Kernel density classification and boosting: an L2 sub analysis

Kernel density estimation is a commonly used approach to classification. However, most of the theoretical results for kernel methods apply to estimation per se and not necessarily to classification. In this paper we show that when estimating the difference between two densities, the optimal smoothing parameters are increasing functions of the sample size of the complementary group, and we provide a small simluation study which examines the relative performance of kernel density methods when the final goal is classification. A relative newcomer to the classification portfolio is “boosting”, and this paper proposes an algorithm for boosting kernel density classifiers. We note that boosting is closely linked to a previously proposed method of bias reduction in kernel density estimation and indicate how it will enjoy similar properties for classification. We show that boosting kernel classifiers reduces the bias whilst only slightly increasing the variance, with an overall reduction in error. Numerical examples and simulations are used to illustrate the findings, and we also suggest further areas of research

Multilane traffic density estimation with KDE and nonlinear LS and tracking with Scalar Kalman filtering

Tezin basılısı, İstanbul Şehir Üniversitesi Kütüphanesi'ndedir.With increasing population, the determination of traffic density becomes very critical in managing the urban city roads for safer driving and low carbon emission. In this study, Kernel Density Estimation is utilized in order to estimate the traffic density more accurately when the speeds of the vehicles are available for a given region. For the proposed approach, as a first step, the probability density function of the speed data is modeled by Kernel Density Estimation. Then, the speed centers from the density function are modeled as clusters. The cumulative distribution function of the speed data is then determined by Kolmogorov-Smirnov Test, whose complexity is less when compared to the other techniques and whose robustness is high when outliers exist. Then, the mean values of clusters are estimated from the smoothed density function of the distribution function, followed by a peak detection algorithm. The estimation of variance values and kernel weights, on the other hand, are found by a nonlinear Least Square approach. As the estimation problem has linear and non-linear components, the nonlinear Least Square with separation of parameters approach is adopted, instead of dealing with a high complexity nonlinear equation. Finally, the tracking of former and latter estimations of a road is calculated by using Scalar Kalman Filtering with scalar state - scalar observation generality level. Simulations are carried out in order to assess theperformanceoftheproposedapproach. Forallexampledatasets, theminimummean square error of kernel weights is found to be less than 0.002 while error of mean values is found to be less than 0.261. The proposed approach was also applied to real data from sample road traffic, and the speed center and the variance was accurately estimated. By using the proposed approach, accurate traffic density estimation is realized, providing extra information to the municipalities for better planning of their cities.Declaration of Authorship ii Abstract iii Öz iv Acknowledgments vi List of Figures ix List of Tables x Abbreviations xi 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Methods to Find Probability Density Function and Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 1.3 Traffic Density Estimation with Kernel Density Estimation . . . . . . . . 4 1.4 The Approaches for Determination of Key Parameters of Traffic Density Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . .5 1.5 Tracking between Estimated Data and New Data . . . . . . . . . . . . . . 6 1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Literature Review 7 2.1 Methodologies Used for Estimation of Traffic Density . . . . . . . . . . . . 7 2.2 An Example Study of Traffic Density Estimation with KDE and CvM . . 9 2.3 Three Complementary Studies for Traffic Density Estimation and Tracking 9 2.4 Comparison of Three Different Nonlinear Estimation Techniques on the Same Problem . . . . . . . . . . . . . . . . . . . . . . . . .10 2.4.1 A Maximum Likelihood Approach for Estimating DS-CDMA Multipath Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.2 Channel Estimation for the Uplink of a DS-CDMA System . . . . 12 2.4.3 A Robust Method for Estimating Multipath Channel Parameters in the Uplink of a DS-CDMA System. . . . . . . . . . . . . . .13 3 The Model 16 3.1 Finding Density Distribution with KDE . . . . . . . . . . . . . . . . . . . 16 3.2 Finding Empirical CDF with KS Test . . . . . . . . . . . . . . . . . . . . 18 3.3 Determination of Speed Centers via PDA . . . . . . . . . . . . . . . . . . 20 3.4 Estimation of Variance and Kernel Weights with Nonlinear LS Method . . 21 3.5 Tracking of Traffic Density Estimation with Scalar Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 Numerical Calculations for Traffic Density Estimation 26 4.1 An Example Traffic Scenario with Five Speed Centers . . . . . . . . . . . 26 4.2 The Estimation of A Real Time Data . . . . . . . . . . . . . . . . . . . . . 29 4.3 Traffic Density Estimation with Different Kernel Numbers . . . . . . . . . 29 5 Examples to Test Tracking Part of the Model 31 5.1 Tracking with the Change only in Mean Values . . . . . . . . . . . . . . . 32 5.2 Tracking with the Change only in Kernel Weights . . . . . . . . . . . . . . 35 5.3 Tracking with the Change in All Three Parameters . . . . . . . . . . . . . 36 6 Assesment 38 7 Conclusion 41 A Derivation of Newton-Raphson Method for the Estimation of Variance Values and Kernel Weights 43 Bibliography 4

