Tracking Edges, Corners and Vertices in an Image

Abstract.  In a range of imaging problems, particularly those where the images are of man-made objects, edges join at points which comprise three or more distinct boundaries between textures. In such cases the set of edges in the plane forms what a mathematician would call a planar graph. Smooth edges in the graph meet one another at junctions, called 'vertices', the 'degrees' of which denote the respective numbers of edges that join there. Conventional image reconstruction methods do not always draw clear distinctions among different degrees of junction, however. In such cases the algorithm is, in a sense, too locally adaptive; it inserts junctions without checking more globally to determine whether another configuration might be more suitable. In this paper we suggest an alternative approach to edge reconstruction, which combines a junction classification step with an edge-tracking routine. The algorithm still makes its decisions locally, so that the method retains an adaptive character. However, the fact that it focuses specifically on estimating the degree of a junction means that it is relatively unlikely to insert multiple low-degree junctions when evidence in the data supports the existence of a single high-degree junction. Numerical and theoretical properties of the method are explored, and theoretical optimality is discussed. The technique is based on local least-squares, or local likelihood in the case of Gaussian data. This feature, and the fact that the algorithm takes a tracking approach which does not require analysis of the full spatial data set, mean that it is relatively simple to implement.     

空间回归模型估计中的最小二乘法

空间回归模型由于引入了空间地理信息而使得其参数估计变得复杂,因为主要采用最大似然法,致使一般人认为在空间回归模型参数估计中不存在最小二乘法。通过分析空间回归模型的参数估计技术,研究发现,最小二乘法和最大似然法分别用于估计空间回归模型的不同的参数,只有将两者结合起来才能快速有效地完成全部的参数估计。数理论证结果表明,空间回归模型参数最小二乘估计量是最佳线性无偏估计量。空间回归模型的回归参数可以在估计量为正态性的条件下而实施显著性检验,而空间效应参数则不可以用此方法进行检验。     

An evaluation of likelihood-based bandwidth selectors for spatial and spatiotemporal kernel estimates

Spatial point pattern data sets are commonplace in a variety of different research disciplines. The use of kernel methods to smooth such data is a flexible way to explore spatial trends and make inference about underlying processes without, or perhaps prior to, the design and fitting of more intricate semiparametric or parametric models to quantify specific effects. The long-standing issue of ‘optimal’ data-driven bandwidth selection is complicated in these settings by issues such as high heterogeneity in observed patterns and the need to consider edge correction factors. We scrutinize bandwidth selectors built on leave-one-out cross-validation approximation to likelihood functions. A key outcome relates to previously unconsidered adaptive smoothing regimens for spatiotemporal density and multitype conditional probability surface estimation, whereby we propose a novel simultaneous pilot-global selection strategy. Motivated by applications in epidemiology, the results of both simulated and real-world analyses suggest this strategy to be largely preferable to classical fixed-bandwidth estimation for such data.     

Local Linear Estimation for Time-Dependent Coefficients in Cox's Regression Models

This article develops a local partial likelihood technique to estimate the time-dependent coefficients in Cox's regression model. The basic idea is a simple extension of the local linear fitting technique used in the scatterplot smoothing. The coefficients are estimated locally based on the partial likelihood in a window around each time point. Multiple time-dependent covariates are incorporated in the local partial likelihood procedure. The procedure is useful as a diagnostic tool and can be used in uncovering time-dependencies or departure from the proportional hazards model. The programming involved in the local partial likelihood estimation is relatively simple and it can be modified with few efforts from the existing programs for the proportional hazards model. The asymptotic properties of the resulting estimator are established and compared with those from the local constant fitting. A consistent estimator of the asymptotic variance is also proposed. The approach is illustrated by a real data set from the study of gastric cancer patients and a simulation study is also presented.     

