Performance analysis using equivariant kernel density estimator in nonlinear mixture

This paper investigates the performance analysis of separation of mutually independent sources in nonlinear models. The nonlinear mapping constituted by an unsupervised linear mixture is followed by an unknown and invertible nonlinear distortion, are found in many signal processing cases. Generally, blind separation of sources from their nonlinear mixtures is rather difficult. We propose using a kernel density estimator incorporated with equivariant gradient analysis to separate the sources with nonlinear distortion. The kernel density estimator parameters of which are iteratively updated to minimize the output independence expressed as a mutual information criterion. The equivariant gradient algorithm has the form of nonlinear decorrelation to perform the convergence analysis. Experiments are proposed to illustrate these results.

Using kernel density estimation to understand the influence of neighbourhood destinations on BMI

OBJECTIVES: Little is known about how the distribution of destinations in the local neighbourhood is related to body mass index (BMI). Kernel density estimation (KDE) is a spatial analysis technique that accounts for the location of features relative to each other. Using KDE, this study investigated whether individuals living near destinations (shops and service facilities) that are more intensely distributed rather than dispersed, have lower BMIs.

STUDY DESIGN AND SETTING: A cross-sectional study of 2349 residents of 50 urban areas in metropolitan Melbourne, Australia.

METHODS: Destinations were geocoded, and kernel density estimates of destination intensity were created using kernels of 400, 800 and 1200 m. Using multilevel linear regression, the association between destination intensity (classified in quintiles Q1(least)-Q5(most)) and BMI was estimated in models that adjusted for the following confounders: age, sex, country of birth, education, dominant household occupation, household type, disability/injury and area disadvantage. Separate models included a physical activity variable.

RESULTS: For kernels of 800 and 1200 m, there was an inverse relationship between BMI and more intensely distributed destinations (compared to areas with least destination intensity). Effects were significant at 1200 m: Q4, β -0.86, 95% CI -1.58 to -0.13, p=0.022; Q5, β -1.03 95% CI -1.65 to -0.41, p=0.001. Inclusion of physical activity in the models attenuated effects, although effects remained marginally significant for Q5 at 1200 m: β -0.77 95% CI -1.52, -0.02, p=0.045.

CONCLUSIONS: This study conducted within urban Melbourne, Australia, found that participants living in areas of greater destination intensity within 1200 m of home had lower BMIs. Effects were partly explained by physical activity. The results suggest that increasing the intensity of destination distribution could reduce BMI levels by encouraging higher levels of physical activity.

Income Inequality in the 1990s: Re-Forging a Lost Relationship?

Using data from March Current Population Surveys we find gains from economic growth over the 1990s business cycle (1989-2000) were more equitably distributed than over the 1980s business cycle (1979-1989) using summary inequality measures as well as kernel density estimations. The entire distribution of household size-adjusted income moved upwards in the 1990s with profound improvements for African Americans, single mothers and those living in households receiving welfare. Most gains occurred over the growth period 1993-2000. Improvements in average income and income inequity over the latter period are reminiscent of gains seen in the first three decades after World War II.

Evaluating machine-learning techniques for recruitment forecasting of seven North East Atlantic fish species

The effect of different factors (spawning biomass, environmental conditions) on recruitment is a subject of great importance in the management of fisheries, recovery plans and scenario exploration. In this study, recently proposed supervised classification techniques, tested by the machine-learning community, are applied to forecast the recruitment of seven fish species of North East Atlantic (anchovy, sardine, mackerel, horse mackerel, hake, blue whiting and albacore), using spawning, environmental and climatic data. In addition, the use of the probabilistic flexible naive Bayes classifier (FNBC) is proposed as modelling approach in order to reduce uncertainty for fisheries management purposes. Those improvements aim is to improve probability estimations of each possible outcome (low, medium and high recruitment) based in kernel density estimation, which is crucial for informed management decision making with high uncertainty. Finally, a comparison between goodness-of-fit and generalization power is provided, in order to assess the reliability of the final forecasting models. It is found that in most cases the proposed methodology provides useful information for management whereas the case of horse mackerel is an example of the limitations of the approach. The proposed improvements allow for a better probabilistic estimation of the different scenarios, i.e. to reduce the uncertainty in the provided forecasts.

Probability density function estimation using orthogonal forward regression

Using the classical Parzen window estimate as the target function, the kernel density estimation is formulated as a regression problem and the orthogonal forward regression technique is adopted to construct sparse kernel density estimates. The proposed algorithm incrementally minimises a leave-one-out test error score to select a sparse kernel model, and a local regularisation method is incorporated into the density construction process to further enforce sparsity. The kernel weights are finally updated using the multiplicative nonnegative quadratic programming algorithm, which has the ability to reduce the model size further. Except for the kernel width, the proposed algorithm has no other parameters that need tuning, and the user is not required to specify any additional criterion to terminate the density construction procedure. Two examples are used to demonstrate the ability of this regression-based approach to effectively construct a sparse kernel density estimate with comparable accuracy to that of the full-sample optimised Parzen window density estimate.

Sparse probability density function estimation using the minimum integrated square error

We develop a new sparse kernel density estimator using a forward constrained regression framework, within which the nonnegative and summing-to-unity constraints of the mixing weights can easily be satisfied. Our main contribution is to derive a recursive algorithm to select significant kernels one at time based on the minimum integrated square error (MISE) criterion for both the selection of kernels and the estimation of mixing weights. The proposed approach is simple to implement and the associated computational cost is very low. Specifically, the complexity of our algorithm is in the order of the number of training data N, which is much lower than the order of N2 offered by the best existing sparse kernel density estimators. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with comparable accuracy to those of the classical Parzen window estimate and other existing sparse kernel density estimators.

