Exploratory methods for the study of incomplete and intersecting shape boundaries from landmark data

Structured spatial point patterns appear in many applications within the natural sciences. Often the points record the location of key features, called landmarks, on continuous object boundaries, such as anatomical features on a human face or on an animal skull. In other situations, the points may simply be arbitrarily spaced marks along a smooth curve, such as on handwritten numbers or letters. Sometimes the points may record the location of clearly visible features from a general structure which has disappeared, such as building foundations at an archaeological site. This paper proposes novel exploratory methods for the identification of structure within point datasets. In particular, points are linked together to form curves which estimate the original shape from which the points are the only recorded information. Nonparametric regression methods are applied to polar coordinate variables obtained from the point locations and periodic modelling allows closed curves to be fitted to circular and elliptical shapes even when data are available on only part of the boundary. Further, the model allows discontinuities to be identified to describe rapid changes in the curves. These generalizations are particularly important when the points represent shapes which are occluded or are intersecting. A range of real-data examples is used to motivate the modelling and to illustrate the flexibility of the approach. The method successfully identifies underlying structure and its output could also be used as the basis for further analysis

Sequential models for time-evolving regression problems with an application to energy demand prediction

In recent years, there has been a dramatic increase in the use of data that are collected over time and hence models with a temporal component, leading to dynamic models, have received increasing attention. The proposed approach uses a general framework which permits many special cases to be considered. Put simply, for each time a parametric observation model is defined with a conditional auto-regressive type model defined relating the parameters at one time to previous parameter values, this is called the evolution equation. Simulation results will be presented investigating estimator properties considering a temporally changing regression problem with results demonstrating improved estimation. The technique will also be applied to a real dataset examining the changing relationship between ambient temperature and electricity consumption in the UK. The fitted model can then be used to predict future demand based on easily obtained temperature forecast information

Two-stage estimation in inverse problems using a combined wavelet thresholding and penalized maximum likelihood

Inverse problems occur in a wide range of practical scientific investigations where the variables of interest are only observed indirectly, such as magnetic and seismic imaging in geophysics, electrical tomography in industrial process monitoring, or PET scanning in medicine. Linear inverse problems can be thought of as highly multivariate regression problems with strong multicollinearity where the aim is to interpret regression parameters-prediction is not of interest. Estimation, to give a fitted model, is known as an inverse problem which can be ill-posed and ill-conditioned, making estimation using least-squares or maximum likelihood unstable or even impossible. Instead, one approach is to introduce additional constraints through a penalty term and a penalized least-squares or penalized maximum likelihood approach taken. The major cause of numerical problems in the estimation is noise in the data and hence using a pre-processing which reduces noise may be helpful. Wavelet thresholding has proven to be highly efficient at separating useful information from noise but there has been very little work considering the use of wavelet methods for inverse problems. Hence it is of great interest to investigate the usefulness of this as an additional step in estimation for inverse problems. In particular a two stage process is proposed combining inversion and wavelet thresholding. The thresholding will be considered as either a pre-inversion or post-inversion filter and the results compared. A simulation investigation is described and reported which compares these two alternative, and also which uses a minimum mean-squared error approach to choose the penalty parameter, in the inversion, and the threshold, in the wavelet thresholding, either sequentially or jointly. The results demonstrate that a combined approach is worthwhile and that for the piecewise constant test function considered, it is better to post-process after the inversion step than it is to use the more intuitive wavelet thresholding pre-processing step for noise reduction before inversion. This new approach hence has the potential to enhance the estimation results in a wide range of applied inverse problems

Bayesian modeling of temperature-related mortality with latent functional relationships

It is common for the mortality rate to increase during periods of extreme temperature and for the minimum mortality rate to depend on factors such as the mean summer temperature. In this paper, local correlation is explicitly described using a generalized additive model with a spatial component which allows information from neighbouring locations to be combined. Random walk and random field models are proposed to describe temporal and spatial correlation structure. Further, joint spatial-temporal modeling is proposed by including a temperature-related mortality term. This will make use of existing data more efficiently and should reduce prediction variability. The methods are illustrated using simulated data based on real mortality and temperature data

Spatially adaptive Bayesian image reconstruction through locally-modulated Markov random field models

The use of Markov random field (MRF) models has proven to be a fruitful approach in a wide range of image processing applications. It allows local texture information to be incorporated in a systematic and unified way and allows statistical inference theory to be applied giving rise to novel output summaries and enhanced image interpretation. A great advantage of such low-level approaches is that they lead to flexible models, which can be applied to a wide range of imaging problems without the need for significant modification. This paper proposes and explores the use of conditional MRF models for situations where multiple images are to be processed simultaneously, or where only a single image is to be reconstructed and a sequential approach is taken. Although the coupling of image intensity values is a special case of our approach, the main extension over previous proposals is to allow the direct coupling of other properties, such as smoothness or texture. This is achieved using a local modulating function which adjusts the influence of global smoothing without the need for a fully inhomogeneous prior model. Several modulating functions are considered and a detailed simulation study, motivated by remote sensing applications in archaeological geophysics, of conditional reconstruction is presented. The results demonstrate that a substantial improvement in the quality of the image reconstruction, in terms of errors and residuals, can be achieved using this approach, especially at locations with rapid changes in the underlying intensity