Transformations for compositional data with zeros with an application to forensic evidence evaluation

In forensic science likelihood ratios provide a natural way of computing the value of evidence under competing propositions such as "the compared samples have originated from the same object" (prosecution) and "the compared samples have originated from different objects" (defence). We use a two-level multivariate likelihood ratio model for comparison of forensic glass evidence in the form of elemental composition data under three data transformations: the logratio transformation, a complementary log-log type transformation and a hyperspherical transformation. The performances of the three transformations in the evaluation of evidence are assessed in simulation experiments through use of the proportions of false negatives and false positives

Uganda and Malawi field pilots of proposed LSMS fisheries module: summary report

While an overwhelming majority of sub-Saharan African countries exhibit serious weaknesses in statistics pertaining to crop and livestock sectors, the deficiencies in terms of nationally representative data on the fishery sector are even more acute. The very little data available on the sector are essentially derived from case studies of selected fisheries, and the limited nationally representative data available are generally derived from a few questions included in the livestock section of household surveys. These do not permit the detailed characterization of the fishery production systems. As a consequence in many countries the decision-makers and planners lack the most basic information about the role and importance of the fisheries sector to their national economy. As part of an initiative called the Living Standards Measurement Study-Integrated Surveys on Agriculture (LSMS-ISA) project, a collaboration was developed between the World Bank and the WorldFish Center to address this situation. This report provides detail on pilot testing of a fisheries module for living standards measurement surveys

Hot Spot mapping - A spatial and methodological approach to analyzing outdoor crimes in Malmö.

This thesis aim to visualize and explore the methodological approach to finding the spatial patterning of crime through a geographical information system as a means for future guidance within spatial crime analysis. The analysis was applied to outdoor crimes for the city of Malmö in the year 2007. As a means for conducting this analysis a casestudy is performed. Both as a review of current police-methodology within crimepreventive work and on two individual geographic locations. After gathering and sorting of data a total of 9876 crime incidents was analysed. Two spatial analyses are performed, the Optimized Hot Spot-analysis tool and the Kernel Density Estimationanalysis tool. The Average Nearest Neighbor-model was applied to the data for further statistical accuracy. The thesis concludes that both tools have their usages. The optimized hot spot analysis was concluded to be of most use when the study area was large whereas the kernel density estimation analysis performed better for finding small variations on smaller study areas. However, they are the most efficient as complementary tools rather than when used as a single-method approach

Exact oracle inequality for a sharp adaptive kernel density estimator

In one-dimensional density estimation on i.i.d. observations we suggest an adaptive cross-validation technique for the selection of a kernel estimator. This estimator is both asymptotic MISE-efficient with respect to the monotone oracle, and sharp minimax-adaptive over the whole scale of Sobolev spaces with smoothness index greater than 1/2. The proof of the central concentration inequality avoids "chaining" and relies on an additive decomposition of the empirical processes involved

Counting with Focus for Free

This paper aims to count arbitrary objects in images. The leading counting approaches start from point annotations per object from which they construct density maps. Then, their training objective transforms input images to density maps through deep convolutional networks. We posit that the point annotations serve more supervision purposes than just constructing density maps. We introduce ways to repurpose the points for free. First, we propose supervised focus from segmentation, where points are converted into binary maps. The binary maps are combined with a network branch and accompanying loss function to focus on areas of interest. Second, we propose supervised focus from global density, where the ratio of point annotations to image pixels is used in another branch to regularize the overall density estimation. To assist both the density estimation and the focus from segmentation, we also introduce an improved kernel size estimator for the point annotations. Experiments on six datasets show that all our contributions reduce the counting error, regardless of the base network, resulting in state-of-the-art accuracy using only a single network. Finally, we are the first to count on WIDER FACE, allowing us to show the benefits of our approach in handling varying object scales and crowding levels. Code is available at https://github.com/shizenglin/Counting-with-Focus-for-FreeComment: ICCV, 201

Measurement of intra-distribution dynamics: An application of different approaches to the European regions

This paper examines the intra-distribution dynamics of per capita income between the European regions for the periods 1980-1993 and 1993-2005. To this end, three approaches are applied: the stochastic kernel approach, the highest conditional density approach and the estimation of mobility measures based on the Markov chain approach. One main conclusion and lesson have been obtained. The conclusion is that, although the distribution exhibits a great persistence, the degree of intra-distribution mobility has been much higher in the first period than in the second. The lesson is that, in dealing with intra-distribution dynamics, the use of different but complementary approaches is highly recommended