Difference between revisions of "User talk:Leedj" - Rhea

Revision as of 14:53, 12 April 2014

Bayes Error for Minimizing Risk
A slecture by ECE student Dennis Lee

Partly based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.

Introduction

Bayes rule for minimizing risk

Example 1: 1D features

Example 2: 2D features

Fig 1: Data for class 1 (crosses) and class 2 (circles). In all cases, Prob(

\omega_1

) = Prob(

\omega_2

) = 0.5. Misclassified points are shown in red. Values of

\mu_1

,

\mu_2

, and

\Sigma

are given in Eqs. (------) - (----------). As the figures show, the separating hyperplanes shift depending on the values of

c_{12}

and

c_{21}

.

Summary and Conclusions

In this lecture we have shown that the probability of error ($Prob \left[ Error \right] $) when using Bayes error, is upper bounded by the Chernoff Bound. Therefore,

Prob \left[ Error \right] \leq \varepsilon_{\beta}

for $\beta \in \left[ 0, 1 \right]$ .

When $\beta =\frac{1}{2}$ then $\varepsilon_{\frac{1}{2}}$ in known as the Bhattacharyya bound.

References

[1]. Duda, Richard O. and Hart, Peter E. and Stork, David G., "Pattern Classication (2nd Edition)," Wiley-Interscience, 2000.

[2]. Mireille Boutin, "ECE662: Statistical Pattern Recognition and Decision Making Processes," Purdue University, Spring 2014.

Questions and comments

If you have any questions, comments, etc. please post them On this page.

@@ Line 15: / Line 15: @@
 Partly based on the [[2014_Spring_ECE_662_Boutin|ECE662 Spring 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]].
 </center>
+----
+== '''Introduction''' ==
+----
+== Bayes rule for minimizing risk ==
+----
+== Example 1: 1D features ==
+----
+== Example 2: 2D features ==
 <center>[[Image:C12_c21.png|600px|thumb|left|Fig 1: Data for class 1 (crosses) and class 2 (circles).
@@ Line 26: / Line 39: @@
 ----
-== '''References''' ==
+== Summary and Conclusions ==
+In this lecture we have shown that the probability of error ($Prob \left[ Error \right] $) when using Bayes error, is upper bounded by the Chernoff Bound. Therefore,
-[2]. [https://engineering.purdue.edu/~mboutin/ Mireille Boutin], "ECE662: Statistical Pattern Recognition and Decision Making Processes," Purdue University, Spring 2014.
+<center><math>Prob \left[ Error \right] \leq \varepsilon_{\beta}</math></center>
+for <math>\beta \in \left[ 0, 1 \right]</math>.
+When <math>\beta =\frac{1}{2}</math> then <math>\varepsilon_{\frac{1}{2}}</math> in known as the Bhattacharyya bound.
 ----
+== References ==
+[1]. Duda, Richard O. and Hart, Peter E. and Stork, David G., "Pattern Classication (2nd Edition)," Wiley-Interscience, 2000.
+[2]. [https://engineering.purdue.edu/~mboutin/ Mireille Boutin], "ECE662: Statistical Pattern Recognition and Decision Making Processes," Purdue University, Spring 2014.
+----
+== Questions and comments==
+If you have any questions, comments, etc. please post them On [[Upper_Bounds_for_Bayes_Error_Questions_and_comment|this page]].