(35 intermediate revisions by 7 users not shown)
Line 1: Line 1:
−
[http://balthier.ecn.purdue.edu/index.php/ECE662 ECE662 Main Page]
+
[[Category:ECE662]]
 +
[[Category:decision theory]]
 +
[[Category:lecture notes]]
 +
[[Category:ECE662]]
 +
[[Category:pattern recognition]]
 +
[[Category:slecture]]
  
−
[http://balthier.ecn.purdue.edu/index.php/ECE662#Class_Lecture_Notes Class Lecture Notes]
+
<center><font size= 4>
 +
'''[[ECE662]]: Statistical Pattern Recognition and Decision Making Processes'''
 +
</font size>
  
 +
Spring 2008, [[user:mboutin|Prof. Boutin]]
 +
 +
[[Slectures|Slecture]]
 +
 +
<font size= 3> Collectively created by the students in [[ECE662:BoutinSpring08_OldKiwi|the class]]</font size>
 +
</center>
 +
 +
----
 +
=Lecture 3 Lecture notes=
 +
Jump to: [[ECE662_Pattern_Recognition_Decision_Making_Processes_Spring2008_sLecture_collective|Outline]]|
 +
[[Lecture 1 - Introduction_OldKiwi|1]]|
 +
[[Lecture 2 - Decision Hypersurfaces_OldKiwi|2]]|
 +
[[Lecture 3 - Bayes classification_OldKiwi|3]]|
 +
[[Lecture 4 - Bayes Classification_OldKiwi|4]]|
 +
[[Lecture 5 - Discriminant Functions_OldKiwi|5]]|
 +
[[Lecture 6 - Discriminant Functions_OldKiwi|6]]|
 +
[[Lecture 7 - MLE and BPE_OldKiwi|7]]|
 +
[[Lecture 8 - MLE, BPE and Linear Discriminant Functions_OldKiwi|8]]|
 +
[[Lecture 9 - Linear Discriminant Functions_OldKiwi|9]]|
 +
[[Lecture 10 - Batch Perceptron and Fisher Linear Discriminant_OldKiwi|10]]|
 +
[[Lecture 11 - Fischer's Linear Discriminant again_OldKiwi|11]]|
 +
[[Lecture 12 - Support Vector Machine and Quadratic Optimization Problem_OldKiwi|12]]|
 +
[[Lecture 13 - Kernel function for SVMs and ANNs introduction_OldKiwi|13]]| 
 +
[[Lecture 14 - ANNs, Non-parametric Density Estimation (Parzen Window)_OldKiwi|14]]|
 +
[[Lecture 15 - Parzen Window Method_OldKiwi|15]]|
 +
[[Lecture 16 - Parzen Window Method and K-nearest Neighbor Density Estimate_OldKiwi|16]]|
 +
[[Lecture 17 - Nearest Neighbors Clarification Rule and Metrics_OldKiwi|17]]|
 +
[[Lecture 18 - Nearest Neighbors Clarification Rule and Metrics(Continued)_OldKiwi|18]]|
 +
[[Lecture 19 - Nearest Neighbor Error Rates_OldKiwi|19]]|
 +
[[Lecture 20 - Density Estimation using Series Expansion and Decision Trees_OldKiwi|20]]|
 +
[[Lecture 21 - Decision Trees(Continued)_OldKiwi|21]]|
 +
[[Lecture 22 - Decision Trees and Clustering_OldKiwi|22]]|
 +
[[Lecture 23 - Spanning Trees_OldKiwi|23]]|
 +
[[Lecture 24 - Clustering and Hierarchical Clustering_OldKiwi|24]]|
 +
[[Lecture 25 - Clustering Algorithms_OldKiwi|25]]|
 +
[[Lecture 26 - Statistical Clustering Methods_OldKiwi|26]]|
 +
[[Lecture 27 - Clustering by finding valleys of densities_OldKiwi|27]]|
 +
[[Lecture 28 - Final lecture_OldKiwi|28]]
 +
----
 +
----
 
LECTURE THEME :
 
LECTURE THEME :
  
−
Bayes classification
+
== Bayes classification ==
  
 
Topics:
 
Topics:
Line 15: Line 62:
  
 
Clarification 2 - All classifiers are defined by hypersurfaces. - Consider a Nearest Neighbor classification problem: The classification of any query is decided by the label of its nearest neighbor. It may not be clear how this classifier can be defined by hypersurface. But we can define separating hypersurfaces which pass mid-way between points of different classes. In the extreme case, there can be a hypersurface for each data point enveloping all queries that will be classified with the label of that data point.
 