Shape-constrained Gaussian process regression for surface reconstruction and multimodal,non-rigid image registration

We present a new statistical framework for landmark ?>curve-based image registration and surface reconstruction. The proposed method first elastically aligns geometric features (continuous, parameterized curves) to compute local deformations, and then uses a Gaussian random field model to estimate the full deformation vector field as a spatial stochastic process on the entire surface or image domain. The statistical estimation is performed using two different methods: maximum likelihood and Bayesian inference via Markov Chain Monte Carlo sampling. The resulting deformations accurately match corresponding curve regions while also being sufficiently smooth over the entire domain. We present several qualitative and quantitative evaluations of the proposed method on both synthetic and real data. We apply our approach to two different tasks on real data: (1) multimodal medical image registration, and (2) anatomical and pottery surface reconstruction.     

Local orthogonal polynomial expansion for density estimation

A local orthogonal polynomial expansion (LOrPE) of the empirical density function is proposed as a novel method to estimate the underlying density. The estimate is constructed by matching localised expectation values of orthogonal polynomials to the values observed in the sample. LOrPE is related to several existing methods, and generalises straightforwardly to multivariate settings. By manner of construction, it is similar to local likelihood density estimation (LLDE). In the limit of small bandwidths, LOrPE functions as kernel density estimation (KDE) with high-order (effective) kernels inherently free of boundary bias, a natural consequence of kernel reshaping to accommodate endpoints. Consistency and faster asymptotic convergence rates follow. In the limit of large bandwidths LOrPE is equivalent to orthogonal series density estimation (OSDE) with Legendre polynomials, thereby inheriting its consistency. We compare the performance of LOrPE to KDE, LLDE, and OSDE, in a number of simulation studies. In terms of mean integrated squared error, the results suggest that with a proper balance of the two tuning parameters, bandwidth and degree, LOrPE generally outperforms these competitors when estimating densities with sharply truncated supports.     

A nonparametric estimation procedure for the Hawkes process: comparison with maximum likelihood estimation

In earlier work, Kirchner [An estimation procedure for the Hawkes process. Quant Financ. 2017;17(4):571–595], we introduced a nonparametric estimation method for the Hawkes point process. In this paper, we present a simulation study that compares this specific nonparametric method to maximum-likelihood estimation. We find that the standard deviations of both estimation methods decrease as power-laws in the sample size. Moreover, the standard deviations are proportional. For example, for a specific Hawkes model, the standard deviation of the branching coefficient estimate is roughly 20% larger than for MLE – over all sample sizes considered. This factor becomes smaller when the true underlying branching coefficient becomes larger. In terms of runtime, our method clearly outperforms MLE. The present bias of our method can be well explained and controlled. As an incidental finding, we see that also MLE estimates seem to be significantly biased when the underlying Hawkes model is near criticality. This asks for a more rigorous analysis of the Hawkes likelihood and its optimization.     

Two step estimation for Neyman-Scott point process with inhomogeneous cluster centers

This paper is concerned with parameter estimation for the Neyman-Scott point process with inhomogeneous cluster centers. Inhomogeneity depends on spatial covariates. The regression parameters are estimated at the first step using a Poisson likelihood score function. Three estimation procedures (minimum contrast method based on a modified K function, composite likelihood and Bayesian methods) are introduced for estimation of clustering parameters at the second step. The performance of the estimation methods are studied and compared via a simulation study. This work has been motivated and illustrated by ecological studies of fish spatial distribution in an inland reservoir.     

Local linear regression with adaptive orthogonal fitting for the wind power application

Short-term forecasting of wind generation requires a model of the function for the conversion of meteorological variables (mainly wind speed) to power production. Such a power curve is nonlinear and bounded, in addition to being nonstationary. Local linear regression is an appealing nonparametric approach for power curve estimation, for which the model coefficients can be tracked with recursive Least Squares (LS) methods. This may lead to an inaccurate estimate of the true power curve, owing to the assumption that a noise component is present on the response variable axis only. Therefore, this assumption is relaxed here, by describing a local linear regression with orthogonal fit. Local linear coefficients are defined as those which minimize a weighted Total Least Squares (TLS) criterion. An adaptive estimation method is introduced in order to accommodate nonstationarity. This has the additional benefit of lowering the computational costs of updating local coefficients every time new observations become available. The estimation method is based on tracking the left-most eigenvector of the augmented covariance matrix. A robustification of the estimation method is also proposed. Simulations on semi-artificial datasets (for which the true power curve is available) underline the properties of the proposed regression and related estimation methods. An important result is the significantly higher ability of local polynomial regression with orthogonal fit to accurately approximate the target regression, even though it may hardly be visible when calculating error criteria against corrupted data.     

