Space–time calibration of radar rainfall data

Motivated by a specific problem concerning the relationship between radar reflectance and rainfall intensity, the paper develops a space–time model for use in environmental monitoring applications. The model is cast as a high dimensional multivariate state space time series model, in which the cross-covariance structure is derived from the spatial context of the component series, in such a way that its interpretation is essentially independent of the particular set of spatial locations at which the data are recorded. We develop algorithms for estimating the parameters of the model by maximum likelihood, and for making spatial predictions of the radar calibration parameters by using realtime computations. We apply the model to data from a weather radar station in Lancashire, England, and demonstrate through empirical validation the predictive performance of the model.     

A flexible and efficient spatial interpolator for radar rainfall estimation

A key challenge in rainfall estimation is spatio-temporal variablility. Weather radars are used to estimate precipitation with high spatial and temporal resolution. Due to the inherent errors in radar estimates, spatial interpolation has been often employed to calibrate the estimates. Kriging is a simple and popular spatial interpolation method, but the method has several shortcomings. In particular, the prediction is quite unstable and often fails to be performed when sample size is small. In this paper, we proposed a flexible and efficient spatial interpolator for radar rainfall estimation, with several advantages over kriging. The method is illustrated using a real-world data set.     

Bayesian Procrustes analysis with applications to hydrology

In this paper, we introduce Procrustes analysis in a Bayesian framework, by treating the classic Procrustes regression equation from a Bayesian perspective, while modeling shapes in two dimensions. The Bayesian approach allows us to compute point estimates and credible sets for the full Procrustes fit parameters. The methods are illustrated through an application to radar data from short-term weather forecasts (nowcasts), a very important problem in hydrology and meteorology.     

A note on the posterior inference for the Yule–Simon distribution

The Yule–Simon distribution has been out of the radar of the Bayesian community, so far. In this note, we propose an explicit Gibbs sampling scheme when a Gamma prior is chosen for the shape parameter. The performance of the algorithm is illustrated with simulation studies, including count data regression, and a real data application to text analysis. We compare our proposal to the frequentist counterparts showing better performance of our algorithm when a small sample size is considered.     

Parameter estimation of state space models for univariate observations

This paper contributes to the problem of estimation of state space model parameters by proposing estimators for the mean, the autoregressive parameters and the noise variances which, contrarily to maximum likelihood, may be calculated without assuming any specific distribution for the errors. The estimators suggested widen the scope of the application of the generalized method of moments to some heteroscedastic models, as in the case of state-space models with varying coefficients, and give sufficient conditions for their consistency. The paper includes a simulation study comparing the proposed estimators with maximum likelihood estimators. Finally, these methods are applied to the calibration of the meteorological radar and estimation of area rainfall.     

Modelling the evolution of a random set utilizing covariate information with applications to nowcasting

ABSTRACT

We propose models that allow us to capture the evolution of objects over time and more importantly, we provide forecasts to describe an object at future unobserved states utilizing information from the current state along with covariate information. We view objects as random sets and proceed to model them in a hierarchical Bayesian framework and estimate the model parameters using a Markov chain Monte Carlo scheme. We illustrate the methodology with an application to nowcasting of severe weather precipitation fields as obtained from weather radar images, where the severe storm cells are treated as random sets and the wind velocity is used to inform the distributions of the model parameters.     

A Basis Function Approach to Position Estimation Using Microwave Arrays

We consider the problem of estimating the bearing of a remote object given measurements on a particular type of non-scanning radar, namely a focal-plane array. Such a system focuses incoming radiation through a lens onto an array of detectors. The problem is to estimate the angular position of the radiation source given measurements on the array of detectors and knowledge of the properties of the lens. The training data are essentially noiseless, and an estimator is derived for noisy test conditions. An approach based on kernel basis functions is developed. The estimate of the basis function weights is achieved through a regularization or roughness penalty approach. Choosing the regularization parameter to be proportional to the inverse of the input signal-to-noise ratio leads to a minimum prediction error. Experimental results for a 12-element detector array support the theoretical predictions.     