A fast algorithm for sparse probability density function construction

A new sparse kernel density estimator is introduced. Our main contribution is to develop a recursive algorithm for the selection of significant kernels one at time using the minimum integrated square error (MISE) criterion for both kernel selection. The proposed approach is simple to implement and the associated computational cost is very low. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with competitive accuracy to existing kernel density estimators.

Sparse density estimation on the multinomial manifold

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion for the finite mixture model. Since the constraint on the mixing coefficients of the finite mixture model is on the multinomial manifold, we use the well-known Riemannian trust-region (RTR) algorithm for solving this problem. The first- and second-order Riemannian geometry of the multinomial manifold are derived and utilized in the RTR algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with an accuracy competitive with those of existing kernel density estimators.

Sparse density estimator with tunable kernels

A new sparse kernel density estimator with tunable kernels is introduced within a forward constrained regression framework whereby the nonnegative and summing-to-unity constraints of the mixing weights can easily be satisfied. Based on the minimum integrated square error criterion, a recursive algorithm is developed to select significant kernels one at time, and the kernel width of the selected kernel is then tuned using the gradient descent algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing very sparse kernel density estimators with competitive accuracy to existing kernel density estimators.

Sparse density estimation on multinomial manifold combining local component analysis

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion combining local component analysis for the finite mixture model. We start with a Parzen window estimator which has the Gaussian kernels with a common covariance matrix, the local component analysis is initially applied to find the covariance matrix using expectation maximization algorithm. Since the constraint on the mixing coefficients of a finite mixture model is on the multinomial manifold, we then use the well-known Riemannian trust-region algorithm to find the set of sparse mixing coefficients. The first and second order Riemannian geometry of the multinomial manifold are utilized in the Riemannian trust-region algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with competitive accuracy to existing kernel density estimators.

Soybean compressibility and bulk density

The objective of this project was to study the influence of surcharge pressure and moisture content on the compressive behavior and bulk density of soybeans. Three varieties were selected with varying dimensions and shapes. Moisture contents of 10.5, 15.0, and 20% were tested at nine surcharge pressures in the range from 0 to 82.8 kPa. Results indicated that the bulk densities of different soybean varieties have similar behavior with respect to pressure level and moisture content but that the magnitude of bulk density was influenced by variety, Bulk density was influenced by both pressure level and moisture content. The four-element Burger model was found to adequately describe the bulk density of soybeans as a function of pressure for all varieties and moisture levels.

Spatial aspects of MRSA epidemiology:A case study using stochastic simulation, kernel estimation and SaTScan

The identification of disease clusters in space or space-time is of vital importance for public health policy and action. In the case of methicillin-resistant Staphylococcus aureus (MRSA), it is particularly important to distinguish between community and health care-associated infections, and to identify reservoirs of infection. 832 cases of MRSA in the West Midlands (UK) were tested for clustering and evidence of community transmission, after being geo-located to the centroids of UK unit postcodes (postal areas roughly equivalent to Zip+4 zip code areas). An age-stratified analysis was also carried out at the coarser spatial resolution of UK Census Output Areas. Stochastic simulation and kernel density estimation were combined to identify significant local clusters of MRSA (p<0.025), which were supported by SaTScan spatial and spatio-temporal scan. In order to investigate local sampling effort, a spatial 'random labelling' approach was used, with MRSA as cases and MSSA (methicillin-sensitive S. aureus) as controls. Heavy sampling in general was a response to MRSA outbreaks, which in turn appeared to be associated with medical care environments. The significance of clusters identified by kernel estimation was independently supported by information on the locations and client groups of nursing homes, and by preliminary molecular typing of isolates. In the absence of occupational/ lifestyle data on patients, the assumption was made that an individual's location and consequent risk is adequately represented by their residential postcode. The problems of this assumption are discussed, with recommendations for future data collection.

The statistics of refractive error maps : managing wavefront aberration analysis without Zernike polynomials

The refractive error of a human eye varies across the pupil and therefore may be treated as a random variable. The probability distribution of this random variable provides a means for assessing the main refractive properties of the eye without the necessity of traditional functional representation of wavefront aberrations. To demonstrate this approach, the statistical properties of refractive error maps are investigated. Closed-form expressions are derived for the probability density function (PDF) and its statistical moments for the general case of rotationally-symmetric aberrations. A closed-form expression for a PDF for a general non-rotationally symmetric wavefront aberration is difficult to derive. However, for specific cases, such as astigmatism, a closed-form expression of the PDF can be obtained. Further, interpretation of the distribution of the refractive error map as well as its moments is provided for a range of wavefront aberrations measured in real eyes. These are evaluated using a kernel density and sample moments estimators. It is concluded that the refractive error domain allows non-functional analysis of wavefront aberrations based on simple statistics in the form of its sample moments. Clinicians may find this approach to wavefront analysis easier to interpret due to the clinical familiarity and intuitive appeal of refractive error maps.

Graph rigidity for near-coplanar structure from motion

Recent algorithms for monocular motion capture (MoCap) estimate weak-perspective camera matrices between images using a small subset of approximately-rigid points on the human body (i.e. the torso and hip). A problem with this approach, however, is that these points are often close to coplanar, causing canonical linear factorisation algorithms for rigid structure from motion (SFM) to become extremely sensitive to noise. In this paper, we propose an alternative solution to weak-perspective SFM based on a convex relaxation of graph rigidity. We demonstrate the success of our algorithm on both synthetic and real world data, allowing for much improved solutions to marker less MoCap problems on human bodies. Finally, we propose an approach to solve the two-fold ambiguity over bone direction using a k-nearest neighbour kernel density estimator.