Estimation of the Reliability of a Stress-Strength System from Power Lindley Distributions

In this paper, we are interested in the estimation of the reliability parameter R = P(X > Y) where X, a component strength, and Y, a component stress, are independent power Lindley random variables. The point and interval estimation of R, based on maximum likelihood, nonparametric and parametric bootstrap methods, are developed. The performance of the point estimate and confidence interval of R under the considered estimation methods is studied through extensive simulation. A numerical example, based on a real data, is presented to illustrate the proposed procedure.     

Inference for Type II bivariate Pólya–Aeppli distribution

In this article, we discuss the estimation of model parameters of the Type II bivariate Pólya–Aeppli distribution using the method of moments and the maximum likelihood method. We also compare some interval estimation methods. We then carry out a Monte Carlo simulation study to evaluate the performance of the proposed point and interval estimation methods. Finally, we present an example to illustrate all the inferential methods developed here.     

Maximum likelihood estimation for directional conditionally autoregressive models

A spatial process observed over a lattice or a set of irregular regions is usually modeled using a conditionally autoregressive (CAR) model. The neighborhoods within a CAR model are generally formed using only the inter-distances or boundaries between the regions. To accommodate directional spatial variation, a new class of spatial models is proposed using different weights given to neighbors in different directions. The proposed model generalizes the usual CAR model by accounting for spatial anisotropy. Maximum likelihood estimators are derived and shown to be consistent under some regularity conditions. Simulation studies are presented to evaluate the finite sample performance of the new model as compared to the CAR model. Finally, the method is illustrated using a data set on the crime rates of Columbus, OH and on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

Bayesian ROC curve estimation under binormality using an ordinal category likelihood

Receiver operating characteristic (ROC) curve has been widely used in medical diagnosis. Various methods are proposed to estimate ROC curve parameters under the binormal model. In this paper, we propose a Bayesian estimation method from the continuously distributed data which is constituted by the truth-state-runs in the rank-ordered data. By using an ordinal category data likelihood and following the Metropolis–Hastings (M–H) procedure, we compute the posterior distribution of the binormal parameters, as well as the group boundaries parameters. Simulation studies and real data analysis are conducted to evaluate our Bayesian estimation method.     

Local likelihood density estimation for interval censored data

The authors propose a class of procedures for local likelihood estimation from data that are either interval‐censored or that have been aggregated into bins. One such procedure relies on an algorithm that generalizes existing self‐consistency algorithms by introducing kernel smoothing at each step of the iteration. The entire class of procedures yields estimates that are obtained as solutions of fixed point equations. By discretizing and applying numerical integration, the authors use fixed point theory to study convergence of algorithms for the class. Rapid convergence is effected by the implementation of a local EM algorithm as a global Newton iteration. The latter requires an explicit solution of the local likelihood equations which can be found by using the symbolic Newton‐Raphson algorithm, if necessary.     

Computational methods for local regression

Local regression is a nonparametric method in which the regression surface is estimated by fitting parametric functions locally in the space of the predictors using weighted least squares in a moving fashion similar to the way that a time series is smoothed by moving averages. Three computational methods for local regression are presented. First, fast surface fitting and evaluation is achieved by building ak-d tree in the space of the predictors, evaluating the surface at the corners of the tree, and then interpolating elsewhere by blending functions. Second, surfaces are made conditionally parametric in any proper subset of the predictors by a simple alteration of the weighting scheme. Third degree-of-freedom quantities that would be extremely expensive to compute exactly are approximated, not by numerical methods, but through a statistical model that predicts the quantities from the trace of the hat matrix, which can be computed easily.     

