Amount Of Smoothing Research Articles

Bayesian Trend Filtering via Proximal Markov Chain Monte Carlo

Proximal Markov chain Monte Carlo is a novel construct that lies at the intersection of Bayesian computation and convex optimization, which helped popularize the use of nondifferentiable priors in Bayesian statistics. Existing formulations of proximal MCMC, however, require hyperparameters and regularization parameters to be prespecified. In this article, we extend the paradigm of proximal MCMC through introducing a novel new class of nondifferentiable priors called epigraph priors. As a proof of concept, we place trend filtering, which was originally a nonparametric regression problem, in a parametric setting to provide a posterior median fit along with credible intervals as measures of uncertainty. The key idea is to replace the nonsmooth term in the posterior density with its Moreau-Yosida envelope, which enables the application of the gradient-based MCMC sampler Hamiltonian Monte Carlo. The proposed method identifies the appropriate amount of smoothing in a data-driven way, thereby automating regularization parameter selection. Compared with conventional proximal MCMC methods, our method is mostly tuning free, achieving simultaneous calibration of the mean, scale and regularization parameters in a fully Bayesian framework. Supplementary materials for this article are available online.

Correct specification of design matrices in linear mixed effects models: tests with graphical representation

Linear mixed effects models (LMMs) are a popular and powerful tool for analysing grouped or repeated observations for numeric outcomes. LMMs consist of a fixed and a random component, which are specified in the model through their respective design matrices. Verifying the correct specification of the two design matrices is important since mis-specifying them can affect the validity and efficiency of the analysis. We show how to use empirical stochastic processes constructed from appropriately ordered and standardized residuals from the model to test whether the design matrices of the fitted LMM are correctly specified. We define two different processes: one can be used to test whether both design matrices are correctly specified, and the other can be used only to test whether the fixed effects design matrix is correctly specified. The proposed empirical stochastic processes are smoothed versions of cumulative sum processes, which have a nice graphical representation in which model mis-specification can easily be observed. The amount of smoothing can be adjusted, which facilitates visual inspection and can potentially increase the power of the tests. We propose a computationally efficient procedure for estimating p-values in which refitting of the LMM is not necessary. Its validity is shown by using theoretical results and a large Monte Carlo simulation study. The proposed methodology could be used with LMMs with multilevel or crossed random effects.

Cohesive parcellation of the human brain using resting-state fMRI

BackgroundThe data burden for resting-state fMRI analysis rises with increasing resolutions available at ultrahigh fields. Therefore, a fundamental preprocessing step in brain network analysis is to reduce the data, usually by performing some kind of data parcellation. Most functional parcellations based on rsfMRI connectivity are synthesized from the dense connectome. In contrast, most network analyses begin by reducing each parcel to a single exemplar time series. This disconnect between parcel formation and usage assumes that parcel exemplars adequately represent their member voxels, which is not always the case for commonly used parcellations. New methodWe propose to parcellate the brain based on parcel cohesion, a measure of similarity between a parcel’s exemplar and its member voxels. A spatially constrained agglomerative hierarchical framework is used to synthesize parcels based on a minimum cohesion threshold, rather than a predetermined number of parcels. ResultsCohesive parcellation generally results in more parcels than existing approaches. The number of parcels scales with the amount of smoothing in preprocessing, yet retains adequate information to extract common intrinsic functional networks. Comparison with previous methodsCohesive parcellation performs better than several widely used anatomical, functional, and data-driven parcellations on the basis of parcel cohesion and comparably using several traditional measures of cluster validity. ConclusionCohesive parcellation ensures that the way parcels are synthesized directly corresponds to the way they are used in subsequent analyses. The resulting parcels are straightforward to interpret and optimal for downstream analysis.

A Bayesian Approach for Penalized Splines with Hierarchical Penalty

Penalized spline has largely been applied in many research studies not limited to disease modeling and epidemiology. However, due to spatial heterogeneity of the data because different smoothing parameter leads to different amount of smoothing in different regions the penalized spline has not been exclusively appropriate to fit the data. The study assessed the properties of penalized spline hierarchical model; the hierarchy penalty improves the fit as well as the accuracy of inference. The simulation demonstrates the potential benefits of using the hierarchical penalty, which is obtained by modelling the global smoothing parameter as another spline. The results showed that mixed model with penalized hierarchical penalty had a better fit than the mixed model without hierarchy this was demonstrated by the rapid convergence of the model posterior parameters and the smallest DIC value of the model. Therefore hierarchical model with fifteen sub-knots provides a better fit of the data.

