Orietta Nicolis

Image Denoising With 2D Scale-Mixing Complex Wavelet Transforms

This paper introduces an image denoising procedure based on a 2D scale-mixing complex-valued wavelet transform. Both the minimal (unitary) and redundant (maximum overlap) versions of the transform are used. The covariance structure of white noise in wavelet domain is established. Estimation is performed via empirical Bayesian techniques, including versions that preserve the phase of the complex-valued wavelet coefficients and those that do not. The new procedure exhibits excellent quantitative and visual performance, which is demonstrated by simulation on standard test images.

Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass‐interaction process

Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper.

L-moments of the Birnbaum-Saunders distribution and its extreme value version: Estimation, goodness of fit and application to earthquake data

ABSTRACT Understanding patterns in the frequency of extreme natural events, such as earthquakes, is important as it helps in the prediction of their future occurrence and hence provides better civil protection. Distributions describing these events are known to be heavy tailed and positive skew making standard distributions unsuitable for modelling the frequency of such events. The Birnbaum–Saunders distribution and its extreme value version have been widely studied and applied due to their attractive properties. We derive L-moment equations for these distributions and propose novel methods for parameter estimation, goodness-of-fit assessment and model selection. A simulation study is conducted to evaluate the performance of the L-moment estimators, which is compared to that of the maximum likelihood estimators, demonstrating the superiority of the proposed methods. To illustrate these methods in a practical application, a data analysis of real-world earthquake magnitudes, obtained from the global centroid moment tensor catalogue during 1962–2015, is carried out. This application identifies the extreme value Birnbaum–Saunders distribution as a better model than classic extreme value distributions for describing seismic events.

Multiresolution analysis of linearly oriented spatial point patterns

The assumption of direction invariance, i.e. isotropy, is often made in the practical analysis of spatial point patterns due to simpler interpretation and ease of analysis. However, this assumption is many times hard to find in real applications. We propose a wavelet-based approach to test for isotropy in spatial patterns based on the logarithm of the directional scalogram. Under the null hypothesis of isotropy, a random isotropic point pattern should be expected to have the same value of the directional scalogram for any possible direction. Monte Carlo simulations of the logarithm of the directional scalogram over all directions are used to approximate the test distribution and the critical values. We demonstrate the efficacy of the approach through simulation studies and an application to a desert plant data set, where our approach confirms suspected directional effects in the spatial distribution of the desert plant species.

Genetic Algorithm in the Wavelet Domain for Large p Small n Regression

Many areas of statistical modeling are plagued by the “curse of dimensionality,” in which there are more variables than observations. This is especially true when developing functional regression models where the independent dataset is some type of spectral decomposition, such as data from near-infrared spectroscopy. While we could develop a very complex model by simply taking enough samples (such that n > p), this could prove impossible or prohibitively expensive. In addition, a regression model developed like this could turn out to be highly inefficient, as spectral data usually exhibit high multicollinearity. In this article, we propose a two-part algorithm for selecting an effective and efficient functional regression model. Our algorithm begins by evaluating a subset of discrete wavelet transformations, allowing for variation in both wavelet and filter number. Next, we perform an intermediate processing step to remove variables with low correlation to the response data. Finally, we use the genetic algorithm to perform a stochastic search through the subset regression model space, driven by an information-theoretic objective function. We allow our algorithm to develop the regression model for each response variable independently, so as to optimally model each variable. We demonstrate our method on the familiar biscuit dough dataset, which has been used in a similar context by several researchers. Our results demonstrate both the flexibility and the power of our algorithm. For each response variable, a different subset model is selected, and different wavelet transformations are used. The models developed by our algorithm show an improvement, as measured by lower mean error, over results in the published literature.

Multi-fractal cancer risk assessment

ABSTRACT From fractal point of view, there exist two types of cancer tissues. Those having multifractal (or fractal) structure, and the rest, for example, Wilms tumors. To support the diagnostics, we shall be aware of this differentiation and we shall use robust methods for measuring of multifractality. The main reason a robust discrimination is needed is the simple fact that histopathological discrimination between mastopathy and cancer is difficult in many cases. As we show in this article, using spectra or energy for multifractal model can be of help. However, it needs robustness since many mastopathic tissues are mimicking cancer ones. This article tackles this multidisciplinary topic. In particular, we provide diffusion model for metabolic heat for both cancer and mastopathy related tissues and novel concept of metabolism related energy.

Discussion of the paper “analysis of spatio-temporal mobile phone data: a case study in the metropolitan area of Milan”

The authors are to be congratulated on a valuable and thought-provoking contribution on the analysis of geo-referenced high-dimensional data describing the use over time of the mobile-phone network in the urban area of Milan, Italy. This is a timely and world-wide problem that opens wide avenues for new methodological contributions. The authors develop a Bagging Voronoi Treelet Analysis which is a non-parametric method for the analysis of spatially dependent functional data. This approach integrates the treelet decomposition with a proper treatment of spatial dependence, obtained through a Bagging Voronoi strategy. In our discussion, we focus on the following points: (i) a mobre general form of the spatio-temporal model proposed in Secchi et al. (Stat Methods Appl, 2015), (ii) alternative methods to approach the smooth temporal functions, (iii) additional methods to reduce the problem of dimension for spatial dependence data, and (iv) comments on the pros and cons of the proposed pre-processing methodology.

2D Anisotropic Wavelet Entropy with an Application to Earthquakes in Chile

We propose a wavelet-based approach to measure the Shannon entropy in the context of spatial point patterns. The method uses the fully anisotropic Morlet wavelet to estimate the energy distribution at different directions and scales. The spatial heterogeneity and complexity of spatial point patterns is then analyzed using the multiscale anisotropic wavelet entropy. The efficacy of the approach is shown through a simulation study. Finally, an application to the catalog of earthquake events in Chile is considered.

Predicting hourly ozone concentrations using wavelets and ARIMA models

In recent years, air pollution has been a major concern for its implications on human health. Specifically, ozone (