Efficient method for identification of cyclic trends in incidence

Various methods for estimating the parameters of the simple harmonic curve and corresponding statistics for testing the significance of the sinusoidal trend are investigated. The locally reasonable method is almost fully efficient when the size of the trend is very small; however, the maximum likelihood method is preferred generally, especially when the trend is not very small. The log likelihood ratio test is more powerful than the R test which is based on locally reasonable estimates. The efficient method and the log likelihood ratio or equivalent tests are the best statistical techniques for identifying the cyclical trend. Thus they are the methods of choice when adequate computing facilities are available.     

Analysis of progressively type-I interval censored competing risks data for a class of an exponential distribution

ABSTRACT

In this paper, we consider some problems of point estimation and point prediction when the competing risks data from a class of exponential distribution are progressive type-I interval censored. The maximum likelihood estimation and mid-point approximation method are proposed for the estimations of parameters. Also several point predictors of censored units such as the maximum likelihood predictor, the best unbiased predictor and the conditional median predictor are obtained. The methods discussed here are applied when the lifetime distributions of the latent failure times are independent and Weibull-distributed. Finally a simulation study is given by using Monte-Carlo simulations to compare the performances of the different methods and one data analysis has been presented for illustrative purposes.     

Relative efficiencies of kernel and local likelihood density estimators

Summary. Local likelihood methods enjoy advantageous properties, such as good performance in the presence of edge effects, that are rarely found in other approaches to nonparametric density estimation. However, as we argue in this paper, standard kernel methods can have distinct advantages when edge effects are not present. We show that, whereas the integrated variances of the two methods are virtually identical, the integrated squared bias of a conventional kernel estimator is less than that of a local log-linear estimator by as much as a factor of 4. Moreover, the greatest bias improvements offered by kernel methods occur when they are needed most—i.e. when the effect of bias is particularly high. Similar comparisons can also be made when high degree local log-polynomial fits are assessed against high order kernel methods. For example, although (as is well known) high degree local polynomial fits offer potentially infinite efficiency gains relative to their kernel competitors, the converse is also true. Indeed, the asymptotic value of the integrated squared bias of a local log-quadratic estimator can exceed any given constant multiple of that for the competing kernel method. In all cases the densities that suffer problems in the context of local log-likelihood methods can be chosen to be symmetric, either unimodal or bimodal, either infinitely or compactly supported, and to have arbitrarily many derivatives as functions on the real line. They are not pathological. However, our results reveal quantitative differences between global performances of local log-polynomial estimators applied to unimodal or multimodal distributions.     

A calibrated imputation method for secondary data analysis of survey data

In practical survey sampling, missing data are unavoidable due to nonresponse, rejected observations by editing, disclosure control, or outlier suppression. We propose a calibrated imputation approach so that valid point and variance estimates of the population (or domain) totals can be computed by the secondary users using simple complete‐sample formulae. This is especially helpful for variance estimation, which generally require additional information and tools that are unavailable to the secondary users. Our approach is natural for continuous variables, where the estimation may be either based on reweighting or imputation, including possibly their outlier‐robust extensions. We also propose a multivariate procedure to accommodate the estimation of the covariance matrix between estimated population totals, which facilitates variance estimation of the ratios or differences among the estimated totals. We illustrate the proposed approach using simulation data in supplementary materials that are available online.     

Asymptotic properties of the estimators of the semi-parametric spatial regression model

Spatial data and non parametric methods arise frequently in studies of different areas and it is a common practice to analyze such data with semi-parametric spatial autoregressive (SPSAR) models. We propose the estimations of SPSAR models based on maximum likelihood estimation (MLE) and kernel estimation. The estimation of spatial regression coefficient ρ was done by optimizing the concentrated log-likelihood function with respect to ρ. Furthermore, under appropriate conditions, we derive the limiting distributions of our estimators for both the parametric and non parametric components in the model.