A Method of Estimating the Partial Power Spectrum of a Bivariate Point Process and an Application to a Neurophysiological Data Set

In this paper, we present a method of estimating the partial power spectrum of a bivariate point process. The method is based on the simple and cross-periodograms which are smoothed by applying a simple averaging weighting scheme in order to improve the properties of the estimate. The asymptotic distribution of the estimate is shown to be approximately normal when the amount of smoothing is large. An illustrative example is presented from the field of Neurophysiology when the complex system of the muscle spindle is affected by a static gamma motoneuron.

A Method for Generating Coherent Spatially Explicit Maps of Seasonal Paleoclimates From Site‐Based Reconstructions

AbstractWe describe a method for reconstructing spatially explicit maps of seasonal paleoclimate variables from site‐based reconstructions. Using a 3‐D‐Variational technique, the method finds the best statistically unbiased, and spatially continuous, estimate of the paleoclimate anomalies through combining the site‐based reconstructions and a prior estimate of the paleoclimate state. By assuming a set of correlations in the error of the prior, the resulting climate is smoothed both from month to month and from grid cell to grid cell. The amount of smoothing can be controlled through the choice of two length‐scale values. The method is applied to a set of reconstructions of the climate of the Last Glacial Maximum (ca. 21,000 years ago) for southern Europe derived from pollen data with a prior derived from results from the third phase of the Palaeoclimate Modelling Intercomparison Project. We demonstrate how to choose suitable values for the smoothing length scales for the data sets used in the reconstruction.

Deep Supervised Hashing Based on Stable Distribution

Recently, the convolutional neural network (CNN)-based hashing method has achieved its promising performance for image retrieval. However, tackling the discrepancy between quantization error minimization and discriminability maximization of the network outputs simultaneously still remains unsolved. Distinguished from the previous works, which only can search an equilibrium point within the discrepancy, we propose a novel deep supervised hashing based on stable distribution (DSHSD) to eliminate the discrepancy with distribution consistency guarantee. First, we utilize a smooth projection function, in which the amount of smoothing is adaptable, to relax the discrete constraint instead of any quantization regularizer. Second, a mathematical connection between the smooth projection and the feature distribution is made to maintain distribution consistency. A relaxed multi-semantic information fusion method is implemented to make hash codes learned to preserve more semantic information and accelerate the training convergence. According to stable distribution, we propose a novel hashing framework to eliminate the discrepancy and support fast image retrieval. The extensive experiments on the CIFAR-10, NUS-WIDE, and ImageNet datasets show that our method can outperform the state-of-the-art methods from various perspectives.

Propose a Comparison Method of the PV Variability Based on roof-top PV solar data of Australia

The use of renewable sources of energy is rising in Australia, and with solar energy becoming the most dominant; the solar (PV) roof-top plant penetration in the electrical energy distribution grid is increasing. As Australia is the sixth largest country in the world consisting of a diverse range of climates, this may be a concern to Distribution Service Operators (DSOs) as the variability in PV power output in different areas, climates/weather and even time of day. This means that DSOs are required to quantify these ‘uncertainties’ for different zones in Australia to aid in the energy planning. This paper will examine PV variability metrics to identify suitable PV variable metric based on purpose of application and propose a method to compare PV variability of large cities in Australia based on historical roof-top PV solar data. This proposed method examined variability metrics and find out suitable variability metric based on purpose of application. The comparative study shows that the PV variability and the amount of smoothing are not equal at all the distribution area in Australia and varies with geographical climatic scenario.

A multi-resolution and adaptive 3-D image denoising framework with applications in medical imaging

Due to recent increase in the usage of 3-D magnetic resonance images (MRI) and analysis of functional magnetic resonance images, research on 3-D image processing becomes important. Observed 3-D images often contain noise which should be removed in such a way that important image features, e.g., edges, edge structures, and other image details should be preserved, so that subsequent image analyses are reliable. Most image denoising methods in the literature are for 2-D images. However, their direct generalizations to 3-D images cannot preserve complicated edge structures well. Because, the edge structures in a 3-D edge surface can be much more complicated than the edge structures in a 2-D edge curve. Moreover, the amount of smoothing should be determined locally, depending on local image features and local signal to noise ratio, which is much more challenging in 3-D images due to large number of voxels. This paper proposes an efficient 3-D image denoising procedure based on local clustering of the voxels. This method provides a framework for determining the size of bandwidth and the amount of smoothing locally by empirical procedures. Numerical studies and a real MRI denoising show that it works well in many medical image denoising problems.