A hierarchical point process with application to storm cell modelling

In environmetrics, interest often centres around the development of models and methods for making inference on observed point patterns assumed to be generated by latent spatial or spatio‐temporal processes, which may have a hierarchical structure. In this research, motivated by the analysis of spatio‐temporal storm cell data, we generalize the Neyman–Scott parent–child process to account for hierarchical clustering. This is accomplished by allowing the parents to follow a log‐Gaussian Cox process thereby incorporating correlation and facilitating inference at all levels of the hierarchy. This approach is applied to monthly storm cell data from the Bismarck, North Dakota radar station from April through August 2003 and we compare these results to simpler cluster processes to demonstrate the advantages of accounting for both levels of correlation present in these hierarchically clustered point patterns. The Canadian Journal of Statistics 47: 46–64; 2019 © 2019 Statistical Society of Canada     

Two-dimensional generalisations of dynamic programming for image analysis

Dynamic programming (DP) is a fast, elegant method for solving many one-dimensional optimisation problems but, unfortunately, most problems in image analysis, such as restoration and warping, are two-dimensional. We consider three generalisations of DP. The first is iterated dynamic programming (IDP), where DP is used to recursively solve each of a sequence of one-dimensional problems in turn, to find a local optimum. A second algorithm is an empirical, stochastic optimiser, which is implemented by adding progressively less noise to IDP. The final approach replaces DP by a more computationally intensive Forward-Backward Gibbs Sampler, and uses a simulated annealing cooling schedule. Results are compared with existing pixel-by-pixel methods of iterated conditional modes (ICM) and simulated annealing in two applications: to restore a synthetic aperture radar (SAR) image, and to warp a pulsed-field electrophoresis gel into alignment with a reference image. We find that IDP and its stochastic variant outperform the remaining algorithms.     

Least squares parameter estimation in multiplicative noise models

A simple multiplicative noise model with a constant signal has become a basic mathematical model in processing synthetic aperture radar images. The purpose of this paper is to examine a general multiplicative noise model with linear signals represented by a number of unknown parameters. The ordinary least squares (LS) and weighted LS methods are used to estimate the model parameters. The biases of the weighted LS estimates of the parameters are derived. The biases are then corrected to obtain a second-order unbiased estimator, which is shown to be exactly equivalent to the maximum log quasi-likelihood estimation, though the quasi-likelihood function is founded on a completely different theoretical consideration and is known, at the present time, to be a uniquely acceptable theory for multiplicative noise models. Synthetic simulations are carried out to confirm theoretical results and to illustrate problems in processing data contaminated by multiplicative noises. The sensitivity of the LS and weighted LS methods to extremely noisy data is analysed through the simulated examples.     

A study of simulated annealing and a revised cascade algorithm for image reconstruction

We describe an image reconstruction problem and the computational difficulties arising in determining the maximum a posteriori (MAP) estimate. Two algorithms for tackling the problem, iterated conditional modes (ICM) and simulated annealing, are usually applied pixel by pixel. The performance of this strategy can be poor, particularly for heavily degraded images, and as a potential improvement Jubb and Jennison (1991) suggest the cascade algorithm in which ICM is initially applied to coarser images formed by blocking squares of pixels. In this paper we attempt to resolve certain criticisms of cascade and present a version of the algorithm extended in definition and implementation. As an illustration we apply our new method to a synthetic aperture radar (SAR) image. We also carry out a study of simulated annealing, with and without cascade, applied to a more tractable minimization problem from which we gain insight into the properties of cascade algorithms.     

Influence functions applied to the estimation of mean rain rate

In this paper we illustrate the usefulness of influence functions for studying properties of various statistical estimators of mean rain rate using space-borne radar data. In Martin (1999), estimators using censoring, minimum chi-square, and least squares are compared in terms of asymptotic variance. Here, we use influence functions to consider robustness properties of the same estimators. We also obtain formulas for the asymptotic variance of the estimators using influence functions, and thus show that they may also be used for studying relative efficiency. The least squares estimator, although less efficient, is shown to be more robust in the sense that it has the smallest gross-error sensitivity. In some cases, influence functions associated with the estimators reveal counterintuitive behaviour. For example, observations that are less than the mean rain rate may increase the estimated mean. The additional information gleaned from influence functions may be used to understand better and improve the estimation procedures themselves.     

The generalized linear chirp process

The linear chirp process is an important class of time series for which the instantaneous frequency changes linearly in time. Linear chirps have been used extensively to model a variety of physical signals such as radar, sonar, and whale clicks (see 1, 5 and 6). We introduce the stochastic linear chirp model and then define the generalized linear chirp (GLC) process as a special case of the G-stationary process studied by Jiang et al. (2006) to model data with time-varying frequencies. We then define GLC(p,q) processes and show that the relationship between stochastic linear chirp processes and GLC(p,q) processes is analogous to that between harmonic and ARMA models. The new methods are then applied to both simulated and actual data sets.     

