Multivariate Kernel Density Research Articles

Fault and performance management in multi-cloud based NFV using shallow and deep predictive structures

Deployment of network function virtualization (NFV) over multiple clouds accentuates its advantages such as flexibility of virtualization, proximity to customers and lower total cost of operation. However, NFV over multiple clouds has not yet attained the level of performance to be a viable replacement for traditional networks. One of the reasons is the absence of a standard based fault, configuration, accounting, performance and security (FCAPS) framework for the virtual network services. In NFV, faults and performance issues can have complex geneses within virtual resources as well as virtual networks and cannot be effectively handled by traditional rule-based systems. To tackle the above problem, we propose a fault detection and localization model based on a combination of shallow and deep learning structures. Relatively simpler detection has been effectively shown to be handled by shallow machine learning structures such as support vector machine (SVM). Deeper structure, i.e., the stacked autoencoder has been found to be useful for a more complex localization function where a large amount of information needs to be worked through to get to the root cause of the problem. We provide evaluation results using a dataset adapted from fault datasets available on Kaggle and another based on multivariate kernel density estimation and Markov sampling.

3D particle shape modelling and optimization through proper orthogonal decomposition

Based on proper orthogonal decomposition (POD), a new method is presented in order to statistically characterize arbitrary particle shapes using an optimal choice of shape functions identified on a set of 1000 digitized railway ballast particles obtained through 3D Scan. The coefficients of the POD expansion enable a description of ballast grains with varying levels of accuracy. On exploiting the knowledge of their statistical distribution we are able, implementing an appropriate multivariate kernel density estimation method, to generate irregular particles with similar morphological features. The description and generation methods are validated by comparing statistical distributions of basic characteristics: surface area, volume, average radius, elongation, flatness, and aspect ratio. Using suitable geometric descriptors defining local curvatures, we identify which surface points might be regarded as forming faces. This shows that the proposed particle generation method is well suited for irregularly shaped granular materials, as a first geometric definition step, before numerical simulations of their collective mechanical properties are carried out by a Discrete Element code dealing with polyhedral shapes. We illustrate this process with the simple case of the assembling of a granular pack from a loose configuration, by one-dimensional compression, using different levels of accuracy in the representation of grain shape.

Thermodynamic analogies for the characterization of 3D human coronary arteries

Abstract The thermodynamics of three-dimensional curves is explored through numerical simulations, providing room for a broader range of applications. Such approach, which makes use of elements of information theory, enables the processing of parametric as well as non-parametric data distributed along the curves. Descriptors inspired in thermodynamic concepts are derived to characterize such three-dimensional curves. The methodology is applied to characterize a sample of 48 human coronary arterial trees and compared with standard geometric descriptors. As an application, the usefulness of the thermodynamic descriptors is tested by assessing statistical associations between arterial shape and diseases. The feature space defined by arterial descriptors is analyzed using multivariate kernel density classification methods. A two-tailed U -test with 95% confidence interval showed that some of the proposed thermodynamic descriptors have different mean values for healthy/diseased left anterior descending (LAD) and left circumflex (LCx) arteries. Specifically: in the LAD, the temperatures based on mean number of intersection points and curvature are larger in healthy arteries ( p p p

Sample size and the multivariate kernel density likelihood ratio: How many speakers are enough?

The likelihood ratio (LR) is now widely accepted as the appropriate framework for evaluating expert evidence. However, an empirical issue in forensic voice comparison is the number of speakers required to generate robust LR output and adequately test system performance. In this study, Monte Carlo simulations were used to synthesise temporal midpoint F1, F2 and F3 values from the hesitation marker um from a set of raw data consisting of 86 male speakers of standard southern British English. Using the multivariate kernel density LR approach, these data were used to investigate: (1) the number of development (training) speakers required for adequate calibration, (2) the number of test speakers needed for robust validity, and (3) the effects of varying the number of reference speakers. The experiments were run over 20 replications to assess the effects of which, as well as how many, speakers are included in each set. Predictably, LR output was most imprecise using small samples. Comparison across the three experiments shows that the greatest variability in LR output was found as a function of the number of development speakers – where stable LR output was only achieved with more than 20 speakers. Thus, it is possible to achieve stable performance with small numbers of test and reference speakers, as long as the system is adequately calibrated. Importantly, however, LRs for individual comparisons may still be substantially affected by the inclusion of additional speakers in each set, even when large samples are used.