Coal resource estimation using Gaussian copula

Coal resource estimation is an important part of coal resource modeling. Kriging, inverse distance weighting and nearest neighborhood are traditional methods used in coal resource estimation. These methods are based on weighted average of local data values but data values are not involved in determination of the weights. For example, kriging uses variogram function, inverse distance weighting Euclidian distance and nearest neighborhood the nearest data. In this study an estimation method based on Gaussian copula is suggested which combines sampling configuration and local data values. The method is applied to coal field located in Thrace region which is at the northwest side of the Turkey. Lower Calorific Value is the variable under study. Following the construction of database and geological solid model of the coal seam, the mean calorific value of each block is estimated by Gaussian copula. The copula estimates are compared with the kriged estimates. The results show that the estimation with Gaussian copula captures local variability well and produces estimates with less amount of smoothing than kriging.

SAR Image Despeckling by the Use of Variational Methods With Adaptive Nonlocal Functionals

In this paper, we focus on the despeckling of synthetic aperture radar (SAR) images by variational methods which introduce nonlocal regularization functionals. To achieve this goal, two models are investigated from different aspects. The first model is derived for the logarithmically transformed (homomorphic) domain of the SAR data, and the other is derived for the original (nonhomomorphic) domain. The statistical properties of the speckle and the log-transformed speckle are analyzed, and the similarity measurements between pixels in the homomorphic domain and nonhomomorphic domain are then derived for constructing the corresponding nonlocal regularization functionals. Meanwhile, in the proposed models, we develop a strategy to adaptively choose the regularization parameters based on both the local heterogeneity information and the noise level of the images, aiming at getting a better balance between the goodness of fit of the original data and the amount of smoothing. A quasi-Newton iteration method is employed to quickly minimize the proposed adaptive nonlocal functionals. Experiments conducted on both simulated images and real SAR images confirm the good performances of the proposed methods, both in reducing speckle and preserving image quality.

Feasibility of reproducible vendor independent estimation of cardiac function based on first generation speckle tracking echocardiography

Background: The clinical approval of speckle tracking echocardiography (STE) as an accepted measure of myocardial strain and of LV function is hindered by the discordance of the results among the vendors. Since echocardiography images are noisy, the measured displacements are smoothed or regularized, an operation affecting the strain results. We introduce an “Error-dependent weighted speckle tracking” (EWST) algorithm that allows sensitivity analysis to the different operations affecting noise and accuracy. The aim here was to study whether by modifying the properties of the post block-matching weighted smoothing in the EWST algorithm it was possible to assess the expected inter-vendor strain differences. Methods and results: 48 echocardiographic clips generated by a software-based phantom were used as “gold standard” for validation of the EWST algorithm. Also, a cohort of 435 normal subjects and a cohort of 47 patients, scanned/re- scanned at 2 frame-rates (~70; ~35), were studied using the EWST. The results were compared to those produced by a commercial product of a leading manufacturer (STELV). Peak global longitudinal strains [PRLS, (%)] and peak regional longitudinal strains [PRLS, (%)] were calculated and compared. Sensitivity to the region (ROI) determination was tested by shifting the apical endocardial boundary. The differences between the measured PGLS and the ground truth produced by the software-based phantom (average ± standard deviation) were 0.4% ± 0.6% and 1.0% ± 0.7% for the EWST and STELV, respectively. Normal values were calculated for 435 subjects: -18.82% ± 2.45%, -20.2% ± 5.6%, -19.62% ± 3.62%, 18.77% ± 4.31% by the EWST, and -21.24% ± 2.91%, -26.5% ± 5.0%, -21.1% ± 3.7%, -18.0% ± 3.9%, by the STELV, respectively, for the PGLS, the peak longitudinal apical, mid-ventricle and basal regions, respectively. A low bias, but significant, was found between PGLS, when calculated for the cohort of 47 patients scanned/re-scanned at 2 frame-rates: -0.80% ± 2.61% and -1.66% ± 2.66% for the EWST and STELV, respectively. When the apex location (and thus the ROI) was shifted, the bias (mm) (average ± standard deviation) relative to the default position was: 0.82 ± 1.04; 0.61 ± 0.72; -1.06 ± 0.75; and -1.87 ± 0.72, for displacement of 5 cm, 2.5 cm, -2.5 cm, and -5 cm, respectively, for the STELV. The EWST proved similarly sensitive to the shifting of the apex location. Conclusions: STE is sensitive to the characteristics and amount of smoothing, as well as to the ROI positioning. Modification of the smoothing can produce different stain results, and different distribution of the regional strains. Thus it is preferable to use automatic determination of the ROI and methods that employ minimal smoothing or regularization.