Performance of Robust RA Estimator for Bidimensional Autoregressive Models

The additive AR-2D model has been successfully related to the modeling of satelital images both optic and of radar of synthetic opening. Having in mind the errors that are produced in the process of captation and quantification of the image, an interesting subject, is the robust estimation of the parameters in this model. Besides the robust methods in image models are also applied in some important image processing situations such as segmentation by texture and image restoration in the presence of outliers. This paper is concerned with the development and performance of the robust RA estimator proposed by Ojeda (1998) for the estimation of parameters in contaminated AR-2D models. Here, we implement this estimator and we show by simulation study that it has a better performance than the classic least square estimator and the robust M and GM estimators in an additive outlier contaminated image model.     

A selection procedure for estimating the number of signal components

This paper proposes a selection procedure to estimate the multiplicity of the smallest eigenvalue of the covariance matrix. The unknown number of signals present in a radar data can be formulated as the difference between the total number of components in the observed multivariate data vector and the multiplicity of the smallest eigenvalue. In the observed multivariate data, the smallest eigenvalues of the sample covariance matrix may in fact be grouped about some nominal value, as opposed to being identically equal. We propose a selection procedure to estimate the multiplicity of the common smallest eigenvalue, which is significantly smaller than the other eigenvalues. We derive the probability of a correct selection, P(CS), and the least favorable configuration (LFC) for our procedures. Under the LFC, the P(CS) attains its minimum over the preference zone of all eigenvalues. Therefore, a minimum sample size can be determined from the P(CS) under the LFC, P(CS|LFC), in order to implement our new procedure with a guaranteed probability requirement. Numerical examples are presented in order to illustrate our proposed procedure.     

On approximate least squares estimators of parameters of one-dimensional chirp signal

Chirp signals are quite common in many natural and man-made systems such as audio signals, sonar, and radar. Estimation of the unknown parameters of a signal is a fundamental problem in statistical signal processing. Recently, Kundu and Nandi [Parameter estimation of chirp signals in presence of stationary noise. Stat Sin. 2008;75:187–201] studied the asymptotic properties of least squares estimators (LSEs) of the unknown parameters of a simple chirp signal model under the assumption of stationary noise. In this paper, we propose periodogram-type estimators called the approximate least squares estimators (ALSEs) to estimate the unknown parameters and study the asymptotic properties of these estimators under the same error assumptions. It is observed that the ALSEs are strongly consistent and asymptotically equivalent to the LSEs. Similar to the periodogram estimators, these estimators can also be used as initial guesses to find the LSEs of the unknown parameters. We perform some numerical simulations to see the performance of the proposed estimators and compare them with the LSEs and the estimators proposed by Lahiri et al. [Efficient algorithm for estimating the parameters of two dimensional chirp signal. Sankhya B. 2013;75(1):65–89]. We have analysed two real data sets for illustrative purposes.     

Spatially varying dynamic coefficient models

In this work we present a flexible class of linear models to treat observations made in discrete time and continuous space, where the regression coefficients vary smoothly in time and space. This kind of model is particularly appealing in situations where the effect of one or more explanatory processes on the response present substantial heterogeneity in both dimensions. We describe how to perform inference for this class of models and also how to perform forecasting in time and interpolation in space, using simulation techniques. The performance of the algorithm to estimate the parameters of the model and to perform prediction in time is investigated with simulated data sets. The proposed methodology is used to model pollution levels in the Northeast of the United States.     

Heteroscedastic Regression Models and Applications to Off-line Quality Control

We discuss in the present paper the analysis of heteroscedastic regression models and their applications to off-line quality control problems. It is well known that the method of pseudo-likelihood is usually preferred to full maximum likelihood since estimators of the parameters in the regression function obtained are more robust to misspecification of the variance function. Despite its popularity, however, existing theoretical results are difficult to apply and are of limited use in many applications. Using more recent results in estimating equations, we obtain an efficient algorithm for computing the pseudo-likelihood estimator with desirable convergence properties and also derive simple, explicit and easy to apply asymptotic results. These results are used to look in detail at variance minimization in off-line quality control, yielding techniques of inferences for the optimized design parameter. In application of some existing approaches to off-line quality control, such as the dual response methodology, rigorous statistical inference techniques are scarce and difficult to obtain. An example of off-line quality control is presented to discuss the practical aspects involved in the application of the results obtained and to address issues such as data transformation, model building and the optimization of design parameters. The analysis shows very encouraging results, and is seen to be able to unveil some important information not found in previous analyses.