Strength of linguistic text evidence: A fused forensic text comparison system

Compared to other forensic comparative sciences, studies of the efficacy of the likelihood ratio (LR) framework in forensic authorship analysis are lagging. An experiment is described concerning the estimation of strength of linguistic text evidence within that framework. The LRs were estimated by trialling three different procedures: one is based on the multivariate kernel density (MVKD) formula, with each group of messages being modelled as a vector of authorship attribution features; the other two involve N-grams based on word tokens and characters, respectively. The LRs that were separately estimated from the three different procedures are logistic-regression-fused to obtain a single LR for each author comparison. This study used predatory chatlog messages sampled from 115 authors. To see how the number of word tokens affects the performance of a forensic text comparison (FTC) system, token numbers used for modelling each group of messages were progressively increased: 500, 1000, 1500 and 2500 tokens. The performance of the FTC system is assessed using the log-likelihood-ratio cost (Cllr), which is a gradient metric for the quality of LRs, and the strength of the derived LRs is charted as Tippett plots. It is demonstrated in this study that (i) out of the three procedures, the MVKD procedure with authorship attribution features performed best in terms of Cllr, and that (ii) the fused system outperformed all three of the single procedures. When the token length is 1500, for example, the fused system achieved a Cllr value of 0.15. Some unrealistically strong LRs were observed in the results. Reasons for these are discussed, and a possible solution to the problem, namely the empirical lower and upper bound LR (ELUB) method is trialled and applied to the LRs of the best-achieving fusion system.

Strength of forensic text comparison evidence from stylometric features: a multivariate likelihood ratio-based analysis

An experiment in forensic text comparison (FTC) within the likelihood ratio (LR) framework is described, in which authorship attribution was modelled with word- and character-based stylometric features. Chatlog messages of 115 authors were selected from a chatlog archive containing real pieces of chatlog evidence used to prosecute paedophiles. Four different text lengths (500, 1000, 1500 or 2500 words) were used for modelling in order to investigate how system performance is influenced by sample size. Strength of authorship attribution evidence (or LR) is estimated with the Multivariate Kernel Density formula. Performance was primarily assessed with the log-likelihood ratio cost (Cllr), but assessments of other metrics, e.g. credible interval and equal error rate, are also given. Taking into account the small number of features used for modelling authorship attribution, results are promising. Even with a small sample size of 500 words, the system achieved a discrimination accuracy of c. 76% (Cllr = 0.68258). With a sample size of 2500 words, a discrimination accuracy of c. 94% (Cllr = 0.21707) was obtained. Larger sample size is beneficial to FTC, resulting in an improvement in discriminability, an increase in the magnitude of the consistent-with-fact LRs and a decrease in the magnitude of the contrary-to-fact LRs. It was found that ‘Average character number per word token’, ‘Punctuation character ratio’, and vocabulary richness features are robust features, which work well regardless of sample sizes. The results demonstrate the efficacy of the LR framework for analysing authorship attribution evidence.

Bandwidth Selection for Multivariate Kernel Density Estimation Using Mcmc

Paper not available. Full text of working paper suppressed by author. We provide Markov chain Monte Carlo (MCMC) algorithms for computing the bandwidth matrix for multivariate kernel density estimation. Our approach is based on treating the elements of the bandwidth matrix as parameters to be estimated, which we do by optimizing the likelihood cross-validation criterion. Numerical results show that the resulting bandwidths are superior to all existing methods; for dimensions greater than two, our algorithm is the first practical method for estimating the optimal bandwidth matrix. Moreover, the MCMC algorithm for bandwidth selection for multivariate data has no increased difficulty as the dimension of data increases.

