1410; Pregnant Women = 708, Postnatal Women = 702) continued accessing services with minor issues concerning referral transport. FLWs reported issues in the provisioning of medicines (P = 0.000) for patients with non-communicable diseases and more ANMs than ASHAs reported it. FLWs commonly experienced challenges in extending services due to community resistance and unavailability of general health services at healthcare facilities, where a greater number of ASHAs faced it (P = 0.000). Both FLWs and beneficiaries (N = 3006; FLWs = 1596, beneficiaries = 1410) demonstrated appropriate COVID-19 knowledge and behavior. Although overwhelmed, the Indian health system performed satisfactorily well during pandemic in terms of essential health services.

Quality assurance for automatically generated contours with additional deep learning

ObjectiveDeploying an automatic segmentation model in practice should require rigorous quality assurance (QA) and continuous monitoring of the model’s use and performance, particularly in high-stakes scenarios such as healthcare. Currently, however, tools to assist with QA for such models are not available to AI researchers. In this work, we build a deep learning model that estimates the quality of automatically generated contours.MethodsThe model was trained to predict the segmentation quality by outputting an estimate of the Dice similarity coefficient given an image contour pair as input. Our dataset contained 60 axial T2-weighted MRI images of prostates with ground truth segmentations along with 80 automatically generated segmentation masks. The model we used was a 3D version of the EfficientDet architecture with a custom regression head. For validation, we used a fivefold cross-validation. To counteract the limitation of the small dataset, we used an extensive data augmentation scheme capable of producing virtually infinite training samples from a single ground truth label mask. In addition, we compared the results against a baseline model that only uses clinical variables for its predictions.ResultsOur model achieved a mean absolute error of 0.020 ± 0.026 (2.2% mean percentage error) in estimating the Dice score, with a rank correlation of 0.42. Furthermore, the model managed to correctly identify incorrect segmentations (defined in terms of acceptable/unacceptable) 99.6% of the time.ConclusionWe believe that the trained model can be used alongside automatic segmentation tools to ensure quality and thus allow intervention to prevent undesired segmentation behavior.

KNOWLEDGE AND PERCEPTION ABOUT COVID-19 AMONGST RESIDENTS IN EDO STATE, NIGERIA

The novel severe acute respiratory syndrome coronavirus (SARS-CoV-2) has challenged public health globally which causes the disease named, by the World Health Organization, coronavirus disease 2019 (COVID-19). Edo State accounts for 3.07% of the total 245,856 cases in Nigeria. The objective of this paper was to assess the knowledge and perception of residents in Edo state, Nigeria toward COVID-19. A sample size of 281 was calculated using the adjusted Cochran formula for infinite sample size using an assumed prevalence of 0.24. A structured questionnaire was designed using Google Forms and distributed through online platforms. Knowledge of respondents was assigned scores and ranked as either good or poor. A total of 307 responses were completed and returned. The mean age of respondents was 37.3 years with a median value of 36. The overall knowledge of respondents was adjudged to be poor in 34.9% (N = 107) and good in 65.1% (N = 200) of respondents respectively. Good knowledge of respondents was significantly associated with an accurate indication of COVID-19 etiology, its spread, natural reservoir, therapeutics, and age groups at risk of infection. The perception that COVID-19 could be a fatal disease with no cure was indicated by 39.7% (N=122) of respondents. The media may serve as a readily accessible source of information but may misinform, disinform and sway public opinions. With the unprecedented ease of information dissemination due to current advancements in technology in recent centuries, there is the need to scrutinize the various mainstream sources of information.