Does estimator choice influence our ability to detect changes in home-range size?

Estimates of home-range size are frequently used to compare areal requirements of animals over time or space. Comparative studies of home-range estimators have highlighted extreme differences among general classes of methods (e.g., polygon-based and kernel density-based estimators) and sensitivity to the choice of various tuning parameters (e.g., amount of smoothing). These studies, however, have largely failed to consider how estimates of home-range size are typically used in applied research. We illustrate simulation-based methods for comparing estimators, which focus on relative differences in home-range size (over time or space), rather than their absolute magnitude. We also consider Global Positioning Technology (GPS) location data from a black bear (Ursus americanus) from northwestern Minnesota, USA, to illustrate the relevance to real-world data applications. In our examples, estimates of home-range size often differed considerably in absolute magnitude. Yet, for relative differences, the choice of home-range estimator was often negligible. Furthermore, choosing the right estimator was less important than other aspects of study design (e.g., number of animals followed). Many questions in ecology focus on changes in space-use patterns (over space or time). For these types of questions, home-range estimators should be evaluated in terms of their ability to detect these spatial and temporal patterns. More importantly, home-range estimation should be seen as a means to an end—i.e., estimators provide indices useful for addressing interesting biological questions or hypotheses—rather than as an end to itself.

Adaptive-Velocity Time-Geographic Density Estimation for Mapping the Potential and Probable Locations of Mobile Objects

Time-geographic density estimation (TGDE) provides a method for generating probability density surfaces for mobile objects. The technique operates by fitting distance-weighted geo-ellipses to each pair of points in a tracking dataset and combining them to create a final density surface. The sizes of the geo-ellipses, and thus the amount of smoothing, are determined by a velocity parameter which specifies how fast the object can travel. In prior formulations of TGDE this maximum-velocity parameter was treated as constant or fixed. This can be problematic if an object displays variable movement patterns, such as alternating between movements and stops. In this research we develop a more robust formulation of time-geographic density estimation, termed adaptive-velocity TGDE, which allows the maximum-velocity parameter to vary for each segment of the space-time path. First, the new mathematical formulation is demonstrated and compared with fixed-TGDE through an application to synthetic pedestrian tracking data. Second, both methods are evaluated in the context of accurately mapping the movement patterns of a homing pigeon tracked by GPS. The results of both applications demonstrate that adaptive-velocity TGDE has the potential to more accurately delineate the trajectories of mobile objects than the fixed-velocity version. Implications of these results in the context of developing a probabilistic time geography for general mobile objects are also discussed.

On the estimation of the structural index from low-pass filtered magnetic data

The estimation of the structural index and of the depth to the source is the main task of many popular methods used to analyze potential field data, such as Euler deconvolution. However, these estimates are unstable even in the presence of a weak amount of noise, and Euler deconvolution of noisy data leads to an underestimation of structural index and depth. We have studied how the structural index and depth estimates are affected by applying low-pass filtering to the data. Physically based low-pass filters, such as upward continuation and integration, have been shown to be the best choice over a range of altitudes (upward continuation) or orders (integration filters), mainly because their outputs have a well-defined physical meaning. In contrast, mathematical low-pass filters require that the filter parameters be tuned carefully by means of several trial tests to produce optimally smoothed fields. The C-norm criterion is a reliable strategy to produce a stabilized vertical derivative, and we discourage Butterworth filters because they tend to a vertical integral filter, for a high cutoff wavenumber, thus complicating the interpretation of the estimated structural index. We found that the estimated structural index and depth to source increase proportionally with the amount of smoothing, unless in the case of overfiltering. In that case, the severe distortion of the original field may cause a decrease of the estimated structural index and depth to source.