FFT-Based Fast Computation of Multivariate Kernel Density Estimators With Unconstrained Bandwidth Matrices

ABSTRACTThe problem of fast computation of multivariate kernel density estimation (KDE) is still an open research problem. In our view, the existing solutions do not resolve this matter in a satisfactory way. One of the most elegant and efficient approach uses the fast Fourier transform. Unfortunately, the existing FFT-based solution suffers from a serious limitation, as it can accurately operate only with the constrained (i.e., diagonal) multivariate bandwidth matrices. In this article, we describe the problem and give a satisfactory solution. The proposed solution may be successfully used also in other research problems, for example, for the fast computation of the optimal bandwidth for KDE. Supplementary materials for this article are available online.

FFT-based fast bandwidth selector for multivariate kernel density estimation

The performance of multivariate kernel density estimation (KDE) depends strongly on the choice of bandwidth matrix. The high computational cost required for its estimation provides a big motivation to develop fast and accurate methods. One of such methods is based on the Fast Fourier Transform. However, the currently available implementation works very well only for the univariate KDE and its multivariate extension suffers from a very serious limitation as it can accurately operate only with diagonal bandwidth matrices. A more general solution is presented where the above mentioned limitation is relaxed. Moreover, the presented solution can be easily adopted also for the task of efficient computation of integrated density derivative functionals involving an arbitrary derivative order. Consequently, bandwidth selection for kernel density derivative estimation is also supported. The practical usability of the new solution is demonstrated by comprehensive numerical simulations.

A cautionary note on the use of hypervolume kernel density estimators in ecological niche modelling

AbstractBlonder et al. (, Global Ecology and Biogeography, 23, 595–609) introduced a new multivariate kernel density estimation (KDE) method to infer Hutchinsonian hypervolumes in the modelling of ecological niches. The authors argued that their KDE method matches or outperforms several methods for estimating hypervolume geometries and for conducting species distribution modelling. Further clarification, however, is appropriate with respect to the assumptions and limitations of KDE as a method for species distribution modelling. Using virtual species and controlled environmental scenarios, we show that KDE both under‐ and overestimates niche volumes depending on the dimensionality of the dataset and the number of occurrence records considered. We suggest that KDE may be a viable approach when dealing with large sample sizes, limited sampling bias and only a few environmental dimensions.

Strength of forensic voice comparison evidence from the acoustics of filled pauses

This study investigates the evidential value of filled pauses (FPs, i.e. um, uh) as variables in forensic voice comparison. FPs for 60 young male speakers of standard southern British English were analysed, drawn from Task 1 of the DyViS corpus (Nolan et al. 2009). The following acoustic properties were analysed: midpoint frequencies of the first three formants in the vocalic portion; ‘dynamic’ characterisations of formant trajectories (i.e. quadratic polynomial equations fitted to nine measurement points over the entire vowel); vowel duration; and nasal duration for um. Likelihood ratio (LR) scores were computed using the Multivariate Kernel Density formula (MVKD; Aitken and Lucy, 2004) and converted to calibrated log10 LRs (LLRs) using logistic-regression (Brümmer et al., 2007). System validity was assessed using both equal error rate (EER) and the log LR cost function (Cllr; Brümmer and du Preez, 2006). The system with the best performance combines dynamic measurements of all three formants with vowel and nasal duration for um, achieving an EER of 4.08% and Cllr of 0.12. In terms of general patterns, um consistently outperformed uh. For um, the formant dynamic systems generated better validity than those based on midpoints, presumably reflecting the additional degree of formant movement in um caused by the transition from vowel to nasal. By contrast, midpoints outperformed dynamics for the more monophthongal uh. Further, the addition of duration (vowel or vowel and nasal) consistently improved system performance. The study supports the view that FPs have excellent potential as variables in forensic voice comparison cases.

A Kernel Density Estimator-Based Maximum A Posteriori Image Reconstruction Method for Dynamic Emission Tomography Imaging.

