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.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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