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# Introduction to Kernel Functions

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In non-parametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. A kernel is a non-negative real-valued symmetric and integrable function K. Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov, quartic (biweight), tricube, triweight, Gaussian, quadratic and cosine. In this presentation we will talk about the properties and applications of kernel functions.

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### Introduction to Kernel Functions

1. 1. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Ten Minute Speech An Overview of Activities Developed in Guided Studies Michel Alves dos Santos Graduate Program in Systems Engineering and ComputingGraduate Program in Systems Engineering and Computing Federal University of Rio de Janeiro - UFRJ - COPPEFederal University of Rio de Janeiro - UFRJ - COPPE Advisors: D.Sc. Ricardo Marroquim & Ph.D. Cláudio Esperança {michel.mas, michel.santos.al}@gmail.com April, 2015April, 2015 «Introduction to Kernels Functions» Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG
2. 2. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Introduction - What’s it? Properties: real, positive, symmetric and integrable. K(x) ∈ R, ∀x ∈ R K(x) ≥ 0, ∀x ∈ R K(−x) = K(x), ∀x ∈ R +∞ −∞ K(x)dx = 1 Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG Indicator Function 1A(x) = 1, x ∈ A 0, x /∈ A A : deﬁnition of a real set
3. 3. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Where’s it Used? Figure: Background Extraction and Data Hiding. First row: Original, Ground Truth, Gaussian, Epanechnikov. Second Row: maps of relevance for data hiding. Best background modeling method, segmentation and clusterization; Functions for machine learning, steganalysis and data hiding. Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG
4. 4. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Be Careful! Some functions may appear but are not kernels! −∞ +∞ K(x)dx > 1 It isn’t symmetric! −∞ +∞ K(x)dx < 1 Pay attention on deﬁnition of kernel function! Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG
5. 5. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Supplement: Some Kernel Functions I Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG
6. 6. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Supplement: Some Kernel Functions II Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG
7. 7. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Supplement: Some Kernel Functions III Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG
8. 8. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Supplement: Some Kernel Functions IV Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG Remember the properties: Real: K(x) ∈ R, ∀x ∈ R Positive: K(x) ≥ 0, ∀x ∈ R Symmetric: K(−x) = K(x), ∀x ∈ R Integrable: +∞ −∞ K(x)dx = 1
9. 9. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Thanks Thanks for your attention! Michel Alves dos Santos - http://www.michelalves.com Michel Alves dos Santos - (Alves, M.) Federal University of Rio de Janeiro E-mail: michel.mas@gmail.com, malves@cos.ufrj.br Résumé: http://lattes.cnpq.br/7295977425362370 http://www.facebook.com/michel.alves.santos http://www.linkedin.com/proﬁle/view?id=26542507 Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG
10. 10. Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities Bibliography M. Alves. Slideshare views, April 2014. URL http://www.slideshare.net/michelalves. B. E. Hansen. Lecture notes on nonparametrics. Spring 2009, University of Wisconsin, 2009. K. Noda. Estimation of a regression function by the parzen kernel-type density estimators. The Institute of Statistical Mathematics, 1975. E. Parzen. On estimation of a probability density function and mode. Annals of Mathematical Statistics, 33:1065–1076, 1962. Y. Soh, Y. Hae, A. Mehmood, R. H. Ashraf, and I. Kim. Performance evaluation of various functions for kernel density estimation. Open Journal of Applied Sciences, 3:58–64, 2013. K. Vopatová. Kernel choice with respect to the bandwidth in kernel density estimates. ACTA UNIVERSITATIS MATTHIAE BELII, 18:47–53, 2011. Michel Alves :: michel.mas@gmail.com Laboratory of Computer Graphics - LCG