A novel maximum a posteriori (MAP) method for dynamic single-photon emission computed tomography image reconstruction is proposed. The prior probability is modeled as a multivariate kernel density estimator (KDE), effectively modeling the prior probability non-parametrically, with the aim of reducing the effects of artifacts arising from inconsistencies in projection measurements in low-count regimes where projections are dominated by noise. The proposed prior spatially and temporally limits the variation of time-activity functions (TAFs) and attracts similar TAFs together. The similarity between TAFs is determined by the spatial and range scaling parameters of the KDE-like prior. The resulting iterative image reconstruction method is evaluated using two simulated phantoms, namely the extended cardiac-torso (XCAT) heart phantom and a simulated Mini-Deluxe Phantom. The phantoms were chosen to observe the effects of the proposed prior on the TAFs based on the vicinity and abutments of regions with different activities. Our results show the effectiveness of the proposed iterative reconstruction method, especially in low-count regimes, which provides better uniformity within each region of activity, significant reduction of spatiotemporal variations caused by noise, and sharper separation between different regions of activity than expectation maximization and an MAP method employing a more traditional Gibbs prior.

Pointwise and uniform convergence of multivariate kernel density estimators using random bandwidths

ABSTRACTWe obtain the rates of pointwise and uniform convergence of multivariate kernel density estimators using a random bandwidth vector obtained by some data-based algorithm. We are able to obtain faster rate for pointwise convergence. The uniform convergence rate is obtained under some moment condition on the marginal distribution. The rates are obtained under i.i.d. and strongly mixing type dependence assumptions.

Estimating the Density of Ethnic Minorities and Aged People in Berlin: Multivariate Kernel Density Estimation Applied to Sensitive Georeferenced Administrative Data Protected Via Measurement Error

SummaryModern systems of official statistics require the timely estimation of area-specific densities of subpopulations. Ideally estimates should be based on precise geocoded information, which is not available because of confidentiality constraints. One approach for ensuring confidentiality is by rounding the geoco-ordinates. We propose multivariate non-parametric kernel density estimation that reverses the rounding process by using a measurement error model. The methodology is applied to the Berlin register of residents for deriving density estimates of ethnic minorities and aged people. Estimates are used for identifying areas with a need for new advisory centres for migrants and infrastructure for older people.

Co-evolution of hydrological components under climate change scenarios in the Mediterranean area

The Mediterranean area is historically characterized by high human pressure on water resources. Today, while climate is projected to be modified in the future, through precipitation decrease and temperature increase, that jointly and non-linearly may affect runoff, concerns about water availability are increasing. For these reasons, quantitative assessment of future modifications in the mean annual water availability are important; likewise, the description of the future interannual variability of some hydrological components such as runoff and evapotranspiration are highly wished for water management and ecosystems dynamics analyses. This study investigates at basin spatial scale future runoff and evapotranspiration, exploring their probability density functions and their interdependence as functions of climatic changes. In order to do that, a parsimonious conceptual lumped model is here used. The model is forced by different future climate scenarios, generated through a weather generator based on a stochastic downscaling of an ensemble of General Circulation Models (GCMs) realizations. The use of the adopted hydrological model, under reliable stochastic future climate scenarios, allows to project future values of evapotranspiration and runoff in a probabilistic framework and, at the same time, the evaluation of their bivariate frequency distributions for changes through the Multivariate Kernel Density Estimation method. As a case study, a benchmark Mediterranean watershed has been proposed (Imera Meridionale, Italy). Results suggest a radical shift and shape modification of the annual runoff and evapotranspiration probability density functions. Possible implications and impacts on water resources management are here addressed and discussed.

