Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities

Ten Minute SpeechAn Overview of Activities Developed in Guided Studies

Michel Alves dos SantosGraduate Program in Systems Engineering and ComputingGraduate Program in Systems Engineering and ComputingFederal 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

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 ∈ A0, x /∈ A

A : definition of a real set

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, GroundTruth, 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

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

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

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

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

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

Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities

ThanksThanks for your attention!

Michel Alves dos Santos - http://www.michelalves.com

Michel Alves dos Santos - (Alves, M.)Federal University of Rio de JaneiroE-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

Federal University of Rio de Janeiro - UFRJ Ten Minute Speech :: Overview of Activities

BibliographyM. 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