Clarification 2 - All classifiers are defined by hypersurfaces. - Consider a Nearest Neighbor classification problem: The classification of any query is decided by the label of its nearest neighbor. It may not be clear how this classifier can be defined by hypersurface. But we can define separating hypersurfaces which pass mid-way between points of different classes. In the extreme case, there can be a hypersurface for each data point enveloping all queries that will be classified with the label of that data point.
 +
 +
To find building blocks "g" or hypersurfaces of a classifier there are two approaches:
 +
 +
# Supervised Learning: Somebody provides a category label in a training set. This collection of examples is used to infer the rules to categorize other examples.
 +
# Unsupervised Learning: Very related to "Clustering", or the "Discovery of groupings". These groupings become the unknown classes that we want to discover in the measurements. Nobody places labels in the training set. In many cases Supervised methods are applied within Unsupervised method to separate known classes within clusters.
 +
 +
==== See Also ====
 +
* Zoubin Ghahramani's tutorial on Supervised, Reinforcement and Unsupervised Learning http://www.inf.ed.ac.uk/teaching/courses/pmr/docs/ul.pdf
 +
 +
===High Dimensional Issues===
 +
The [[Curse_of_Dimensionality_Old_Kiwi|curse of dimensionality]] starts at d>17-23. There are no clusters or groupings of data points when d>17. In practice each point turns to be a cluster on its own and as a result this explodes into a high dimensional feature vectors which are impossible to handle in computation.
 +
 +
== Bayes rule ==
 +
 +
Bayes rule addresses the predefined classes classification problem.
 +
Given value of X for an object, assign one of the k classes to the object
 +
 +
Bayes rule for discrete feature vector
 +
Bayes rule is to do the following: Given x=x, choose the most likely class <math>E{\lbrace}w_1,...,w_k{\rbrace}</math>
 +
 +
<math>w: E{\lbrace}w_1,...,w_k{\rbrace}</math>
 +
ie. choose <math>w_i</math> such that the <math>P(w_i|x) \geq P(w_j|x), {\forall}j</math>
 +
 +
<math>posterior = \frac{(likelihood)(prior)}{(evidence)}</math>
 +
 +
<math>posterior = P(w_i|x)= \frac{p(x|w_i)P(w_i)}{P(x)}</math>
 +
 +
Bayes rule: choose the class <math>w_i</math> that maximizes the  <math>p(x|w_i)P(w_i)</math>
 +
 +
Example: Given 2 class decision problems <math>w_1 = </math> women & <math>w_2 </math>= men, <math>L = hair length </math>
 +
choose <math>w_1</math>, if <math>P(w_1|L) \geq P(w_2|L)</math>
 +
else choose <math>w_2</math>
 +
or
 +
 +
choose <math>w_1</math> if <math>p(L|w_1)P(w_1)>p(L|w_2)P(w_2)</math>
 +
 +
else choose <math> w_2 </math>
 +
 +
Minimum probability of error is the error made when <math> w = w_2 </math> and decided <math> w_1 </math>
 +
 +
Special cases <br>
 +
If <math> P(w_1) = P(w_2)</math> <br>
 +
<math>p(x|w_1)P(w_1) \geq p(x|w_2)P(w_2), {\forall j}</math><br>
 +
<math>p(x|w_1) \geq p(x|w_2)</math> decision is based on the likelihood<br>
 +
<br>
 +
-If <math>p(x|w_1)=p(x|w_2)</math><br>
 +
<math>p(x|w_1)P(w_1) \geq p(x|w_2)P(w_2), {\forall j}</math><br>
 +
<math>P(w_1) \geq P(w_2)</math> decision is based on the prior<br>
 +
----
 +
Previous: [[Lecture_2_-_Decision_Hypersurfaces_OldKiwi|Lecture 2]]
 +
Next:  [[Lecture_4_-_Bayes_Classification_OldKiwi|Lecture 4]]
 +
 +
[[ECE662:BoutinSpring08_OldKiwi|Back to ECE662 Spring 2008 Prof. Boutin]]