Process monitoring using kernel density estimation and Bayesian networking with an industrial case study

Principal component analysis has been widely used in the process industries for the purpose of monitoring abnormal behaviour. The process of reducing dimension is obtained through PCA, while T-tests are used to test for abnormality. Some of the main contributions to the success of PCA is its ability to not only detect problems, but to also give some indication as to where these problems are located. However, PCA and the T-test make use of Gaussian assumptions which may not be suitable in process fault detection. A previous modification of this method is the use of independent component analysis (ICA) for dimension reduction combined with kernel density estimation for detecting abnormality; like PCA, this method points out location of the problems based on linear data-driven methods, but without the Gaussian assumptions. Both ICA and PCA, however, suffer from challenges in interpreting results, which can make it difficult to quickly act once a fault has been detected online. This paper proposes the use of Bayesian networks for dimension reduction which allows the use of process knowledge enabling more intelligent dimension reduction and easier interpretation of results. The dimension reduction technique is combined with multivariate kernel density estimation, making this technique effective for non-linear relationships with non-Gaussian variables. The performance of PCA, ICA and Bayesian networks are compared on data from an industrial scale plant.

Consistency of multivariate density estimators using random bandwidths

Asymptotic properties of a kernel density estimator using a random bandwidth are difficult to establish. Under some assumptions we prove the L1 consistency of a class of multivariate kernel density estimators using different bandwidth vector selectors. The expected L1 distance between such an estimator and the density is also shown to converge to zero. Our results hold even when the marginal densities are heavy-tailed. As a special case, we propose a simple estimator that depends on only one parameter, irrespective of the dimension. Its L1 distance from the density goes to zero, exponentially. Simulations suggest that this estimator performs well in terms of the integrated squared error as well.

Online clutter estimation using a Gaussian kernel density estimator for multitarget tracking

In this study, the spatial distribution of false alarms is assumed to be a non-homogeneous Poisson point (NHPP) process. Then, a new method is developed under the kernel density estimation (KDE) framework to estimate the spatial intensity of false alarms for the multitarget tracking problem. In the proposed method, the false alarm spatial intensity estimation problem is decomposed into two subproblems: (i) estimating the number of false alarms in one scan and (ii) estimating the variation of the intensity function value in the measurement space. Under the NHPP assumption, the only parameter that needs to be estimated for the first subproblem is the mean of false alarm number, and the empirical mean is used here as the maximum likelihood estimate of that parameter. Then, for the second subproblem, an online multivariate local adaptive Gaussian kernel density estimator is proposed. Furthermore, the proposed estimation method is seamlessly integrated with widely used multitarget trackers, like the joint integrated probabilistic data association algorithm and the multiple hypotheses tracking algorithm. Simulation results show that the proposed KDE-based method can provide a better estimate of the false alarm spatial intensity and help the multitarget trackers yield superior performance in scenarios with spatially non-homogeneous false alarms.

K-Co-occurrences Density Map Estimation

Detecting and visualizing structures in spatial and spatio-temporal information is often the primer interest of the data scientist before going further into a statistical modeling. For labeled point patterns data, multiple co-occurrences based entropy and distance-ratios based entropy indices have proven to be useful to a global assessment or as providing hot-spot maps when localizing the statistical indices. As local information, the co-occurrences of the labeled points or of a statistic of interest (e.g., distance-ratios) are nonetheless the crucial information. This paper focuses on estimating the spatial or spatio-temporal distribution of these multiple co-occurrences. A non-parametric estimation of the co-occurrence density for sample data from a qualitative or quantitative process is built from the multivariate kernel density framework introducing a penalization linked to a co-occurring event. Direct applications as well as potential uses in spatial or spatio-temporal regression are briefly explored with examples using Twitter data on health related to seasonal diseases and using citizen science data on invasive species.

Emotional Profiling of Locations Based on Social Media

Abstract Social Media is increasingly becoming an integral part of our lives and a place where an ever growing portion of our daily communication takes place. As we communicate, we reveal our emotions and this emotional chronicle is kept in our Social Media history. As the access to Internet became more pervasive, Social Media platforms could also store the location where the interactions took place, enabling the analysis of the emotions in these locations. Pursuing this idea, we suggest a method to create the emotional profile of a location based on the long-term emotional rating of the geo-localized SM interactions. In this paper we present our method based on a multivariate kernel density function of SM interactions on a Russell's inspired circumplex plane, explain how we extract the emotions from Social Media Interactions relying on a modified version of extended Affective Norms for English Words and validate our approach with real-life locations.