Ronny Vallejos

Assessing the association between two spatial or temporal sequences

This paper deals with the codispersion coefficient for spatial and temporal series. We present some results and simulations concerning the codispersion coefficient in the context of spatial models. The results obtained are immediate consequences of the asymptotic normality of the sample codispersion coefficient and show certain limitations of the coefficient. New simulation studies provide information about the performance of the coefficient with respect to other coefficients of spatial association. The behavior of the codispersion coefficient under additively contaminated processes is also studied via Monte Carlo simulations. In the context of time series, explicit expressions for the asymptotic variance of the sample version of the coefficient are given for autoregressive and moving average processes. Resampling methods are used to compute the variance of the coefficient. A real data example is presented to explore how well the codispersion coefficient captures the comovement between two time series in practice.

Robust image modeling on image processing

Abstract This paper is concerned with robust models for representing images. The robust methods in image models are also applied to some important image processing situations such as segmentation by texture and image restoration in the presence of outliers. We consider a non-symmetric half plane (NSHP) autoregressive image model, where the image intensity at a point is a linear combination of the intensities of the eight nearest points located on one quadrant of the coordinate plane, plus an innovation process. Robust estimation algorithms for different outlier processes in causal autoregressive models are developed. These algorithms are based on robust generalized M (GM) estimators. Theoretical properties of the robust estimation algorithms are presented. The robust estimation algorithm for causal autoregressive models is applied to image restoration. The restoration method based on robust image model cleans out the outliers without involving any blurring of the image. Experimental results show that the quality of images restored by the model-based method is superior to the images restored by other conventional methods.

Study of spatial relationships between two sets of variables: a nonparametric approach

We propose a new method for estimating a codispersion coefficient to quantify the association between two spatial variables. Our proposal is based on a Nadaraya–Watson version of the codispersion coefficient through a suitable kernel. Under regularity conditions, we derive expressions for the bias and mean square error for a kernel version of the cross-variogram and establish the consistency of a Nadaraya–Watson estimator of the codispersion coefficient. In addition, we propose a bandwidth selection method for both the variogram and the cross-variogram. Monte Carlo simulations support the theoretical findings, and as a result, the new proposal performs better than the classic Matherons estimator. The proposed method is useful for quantifying spatial associations between two variables measured at the same location. Finally, we study forest data concerning the relationship among the tree height, basal area, elevation and slope of Pinus radiata plantations. A two-dimensional codispersion map is constructed to provide insight into the spatial association between these variables.

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 multivariate geostatistical approach for landscape classification from remotely sensed image data

This paper proposes a methodology to address the classification of images that have been acquired from remote sensors. One common problem in classification is the high dimensionality of multivariate characteristics. The methodology we propose consists of reducing the dimensionality of the spectral bands associated with a multispectral satellite image. Such dimensionality reduction is accomplished by the use of the divergence of a modified Mahalanobis distance. Instead of using the covariance matrix of a multivariate spatial process, the codispersion matrix is considered which have some desirable asymptotic properties under very precise conditions. The consistency and asymptotic normality hold for a general class of processes that are a natural extension of the one-dimensional spatial processes for which the asymptotic properties were first established. The results allow the selection of a set of spectral bands to produce the highest value of divergence. Then, a supervised maximum likelihood method using the selected spectral bands is employed for landscape classification. An application with a real LANDSAT image is introduced to explore and visualize how our method works in practice.

Testing for the absence of correlation between two spatial or temporal sequences

The purpose of this paper is to elucidate the problem of testing for the absence of correlation between the trajectories of two stochastic processes. It is assumed that the process is homogeneous on a pre-specified partition of the index set. The hypothesis testing methodology developed in this article consists in estimating codispersion coefficients on each subset of the partition, and in testing for the simultaneous nullity of the coefficients. To this aim, the Mahalanobis distance between the observed and theoretical codispersion vectors is used to define a test statistic, which converges to a chi-square distribution under the null hypothesis. Three examples in the context of signal processing and spatial models are discussed to point out the advantages and limitations of our proposal. Simulation studies are carried out to explore both the distribution of the test statistic under the null hypothesis and its power function. The method introduced in this paper has potential applications in time series where it is of interest to measure the comovement of two temporal sequences. The proposed test is illustrated with a real data set. Two signals are compared in terms of comovement to validate two confocal sensors in the context of biotechnology. The analysis carried out using this technique is more appropriate than previous validation tests where the mean values were compared via t test and Wilcoxon signed rank test ignoring the correlation within and across the series.

Measure of similarity between images based on the codispersion coefficient

We propose to use the codispersion coefficient to define a measure of similarity between images. This coefficient has been widely used in spatial statistics to quantify the association between two spatial processes, and here we explore its capabilities in an image processing context is mathematically simple to compute and possesses good statistical properties. The new measure takes into account the spatial association in a specific direction h between a degraded image and the original unmodified image. Three applications are developed to illustrate the capabilities of our proposal. The defined measure captures the spatial association produced by fitting AR-2D processes with different window sizes. It is able to distinguish the levels of similarity between two images for specific directions in two-dimensional space. Finally, it detects stochastic resonance when an image is transmitted by a nonlinear device.

A new image segmentation algorithm with applications to image inpainting

This article describes a new approach to perform image segmentation. First an image is locally modeled using a spatial autoregressive model for the image intensity. Then the residual autoregressive image is computed. This resulting image possesses interesting texture features. The borders and edges are highlighted, suggesting that our algorithm can be used for border detection. Experimental results with real images are provided to verify how the algorithm works in practice. A robust version of our algorithm is also discussed, to be used when the original image is contaminated with additive outliers. A novel application in the context of image inpainting is also offered.

Bayesian analysis of contaminated quarter plane moving average models

This article deals with Bayesian analysis of quarter plane moving average (MA) models observed on a rectangular part of a lattice. We present some properties concerning the autocorrelation function of MA models. These properties relate correlation parameters with the original model parameters providing much more understandable interpretation of results concerning the model. Simulation experiment is developed to explore the sensitivity of the posterior distribution when the process is contaminated with innovation and additive contamination. We show by simulation that the correlation structure of the model is seriously affected when the process contains additive contamination. We then propose a more general class of MA models which automatically deals with the contamination phenomenon [contaminated MA (CMA) model]. Also, we establish theoretical properties of the correlation function analogous with those in the previous model. Finally, we consider two applications of the CMA model. The results obtained in numerical examples show the goodness of the CMA model under contaminated data.

The application of a general time series model to floodplain fisheries in the Amazon

Time series analysis is a common tool in environmental and ecological studies to construct models to explain and forecast serially correlated data. There are several statistical techniques that are used to deal with univariate and multivariate (more than one series) chronological patterns of fisheries data. In this paper, an additive stochastic model is proposed with explicative and predictive features to capture the main seasonal patterns and trends of a fisheries system in the Amazon. The model is constructed on the assumption that the multivariate response variable - vector containing fishery yield of eight periodic species and the total fishery yield - can be decomposed into three terms: an autoregression of the response vector, an exogenous environmental variable (river level), and a seasonal component (significant frequencies obtained by using spectral analysis and the periodogram indicating the regularity of periodic cycles in the natural and fisheries system). The estimation procedure is carried out via maximum likelihood estimation. The model explained, on average, 78% of the variability in yield of the study species. The model represents the optimal solution (minimum mean square mean error) among the class of all multivariate autoregressive processes with exogenous and seasonal variables. Predictions for one period ahead are provided to illustrate how the model works in practice.