Latest revision as of 11:17, 10 June 2013


ECE662: Statistical Pattern Recognition and Decision Making Processes

Spring 2008, Prof. Boutin

Slecture

Collectively created by the students in the class

Lecture 3 Lecture notes

Jump to: Outline| 1| 2| 3| 4| 5| 6| 7| 8| 9| 10| 11| 12| 13| 14| 15| 16| 17| 18| 19| 20| 21| 22| 23| 24| 25| 26| 27| 28

LECTURE THEME :

Bayes classification

Topics:

Clarification 1 - Difference between Hyperplane and Hypersurface: In simple terms, a hypersurface is any n-1 dimensional surface in n-dimensional space, while hyperplane is a flat hypersurface.

Example Hypersurface example OldKiwi.jpg

Clarification 2 - All classifiers are defined by hypersurfaces. - Consider a Nearest Neighbor classification problem: The classification of any query is decided by the label of its nearest neighbor. It may not be clear how this classifier can be defined by hypersurface. But we can define separating hypersurfaces which pass mid-way between points of different classes. In the extreme case, there can be a hypersurface for each data point enveloping all queries that will be classified with the label of that data point.

To find building blocks "g" or hypersurfaces of a classifier there are two approaches:

  1. Supervised Learning: Somebody provides a category label in a training set. This collection of examples is used to infer the rules to categorize other examples.
  2. Unsupervised Learning: Very related to "Clustering", or the "Discovery of groupings". These groupings become the unknown classes that we want to discover in the measurements. Nobody places labels in the training set. In many cases Supervised methods are applied within Unsupervised method to separate known classes within clusters.

See Also

High Dimensional Issues

The curse of dimensionality starts at d>17-23. There are no clusters or groupings of data points when d>17. In practice each point turns to be a cluster on its own and as a result this explodes into a high dimensional feature vectors which are impossible to handle in computation.

Bayes rule

Bayes rule addresses the predefined classes classification problem. Given value of X for an object, assign one of the k classes to the object

Bayes rule for discrete feature vector Bayes rule is to do the following: Given x=x, choose the most likely class $ E{\lbrace}w_1,...,w_k{\rbrace} $

$ w: E{\lbrace}w_1,...,w_k{\rbrace} $ ie. choose $ w_i $ such that the $ P(w_i|x) \geq P(w_j|x), {\forall}j $

$ posterior = \frac{(likelihood)(prior)}{(evidence)} $

$ posterior = P(w_i|x)= \frac{p(x|w_i)P(w_i)}{P(x)} $

Bayes rule: choose the class $ w_i $ that maximizes the $ p(x|w_i)P(w_i) $

Example: Given 2 class decision problems $ w_1 = $ women & $ w_2 $= men, $ L = hair length $ choose $ w_1 $, if $ P(w_1|L) \geq P(w_2|L) $ else choose $ w_2 $ or

choose $ w_1 $ if $ p(L|w_1)P(w_1)>p(L|w_2)P(w_2) $

else choose $ w_2 $

Minimum probability of error is the error made when $ w = w_2 $ and decided $ w_1 $

Special cases
If $ P(w_1) = P(w_2) $
$ p(x|w_1)P(w_1) \geq p(x|w_2)P(w_2), {\forall j} $
$ p(x|w_1) \geq p(x|w_2) $ decision is based on the likelihood

-If $ p(x|w_1)=p(x|w_2) $
$ p(x|w_1)P(w_1) \geq p(x|w_2)P(w_2), {\forall j} $
$ P(w_1) \geq P(w_2) $ decision is based on the prior

Previous: Lecture 2 Next: Lecture 4

Back to ECE662 Spring 2008 Prof. Boutin

Retrieved from "https://www.projectrhea.org/rhea/index.php?title=Lecture_3_-_Bayes_classification_OldKiwi&oldid=56556"
Shortcuts
Help Main Wiki Page Random Page Special Pages Log in
Search

Alumni Liaison

Followed her dream after having raised her family.

Ruth Enoch, PhD Mathematics