Surface models and the spatial structure of population variables: Exploring smoothing effects using Northern Ireland grid square data

Where areal units used to report population counts from Censuses and other sources are incompatible, direct comparison of counts is not possible. To enable such comparisons, a wide variety of areal interpolation and surface modelling approaches have been developed to reallocate counts from one zonal system to another or to a regular grid. The particular characteristics of individual variables, representing population sub-groups, mean that the most accurate results for each sub-group may be obtained using quite different approaches, or different model parameters. This paper seeks to assess how the degree of smoothing associated with population surface modelling relates to the accuracy of predictions made using two variables in Northern Ireland – the number of Catholics and persons with a limiting long term illness (LLTI). The study makes use of counts for 2001 released for output areas (OAs) and wards to generate population grids with 100m square cells. The accuracy of the predictions is then systematically assessed using counts released for 100m grid cells as an additional output from the 2001 Census. The results show that the amount of smoothing and the spatial structure of the variables are related to the prediction errors and this suggests that use of information on the spatial structure of variables is likely to provide benefits, in terms of accuracy of population reallocations, over common areal weighting approaches.

Optimal Choice of Splines and Knots in TPSPLINE and TRANSREG Procedures

Multivariate functions observed with noise can be approximated by thin-plate smoothing splines. It does not depend on the assumptions of the parametric model. The amount of smoothing can be judged by generalized cross validation. If there is no prior knowledge about the model and the data is unable to represent a model with fixed number of parameters then TPSPLINE procedure is appropriate. With the increase in sample size, the model space also increases but this situation can be handled by thin-plate smoothing spline i.e. it is suitable for complicated situations. We worked on optimal choice of knots in TPSPLINE procedure. The procedure has been demonstrated by taking real life data set. The TRANSREG procedure which is applicable to many models including ordinary, multiple, multivariate regression with variable transformations is also discussed. It can also fit regression functions with smooth, spline or penalized B-splines. This procedure uses the method of alternating least-squares, that is, finding least-square estimates of the model parameters given the current scoring of the data, and then finding least-square estimates of the scoring parameters based on the current set of model parameters. In this paper, the fitting of the model through penalized B-splines using splines for AICC, AIC, CV, SBC and GCV criterion have been discussed and compared by taking a real life data set.

Simultaneous estimation of quantile curves using quantile sheets

The results of quantile smoothing often show crossing curves, in particular, for small data sets. We define a surface, called a quantile sheet, on the domain of the independent variable and the probability. Any desired quantile curve is obtained by evaluating the sheet for a fixed probability. This sheet is modeled by $$P$$ -splines in form of tensor products of $$B$$ -splines with difference penalties on the array of coefficients. The amount of smoothing is optimized by cross-validation. An application for reference growth curves for children is presented.

Optimum smoothing parameter selection for penalized least squares in form of linear mixed effect models

In this article, we discuss the penalized least squares problem which has many advantageous computational properties. This method based on penalized spline (P-spline) smoothing can be formulated to fit into a linear mixed effects model framework. The most important issue in the implementation of this method is to specify the amount of smoothing. In an attempt to address the strategy of optimum amount of smoothing, this article provides a comparative study for different methods (or criteria) of choosing the optimum smoothing parameter: an improved version of the Akaike information criterion (AICc); generalized cross-validation (GCV); cross-validation (CV); Mallows’ C p criterion; risk estimation using classical pilots (RECP) and restricted maximum likelihood (REML). In order to explore and compare the performance of these methods, a simulation study is performed for data sets with different sample sizes. As a result of simulation, the appropriate selection criteria are provided for a suitable smoothing parameter selection.

Investigation of a method for the three-dimensional rigid body dynamics inverse problem

The three-dimensional rigid body dynamics inverse problem is cast as a discrete optimal control problem and solved using dynamic programming. The optimal control problem uses a forward dynamic model to provide estimates of unknown forces while matching noisy measurement histories. An L curve analysis is used to objectively select the amount of smoothing by trading off the magnitude of the estimated unknown forces with the fit to the noisy measurements. The forward dynamic model is derived using finite-element methodology and accounts for the inertia and mass properties of rigid bodies with the use of natural coordinates. The advantages of this forward model are that the large displacements’ nonlinearities can be routinely represented in the inverse problem and that it also allows the use of an exact energy conserving method to numerically integrate the equations of motion. A numerical example of a large displacement three body model is included to demonstrate the performance of the methodology.