Supervised Multivariate Learning with Simultaneous Feature Auto-Grouping and Dimension Reduction

Abstract Modern high-dimensional methods often adopt the ‘bet on sparsity’ principle, while in supervised multivariate learning statisticians may face ‘dense’ problems with a large number of nonzero coefficients. This paper proposes a novel clustered reduced-rank learning (CRL) framework that imposes two joint matrix regularizations to automatically group the features in constructing predictive factors. CRL is more interpretable than low-rank modelling and relaxes the stringent sparsity assumption in variable selection. In this paper, new information-theoretical limits are presented to reveal the intrinsic cost of seeking for clusters, as well as the blessing from dimensionality in multivariate learning. Moreover, an efficient optimization algorithm is developed, which performs subspace learning and clustering with guaranteed convergence. The obtained fixed-point estimators, although not necessarily globally optimal, enjoy the desired statistical accuracy beyond the standard likelihood setup under some regularity conditions. Moreover, a new kind of information criterion, as well as its scale-free form, is proposed for cluster and rank selection, and has a rigorous theoretical support without assuming an infinite sample size. Extensive simulations and real-data experiments demonstrate the statistical accuracy and interpretability of the proposed method.

Marginalized Graph Self-Representation for Unsupervised Hyperspectral Band Selection

Unsupervised band selection is an essential step in preprocessing hyperspectral images (HSIs) to select informative bands. Most existing methods exploit the spatial information from the entire HSI while ignoring the difference between diverse homogeneous regions. Moreover, traditional methods utilize the limited size of data for model training that may result in degraded generalization performance. In this article, we propose a marginalized graph self-representation (MGSR) method for unsupervised hyperspectral band selection. To explore the spatial information from diverse homogenous regions, MGSR generates the segmentations of an HSI by superpixel segmentation and records the relationships between adjacent pixels of the same segmentation in a structural graph. Meanwhile, to improve the generalization and robustness, infinite corrupted samples are obtained from the original pixels by introducing noises in spectral bands for model training. To solve the proposed formulation, we design an alternating optimization algorithm to marginalize out the corruption and search for the optimal solution. Experimental studies on HSI datasets demonstrate the effectiveness of the proposed MGSR and the superiority over the state-of-the-art methods. The source code is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/ZhangYongshan/MGSR</uri> .

Correction to “Planning With Learned Dynamics: Probabilistic Guarantees on Safety and Reachability Via Lipschitz Constants” [Jul21 5129-5136

We wish to make the following corrections and clarifications to our manuscript [1]. For a version of the manuscript that has these changes integrated into the text, please see [2]. •In [1], the method is claimed to provide safety guarantees with probability $\rho$; this probability should instead be $\rho ^3$. This results from the Lipschitz constant estimation approach described in Sec. IV.B of [1] returning valid estimates with probability $\rho$. Our method requires three Lipschitz constants: $L_{f-g}$ (the Lipschitz constant of the model error), $L_{g_0}$, and $L_{g_1}$ (the Lipschitz constants for each term in the learned control-affine dynamics). If independent samples are used to estimate each constant, and each estimate is valid with probability $\rho$, the probability of all three estimates being valid is $\rho ^3$. We note that it is possible to achieve the original safety probability of $\rho$ if one can obtain guaranteed overapproximations of $L_{g_0}$ and $L_{g_1}$. In [1], we model $g_0$ and $g_1$ with neural networks, and there is an expanding body of work for more scalably obtaining a guaranteed (though potentially loose) overestimate of the Lipschitz constant for large networks, e.g. [3]. Moreover, this probability $\rho ^3$ holds exactly only in the limit of infinite samples for estimating each Lipschitz constant. This is due to the Fisher-Tippett-Gnedenko theorem [4], which justifies our Lipschitz estimation method, being an asymptotic result. •The Weibull distribution we fit in Sec. IV.B of [1] is specifically the three-parameter reverse Weibulldistribution, which for a random variable $W$ has the cumulative distribution function \begin{equation*} F_W(w) = {\begin{cases}\exp (-(\frac{\gamma - w}{\alpha })^\beta), & \text {if } \; w < \gamma \\ 1, & \text {if }\; w \geq \gamma, \end{cases}} \end{equation*} where $\alpha$, $\beta$, and $\gamma$ are the scale, shape, and location parameters, respectively. •In line 6 of Algorithm 2 in [1], if $L_{f-g} \geq 1$, then failure should be returned (as our planner requires $L_{f-g} < 1$). •There is some notational confusion in [1] between $r$ (the radius of the balls used to define the trusted domain $D$) and $b_T$ (the dispersion of the dataset within $D$); please see [2] for the corrections.