A statistical approach to the inclusion of electrode contact impedance uncertainty in electrical tomography reconstruction

Electrical tomography is a visualisation tool used for industrial process monitoring. The complete electrode model relates an unknown conductivity field with the measurements but also involves unknown electrode contact impedances. Here, a real data analysis shows that the contact impedances vary spatially and with time. Then, the main reconstruction process is repeated using contact impedance values drawn at random from a fitted contact impedance distribution, and a model average calculated as the final reconstruction. This additional source of variation can then be appropriately accounted for thereby preventing overly optimistic assumption in subsequent decision making

Classification of multiple time signals using localized frequency characteristics applied to industrial process monitoring

A general framework for regression modeling using localized frequency characteristics of explanatory variables is proposed. This novel framework can be used in any application where the aim is to model an evolving process sequentially based on multiple time series data. Furthermore, this framework allows time series to be transformed and combined to simultaneously boost important characteristics and reduce noise. A wavelet transform is used to isolate key frequency structure and perform data reduction. The method is highly adaptive, since wavelets are effective at extracting localized information from noisy data. This adaptivity allows rapid identification of changes in the evolving process. Finally, a regression model uses functions of the wavelet coefficients to classify the evolving process into one of a set of states which can then be used for automatic monitoring of the system. As motivation and illustration, industrial process monitoring using electrical tomography measurements is considered. This technique provides useful data without intruding into the industrial process. Statistics derived from the wavelet transform of the tomographic data can be enormously helpful in monitoring and controlling the process. The predictive power of the proposed approach is explored using real and simulated tomographic data. In both cases, the resulting models successfully classify different flow regimes and hence provide the basis for reliable online monitoring and control of industrial processes

Birnbaum–Saunders autoregressive conditional duration models applied to high-frequency financial data

Modern financial markets now record the precise time of each stock trade, along with price and volume, with the aim of analysing the structure of the times between trading events—leading to a big data problem. In this paper, we propose and compare two Birnbaum–Saunders autoregressive conditional duration models specified in terms of time-varying conditional median and mean durations. These models provide a novel alternative to the existing autoregressive conditional duration models due to their flexibility and ease of estimation. Diagnostic tools are developed to allow goodness-of-fit assessment and to detect departures from assumptions, including the presence of outliers and influential cases. These diagnostic tools are based on the parameter estimates using residual analysis and the Cook distance for global influence, and different perturbation schemes for local influence. A thorough Monte Carlo study is presented to evaluate the performance of the maximum likelihood estimators, and the forecasting ability of the models is assessed using the traditional and density forecast evaluation techniques. The Monte Carlo study suggests that the parameter estimators are asymptotically unbiased, consistent and normally distributed. Finally, a full analysis of a real-world financial transaction data set, from the German DAX in 2016, is presented to illustrate the proposed approach and to compare the fitting and forecasting performances with existing models in the literature. One case related to the duration time is identified as potentially influential, but its removal does not change resulting inferences demonstrating the robustness of the proposed approach. Fitting and forecasting performances favor the proposed models and, in particular, the median-based approach gives additional protection against outliers, as expected

Horizon Detection in Seismic Data: An Application of Linked Feature Detection from Multiple Time Series

Seismic studies are a key stage in the search for large scale underground features such as water reserves, gas pockets, or oil fields. Sound waves, generated on the earth’s surface, travel through the ground before being partially reflected at interfaces between regions with high contrast in acoustic properties such as between liquid and solid. After returning to the surface, the reflected signals are recorded by acoustic sensors. Importantly, reflections from different depths return at different times, and hence the data contain depth information as well as position. A strong reflecting interface, called a horizon, indicates a stratigraphic boundary between two different regions, and it is the location of these horizons which is of key importance. This paper proposes a simple approach for the automatic identification of horizons, which avoids computationally complex and time consuming 3D reconstruction. The new approach combines nonparametric smoothing and classification techniques which are applied directly to the seismic data, with novel graphical representations of the intermediate steps introduced. For each sensor position, potential horizon locations are identified along the corresponding time-series traces. These candidate locations are then examined across all traces and when consistent patterns occur the points are linked together to form coherent horizons

Birnbaum-Saunders spatial modelling and diagnostics applied to agricultural engineering data

Applications of statistical models to describe spatial dependence in geo-referenced data are widespread across many disciplines including the environmental sciences. Most of these application assume that the data follow a Gaussian distributions. However, in many of them the normality assumption, and even a more general assumption of symmetry, are not appropriate. In non-spatial applications, where the data are uni-modal and positively skewed, the Birnbaum-Saunders distribution has excelled. This paper proposes a spatial log-linear model based in the Birnbaum-Saunders distribution. Model parameters are estimated using the maximum likelihood method. Local influence diagnostics are derived to assess the sensitivity of the estimators to perturbations in the response variable. As illustration, the proposed model and its diagnostics are used to analyse a real-world agricultural data-set, where the spatial variability of phosphorus concentration in the soil is considered- which is extremely important for agricultural management