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 Computing Graduate Program in Systems Engineering and Computing Federal University of Rio de Janeiro - UFRJ - COPPE Federal University of Rio de Janeiro - UFRJ - COPPE Advisors: D.Sc. Ricardo Marroquim & Ph.D. Cláudio Esperança {michel.mas, michel.santos.al}@gmail.com April, 2015 April, 2015 «Introduction to Kernels Functions» Michel Alves :: [email protected]Laboratory of Computer Graphics - LCG
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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
April, 2015April, 2015«Introduction to Kernels Functions»
Michel Alves :: [email protected] 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 :: [email protected] 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 :: [email protected] 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 :: [email protected] 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 :: [email protected] 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 :: [email protected] 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 :: [email protected] 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 :: [email protected] 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
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 :: [email protected] Laboratory of Computer Graphics - LCG
